class field theory - its centenary and prospect

ADVANCED STUDIES IN PURE MATHEMATICS 30 Chief Editor: Eiichi Bannai (Kyushu University)

Class Field Theory - Its Centenary and Prospect

Edited by

Katsuya Miyake (Tokyo Metropolitan Univ.)

Mathematical Society of Japan

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2000 Mathematics Subject Classification. Primary llR37; Secondary llF70, llH99, llM06, llR23, llR39, llR42, llR56, llS37, 01A60.

Advanced Studies in Pure Mathematics 30 Chief Editor

Eiichi Bannai (Kyushu University)

Editorial Board of the Series

HITOSHI ARAI EIICHI BANNAI YOSHIKAZU GIGA (Univ. of Tokyo) (Kyushu Univ.) (Hokkaido Univ.)

KI-ICHIRO HASHIMOTO KAZUYA KATO TOSHITAKE KOHNO (Waseda Univ.) (Univ. of Tokyo) (Univ. of Tokyo)

YOICHI MIYAOKA SEIKI NISHIKAWA JUNJIRO NOGUCHI (Kyoto Univ.) (Tohoku Univ.) (Univ. of Tokyo)

TOSHIYUKI TANISAKI (Hiroshima Univ.)

PRINTED IN JAPAN by Tokyo Shoseki Printing Co., Ltd.

Preface

The Seventh MSJ International Research Institute of the Mathe- matical Society of Japan was held in Tokyo for ten days from June 3rd to 12th, 1998. The theme was 'Class Field Theory - its Centenary and Prospect', which is taken as the title of the book. The program of this international conference is attached at the end of the preface. This volume is a collection of articles contributed by the speakers of the conference. All but a few of them are full scale papers. Some of them are expository on those subjects which are of central issues of algebraic number theory, and are prepared by the leading experts; they contains important and interesting problems with extensive references. Some of them are historical, and vividly explain how number theorists were motivated and exchanged their mathematical ideas.

In 1920 Takagi published the complete version of his class field theory as 'Ueber eine Theorie des relativ Abel'schen Zahlkorpers' in J . Coll. Sci. Tokyo, vol. 41. Chapter V of it is devoted to an affirmative solution to 'Kronecker's youth-dream'. This problem asks, roughly speaking, whether all abelian extensions of an imaginary quadratic number field could be obtained by the singular moduli and special values of elliptic functions which have complex multiplication by the elements of the quadratic field; and it was reformulated in a general frame-work by Hilbert as the twelfth problem of his celebrated 23 problems. There is another problem behind class field theory: that is, the principal ideal theorem which was finally proved by Furtwangler in 1930 based on Artin's general reciprocity law. Artin established this significant result in his short paper, 'Beweis des allgemeinen Reziprozitatsgesetzes', in Abh. Math. Sem. Univ. Hamburg, vol. 5 (1927). Like the former problem, the origin of the latter goes back to Kronecker. We may say, however, that the Takagi-Artin class field theory has its direct origins in several papers by Weber and by Hilbert published in 1897-1898:

H. Weber, Ueber Zahlengruppen in algebraischen Korpern I, Math. Ann. 48 (1897), 433-473; 11, 49 (1897), 83-100; 111, 50 (1898), 1-26;

D. Hilbert, Die Theorie der algebraischen Zahlkorper, Jber. Deutschen Math.-Ver.4 (1897), 175-546; ~ b e r die Theorie der relativ-Abelschen Zahlkorper, Nachr. Akad. Wiss. Gottingen 5 (l898), 377-399;

(cf. e.g. K. Miyake, The Establishment of the Takagi-Artin Class Field Theory, in T h e Intersect ion of History and Mathematics (ed. J. W.

Dauben et al) , Birkhauser Verlag, Basel-Boston-Berlin, 1994, pp. 109- 128). This is one of the reasons why the editor of this volume chose the above stated theme of the Seventh MSJ International Research Institute in 1998 when he was appointed as organizer.

After the marvelous results of Takagi and Artin, the next act of the drama of class field theory was played by H. Hasse. In this energetic performance, he established his reciprocity law in the form which beautifully embodied the Local-Global Principle in class field theory (cf. e.g. G. Frei's article in this volume). Under his influence F. K. Schmidt could demonstrate local class field theory in 1930 based on the global theory. Then Chevalley gave an 'arithmetic proof' to class field theory, and also introduced idhles (cf. e.g. C. Chevalley, La thborie du corps de classes, Ann. Math. 41 (1940), 394-418). And finally in 1950, K. Iwasawa and J. Tate independently developed functional analysis over idele groups of algebraic number fields to present zeta- and L-functions with functional equations in an elegant manner. Here ideles became a natural and basic concept of algebraic number theory. It should also be mentioned that A. Weil gave an important impetus by introducing the ring of adkles in his lecture notes, Adhles and algebraic groups, at Princeton in 1959.

In 1951 and 1952, G. Hochschild, T. Nakayama and J. Tate successfully gave a cohomological description to class field theory. In his lectures at Princeton in 1951-52, E. Artin treated class field theory with this method. It enabled him to give an axiomatic presentation, 'class formation'. Then extensive studies were carried out by Y. Kawada with the partial help of I. Satake.

It would be too much for me to give even a brief sketch of the development of the theory of automorphic functions and forms. I would like, however, to point out just two significant works. Mainly motivated by Hilbert's twelfth problem, E. Hecke developed his works on 'Hilbert modular functions' of two variables in his two papers published in Math. Ann. 71 (1912) and ibid. 74 (1913); the title of the second paper is ' ~ b e r die Konstruktionen relativ-Abelscher Zahlkiirper durch Modulfunktionen von zwei Variabeln'. Here he deals with SL2 over the ring of integers of a real quadratic number field which discontinuously acts on a direct product of two copies of the complex upper half plane. The other work I point out here is one of C. L. Siegel's works, Symplec- tic Geometry, Amer. J. Math. 65 (1943). In this paper he worked on Sp, over the rational number field and established the analytic theory of moduli of (polarized) Abelian varieties. There followed extensive works on automorphic functions and forms in many variables with respect to semi-simple and reductive algebraic groups over algebraic number fields. Weil's lecture notes referred above should also be mentioned here.

Towards the end of the 1960's there appeared three important works by K. Iwasawa, by G. Shimura and by R. P. Langlands. Concerning the first one and the third, we have splendid articles by R. Greenberg and by B. Casselman, respectively, in this book. Let me just point out one of Shimura's works here:

G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. Math. 91 (1970), 144-222; 11, ibid. 92 (1970), 528-558.

This paper represented a modern summit in the language of adele geometry of which Kronecker once dreamed and then Hilbert anticipated with his twelfth problem.

On the history of Hilbert's twelfth problem, N. Schappacher gave two interesting lectures in our conference as you will see from the program at the end of the preface. However, we could not include his article in this book. For those who are interested in it, I refer to his article, On the history of Hilbert's twelfth problem: a comedy of errors, MatQriaux pour l'histoire des mathdmatiques au XXe sikcle (Nice, 1996), 243-273, Sbmin. Congr., 3, Soc. Math. France, Paris, 1998. In our conference John Coates gave two inspiring lectures on Iwasawa theory of elliptic curves for graduate students and young mathematicians at the request of the organizers. Those who are interested in the subject are recom- mended to see his article, Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication, Arithmetic theory of elliptic curves (Cetraro, 1997), 1-50, Lecture Notes in Math., 1716, Springer, Berlin, 1999.

I have mentioned only a strictly limited number of important works on those topics which are closely related with class field theory. It is, of course, apparent that this list is far from sufficient. I certainly failed to mention such important topics as class field theory in positive characteristic, higher dimensional class field theory, and so on. (As for the first theme, we are happy to include an article of P. Roquette in this volume.) I just hope this humble review of mine may help younger number theorists who might be interested in this volume.

Acknowledgement

The chief organizer of the international conference and the editor of this volume would like to express his gratitude to those who financially supported these scientific activities:

Mathematical Society of Japan, Japan Association for Mathematical Sciences, Waseda University Advanced Research Institute for Science and En- gineering, Tokyo Metropolitan University, IBM Japan, Ltd., Nissan Science Foundation, Mr. Kenshiro Koto, Director and General Manager, Tokyo Gas Co., Ltd., TAMAT Number Theory Association, and several anonymous personae.

We were also partially supported by

the Grant-in-Aid for Scientific Research (A) No. 08304004, Ministry of Education, Science, Sports and Culture,

the Grant-in-Aid for Scientific Research (A) No. 10304004, Japan Society for the Promotion of Science, and

the Grant-in-Aid for Scientific Research (B) No. 11440013, Japan Society for the Promotion of Science.

December 25, 2000 Katsuya Miyake, Editor

All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

The Seventh International Research Institute of the Mathematical Society of Japan

- Class Field Theory - its Centenary and Prospect

DATE: ARRIVAL DAY Tuesday, June 2, 1998 TALKS BEGIN Wednesday, June 3, 1998 TALKS END Friday, June 12, 1998

(There will be no talks in the afternoon of Saturday, June 6, and on Sunday, June 7.)

PLACE: Building 55, School of Science and Engineering, Waseda University Okubo, Shinjuku, Tokyo 169, Japan

PROGRAM:

June 3

10: 10 Opening Address 10:20 - 10:50 Shokichi Iyanaga (Japan Acad.)

Memories of Professor Teiji TAKAGI 11:OO - 12:OO Gunther Frei (Univ. Laval)

How H. Hasse was led to the Local-Global Principle, the Reciprocity Laws and Local Class Field Theory.

13:30 - 14:20 Noriyuki Otsubo (Univ. Tokyo) Recent Progress on the Finiteness of Torsion Algebraic Cycles

14:30 - 15:20 Kanetomo Sato (Tokyo Inst. Tech.) Finiteness of a certain Motivic Cohomology of Varieties over Local and Global Fields

15:50 - 16:40 Moulay Chrif Ismaili (Univ. Oujda) The Capitulation Problem for Certain Number Fields

16:50 - 17:50 Hiroshi Suzuki (Nagoya Univ.) On the Capitulation Problem

June 4

1 l: l5 - 12: 15 Peter Stevenhagen (Univ. Amsterdam) Generating Class Fields using Shimura Reciprocity

13:45 - 14:45 Takeshi Saito (Univ. Tokyo) Modular Forms and padic Hodge Theory

15:40 - 16:40 Spencer Bloch (Univ. Chicago) Higher Theta Functions

16:50 - 17:50 Kazuya Kato (Univ. Tokyo) Log Hodge Structures and Classifying Spaces

June 5

10:OO - 11:OO Shuji Saito (Tokyo Inst. Tech.) Generalization of the Theorem of Brauer-Hasse-Noether

11:15 - 12:15 Ivan Fesenko (Univ. Nottingham) Neukirch Construction of the Reciprocity Map

13:45 - 14:45 Masakazu Yamagishi (Nagoya Inst. Tech.) A Survey of pExtensions

15:15 - 16:15 Helmut Koch (Humboldt-Univ., Berlin) History of the Theorem of Shafarevich in the Theory of Class Formations

16:30 - 17:30 Thong Nguyen Quang Do (Univ. Franche-Comtk) Galois Module Structure of Class-Formations and free pro-pExt ensions

18:30 - 20:30 RECEPTION at Avaco

June 6

10:OO - 11:OO Peter Roquette (Univ. Heidelberg) Class Field Theory in Characteristic p: its Origin and History

11:15 - 12:15 Tsuneo Tamagawa (Yale Univ.) Class field theory is easy: Hazewinkel-Iwasawa Theory of Class Fields of Function Fields over Finite Fields

June 8

10:OO - 10:50 Winfried Kohnen (Univ. Heidelberg) Class Numbers of Imaginary Quadratic Fields

11:OO - 11:50 Ryotaro Okazaki (Doshisha Univ.) On Parities of Relative Class Numbers of Certain CM-Extensions

13:20 - 14:lO Akito Nomura (Kanazawa Univ.) On Embedding Problems with restricted Ramifications

14:20 - 15: 10 Sang Geun Hahn (KAIST) Some Congruences for Binomial Coefficients

15:40 - 16:40 Florian Pop (Univ. Bonn) On the Absolute Galois Group of the Rationals

June 9

10:OO - 11:OO Ralph Greenberg (Univ. Washington) Iwasawa Theory - Past and Present

11:lO - 12:OO Manabu Ozaki (Waseda Univ.) Iwasawa Invariants of Z,-Extensions over an imaginary Quadratic Field

13:30 - 14:20 Masato Kurihara (Tokyo Metropolitan Univ.) Iwasawa Theory and Ideal Class Groups of large Number Fields

14:30 - 15:20 Hiroaki Nakamura (Tokyo Metropolitan Univ.) Artin Braid Groups and Families of Elliptic Curves

15:50 - 16:40 Hisao Taya (Tohoku Univ.) On padic Zeta Functions and Class Groups of Z,- Extensions of certain totally real Fields

16:50 - 17:50 John H. Coates (Univ. Cambridge) Iwasawa Theory of Elliptic Curves without Complex Multiplication I

June 10

10:OO - 11:OO Norbert Schappacher (Univ. Strasbourg) On the History of Hilbert's 12th Problem, 1900-1932

11:15 - 12:15 William Casselman (Univ. British Columbia) Early Years of the L-group

13:45 - 14:45 Norbert Schappacher (Univ. Strasbourg) Complex Multiplication in Arithmetic Algebraic Geometry: Hecke Characters, Abelian Varieties, and Motives

15:15 - 16:15 Roland Gillard (Institut Fourier) Obstruct ions in Deformation Problems

16:30 - 17:30 Thomas Zink (Univ. Bielefeld) P-adic Periods and Displays

June 11

10:OO - 11:OO Rend Schoof (Univ. Roma 11) Abelian Varieties over Number Fields with good Reduction everywhere

11:15 - 12:15 Hiromichi Yanai (Aichi Inst. Tech.) Exceptional Hodge Cycles and unramified Class Fields

13:45 - 14:45 David R. Kohel (Univ. Singapore) Hecke Module Structure of Quaternions

15:15 - 16:15 Ram Murty (Queen's Univ. at Kingston) What is known about Artin L-Functions ?

16:30 - 17:30 Ichiro Satake (Univ. Chuo) Classification Theory of Semisimple Algebraic Groups

I

I June 12

10:OO - 11:OO John H. Coates (Univ. Cambridge) Iwasawa Theory of Elliptic Curves without Complex Multiplication I1

11:15 - 12:15 Jean Cougnard (Univ. Caen) On Rings of Integers which are stably free and non free H8 x C2 Modules

13:45 - 14:45 M. Morishita (Kanazawa Univ.) and T. Watanabe (Osaka Univ.)

Adele Geometry of Numbers 15:05 - 16:05 Takashi Ono (Johns Hopkins Univ.)

On Shafarevich-Tate Sets 16:15 - 17:15 Yoshihiko Yarnamoto (Osaka Univ.)

The Canonical Power Series associated with Elliptic Curves over Q

17:30 - 18:30 Farewell Party

CONTENTS

Shokichi IYANAGA - Memories of Professor Teiji Takagi

Maruti Ram MURTY - On Artin L-Functions

Giinther FREI - How Hasse was led to the Theory of Quadratic Forms, the Local-Global Principle, the Theory of the Norm Residue Symbol, the Reciprocity Laws, and to Class Field The- ory 31

Ivan FESENKO - Nonabelian Local Reciprocity Maps 63

Akito NOMURA - Embedding Problems with restricted Ramifica- tions and the Class Number of Hilbert Class Fields 79

Helmut KOCH - The History of the Theorem of Shafarevich in the Theory of Class Formations 87

Masakazu YAMAGISHI - A Survey of p-Extensions 107

Thong NGUYEN QUANG DO - Galois Module Stucture of pClass Formations 123

Thomas ZINK - A Dieudonnk Theory for p-Divisible Groups 139

Peter STEVENHAGEN - Hilbert 's 12t h Problem, Complex Multipli- cation and Shimura Reciprocity 161

David R. KOHEL - Hecke Module Structure of Quaternions 177

Ichiro SATAKE - On Classification of Semisimple Algebraic Groups 197

Bill CASSELMAN - The L-Group 217

Roland GILLARD - Groupe des Obstructions pour les Dkformations de Representations Galoisiennes 259

Renk SCHOOF - Abelian Varieties over Q(&) with Good Reduc- tion Everywhere 287

Hiromichi YANAI - Hodge Cycles and Unramified Class Fields 307

Noriyuki OTSUBO - Recent Progress on the Finiteness of Torsion Algebraic Cycles 313

Kanetomo SATO - Finiteness of a certain Motivic Cohomology -. Group of Varieties over Local and Global Fields 325 Z

Advanced Studies in Pure Mathematics 30, 2001 Class Field Theory - Its Centenary and Prospect pp. 1-11

Ralph GREENBERG - Iwasawa Theory - Past and Present 335

Manabu OZAKI - Iwasawa Invariants of Zp-Extensions over an Imaginary Quadratic Field 387

Hisao TAYA - On p-Adic Zeta Functions and Class Groups of Zp- Extensions of certain Totally Real Fields 401

Winfried KOHNEN - Class Numbers of Imaginary Quadratic Fields 415

Ryotaro OKAZAKI - On Parities of Relative Class Numbers of certain CM-Extensions 419

Sang Geun HAHN and Dong Hoon LEE - Some Congruences for Binomial Coefficients 445

Jean COUGNARD - Stably Free and Not Free Rings of Integers 463

Mohammed AYADI, Abdelmalek AZIZI and Moulay CHRIF ISMAILI - The Capitulation Problem for certain Number Fields 467

Hiroshi SUZUKI - On the Capitulation Problem 483

Masanori MORISHITA and Takao WATANABE - Adele Geometry of Numbers 509

Takashi ONO - On Shafarevich-Tate Sets 537

Peter ROQUETTE - Class Field Theory in Characteristic p, its Origin and Development 549

Memories of Professor Teiji Takagi

Shokichi Iyanaga

I am particularly sensible of the honor of being given this chance of speaking today at this international meeting on the memories of Profes- sor Takagi, Japanese mathematician who founded the class field theory, as well known to all of you. Personally, he was an unforgettable teacher for me to whom I owe, so to speak, my whole existence as a mathematician. Not only myself, the Japanese community of mathematicians as a whole, owes much to him, I believe. May I remind you that he was born in 1875 and deceased in 1960? I was born in 1906 so that I have a privilege of having lived 54 years on this earth with him, but more precisely I met him for the first time at a class-room at the University of Tokyo when I had 20 years of age. Thus I could share with him more than 30 years in mutual acquaintance, which have left me numerous personal memories, on which I shall touch in the present talk, together with the role of Takagi in the development of mathematics in Japan, a more important subject. But the main theme of my talk of today should be, I believe, to trace how he came to found the class field theory. With this in mind, I have made the following short chronological table of his life. (+ next page) This table contains only five titles of Takagi's work: three in German, one in French and one in Japanese, the last one is of his talk on Remi- niscences and Perspectives, particularly important for the history of the class field theory. Takagi published many other papers and books, many of which only in Japanese, which contributed greatly to the progress of Japanese mathematics. His Collected Papers, gathering all his papers published in European languages, ran into the second edition; the first edition being sold out, the second augmented edition is now on sale at bookstores. It has an appendix containing Iwasawa's "On papers on Takagi in number theory" and if you permit me to mention, also my

Received September 4, 1998.

2 S. Iyanaga

Year Age Event

Born on April 21 in Gifu prefecture Studied in the Third High School in Kyoto Studied in Todai (the University of Tokyo) with Dairoku Kikuchih, Rikitaro Fujisawa Studied in Berlin University with Frobenius, H.A.Schwarz Studied in Gottingen University with Klein, Hilbert Returned to Todai (associate professor) " ~ b e r die im Bereiche der rationalen complexen Zahlen Abe17schen Zahlkorper" Dr.Sc. (Todai) Professor at Todai World War I " ~ b e r eine Theorie des relativ Abel'schen Zahlkorpers" ICM in Strassburg "Sur quelques thhorkmes gknkraux de la thkorie des nombres algkbriques" " ~ b e r das Reciprocitiitsgesetz in einem beliebigen algebraischen Zahlkorper" Member of Japan Academy ICM in Ziirich World War I1 (Japan involved in 1941-45) Awarded a Cultural Medal "Kaiko to Tembo" (Reminiscences and Perspectives) International Symposium on Algebraic Number Theory, Tokyo-Nikko Deceased on February 28 in the Hospital of Todai

article "On the life and works of Teiji Takagi." I show you two portraits of Takagi, taken from both editions of his Collected Papers. This one from the first edition dates from the 1920's. When I met him for the first time, he looked like this. The other one dates from the time after the World War 11. Both are accompanied with his signatures in Chinese

Memories of Professor Teiji Takagi 5

and Latin characters, which are the same, taken from the first edition. You see that Takagi had a very beautiful handwriting.

Now let me give some comments on the table I have just shown you. The year of birth of Takagi, 1875 A.D., is called in Japan "Meiji 8 nen" or the 8th year of Meiji, i.e. the 8th year since the coronation of Emperor Meiji, 1868, around which time the modernization was started in Japan. The modern Japan was thus still in a very young age. Gifu prefecture is in a mountainous region in the central Japan. He was born in an agricultural area of this prefecture, finished the primary school a t his native village, the secondary school at Gifu, chief town of the prefecture, the High School, i.e. the senior secondary school at Kyoto and then he came to Tokyo to study in the unique university in Japan at that time. It was called just the "Imperial University", but renamed the "Tokyo Imperial University" when another Imperial University was created in Kyoto in 1897, renamed again "the Tokyo University" dropping the word "Imperial" after World war 11. In Japanese it is called "Tokyo Daigaku" or "Todai" in abbreviation.

Professors Dairoku Kikuchi and Rikitaro Fujisawa, representing the first and the second generations of mathematicians in modern Japan, taught then at Todai. Kikuchi had been educated in England from his very early age, and returning to Japan just after graduation from Cam- bridge, was named immediately professor at Todai. Later he was named president of Todai and other universities, minister of education and en- nobled baron. Fujisawa studied in British and German universities and obtained his Ph.D. at German university of Strassburg. He transferred to Japan the atmosphere of research in German universities, but in later years, he became a senator and more interested in administrative affairs. We had to wait until the third generation to find a Japanese mathematician who created such a work as the class field theory, which attracted the attention of the world's mat hematical community.

Fujisawa had studied in Berlin and received influence of Kronecker; thus he knew the importance of algebra. Takagi joined his seminar and studied about abelian equations by the French text-book of Serret. The first 2 volumes of German text-book of H. Weber arrived at Tokyo toward that time, which Takagi read with great interest. He learned also in Todai the theory of elliptic functions in the style of Jacobi. After graduation, he continued his study for one year in the graduate school, then he was sent to Germany by the government. He went first to Berlin,

6 S. Iyanaga Memories of Professor Teiji Takagi 7

but Kronecker, Weierstrass and Kummer were already dead and Schwarz was old. He found interest in the lively lecture of young Frobenius. Then he went to Gottingen, where Klein and Hilbert were at the top of their activities. In attending the lecture of Klein and the weekly meeting of mathematicians from all parts of Europe, Takagi realized that the mathematics he had learned in Tokyo was about half a century behind the mathematics which was studied by contemporary researchers. He had then certainly studied for himself the Zahlbericht of Hilbert. Appar-

ently, he did not attend the ICM at Paris in 1900, but I am sure that he was interested in the 23 mathematical problems presented there by Hilbert, particularly in those inconnection with Kronecker's Jugendraum (dream in his younger day.) He chose as the subject of his research, a special case of this problem when the ground field is the Gaussian field Q(-). When he told this to Hilbert, he was encouraged by Hilbert's response "That's a fine subject!" as he related us later. He succeeded to bring this research to a happy end, and presented it to Todai as his doctoral thesis: "On the abelian field on the field of rational complex numbers" which was accepted of course. In its preface, we see a word of acknowledgement by Takagi to Hilbert .

Takagi was nominated to an associate professorship of Todai while he was in Gottingen, then to a professorship two years later in 1904, in which position he remained 32 years until 1936, the year in which he attained the age limit of 60 years fixed at Todai. The World War I took place during these years which interrupted the scientific correspondence between Europe and Japan. It is during this period that Takagi worked on his research in the class field theory. In 1920, two years after the end of War, he could complete his results in his important paper: "On the theory of relatively abelian number fields" whose contents are just Takagi's class field theory. The first ICM after the War was organized at Strassburg in the same year. Takagi attended it and gave a report in French "On some general theorems in the theory of algebraic numbers" which appeared in the Proceedings of this Congress as an article of only 4 pages, containing, however, in addition to all the essential results of his theory, a presentation of an important problem to extend this theory to the general galois, not necessarily abelian, case; which is still unsolved today. Unfortunately, the reconciliation between France and Germany was not yet perfect at that time, and the German mathematicians were not invited to this Congress. As the algebraic number theory was not

much studied then outside Germany, the report of Takagi at Strassburg Congress did not have an immediate response. I suspect, however, that the problem posed by Takagi at that occasion attracted the attention of Artin, who had an idea to introduce a new kind of L-series for galois extensions generalizing Hecke's L-series for abelian extensions. Two years later in 1922, Takagi published his paper "On the law of reciprocity for arbitrary algebraic number fields" based on the results of his theory. Suggested by the form of the reciprocity law given in this paper, Artin has found the form of the general reciprocity law which retains his name, from which the classical reciprocity laws for power residues would follow as simple corollaries and the new kind of L-series for galois extensions would be expressible by Hecke's L-series. Thus Artin published in 1924 his idea of new L-series together with the formulation of his general reciprocity law, proposed then as a conjecture. As well-known this law makes the isomorphism theorem of Takagi's class field theory more precise by means of the "Frobenius automorphism", concerning

which, F'robenius had given a conjecture in 1896. This conjecture was proved by Tschebotareff in 1926 using the so-called "crossing method with cyclotomic extensions" as was remarked by Schreier in the same year and which enabled Artin to prove his general reciprocity law by the same method in 1927. As soon as his proof appeared in Hamburger Ab- handlungen, Takagi made a review article in the newly published journal (in Japanese) of Physico-Mathematical Society of Japan in which he expressed his admiration to Artin's paper as one of the most prominent results in recent times in the algebraic number theory. This was, so to speak, the corner stone of Takagi-Artin's class field theory.

On the other hand, the German mathematical Society asked Hasse to give a report on the class field theory in its annual meeting in 1925, which was published in 1926 in a written form, followed by a more detailed description with proofs published in the next year. This "Klassen- korperbericht" of Hasse contributed to the greater diffusion of this theory in world's mathematical community. (After the discovery by Artin of the general reciprocity law, the second part of this report appeared in 1930.)

May I speak now a little of myself ? I entered Todai as a student in 1926, just the year in which the first part of Hasse's report arrived

in Japan. I knew of course nothing of the class field theory when I entered Todai, but I was immediately fascinated by Takagi's lecture and

8 S. Iyanaga Memories of Professor Teiji Takagi 9

his personality, and after following his lessons for two years, I began to understand what was said in Hasse's report. In the third and last student year, each student had to join a seminar of a professor, in which he had to begin his research. I requested then Professor Takagi to join his seminar and was delighted in learning that I was admitted by him. Thus I began my research work in 1929, the year of my graduation. I remained in the graduate school for one and half more years after which I went over to Hamburg to study with Artin, where I had a good luck of making acquaintance with Chevalley.

I stayed in Europe in 1931-34, during which period an ICM was organized in Ziirich in 1932. Takagi was invited there as one of the vice- presidents (together with mathematicians like Hilbert and Hadamard. The president was, by the way, R. Fueter, veteran in the field of complex multiplications.) I participated in this Congress together with Chevalley who made acquaintance with Takagi at this occasion and handed over a copy of his thesis to Takagi after a suggestion of Artin. His thesis, in which he tried to arithmetize the proof of the class field theory (without, however, attaining then a complete success which he achieved later in 1940) appeared thus in 1933 in the Journal of the Faculty of Science of Todai, the same journal in which Takagi's main work had appeared in 1920.

In 1935, I was named associate professor at Todai and charged with direction of exercise of Takagi's lecture for the first year students on infinitesimal calculus. I was lucky, to make acquaintance with excellent students in this class: Kunihiko Kodaira, Yukiyosi Kawada, Kiyosi Ito ... who became later well-known mathematicians. They followed Takagi's lecture only for one year, but I am sure that they received his deep influence. His lecture was, by the way, published as a well-known book (in Japanese) "Kaiseki Gairon" (Course of Analysis) and served greatly for raising upward the general level of Japanese mathematics.

In Germany, the nazis took the power in 1933, and the exodus of European intellectuals to America began toward that time. Still earlier, the Japanese army had started its invasive maneuvers in China. The world in the 1930 was thus rather in a tumultuous situation. The World War I1 began in Europe in 1939, in which Japan engaged formally in December 1941. In 1940, Japan was still before the formal engagement in the War, and celebrated the so-called 2600th year of its foundation after traditional history. In that year, the government of Japan decerned

Takagi with a Cultural Medal, as its highest recognition for cultural achievement, as it noticed the respect and honor paid to him by the international scientific community.

I am sure that Takagi was well aware of the importance of what he had done, but we have never heard him speaking boastfully of his proper work. However, when we requested him just after he received the Cultural Medal, to talk on the memories of his research and of his views on the future of mathematical studies in Japan, he was willing to accept our proposal and gave a very interesting and important lecture: "Reminiscences and Perspectives (in Japanese: Kaiko to Tembo)" at the Department of Mathematics at Todai in December 1940. Fortunately, the content of this lecture was noted and published in 1941 and we can read it now in an appendix to his another masterpiece "Topics from the history of mathematics in the 19th century (in Japanese: Kinsei Sugaku Shidan)" in the Iwanami Collection of Classical Books together wirh comments by Mitsuo Sugiura. I have given an English translation of some passages of this lecture in my article "On the life and works of Teiji Takagi" in the second edition of his Collected Papers, but otherwise there has been only text in Japanese of this lecture until we could find quite recently a French text in an article entitled "Takagi Teiji et la dkcouverte de la thkorie du corps de classes" by Dr. Pierre Kaplan, Professor at the Faculty of Science of the University of Nancy, versed also in japanese language and staying in Japan, more precisely in Ebisu, Tokyo, since two years as French director of the Maison franco-japonaise, in the review "Ebisu" 110.16 published by this Maison. To those among you who are more familiar with French than with Japanese language, I would like to warmly recommend to look at that article. I would like also to read some passages of that lecture by Takagi in my English translation later in this talk.

In 1945 toward the end of World War 11, the house close to Todai, where Takagi's family had lived since 1902, was burnt down by American bombardment. Takagi was forced to retire to his native village in Gifu prefecture, but he came back to Tokyo after 2 years, and lived together with the family of his eldest son. In 1952, he had to endure the sad event of the death of his wife, who had given him a prosperous family. I believe his second portrait dates from that period. In 1955, ten years after the end of the War, we could organize an International Symposium on Algebraic Number Theory, Tokyo-Nikko, where we could invite ten

10 S. Iyanaga

mathematicians from outside Japan, including Artin, Chevalley, and Weil. Takagi attended this Symposium as honorary chairman. He did not give any official talk, but we were happy to see him enjoy private conversations with these mat hemat icians.

Five years later in 1960, we had to see him off from this earth at the age of 85 years. A few days after he had been hospitalized at the hospital of Todai stricken by a cerebral apoplexy, he went calmly away. His funeral service was very beautifully served without religion.

Now I would like to read some passages of his lecture "Reminiscences and perspectives" in my English translation as far as the time permits.

First from his reminiscences of his studies in Gotingen around 1900: "At the time when I studied in Germany, Gottingen was perhaps the only place in the world where research on algebraic number theory was going on. Thus, when I told Hilbert that I wanted to study this theory, Hilbert did not seem to believe me immediately ... He invited me one day to follow him on his way home. During our talk, I told him that I was studying the special case of "Kroneckers Jugendtraum" where the ground field is the Gaussian field, i.e. I was dealing with complex multiplication of the lemniscate function. He said to me: 'Oh! that's fine,' and stopped at the corner of the street crossing with Wilhelm- Weber Strasse where he drew on the earth two figures, one of a square and another of a circle, figures related to lemniscate function, which we find in the work of Schwarz, saying 'you certainly know this as you have studied with Schwarz.' I remember that place even now."

Next, from the reminiscences of the time of World War I: "I am of a nature which needs a stimulus in order to work. There are now quite a number of Japanese mathematicians, but in these days, we had few colleagues. Neither had I heavy duties. You might imagine that I did research on class field theory in those carefree days but it was not quite SO.

The World War I started in 1914. This gave me a stimulus, so to say a negative stimulus. No scientific message reached us from Europe for four years. Some said that this would mean the end of Japanese science. Some newspaper articles showed 'sympathy' with Japanese professors from losing their 'jobs.' This made me aware of the obvious truth that every researcher should make research for himself, independently of others. Possibly I would have done no research for myself, but for World War I."

Memories of Professor Teiji Takagi 11

Lastly, on the class field theory: "Concerning the class field theory, I should confess that I had been misled by Hilbert. Hilbert considered only unramified class fields. From the standpoint of the theory of algebraic functions which are defined by Riemann surfaces, it is natural to limit considerations to unramified cases. I do not know precisely whether Hilbert himself stuck to this constraint, but anyway, what he had written induced me to think so. However after the cessation of scientific exchange between Japan and Europe owing to World War I, I was freed from that idea and suspected that every abelian extension might be a class field, if the latter is not limited to the unramified case. I thought at first this could not be true. Were this to be false, the idea should contain some error. I tried my best to find this error. At that period, I almost suffered from a nervous breakdown. I dreamt often that I had resolved the question. I woke up and tried to recover the reasoning, but in vain. I made my utmost effort to find a counterexample to the conjecture which seemed all too perfect. Finally, I made up my theory confirming this conjecture, but I could never get rid of the doubt ..."

The talk of Professor Takagi continues still further, but I am afraid my time is running out. I have now to stop here my talk. Thank you very much for your kind attention!

12-4, Otsuka 6-chome Bunkyo-ku, Tokyo, 11 2-001 2 Japan


On Art in L-Funct ions

Maruti Ram Murtyl

1. Introduction.

The celebrated conjecture of Emil Artin about the holomorphy of his non-abelian L-series has inspired a vast amount of development in number theory, algebraic geometry and representation theory. There exist at least four different programs to approach this conjecture. Most notable amongst these is the famous Langlands program. One of the objectives of the Langlands program has been to create the theoretical framework with which to attack Artin's conjecture. The notions of base change, automorphic induction and converse theory provide the conceptual tools to be developed and applied towards this goal. There are already excellent descriptions of this approach in the literature, such as Gelbart [Gel, Murty [Mull, Prasad and Yogananda [PY] and the recent paper by Rogawski [Rog]. Therefore, we shall not deal with this approach in this survey.

A second method is the program initiated by Serre [Se2]. Indeed, Khare [Kh] has recently shown that Serre's conjectures imply Artin's conjecture for two-dimensional, complex, odd representations over Q. After the spectacular success of Wiles, Buzzard and Taylor [BT] proved a theorem that makes a significant advance towards the A5 case of Artin's conjecture. We will refer the reader to these papers as well as [ST] for an insight into these new padic methods. In this paper, we shall focus more on the analytic aspects of Artin L-series and describe the approach implied by the recently formulated conjectures of Selberg concerning general L-functions with Euler products and functional equations. At the end of the paper, we discuss certain group-theoretic considerations

Received November 12, 1998 Revised March 2, 1999

'Research partially supported by NSERC and a Killam Research Fellowship.

14 M. R. Murty

that give some information on the location of poles (if any) of Artin L-functions. These are the third and fourth methods implied above.

The Selberg conjectures predict a certain 'orthogonality' principle and a remarkable unique factorization theorem for decomposing general L-functions into 'primitive' functions from which both the Artin conjecture and the Langlands reciprocity conjecture (at least in the solvable case) follow. Perhaps a final resolution of Artin's conjecture will involve a marriage of the two approaches.

The group-theoretic approach initiated by Heilbronn [HI and Stark [St] fuses group theory and analytic number theory and has been successfully developed by Murty [VKM, Mu] and Foote-Murty [FM]. The new results of this paper are contained in section 3 where we push this theme further and refine the results of Foote-Murty [FM].

The notion of an Artin L-function can be defined for any global field. In the case of a function field over a finite field, the holomorphy of Artin L-series is known and is due to Weil [W]. However, the reciprocity conjecture of Langlands has been settled only for dimensions 1 (class field theory) and 2 (Drinfeld theory). It would be interesting to formulate the Selberg approach in the function field context. In this paper, we shall deal exclusively with the number field setting.

Let Klk be a Galois extension of algebraic number fields with Gal(K/k) = G. Let V be a finite dimensional vector space over C and 4 : G+GL(V) a representation. An Artin L-function is a meromorphic

attached to this data. For each prime ideal p of K , let

I, = {a E G : a(x) = x (mod p))

be its inertia group and

its decomposition group. The inertia group is a normal subgroup of the decomposition group and the quotient D,/I, is cyclic generated by the Frobenius automorphism a, . This automorphism has the property that

a,(x) G xN(P) (mod p)

where p = p n k and N is the absolute norm from k to Q. For any pl p, the F'robenius elements a, are well-defined modulo I, and are all conjugate.

When the inertia is trivial, which is the case for all p unramified in K, the conjugacy class of a, is called the Artin symbol a,, . In all cases, it is a well-defined conjugacy class modulo inertia. The Artin L-function is defined as

L(s, 4; Klk) = n det (1 - ~ ( ~ , ) N ( ~ ) - ~ I V ~ ~ ) - ~ . P

We may sometimes abbreviate this as L(s, 4). Since x = Tr determines q5 upto equivalence, we also write L(s, X; Klk) for L(s, 4; Klk) .

Artin [A] showed that these L-functions satisfy the following functorial properties:

If H is a subgroup of G and T is a representation of H, then

We have the famous Artin's Conjecture: for any irreducible 4 # 1, L(s, 4; Klk) extends to an entire function of s. In the case 4 is one-dimensional, Artin proved his conjecture by establishing what is now called the Artin reciprocity law. This states that if 4 is one- dimensional, there is a Hecke character ~4 of k such that

This theorem is considered to be one of the masterpieces of class field theory. It embodies all the classical reciprocity laws such as quadratic, cubic and higher power reciprocity laws.

Brauer[B] showed that L(s, 4; KIk) extends to a meromorphic function for all s E C using his famous induction theorem and Artin's reciprocity law. Brauer's induction theorem states that any character x of a finite group can be written as

where Hi's are nilpotent and $i are one-dimensional and the ni are

16 M. R. Murty

integers. This immediately implies Brauer's theorem since

L(s, X . Klk) = n L(S, lndEidi; ~ l k ) ~ ' i

= n L(S, h; K I K ~ ~ by (2) i

= ( s + ) by Artin reciprocity. i

Brauer's theorem also implies Artin's conjecture if G is nilpotent or supersolvable since in these two cases every character of G is monomial (see for example Serre [Ser]).

In the case of two-dimensional representations we have the important Langlands-Tunnel1 t heorem: if 4 is 2-dimensional with solvable image, then Artin's conjecture is true and in this case L(s, 4, Klk) is equal to the (Jacquet-Langlands) L-function of an automorphic r e p resentation of GL2(Ak), where Ak is the adele ring of k.

The Langlands reciprocity conjecture is also referred to as the Strong Artin Conjecture in the literature. This conjecture forecasts that for each irreducible representation 4 of G of degree n, there exists a cuspidal automorphic representation 7r = 7rd of GLn(Ak) such that

Some progress has been made towards this conjecture. For instance, there is the theorem of Arthur and Clozel [AC]: the strong Artin conjecture is true for nilpotent extensions. There is a recent theorem of Yuanli Zhang [Z]: the strong Artin conjecture is true if G is a Frobenius group. Some examples of the "icosahedral" case of Artin's conjecture are due to J. Buhler [B].

The strong Artin conjecture associates an automorphic form for each complex linear representation of GQ. In the other direction, there is the Deligne-Serre theorem: given any newform f of weight 1 and level N, there is a two dimensional GQ representation 4 such that

The converse theorem in the simplest case as described by Weil [W2] shows that Artin's conjecture for two dimensional Galois representations implies the strong Artin conjecture. A suitable converse theory would give in general that Artin's conjecture implies strong Artin conjecture.

See for example the paper by Ramakrishnan [R] for a discussion of this theme.

52. Analytic theory of Artin L-series.

We will now study the behaviour of Artin L-series at a given point in complex plane. These ideas originate with Heilbronn [HI, Stark [St] and Kumar Murty [Mu] and involve a beautiful interplay of character theory of finite groups and the Artin L-function formalism from which many important deductions such as the Aramata-Brauer theorem can be made. The most important of these deductions seems to be the relationship between the zeros of the Dedekind zeta function and Artin L-series.

Fix so E C. We want to study the behaviour of Artin L-functions at so. Let G = Gal(K/lc) and H a subgroup of G. We define the function

where n+ = ord,,,, L(s, 11; K/KH).

Since L(s, $; K/KH) = L(s, IndE$; K/k) we get

Proposition 1 (Heilbronn-Stark) .

Proof. By F'robenius reciprocity,

This completes the proof.

18 M. R. Murty

Now recall the Artin-Takagi factorization:

We can use this to deduce:

Corollary 2. If H is abelian,

l e ~ ( g ) l I ord,=soC~(s).

Theorem 3 (Kumar Murty).

Proof. We have,

This completes the proof.

Corollary 4. CK (s)L(s, 4; Klk) is regular for s # 1

Proof. Let us write x for the character of the representation 4. By the inequality of Theorem 3, we have lnxl 5 ord,,,, CK (s) from which the result is immediate.

Corollary 5. L(s, 4; Klk) is analytic and non-zero for Re(s) = 1.

Proof. Since the Dedekind zeta function doesn't vanish on the line Re(s) = 1, we get immediately that every Artin L-function is regular on the line Re(s) = 1.

Corollary 6 ( Aramata-Brauer). CK (s)/Ck (s) is entire.

Proof. Since the trivial character corresponds to the zeta function of the base field, the result is immediate from Theorem 3.

Corollary 7. All the poles (if any) of an Artin L-function L(s, 4; K/k) are contained in the zeros of CK (s) .

Proof. This is clear from Theorem 3.

Corollary 8 (Stark). If s = so is a zero of CK (s) of order 5 1, then any Artin L-function is analytic at s = so.

Proof. Any simple zero of CK ( s ) must arise from at most one character. If this comes from a pole of an Artin L-series attached to a non- abelian character, then the Artin-Takagi facatorization gives a contradiction, for then we would have a pole of CK(s) at a point s unequal to 1 and this is a contradiction. Thus the zero must come from an abelian L-series and by the reciprocity law, the L-series is a Hecke L-series which is entire. This completes the proof.

Corollary 9 (Foote-K. Murty). If K / k is solvable with group G and s = so is a zero of CK(s) of order 5 p - 2 where p is the second smallest prime divisor of (GI, then any Artin L-function L(s, 4; K/k) is analytic at s = so.

Proof. We refer the reader to [KM].

Corollary 10. If RH holds for CK(s), then Artin L-functions are analytic for Re(s) > 112.

Proof. This is clear from Theorem 3.

In section 3 below, we will discuss variations on this theme.

Brauer's theorem also implies non-vanishing of Artin L-functions on the line Re(s) = 1 since Hecke's L-functions don't vanish there. However, the non-vanishing of Artin L-functions can also be deduced from Theo- rem 3 above. The classical trigonometric inequality 3+4 cos O+cos 20 2 0 can be used to show the non-vanishing of the Dedekind zeta function on

20 M. R. Murty

the line Re(s) = 1. From Theorem 3, the holomorphy and non-vanishing of Artin L-functions on the line Re(s) = 1 is now immediate. (The reader may consult the monograph [MM2] for details.) By the Wiener-Ikehara Tauberian theorem, this implies

as x-oo and x irreducible # 1.

One can also derive the Chebotarev density theorem from this. For each conjugacy class C of G, set

nc(x) = #{N(p) 5 x : p unramified, op E C }

and

so that the orthogonality relations give

where g c E C. We then have the Chebotarev density theorem. As

where ~ ( x ) is the number of primes 5 x. This theorem generalizes Dirich- let's theorem about primes in arithmetic progressions. One important consequence of the Chebotarev density theorem is that if $1 and 4 2 are two representations of G and

for all but finitely many ideals p of I c , then 41 and 4 2 are isomorphic.

In many questions of analytic number theory, one needs a stronger version of this theorem with effective error terms. The relationship between the effective versions of Chebotarev density theorem and Artin's conjecture is exemplified by the following theorem.

Theorem 11 (Effective Chebotarev) : (K. & R. Murty, Sa- radha [MMS]). Assume Artin's conjecture and that CK(s) has all its

zeros on Re(s) = 112, then

where

and P(K/k) is the set of rational primes p for which there is a prime ideal plp in k which ramifies in K.

The above discussion leads to the following question. Does the Rie- mann hypothesis for Dedekind zeta functions imply Artin's conjecture? K. Murty's theorem shows that if there are any poles, they must be on the line Re(s) = 112. It seems difficult to answer this question. How- ever, recently in joint work with A. Perelli, it was proved that a 'pair- correlation' conjecture for zeros of functions in the Selberg class leads to a proof of the Selberg conjectures. This in turn implies Artin's conjecture and the Langlands reciprocity conjecture (in the solvable case) as described by Murty MU^].

We begin by describing the Pair Correlation Conjecture. As- sume RH for the Riemann zeta function. Recall the pair correlation conjecture formulated by Montgomery [Mo]. Write 112 + iy for a typical zero of [(s). If 0 $ [a,P] and T-oo, then

This is consistent with the pair-correlation function of eigenvalues of a random Hermitian operator.

Selberg's Class consists of Dirichlet series

which are absolutely convergent in Re(s) > 1 such that (s - l )mF(s) admits an analytic continuation to an entire function for some nonnegative integer m, and satisfies a functional equation and an Euler

22 M. R. Murty On Artin L-Functions

product condition. More precisely, the functional equation has the form:

where Iwl = 1, Q > 0, and Xi > 0, Re(pi) 2 0 and the Euler product is of the form

F(4 = n FP(s)

P

with

and b(pk) = O(pke) for some 0 < 8 < 112. In addition, a l (F) = 1 and a,(F) << n'.

An element F in the Selberg class S is called primitive if it cannot be factored as a product of two elements non-trivially. Selberg [Sell proved that every element of S can be written as a product of primitive elements. Is this factorization unique? Selberg conjectures yes.

For two primitive functions Fl and F2, Selberg conjectures

This conjecture implies the unique factorization conjecture (see [CG] and MU^]).

Theorem 12 (R. Murty). Selberg7s conjecture implies Artin7s conjecture and for K/Q solvable, it implies the strong Artin conjecture.

Proof. We refer the reader to [Mu21 for details. We outline the highlights below. By the classical factorization of Artin and Takagi, we have, on the one hand,

and on the other, by the result of Arthur and Clozel [AC]

Selberg conjectures imply each L(s, 4; K/Q) is entire and primitive. (This is an application of Chebotarev density theorem.)

Selberg conjectures also imply L(s,7ri) are primitive and so by unique factorisation, we get that each L(s, 4; K/Q) is an L(s, ri) which is the strong Artin conjecture. This proves Theorem 12.

For a primitive function F, it seems reasonable to conjecture that if

0 4 [a, PI, then

There should be very little correlation between zeros of two distinct primitive functions.

Turning to the analogue of the pair correlation function of Mont- gomery, for functions in S, we define for F, G E S, the correlation function

7r -TF,G(~, T) = - ~ ~ ~ ( Y F - Y c )

T log X ~ ( Y F - YG) - T ~ Y F , Y G ~ T

d where w(u) = 4/(4 + u2), dF = 2 x i = , Xi , X = Td', 112 + iyF and 112 + iyG run through the zeroes of F and G respectively. Now assume the GRH for each F E S . The pair correlation conjecture is that for F, G primitive functions of S, we have

uniformly for a in any bounded interval. (Here 6F,c = 1 if F = G and 0 otherwise.) This conjecture implies that almost all zeros of a primitive function are simple [MP].

Theorem 13 (R. Murty and A. Perelli). Assume that each el- \'I

ement of the Selberg class satisfies the ~ i e m a n k hypothesis. In addition,

24 M. R. Murty On Artin L-Functions

suppose a pair correlation conjecture for primitive elements. Then Sel- berg's conjectures are true and thus the unique factorization conjecture follows.

Therefore, RH and a form of the pair correlation conjecture for elements of the Selberg class imply Artin's conjecture and the strong Artin conjecture for solvable Galois extensions of Q.

$3. Variations.

The above discussion can be generalised in various ways. We begin by describing a general formalism first outlined in [ M M l ] . Let G be a finite group. For every subgroup H of G and complex character $ of H, we attach a complex number n(H, $) satisfying the following properties:

Defining

where the sum is over all irreducible characters $ of H leads to the generalisation of the Heilbronn-Stark lemma above. Namely, OG I H = OH. And this leads, in a purely formal way to the fundamental inequality

whenever n ( H , $) >_ 0 for every cyclic subgroup H of G and reg denotes the regular representation. The formalism can be applied to the case G is the Galois group of a normal extension K/k and n(H, $) is the order of the zero at s = so of the Artin L-series attached to the Galois extension K / K ~ . It can also be applied to the situation when E is an elliptic curve over k and n(H, $) corresponds to the order of the zero at s = so of a twist by $ of the L-function of the elliptic curve over K ~ . This was addressed in [ M M l ] and we refer the reader to that paper for further details. We will sharpen the above discussion in the following way.

It is clear that

Since OG I H = OH, we observe that

If we suppose that for every cyclic H , we have n(H, 1) 2 n(G, l ) , then

This proves:

Theorem 14. Suppose n(H, 1) 2 n(G, 1) for every cyclic H. Then

C I+, x)12 5 (n(G, reg) - n(G, 1N2. x# 1

Applying this to the context of zeros discussed in the previous section gives the following sharper form of Theorem 2.

Corollary 15. Assume [ K ~ ( s ) / C k ( s ) is regular at s = so for every cyclic subgroup H of G. Then,

The hypothesis of Corollary 15 is satisfied when K/k is solvable (see [Mu], Uchida [U] or van der Waall [VW] ) . This gives

Theorem 16. Let K/k be solvable. Then,

Another variation is obtained by combining two functions in the above discussion. Suppose we have numbers ni(H, $) satisfying (3) and (4). Let

Then, we easily see directly, or by applying the Cauchy-Schwarz inequal-

26 M. R. Murty On Artin L-Functions 27

Proposition 17. Suppose n i ( H , +) 2 0 for every cyclic subgroup H of G and every irreducible + of H . Then,

Applied to zeros, this leads to

Corollary 18. For so, sh E @, we have

The corresponding version of Theorem 14 is:

Theorem 19. Suppose n i ( H , 1 ) 2 ni (G, 1) and ni (H, $) 2 0 for every cyclic subgroup H of G. Then

Corollary 20. Suppose CKH ( s ) /Ck(s ) is regular at s = so, sh for every cyclic subgroup H of G. Then

Corollary 21. If K / k is solvable, then

An interesting special case is if sh = 1 - so. Then, by the functional equation, nx(l - so) = nT(so) so we deduce:

Corollary 22. If Kllc is solvable, then

To conclude, let us observe that Theorem 16 implies that all the poles (if any) of a non-abelian Artin L-function corresponding to a solvable extension K / k are contained in the zeros of CK(s)/Ck(s). One can

also deduce the same result provided that the base change of a zeta function to any arbitrary finite extension of Q does indeed correspond to an automorphic representation as predicted by Langlands. At present, this is not known.

Acknowledgements. I would like to thank Hershy Kisilevsky, Ku- mar Murty, C.S. Rajan and Y. Petridis for their comments on a preliminary version of this paper.

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[Zh] Y. Zhang, A reciprocity law for some F'robenius extensions, Proceed- ings of the Amer. Math. Society, 124 no.6 (1996), 1643-1648.

Department of Mathematics Queen's University Kingston, K7L 3N6 Ontario, Canada E-mail address: murtyamast . queensu . ca


How Hasse was led to the Theory of Quadratic Forms, the Local-Global Principle, the Theory of the Norm Residue Symbol, the Reciprocity Laws, and to

Class Field Theory

Giint her Frei

1 Representation by Rational Quadratic Forms over Q.

1. Hasse began his studies during the First World War on the 27th of September 1917 at the University in Kiel, where Otto Toeplitz was his principal teacher. After the war Hasse moved to Gottingen where he registered a t the Georg-August University on the 16th of December 1918. At that time Gottingen was the center of mathematical research, not only in Germany but worldwide. The three main chairs for pure mathematics were occupied by Hilbert, Hecke (when Hecke left to Ham- burg in 1920, Courant, who was Extraordinarius since 1918, became Hecke7s successor) and Landau (see [Scha- 19901). Emmy Noet her was Extraordinaria (associate professor). When Hecke, Hasse7s most influen- tial teacher, was appointed to the newly founded University of Hamburg in the spring of 1920, Hasse decided to leave Gottingen. He exmatricu- lated on the 23rd of March 1920 and moved to Marburg in order to study under Kurt Hensel the theory of padic numbers, introduced by Hensel in a short note in 1897 (see [He-18971). This decision was taken after Hasse had acquired, while still in Gottingen, Hense17s book "Zahlenthe- orie7' (see [He-1913)] on the 20th of March 1920. In this book Hensel developed in more detail the theory of padic numbers for the rational numbers Q. He had already presented a thorough introduction to algebraic number theory and padic and, more generally, to T-adic numbers, for an algebraic number field K with respect to a prime divisor p dividing n E K exactly to the first power, in the book "Theorie der algebmischen Zahlen" (see [He-19081).

Received July 12, 1998. Revised December 9, 1998.

How Hasse was led to 33

2. Hasse registered at the University of Marburg on the 11th of May 1920 in order to acquire the last two out of a total of eight semesters required before he was allowed to graduate as a Doctor of Philosophy. Already at the end of May 1920 Hensel suggested to Hasse, as a research subject for the doctoral dissertation, to continue the investigations begun by Hensel in the last chapter (Zwolftes Kapitel) of his book "Zahlen- theorie" (see [He-19131) on the conditions under which an integer or a rational number can be represented by a binary (see [He-19131, Zwolftes Kapitel, 53 and 57) or ternary (see [He-19131, Zwolftes Kapitel, 54) rational quadratic form (i.e. with rational coefficients) over the rational numbers Q. Hasse was to examine whether the necessary conditions, given by Hensel by means of padic numbers (see [He-19131, pp. 312-314 and pp. 326-336) for the representation of an integer (or a rational number) m by a rational diagonal1 binary quadratic form f (x, y) = ax2 + byz or a rational diagonal ternary quadratic form f (x, y) = ax2 + by2 + cz2 with rational numbers x, y, z E Q, are also sufficient and he was to look for analogous conditions for quaternary quadratic forms.

3. Already on the 26th of May 1921 Hasse graduated as "Doc- tor Philosophiae" (Ph.D.) with the thesis entitled "Zur Theorie der quadmtischen Formen, insbesondere ihrer Darstellbarkeitseigenschaften im Bereich der rationalen Zahlen und ihrer Einteilung in Geschlechter" , where Hasse solved the problem posed by Hensel completely. Not only did he solve it for ternary and quaternary rational quadratic forms but for any rational quadratic form in n variables over the rational numbers.

The clue for the solution was suggested to Hasse by Hensel (see [Ha- 19751, Volume I, pp. VIII-IX) on a postcard dated 2nd of October 1920. Hasse soon discovered that the solution for binary quadratic forms is given by a reduction principle going back to Lagrange. He found it in the lectures on number theory by Dirichlet, edited by Dedekind, " Vorlesun- gen uber Zahlentheorie", Vierte Auflage, Vieweg, Braunschweig, 1894 (see [DD-18941, $157, in particular p. 428). It is possible that Hasse was led to this reference by a remark made by Minkowski in [Mi-18901, p. 13 (Ges. Abh., Bd. 1, p. 227).

4. Dedekind says there on p. 422, $156 and on p. 428, $157, that

Theorem 1. A necessary condition for the existence of a proper solution x, y, z E Z, that is with x, y, z relatively prime, to the equation ax2 + by2 + cz2 = 0 (with a, b, c E Z, square free and relatively prime)

' ~ n ~ rational quadratic form is rationally equivalent to a diagonal quadratic form (see [He-19131, p. 296).

is that -bc, -ca, -ab are quadratic residues for a , b, c respectively and that a , b, c cannot all have the same sign,

and then continues on p. 428, $157 by saying that, because of a reduction principle going back to Lagrange one can prove that

Theorem 2. The conditions in Theorem 1 are also suftcient.

Namely, he writes "Mit Hulfe einer Reductionsmethode, welche im Wesentlichen von

Lagrange herruhrt, l&st sich nun wirklich beweisen, dass also folgender Satz besteht:

Sind a, b, c drei von Null verschiedene und durch lcein Quadrat theil- bare relative Primzahlen, welche nicht alle dasselbe Vorzeichen haben, und sind die Zahlen -bc, -ca, -ab resp. quadratische Reste der Zahlen a, b, c, so ist die Gleichung ax2 + by2 + cz2 = 0 eigentlich losbar."

Dedekind refers to - Lagrange: Sur la solution des probl2mes inde'terminks du second degrk. M6m. de 1'Acad. de Berlin, tome XXIII, 1769. (Buvres de Lagrange, tome 11, 1868, p. 375). - Lagrange: Additions a m ~le'ments d7Alg6bre par L. Euler, 5. V. - Legendre: The'orie des Nombres, 3me kdition, tome I, $5. 111, IV. - Gauss: Disquisitiones Arithmeticae, artt. 294, 295.

As a matter of fact, this Theorem by Lagrange was the cornerstone for Legendre's proof of the quadratic reciprocity law (see [Fr-19941, pp. 73-74 and compare also with Hensel's proof in [He-19131, pp. 324- 325). However, Legendre's proof was correct, as Gauss pointed out (see [Ga-18011, art. 151 and artt. 296-297), only in two out of eight cases. In the other six cases Legendre made use of the theorem (proved only later by Dirichlet in 1837) that an arithmetic progression prime to a given modulus contains at least one prime number (see [Fr-19941, p.74).

Dedekind makes use of this theorem in order to prove Gauss' fundamental theorem stating that each proper class of proper integral binary quadratic forms of a given discriminant d lying in the principal genus is the square of a proper class (of binary quadratic forms) of discriminant d.

5. Dedekind's proof of Theorem 2, based on Lagrange's reduction, 2 runs by induction on what Dedekind calls the index. If f (x, y) = ax +

by2 + cz2 is the given quadratic form for which one can suppose that la( 5 lbl 5 Icl, then the one of the three numbers out of lab(, (ca(, (bcl

.' which lies between the other two is called the index J of f , that is J = Ical. Dedekind first shows that the theorem is correct if the index Jisequal to 1, because then la1 = Ibl = I c I = 1. Hence x = y = 1, z = O

34 G. Frei How Hasse was led t o 35

gives a solution if, for instance, a and b have different signs. Thereafter he proceeds as follows.

If J 2 2, then the condition that -ab is a quadratic residue of c implies ar2 + b = CC for integers T and C with 11.1 5 +lcl and (CI < J. If a' is the greatest common divisor of ar2, b and cC (if C # 0), we put b' := and c' := - where y2 is the biggest square contained

in . Then a', b', c' satisfy all the conditions of Theorem 1, but the

corresponding equation a'x12 + bly12 + c 'z '~ = 0 has an index J' which is strictly smaller than J, hence by induction admits a solution x', y', z' E Z. From this solution one can construct a solution x, y, z E Z of the original equation ax2 + by2 + cz2 = 0.

6. When Hasse wrote to Hensel that he found the solution to the problem by means of a reduction going back to Lagrange and presented by Dedekind in $157 of the fourth edition (1894) of the lectures on number theory by Dirichlet and augmented by Dedekind, but that he could not see any connection with the padic numbers, Hensel wrote back on this postcard (see [Ha-19751, Volume I, pp. IX) , given here in a free translation:

He [that is Hensel] always thought that there is the following underlying question. If one knows that an analytic function has a rational character [i.e. is rational, that is admits a Laurent series with finite principal part] at each place, then it is a rational function. If one knows that a number has a rational character [i.e. is rational, that is admits a padic expansion with finite principal part] for each finite prime p and for the infinite prime p,, then this does not imply that it is a rational number. How does this have to be completed?

Hensel had already began to examine this question at the end of his book of 1913 (see [He-19131, Zwolftes Kapitel, $7, pp. 336-337).

This remark made Hasse realize that the conditions in Lagrange's theorem (Theorem I), namely that

-bc, -ca, -ab are quadratic residues for a , b, c respectively and that a, b, c cannot all have the same sign, where a, b, c are square free integers relatively prime to each other,

can be interpreted as follows with the help of Hensel's criteria given in Hensel's book of 1913 for the representation of 0 by a ternary diagonal quadratic form in the field of padic numbers (see [He-19131, Zwolftes Kapitel, 54, in particular p. 312) :

The ternary quadratic f o m j(x, y) = ax2 + by2 + cz2 represents 0 (non trivially) over the p-adic numbers for each prime p.

The criteria actually given in [He-19131, p. 312 has the following form:

Theorem 3. The ternary quadratic form f (x, y) = ax2 + by2 + cz2 with rational numbers a, b, c not all of even , resp. of odd order, represents 0 (non trivially) over the p-adic numbers for an odd prime p i j and only if at least one of the three Legendre symbols

is equal to one.

A similar criterion is given for the prime p = 2.

Hence Theorem 1 and Theorem 2 can be given the following form, which, according to Dedekind's proof, is now obtained by means of La- grange's reduction (see [Ha-1923a1, (11) F'undamentalsatz, pp. 130-131):

Theorem 4. The ternary quadratic form f (x, y) = ax2 + by2 +cz2 represents 0 (non trivially) over the rational numbers Q if and only if it represents 0 (non trivially) over the p-adic numbers Q(p) for each prime p (finite and infinite).

This is how Hasse was led to the general Local-Global-Principle, called Fundamental Theorem by Hasse, for the representation by quadratic forms (see [Ha-1923a1, (11) F'undamentalsatz, p. 130):

Theorem 5. A rational number m f Q is represented by a rational quadratic form f over the rational numbers Q if and only if m is represented by f over all p-adic fields (i. e. completions) Q(p) for p finite and infinite.

The solution of Hensel's representation problem was thus reduced to (1) the proof of Theorem 5 and to (2) give criteria for the representation of m E Q by f in Q(p) for each prime p (finite and infinite).

7. The results of Hasse's thesis were published in Crelle's Journal (Journal fiir die reine und angewandte Mathematik), of which Hensel was the chief editor, under the title " ~ b e r die Darstellbarkeit von Zahlen dzlrch quadratische Fomen zm Korper der rationalen Zahlen" (Crelle 152 (1923), 129-148; see [Ha-1923al).

36 G. Frei How Hasse was led to 37

There Hasse first remarks that the representation of a rational number m E Q by a quadratic form can be reduced to a representation of 0 by another quadratic form (see [Ha-1923a1, (IV), p. 133 and also [He-19131, Zwolftes Kapitel, $5, p. 314 and $6, p. 325):

Theorem 6. A rational number m E Q, m # 0, is represented by a mtional n-ary quadratic form f = f (xl, ... , x,) over the mtional numbers Q, resp. over the field of p-adic numbers Q(p), if and only if 0 is represented (non trivially) by the n + 1-ary quadratic form F =

f (xl, ..., x,) - mx2 over Q, resp. over Q(P).

Then Hasse interprets Hensel's results for binary quadratic forms (see [Ha-1923a], Satz 4, p. 135; see also [He-1913], p. 312):

Theorem 7. A ternary rational quadratic form f represents 0 over Q(p) if and only if a diagonal form fo = alx2 + azy2 + a3z2 rationally equivalent to f has the trivial symbol, now called Hasse symbol:

The symbol ( y ) , now called (quadratic) Hilbert symbol, was introduced by Hilbert in his 'Zahlbericht' (see [Hi-18971, $64) as a norm symbol modulo p. But later it was defined by Hensel with the help of his padic numbers (see [He- 19131, Zwolftes Kapitel, 55, p. 315):

Definition 8.

+I if b = x2 - ay2 is solvable with x, y E Q(p), -1 otherwise.

In the case of a binary quadratic form Hensel called cp := ($) the

character with respect to p of the binary quadratic form f = ax2 + bxy + cY2 with discriminant d := b2 - 4ac (see [He-19131, Zwolftes Kapitel, 57, p. 327), a term taken from Gauss7 theory of binary (integral) quadratic forms (see [Ga-180 11, art. 230 or also [F'r- 19791).

The notation cp was chosen by Hasse probably following Minkowski who introduced the notation Cp to denote the Minkowski invariant for rational quadratic forms. This invariant Cp takes on the values +1 or -1 for each prime p and it is invariant under rational equivalence (see [Mi-18901, p. 6; Ges. Abh., Bd. 1, p. 220). Hasse was always very careful in choosing his notations.

For the Hasse symbol one has the Product Theorem (see [Ha-1923a], formula (4.), p. 135 and also [He-19131, Zwolftes Kapitel, 56, pp. 321- 322) and compare with [Mi-18901, formula (7), p. 18; Ges. Abh., Bd. 1, p. 232):

Theorem 9.

the product taken over all primes (finite and infinite).

This property is due to Hilbert's Reciprocity Law for the Hilbert symbol (see [Hi-18971, $69, Hilfssatz 14 and [He-19131, Zwolftes Kapitel, 56, pp. 321-322):

Theorem 10.

the product taken over all primes (finite and infinite)

Then Hasse derives the Local- Glo bal- Principle for the representation of 0 by a ternary rational quadratic form (Fundamentalsatz) and hence solves Hensel's problem for binary rational quadratic forms (see [Ha- 1923a], Satz 6, p. 135):

Theorem 11. A ternary rational quadratic form f represents 0 (non trivially) over the rational numbers Q if and only iff represents 0 (non trivially) over the p-adic numbers Q(p) for all primes p (finite and infinite), that is if and only if cp(f) = 1 for allp.

For the proof Hasse follows Dedekind7s proof of Theorem 2 step by step, that is he uses the reduction procedure of Lagrange.

Then Hasse shows that this Local-Global-Principle for ternary rational quadratic forms with respect to the representation of 0 is true for any n-ary rational quadratic form (see [Ha-1923a1, Satz 14, p. 142 and Satz 21, p. 146).

92. Equivalence of Rational Quadratic Forms over Q.

1. Already on the 8th of December 1921 Hasse was in a position to

;* hand in his habilitation paper which was dedicated to the problem of equivalence of rational quadratic forms over the rational numbers. He was led to this problem by his thesis where he already had to consider equivalence of quadratic forms over Q and over Q(p) (see [Ha-1923al).


In the thesis he also already found some necessary conditions for the equivalence of two quadratic forms over Q and over Q(p) (e.g. [Ha-1923al 5§4,5,7,14 and Tabellen 1) and 2)). In order to determine the equivalence over Q for two rational quadratic forms, Hasse again made use of Hensel's padic numbers and discovered another Local- Glo bal-Principle (see [Ha- l923b], (II.), p. 208):

Theorem 12. Two rational quadratic forms f and g are equivalent over the rational numbers Q if and only if they are equivalent over the p-adic numbers Q(p) for all primes p (finite and infinite).

Hence the rational equivalence of two rational quadratic forms was reduced to (1) the proof of Theorem 12 and to (2) give criteria or invariants for the equivalence of two rational quadratic forms f and g over Q(p) for all primes p (finite and infinite).

2. Inspired by Minkowski's work on integral quadratic forms (see [Ha-1923b1, pp. 206-208 and [Mi-18901, Ges. Abh., Bd. 1, p. 219ff.) and starting from his own results presented in his thesis, Hasse was able to give a full set of invariants for the equivalence of rational quadratic forms over Q(p). To the already known invariants, i.e. the number of variables n, the discriminant d and the Sylvester index c,, = J of a rational quadratic form f , Hasse added the now so called Hasse symbol c,(f) (see [Ha-1923b1, pp. 216-217 and [O'M-19631, p. 167) to obtain a full set of invariants. This Hasse symbol is a product of Hilbert symbols (see [Ha-1923b1, formula (23.), p. 217) and hence is expressible in terms of padic numbers because of Hensel's general definition (Definition 8) of the Hilbert symbol by means of padic numbers. It satisfies the Product Law (see [Ha-1923b1, formula (30.), p. 219):

Theorem 13.

the product taken over all primes (finite and infinite).

These results were published in Volume 152 (1923) in Crelle's Jour- nal, pp. 205-224 (see [Ha-1923b]), in the same volume which contained already the results of his thesis (pp. 129-148).

53. Rational Quadratic Forms over a Number Field K. Qua- dratic Norm Residue Symbol in K.

1. In the following Volume 153 (1924), pp. 12-43, Hasse extended his discoveries on the necessary and sufficient conditions for the representation of a rational number by a rational quadratic form over Q and the equivalence of rational quadratic forms over Q, first to the more general case of symmetric matrices with coefficients in Q (see [Ha-1924a]), and then in two other papers in the same volume (pp. 113-130 and pp. 158- 162) to the case where the ground field Q is replaced by any algebraic number field K (see [Ha-1924~1 and [Ha-1924dl).

2. In the last two papers Hensel's p-adic numbers for a number field K and the Local-Global-Principle again turn up as the fundamental tool for the investigations. In addition, completely new tools had to be introduced, namely the theory of Hilbert and F'urtwangler on Hilbert's quadratic norm residue symbol (y) in a number field K, Weber's generalization (1897) to number fields K of Dirichlet's theorem on the existence of primes in arithmetic progressions and the (General) Quadratic Reciprocity Law by Hilbert in a number field K , that is

Theorem 14.

where p runs over all prime divisors in K (finite and infinite).

They play a central and crucial r6le in Hasse's paper on the representation theory of quadratic forms in a number field K (see [Ha-1924~1, in particular p. 114).

Hasse needs the quadratic Hilbert reciprocity law in K in order to establish the Local-Global-Principle for the representation of 0 by a quaternary rational quadratic form over K.

In the case of the representation of 0 by a ternary rational quadratic form over K the Local-Global-Principle is obtained from the following (norm) theorem of Furtwangler (see [Ha- 1924~1, 55, p. 122 and [F'u-19 131, Satz 118, p. 429):

Theorem 15. Let a, P be two integers in a number field K. If

for all prime divisors p in K (finite and infillite), then ,O is a relative norm of a number in K(+).

This theorem is the very last theorem in the series of three papers by Furtwangler on the reciprocity laws (see [Fu-19091 , [Fu-19121, [Fu-19131). Furtwangler deduced it by following the footsteps of Gauss (see [Ga- 18011, artt. 262, 286,287), namely from the Hilbert quadratic reciprocity law in number fields K via the theorem on the existence of genera and the generalization to K of what we consider today as one of the main theorems of class field theory of quadratic extensions. It asserts that each (proper) class of (proper integral) quadratic forms in the principal genus (with given discriminant) is the square of a (proper) class.

As for the criteria for the representability in a local field K(p) of 0 by a binary, ternary or quaternary quadratic form Hasse heavily builds on his paper [Ha-1924b1, his joint paper with Hensel [HH-19231 as well as on Hensel's paper [He-19221 on the explicit description of the norm residue symbol ( ).

3. So we see that Hasse's generalization of his theory of rational quadratic forms over Q to quadratic forms over a number field K naturally led him to look for an explicit determination of the (quadratic) norm residue symbol in K , to study the quadratic Hilbert reciprocity law in K and to make use of Weber's generalization to number fields K of Dirichlet's theorem on primes in arithmetic progressions. All these theorems are closely related to class field theory over K.

In fact, Furtwangler already in 1909 stressed the close connection between the reciprocity law in K and the theory of the (Hilbert) class field over K (see [Fu-19091, p. 5) and he made the interesting remark (see [Fu-19091, p. 2) that it was the quadratic reciprocity law in number fields K whose class number is divisible by 2 that led Hilbert to sketch a general theory of class fields for relatively abelian number fields (see [Hi-19021) on the basis of his theory of relatively quadratic fields (see [Hi-18991).

So it seems that Takagi must have misunderstood Hilbert's intention when he says that Hilbert misled him on how to develop class field theory (see [Ka-19771, p. 4).2

4. Let us add what Hasse himself has to say in his paper [Ha-1924~1 on p. 114 (the number field is called k instead of K):

"Meine Entwicklungen fui3en vor allem auf den Satzen uber das quadratische Hilbertsche Normenrestsymbol ( ) , die von Hilbert-

Furtwangler in deren Arbeiten uber die Reziprozitatsgesetze und Klas- senkorper erhalten sind und neuerdings von Herrn Hensel und mir

'1 would like to thank Pierre Kaplan for bringing Takagi's complaint to my attention.

How Hasse was led to 4 1

auf Grund der Henselschen Methoden in der algebraischen Zahlentheo- rie von anderen Grundlagen ausgehend behandelt und erweitert werden. Insbesondere lege ich das allgemeine quadratische Reziprozitatsgesetz in der Hilbertschen Fassung

zugrunde, das zum Beweis meines Prinzips (I.)3 verwendet wird, ebenso zu demselben Zweck den auf algebraische Korper verallgemeinerten Satz von den Primzahlen in einer arithmetischen Reihe, also die Tatsache, daB in jeder Idealklasse im allgemeinsten Sinne eines algebraischen Zahl- korpers unendlich viele Primideale vorhanden sind. Die Notwendigkeit - der Verwendung der in diesen beiden Satzen steckenden transzendenten4 Methoden zum Beweis des Prinzips (I.) im Gegensatz zu allen ubrigen, rein arithmetischen Entwicklungen dieser Arbeit scheint mir in der Natur der Sache zu liegen. Es sol1 aus der Moglichkeit gewisser Beziehungen fur jeden einzelnen Primteiler p von k auf das Bestehen dieser Beziehungen in k selbst, d. h. fur die Gesamtheit aller p geschlossen werden. Es ist daher naturlich, dai3 hierbei Betrachtungen uber die "Dichtigkeit" von Primteilern p gewisser Eigenschaften hineinspielen, wie sie doch den genannten transzendenten Beweisen eigentiimlich sind."

J

' $4. 1-th Degree Norm Residue Symbol in a Number Field K. 1

1. Almost on the same day when Hasse handed in his habilitation $ paper, Hensel had finished his article on norm residues in general rela- y ~ ;

tively abelian number fields which we already mentioned (see [He-1922]), namely on the 5th of December 1921. Hilbert had developed this the-

' ory in the case where the ground field K is an algebraic number field and the corresponding (relative) field extension L / K is a quadratic or a

, Kummer extension. In this paper, Hensel extended Hilbert's theory to the case where L/K is any finite abelian extension of algebraic number fields (see [He- 19221).

2. Hensel first gives a new definition of a norm residue (see [He- 19221, p. 2):

Definition 16. Let L / K be a finite algebraic extension of algebraic number fields, p a prime divisor in K and !J3 a prime divisor of p

3that is, the Local-Global-Principle 4that is, analytic


lying in L. Let K(p) be the completion of K with respect to p and L(Y) the completion of L with respect to Y.

Then a number ,6 E K K(p) is called a norm residue of L with respect to p, if there exists a number B E L(Y) such that 0 = N (B), where N = NL(S)IK(p) denotes the norm from L ( p ) to K(p).

This definition coincides with the definition given by Hilbert for a norm residue in relatively quadratic extensions L/K or in a Kummer extension LIK, so called by Hilbert if K = Q(C), where (' is a primitive 1-th root of unity for a prime 1 and L = K(f i ) , where a is an algebraic integer in K which is not the 1-th power of another integer in K (see [Hi-18991, $7 and [Hi-18971, $129):

Definition 17. Let L/K be a finite algebraic extension of algebraic number fields, p a prime in K, OK the ring of integers in K and or; the ring of integers in L.

Then an integer P E OK is called a norm residue of the field L with respect to p, if for any power pn, n E N, of p there exists an integer B E or; such that r N(B) modulo pn, where N = NLIK denotes the norm from L to K.

3. Then Hensel determines the norm residues with respect to a prime divisor p in the case where K contains the 1-th roots of unity for a prime 1 and L = K(*) for an integer a E K which is not the 1-th power of another integer in K and p does not divide 1. This is done by means of well chosen, what we will call (multiplicative) Hensel bases, but what Hensel calls fundamental systems (for the multiplicative representation) or multiplicative fundamental systems (see [He-19221, p. 5 and [He-19161, pp. 205-206 and also [HH-19231, formula (I) , p. 264 and also [Ha-19691, $15, 1.) for the local fields K(p) and L ( p ) (see [He-19221, $2, in particular equations (7) and (7a)).

For this field extension K(*)/K Hensel also defines explicitly a norm residue symbol

where E K C K(p), C is a primitive l-th root of unity and X is a rational integer obtained in terms of the exponents of a and P with respect to a multiplicative Hensel basis properly chosen in K (p) , or more precisely, as a 2 x 2 determinant of the first two exponents, namely the order number e and the index f (see [Ha-1924b1, p. 76), of a and of P if 1 # 2, and a modification of the determinant if 1 = 2 (see [He-19221, equation (1 I), p. 9).

For this explicit norm residue symbol Hensel can establish what he calls the Permutation Law (see [He-19221, equation (12), p. 10 and compare with [He-19131, formula (6), p. 316):

Theorem 18.

and the Decomposition Law (see [He-19221, equation (12), p. 10 and compare with [He-19131, formula (12"), p. 318):

Theorem 19.

It follows from Hense17s definition of the symbol { } that the following property holds (see [He-19221, Theorem (C) , p. 6) :

Theorem 20.

if and only if /3 is a norm residue of L = K(f i ) with respect to p.

Hence Theorem 18 implies that P is a norm residue of K( f i ) with respect to p if and only if a is a norm residue of K ( G ) with respect to p (see [He-19221, p. 10).

In the case where K is the 1-th cyclotomic field K = Q(Cl) Hilbert had already introduced in $131 of his Zahlbericht [Hi-18971 an explicit norm residue symbol {k } for any prime divisor q in K as a certain 1-th

4 root of unity, and he had shown in $133 for a prime divisor q not dividing 1 (and also in certain cases if q divides 1) that this Hilbert norm residue symbol is equal to 1 if and only if P E K is a norm residue of L = K(&) with respect to q (see [Hi-18971, Satz 151). It follows immediately from Hilbert's definition of this Hilbert norm residue symbol that it satisfies the Permutation Law and the Decomposition Law (see [Hi-18971, $131, formulae (80) and (83)).

At the end of his paper Hensel mentions (see [He-19221, p. 10) that one gets the same results also in the most difficult but also most important and essential case where p is a divisor of I, and he announces that he will treat this case in a second paper.

How Hasse was led t o

4. This second paper was presented to the Mathematische Annalen on the 1st of May 1923 (see [HH-19231). However, it appears as a joint paper of Hensel and Hasse and is a result of joint discussions, as the authors mention at the very beginning of their article. So it becomes already clear, and this is what we will see in more detail, that Hasse's occupation with the norm residue symbol for a number field K , first in the quadratic case and then for odd prime degrees 1 , began in the year 1922, very probably under the influence of Hensel. The quadratic norm residue symbol was, in fact, needed for his theory of representing numbers by quadratic forms in the local fields K(p), which we discussed before(see [Ha-1924~1). .

Soon thereafter, or even at the same time, Hasse must have become aware (maybe also under the influence of Hensel) of the importance of the (quadratic) reciprocity law for the norm residue symbol in the form given by Hilbert. We have seen that this law played a crucial r6le in his paper on the representation by quadratic forms in algebraic number fields, published in Crelle's Journal 153 (1924) (see [Ha-1924~1).

5. In that year 1922 Hasse obtained his habilitation on the 28th of February at the University of Marburg and then had to do his "Vorberei- tungszeit" (preparation period for teaching at colleges, required for future college teachers) and pass the pedagogical examination for the "Lehramt an hoheren Schulen", which he did on the 6th of July 1922. On the 25th of September he was awarded a research grant (Forschungs- stipendium) of 308 000 Mark from the "Notgemeinschaft fur die deutsche Wissenschaft" for one year "zur Fortfuhrung der Arbeiten auf dem Ge- biete der hoheren Zahlentheorie", and already for the autumn semester 1922123 he was offered a paid lectureship (bezahlter Lehrauftrag) for Geometry at the University of Kiel. This offer was arranged by his former teacher in Kiel, Otto Toeplitz, whom Hasse had met at the annual meeting of the DMV (Deutsche Mathematiker-Vereinigung), held in Leipzig from the 17th until the 24th of September 1922. It was Hensel who suggested to Hasse that he accompany him to this meeting. There Hasse must also have met Emil Artin (probably for the first time), who delivered a lecture on quasi-ergodic geodesic orbits in the complex u p per half plane 'H (or more precisely on 'H/SL(2, Z)) with respect to the Poincark metric, entitled " ~ b e r einen Fall von geodiitischen Linien mit quasiergodischem Verlauf " (see also [ Ar- 1924bl).

6. At any rate, by the first of May 1923 Hasse was fully in possession of the theory of norm residues as developed by Hilbert (see [Hi-18991) and Hensel (see [He-19221) and the reciprocity laws for the

1-th power and 1"-th power residues as developed by Hilbert (see [Hi- 18991) and Furtwangler (see [Fu- 19041, [Fu- 19091, [Fu- 19121, [Fu- 19 131). He must also already have known Takagi's long and fundamental paper on class field theory of 1920 " ~ b e r eine Theorie des relativ Abel'schen Zahlkorpers" (see [Ta-1920]), since all these papers are explicitly mentioned in a footnote on p. 262 and then explicitly referred to on p. 263 of the joint publication [HH-19231.

We will now make these assertions more precise. Let us begin with what the authors have to say on the pages 262-263 in [HH-19231 (the extension LIK is now called Klk and p and I denote prime divisors in k) :

"Wahrend die Resultate fur den in N.R.' behandelten Fall eines zu 1 primen p im wesentlichen mit den Satzen der Hilbert-Furtwanglerschen ~ h e o r i e ~ ubereinstimmen und sich von jenen nur durch die u. E. natur- gemai3ere Behandlungsweise unterscheiden, was schon in der vie1 ein- facheren Definition des Normenrestcharakters (N.R., S. 2)7 deutlich zum Ausdruck kommt, werden die Ergebnisse dieser Arbeit erheblich uber die entsprechenden Resultate Hilberts und F'urtwanglers hinausgehen. Das dortige Hauptresultat, das in dem Satz enthalten istg:

Geht I nicht i n der Relativdislmminante von K auf, so sind alle zu I primen Zahlen von k Nomenres te von K nach I . I m anderen Falle bilden die zu [ primen Normenreste von K nach I eine Untergruppe vom Index 1 aller zu I primen Restklassen nach jedem genugend hohen Modul Ig ( e s genugt stets g 2 6, wenn 1 genau durch Ie teilbar),

gibt namlich nur eine Abzahlung der zu I primen Normenreste. Wir werden hier erstens das Resultat in vollster Allgemeinheit, d. h. auch fiir zu [ nicht prime Zahlen erhalten und zweitens die betreffende Untergruppe der Normenreste genau angeben.

Das erhaltene Ergebnis wird von dem jungeren von uns in einigen weiteren Arbeiten zur Aufstellung einer systematischen Theorie der quadratischen Formen in einem algebraischen Korper k19 sowie zu be- merkenswerten Verallgemeinerungen der bekannten Furtwanglerschen

z', YF 5that is, in (He19221

ere Hasse and Hensel refer to [Hi-18991, [Fu-19041, [F'u- 19091, [Fu- 19121, [Fu- 19 131 and [Ta- 19201.

7that is, in [He-19221, p. 2 ere Hasse and Hensel refer explicitly to [F'u-19041, p. 47 and to [Ta-

: 1920], Satz 9, p. 28. in [Ha-1924~1


Reziprozitat sgesetze fur 1- te Potenzreste in k auf nichtprimare Zahlen verwendet werden lo."

and further down on p. 267:

"Die hier angewandte Methodel' erijffnet ferner die Moglichkeit, zu einer in der Hilbert-Furtwanglerschen Theorie keinen Platz findenden, direkten Definition und expliziten Formel fur das den Normenrestcharak- ter von p in bezug auf k ( G ) ausdruckende Hilbertsche Normenrestsym- bol (v) zu gelangen, und so eine neue Grundlage fur die Behandlung des allgemeinsten Reziprozitatsgesetzes fur die 1-ten Potenzreste in k zu schaffen, worauf der jiingere von uns in einigen weiteren Arbeiten einzugehen gedenkt ." '

7. We can gain some more insight on the sequence of events that led Hasse to the norm residue symbols and to the reciprocity laws from the most interesting correspondence between Kurt Hensel and Helmut Hasse.13 It is fascinating to watch Hasse progressing in a very short period to the center of number theory, starting out from the theory of quadratic forms, proceeding to the theory of norm residues and then to the laws of reciprocity and finally ending up in class field theory.

On the 2nd of February 1923 Hasse wrote from Kiel, where he held a lectureship since October 1922, that he had just settled the theory of the norm residue symbol for a prime divisor I dividing the prime degree 1 = [K : k], where K = k(*), for the moment for 1 = 2, in the way they discussed when they met last time, that is, during the Christmas vacation 192214, namely to give an explicit expression for the norm residue symbol by means of a Hensel basis:

"Zu meiner grofien Freude kann ich Ihnen heute mitteilen, dafi mir soeben gelungen ist, die Theorie des Normenrestsymbols fur einen Prim- teiler I in dem in unserer letzten Besprechung formulierten Sinn zu einem befriedigenden Abschluss zu bringen. Da ich uberzeugt bin, dafi Sie an diesem Resultat eine ebenso groi3e Freude haben werden, als ich selber, schreibe ich Ihnen sogleich, kaum 5 Minuten nach dem letzten Feder- strich. - Es handelte sich darum, auch im Falle eines Primteilers I

'Opublished in [Ha-1924fl '' that is, the method of the systematic use of Hensel bases for k and K,

or more precisely for k ( p ) and K(!Q), where !Q in K is a divisor of p in k, 12published in [Ha-1924b], [Ha-1924e], [Ha-1925~1 and [Ha-1924fl. 1 3 ~ h i s correspondence is kept, according to Hasse's will, at the University

Library in Gottingen. 14see the postcard from Hasse to Hensel of 10.2.1923.

eine Darstellung des Normensymbols anzugeben, die seinen Wert unmittelbar aus der Exponentialdarstellung zu erschliefien gestattet. Ich habe die Untersuchung vorlaufig fiir den Fall 1 = 2, d. h. des quadratischen Symbols durchgefiihrt, da mir fur ungerades 1 die von Ihnen gefunde- nen Resultate noch nicht in dem erforderlichen MaBe zur Verfugung st anden."

8. F'rom [Ha-1924bl we can see more precisely what Hasse was doing in the case 1 = 2, while following Hensel's idea to deduce the value of the norm residue symbol directly from the exponents of a well chosen multiplicative presentation of a and P with respect to a Hensel basis in k(I). He gets:

where L is a symmetric bilinear form (or - what amounts to the same, since 1 = 2 - a skew-symmetric bilinear form) of the exponents of a and /3 with respect to a Hensel basis in k(I) (see [Ha-1924b], Hauptsatz, p. 77). The existence of such a form is deduced directly from the decomposition and permutation law for the quadratic norm residue symbol (y), due to the fact that in the quadratic case the latter simply takes the form ( q ) = (y) (see [Ha-1924b], Satz 1, p. 80).

As for the case 1 # 2, 1 prime, Hasse says in that same letter that the conclusions are the same, except for the crucial permutation law for which he would like to have a more direct proof. By this Hasse meant a proof which does not make use of the 1-th degree Hilbert reciprocity law in k. Thereby he was alluding to Furtwangler who was building on the Hilbert reciprocity law in k in order to define the 1-th degree norm residue symbol ( q ) for a prime divisor I dividing the degree 1 and to deduce the permutation law for it from the fact that the definition and the permutation law were already established for the symbols ( ) with prime divisors p not dividing 1 (see [Fu-19041, $14, p. 37). This direct proof was given a little later in [Ha-1925al in a rather roundabout way.

9. In [Ha-1924el Hasse accomplished the task of deducing, similarly as in the case 1 = 2, the existence of a unique skew symmetric bilinear form L modulo 1 from the decomposition law

and the permutation law


such that the norm residue symbol (q) can be expressed as

for a fixed primitive 1-th root of unity C (see [Ha-1924e], Satz 3, p. 186). The skew-symmetric bilinear form L is again given in terms of the

exponents of a and ,O with respect to a Hensel basis in k(I) (see [Ha- 1924e], Satz l, p. 186).

10. The values of ( ) = cL for a prime p not dividing 1 and

(G) I = <lL' for a prime I dividing 1 are uniquely determined, except for the choice of C and C'. In the case where p does not divide 2, C can be normalized in a natural way by the choice of an appropriate Hensel basis, but in the case where I divides 1, this is not possible for C'. However, Hasse is able to obtain a normalization also in this case by postulating the Hilbert reciprocity law in k for degree 1:

for all a, ,b' E k, where the product is taken over all pime divisors q in k (see [Ha-1924e1, pp. 189-191).

Hence this normalization is subject to the validity of the Hilbert reciprocity law for the prime degree 1, for which, in fact, proofs had already been given by Furtwangler (see [Fu-19121, Satz 16, p. 385) and by Takagi (see [Ta-19221, $11; or [Ta-19731, p. 209) by transcendental (i.e. analytic) means. l5

11. Later Hasse went back again and again to this fundamental problem of giving an explicit and canonical description of the norm residue symbol, e.g. [Ha-19301 and [Ha-19331, where he introduced the norm residue symbol as a fundamental 2-cycle and herewith prepared the way for the cohomological formulation of the Artin-Tate reciprocity law and the fundamental theorems of class field theory.

12. These results on the explicit determination of the I-th degree norm residue symbol enabled Hasse to give strong improvements of the explicit reciprocity laws for 1-th powers as established by Hilbert and Furtwangler (see [Ha-1924fl).

15 Hasse in [Ha-1924el on p. 191 gives an incorrect title in his reference to Takagi.

13. Hasse then continues in still the same letter of the 2nd of Feb- ruary 1923 referring to the planned but not yet published papers [Ha- 1925a] and [Ha-1924el we just discussed:

"Ich werde mir diese ubertragung16 noch einmal griindlich fiber- legen, kann aber vorlaufig nicht ausfuhrlich vorgehen, da mir die Grund- lagen, namlich Ihre Entwicklungen uber diesen Fall noch nicht ganz im Kopf sind. Ich wurde mich sehr freuen, wenn ich bei Gelegenheit meiner Anwesenheit in Marburg im Marz ausfuhrlich mit Ihnen uber den Fall sprechen konnte.

Es ware mir sehr lieb, wenn es sich machen liefie, dai3 mein heutiges Resultat vielleicht noch im Anschlui3 an Ihre demnachst erscheinende Arbeit uber dasselbe Problem veroffentlicht werden konnte17. Sollten Sie damit einverstanden sein, so mochte ich mir die Bitte erlauben, dai3 Sie mit der Einsendung Ihres Manuskriptes - ich meine von Ihnen gehort zu haben, dafi Sie es bei Springer18 drucken lassen wollen - warten, bis zu meinem Kommen nach Marburg, damit ich mir noch die notigen Angaben betr. Riickverweisung auf Stellen Ihrer Arbeit holen kann. Ich wurde dann selbstverstandlich unmittelbar nach Semesterschlufi, also etwa am 5. Marz in Marburg sein, wo sich ubrigens auch meine Braut seit einigen Tagen wieder eingefunden hat.

Die Fortsetzung meiner Habilitationsschrift, die bisher noch nicht zum Druck gegeben ist, habe ich nunmehr fertig gestellt, und darf sie Ih- nen wohl bei dieser Gelegenheit dann noch mitbringen, ebenso eine oder zwei weitere Arbeiten uber quadr. Formen in algebraischen Korpern, die ich nachster [?I Tage fertigstellen werde. - "

Hasse then finishes his letter by saying:

"In der Hoffnung Ihnen mit der Mitteilung meines Resultats eine Freude gemacht zu haben, mochte ich Ihnen gleichzeitig meinen herz- lichsten Dank aussprechen fur Ihre so schone und wertvolle Anregung, mich mit diesen Fragen zu beschaftigen. Ich war die letzten Tage wie im Fieber dabei, und meine grofie Freude uber den glucklichen Erfolg konnen Sie sich kaum vorstellen. Die Zahlentheorie birgt doch wahrlich die schonsten Schatze in der Mathematik!"

14. From this letter we can infer that, indeed, Hasse's occupation with the theory of norm residues was initiated by Hensel, that Hasse and Hensel planned to publish their results on the norm residues for the case

16namely from 1 = 2 to 1 # 2 17~asse's result appeared in [Ha-1924bl and Hensel's paper was published

as the joint paper [HH-19231. ''that is, in the Mathematische Annalen

50 G. Frei How Hasse was led to 51

when the prime divisor I in k divides the prime degree 1 = [k(&) : k] in two separate papers in the Mathematische Annalen, but that they must have decided, when they met in Marburg in the beginning of March 1923, to publish a joint paper (see the letter of 21st of March 1923).

15. This letter of the 2nd of February 1923 was followed by a long series of letters and postcards from Hasse to Hensel written in short intervals and announcing a whole series of new discoveries, testifying that Hasse had reached an extremely productive period1g.

On a postcard dated the 10th of February 1923, Hasse first refers to Hensel's investigations on the computation of the norm residue symbol in the case p = I by means of a Hensel basis, a problem they discussed during the Christmas vacation.

On a postcard dated from the 14th of February 1923, Hasse announces that he believes that he can now prove the permutation law for the norm residue symbol (y) with I dividing 1 for an odd prime 1 (see [Ha-1925al):

"den Vertauschungssatz fur ungerades 1 glaube ich jetzt beweisen zu konnen. Ich muss es in den nachsten Tagen ma1 mit meinen neuartigen Gedanken probieren."

$5. Explicit Reciprocity Laws

1. Then two days later Hasse begins to study the explicit reciprocity laws to which he was probably led by the article of Furtwangler [Fu- 19041, where Furtwangler deduces the general reciprocity law for the 1-th power residues and the two complementary laws for primary and hyperprimary numbers in a number field k from the Hilbert reciprocity law for norm residues of degree 1 in k at the end of his article (see [Fu-19041, $17, p. 47).

First Hasse reports in a letter (Kiel, 16th of February 1923) on a new discovery about the quadratic Hilbert symbol (w) where I is a prime divisor in k dividing 2:

"Schon wieder kann ich Ihnen ein schones Resultat mitteilen, das mir gestern und heute zugefallen ist. Meine Bemuhungen richteten sich zuniichst auf das spezielle Hilbertsche Symbol (w) in einem beliebigen algebraischen Korper k, wenn I ein Teiler der 2 ist. Dieses Symbol interessierte mich besonders, da es schon bei der Reduktion der Form L

''6 of the 24 papers published in Volume 153 (1924) and 5 out of 24 papers published in Volume 154 (1925) of Crelle's Journal were written by Hasse.

eine gewisse Rolle spielte, dann aber auch in der Theorie der quadratischen Formen eine Rolle spielt, wie Sie sich wohl aus meiner Habilita- tionsschrift entsinnen, wo die Symbole (F) in den Invarianten hiiufig vorkommen."

Hasse there determines the value (-';-I) explicitly with Hensel's method of using fundamental multiplicative systems (Hensel bases) for the ground field k and for the corresponding extension field K and ob- t ains:

where e is the ramification order (ramification index) and f the degree of (the residue field of) I in k. This value is compatible with the quadratic Hilbert reciprocity law in k:

(over all primes q , finite and infinite).

2. More generally Hasse is able to determine the symbol (v), for a E k and a - 1 (mod 2), explicitly with the help of the local trace SI of 9 in k(I), that is the trace from k(0 to Q(2):

and then obtains a part of the first complementary law of the quadratic reciprocity in lc, namely

where S denotes the global trace in k, that is the trace from k to Q. He then expresses the hope that he will obtain in a similar way the whole quadratic reciprocity law in k, e.g. the analogy of the second complementary law for algebraic number fields k.

His results appeared as special cases published in [Ha-1924fl (for (q) see [Ha-1924f], p. 201 and also [Ha-1924~1, p. 127; for (v) and (2) see [Ha-1924f], (4a.), p. 194, where the more general case ( y) with I a divisor of any given prime 1 and < a primitive 1-th root of unity is treated).

3. Already on the next day Hasse writes enthusiastically on a postcard (Kiel, 17.2.1923) that he was able to extend quite considerably his


special results on (w) and (9) towards a general quadratic reciprocity law for any number field k and to establish the full analogy with the case of the rational number field Q by applying Hensel's method of making use of a Hensel basis in the corresponding local field k(I):

"Ich scheine in einer sehr glucklichen Epoche meines Lebens ange- langt zu sein, denn ganz plotzlich offnen sich mir tausend Tore, durch die neue, schone Erkenntnisse einstromen. Im AnschluB an mein spezielles Resultat uber (w) und (F) habe ich nunmehr mit derselben, d. h. mit Ihren schonen Fundamentalsystemen fur die mult[iplikative] Darstellung, das gesamte quadratische Reziprozitatsgesetz fur beliebige algebraische Zahlkijrper sehr wesentlich erweitert und nun erst die volle Analogie mit dem rationalen Zahlkorper hergestellt."

and he ends:

"Ich erzahle Ihnen bald ausfuhrlich, welch' einfache Beweise dieser Satze sich mit Ihren so handlichen Fundamentalsystemen geben lassen. Mit Stolz und Dank sehe ich zu Ihnen als dem Schopfer dieser Methoden auf."

4. Eleven days later, on the 28th of February 1923, he reports that he succeeded in extending his results on the quadratic reciprocity law to the reciprocity law for the I-th power residues in a field k, just before he leaves to Hamburg, where he is going to present a lecture about his new results on the reciprocity laws:

"Eben vor meiner Abreise noch die erfreuliche Mitteilung, daB das Reziprozitatsgesetz fur 1-te Potenzreste sich ebenso schon erweitern laBt, wie ich es Ihnen neulich fur die quadratischen Reste schrieb. Naheres bald mundlich. Ich komme am 6. Marz nach Marburg und wurde mich sehr freuen, wenn Sie mich (am besten durch Nachricht an meine Schwie- germutter) wissen lieBen, wann ich Sie zu Hause antreffe. Morgen trage ich in Hamburg meine Resultate iiber die Reziprozitatsgesetze vor."

It must have been on this occasion (1st of March 1923) that Hasse and Artin began to discuss the reciprocity laws which eventually led to two joint papers " ~ b e r den zweiten Ergiinzungssatz zum Reziprozitits- gesetz der I-ten Potenzreste im Korper kc der I-ten Einheitswurzeln in Oberlcorpern von kc" (Crelle 154 (1925), 143-148) (see [AH-1925]), to which the first letters of their correspondence, beginning on the 9th of July 1923, were dedicated, and "Die beiden Erganzungssatze zum Rezzprozitatsgesetz der In-ten Potenzreste im Korper der In-ten Ein- heitswurzeln" (Abh. Math. Sem. Hamburg 6 (1928), 146-162) (see [AH- 19281).

5. From the publication [Ha-1924f1, Hasse's very first paper dedicated to the explicit reciprocity laws, written up on the 30th of June 1923 and presented to Crelle's Journal, we can see that Hasse found the following form of the general reciprocity law and the two complementary laws for the 1-th power residues in any number field k containing the I-th roots of unity for a prime number 1 and in cases where a and /3 are non-primary numbers in k. The laws were obtained by making systematic use of the explicit formulae for the norm residue symbols, derived by Hensel and Hasse with the help of well chosen Hensel bases in the corresponding local fields. They are completely analogous to the quadratic reciprocity laws in Q (see [Ha- 1924f1, p. 194):

Theorem 21. If k is a number field containing the I-th roots of unity for a prime number 1 and a, P E k, then

(2) (6) = pW) , z f a = l m o d l .

(3) (k) = cS(=), if a = 1 mod llo,

where (-) denotes the Legendre symbol generalized to the 1-th power residues in k, S denotes the trace from k to Q, 5 is a fied primitive I-th root of unity, Xo = 1 - C and lo = (1 - 6) = (Ao) is the corresponding principal divisor in Q([ ) C k.

In his proof of Theorem 21, Hasse starts out from the Hilbert reciprocity law for norm residues of degree 1 in k, namely

proved by Furtwangler in 1904 and 1912 (see [Fu-19041, Satz 47, p. 46 and [Fu-19121, p. 347), in order to deduce the three relations

(1') (;)($)-I = n l = , ( F ) , if a , p and 1 are prime to each other;

(2') (i) = n:=, ( ), if a is prime to 1 and ( E ) = a', where a is an ideal in k:

(3') ($) = n l = , ( ), if a is prime to I and (A) = n:=, liaa a' where 1 = Ilel . . . lteYs the prime decomposition of I in k.

Then Hasse evaluates the norm residue symbols on the right hand side by means of his explicit formula for the I-th degree norm residue symbol ( y ) for a prime divisor li of 1.

6. In the case of the second complementary law (3') Hasse also makes use of an argument he learnt from Artin, when I = 2,3, an argument he

56 G. Rei

dividing the degree 1 = [k(f i ) : k], Hasse and Hensel must have decided to publish their findings not in two separate papers, each signed by their respective author, but jointly in one paper. Hasse took over the task to write up the article as can be seen from a letter from Hasse to Hensel on the 21st of March 1923:

"Beiliegend ubersende ich Ihnen das nunmehr fertiggestellte Manu- skript unserer gemeinsamen Arbeit und mochte Sie freundlichst bitten, es noch einmal genau durchzulesen. Ich habe es mit groDer Liebe und Sorgfalt bis ins Einzelne durchdacht und eigentlich jedes Wort und jeden Passiis einer reiflichen Erwagung unterzogen, wie es wohl am besten sei, zu schreiben. So glaube ich nunmehr eine nach allen Seiten hin ausgeglichene und passende Form gefunden zu haben und bin gespannt, Ihr Urteil dariiber zu vernehmen.

[.-I - dai3 die Arbeit in gemeinsamen Besprechungen zwischen uns ent-

standen ist, konnen wir wohl schreiben. Wenn auch jeder von uns wesentliche Punkte allein gefunden hat, so ist doch die jetzt vorliegende Form als eine schone Durchdringung unserer beiderseitigen Ideen anzuse- hen. Ich fur mein Teil kann nur versichern, dai3 mein Anteil zum groi3ten Teil Ihren wertvollen Anregungen bei unseren Besprechungen Weihnach- ten (ja auch schon vorigen Sommer) und kurzlich zu verstehen ist.

[.-I Was nun die Bezeichnungen anbelangt, so haben mir diese am meis-

ten Nachdenken verursacht. Ich weiss, wie sehr eine gute Bezeich- nungsweise die Verstandlichkeit und Beliebtheit einer Arbeit zu heben vermag .

[.-I Ich habe das Manuskript, bis auf einige bei der Korrektur noch

einzufugende Zitate gleich druckfertig gemacht. Wenn Sie es fur gut befinden, schicken Sie es wohl bald ein. Mir liegt sehr an einem baldigen Erscheinen, da eben alle weiteren Arbeiten von mir darauf fuDen.

[-.I Zu weiteren mathematischen uberlegungen bin ich noch nicht ge-

kommen, und werde (wohl?) auch bis Mitte Sommer wenig machen konnen. Denn jetzt bin ich ohne Literatur und [darlan meine Vorlesun- gen fur den Sommer [zu] praparieren. Pfingsten wird meine Hochzeit sein. Nichtsdestotrotz behalte ich unsere groi3en Ziele, vor allem einen vernunftigen Beweis des Reziprozitatsgesetzes dauernd im Auge."

On the 31st of March 1923 Hasse is able to announce that he proved the permutation and decomposition law for [ a divisor of 1 by means of group theory (subgroups of the norm residues) (see [Ha-1925al).


In a letter of 21st of April 1923 Hasse says that he is planning to prepare an exposition of Takagi7s class field theory for a course:

"AuBerdem habe ich gerade die Ausarbeitung eines Kollegs uber die Klassenkorpertheorie von Takagi vor, die ich mit unseren Methoden sehr schon und einfach darstellen kann."

and further down:

"Ich habe in den letzten Tagen auf verschiedenen allabendlichen Spazierfahrten per Rad einmal ganz grundlich uber das allgemeine Nor- menrestproblem nachgedacht, dessen Grundlegung Sie kurzlich gaben. Zunachst etwas Spezielles: Unsere gemeinsame Arbeit setzt die Exis- tenz der 1-ten Einheitswurzeln in k voraus, gibt also keine vollstandige Theorie (mit dem fruheren fur p zusammen) des Normenrestproblems fur reine, absolut zyklische Korper von Primzahlgrad 1. Nun habe ich in der Takagischen Arbeit die ganz entsprechende Untersuchung fur ganz beliebige relativ zyklische Korper von Primzahlgrad gefunden, also auch solche, die durch nicht reine Gleichungen definiert werden, z. B. den kubischen Korper, wenn er Galoissch ist, etz. Takagi's Methoden sind nicht sehr schon, [. ..]

Seine Methoden sind A-adisch und es ist eine Klei,nigkeit, wie Sie aus obiger Stichprobe sehen, sie zu unseren Zielen zu vervollstandigen, d. h. mittels multiplikativer Normalform das kritische Element zu charakter- isieren und so die Anwendung auf hohere Reziprozitatsgesetze in der Art, wie ich sie vorhabe, vorzubereiten. -

Die Takagischen Resultate stellen sich also als der erste Schritt zu . der von Ihnen beabsichtigten Verallgemeinerung des Normenrestprob-

lems auf beliebige Relativkorper dar."

Hasse then sketches how to proceed in the general case of an arbitrary (relative) extension Klk in oder to solve the norm residue problem, that is to determine whether an element in k is a norm residue or not and to determine the subgroup of norm residues and its index. He proposes to study the local structure, i. e. to analyze stepwise the tower of fields "climbing" from k(1) over the inertial, the first and the higher ramification fields up to K(C), where C is a prime divisor of [ lying in K.

In a letter, dated 23rd of April 1923, Hasse comes back to his ideas for a very general norm residue theory in arbitrary (relative) extensions Klk and for 'the most general' reciprocity law in Klk:

"In meinem Briefe vom Sonnabend vergd ich noch zu sagen, daB ich nach meinen "uberfliegenden" uberlegungen das "allgemeinste Nor-

L menrestsymbol" fiir aufkrordentlich wertvoll halte. Ich glaube, dai3 eine ganz neue Sorte von Reziprozitatsgesetzen auf diese Weise erschlossen

I

L

58 G. Frei

werden kann, von denen dann die bisherigen nur Spezialfalle sind. Als oberstes Ziel mochte ich den Nachweis von

hinstellen, was dann das "allgemeinste" Reziprozitatsgesetz fur beliebige Relativkikper darstellen wiirde. Durch Satze uber die Auswertung von { $ } wiirden vermutlich Reziprozitatsgesetze uber die m-ten Potenzreste (m beliebige ganze Zahl), aber noch vielmehr, namlich Satze uber "nicht Galoissche Kongruenzreste" folgen, die sich vermutlich in Reziprozitaten zwischen den Relativdiskriminanten auBern und womoglich zu schonen Diskriminantensatzen uberhaupt fuhren. Dieses Ziel ist aber noch in weiter Ferne. Denn dazu ist auBer der Bestimmung des Symbols ($1 noch ein ganz entsprechendes Gebaude fur beliebige Korper notwendig, wie das Hilbert-F'urtwangler-Takagische fur Abelsche Korper."

It is not clear whether Hasse actually taught the planned course on Takagi's class field theory, but he did write up an exposition of that theory, partly in order to prepare himself for the study of the norm residue theory and the reciprocity laws in non-Galois extensions Klk. He gave a report on this exposition at the meeting of the German Mathematical Society (DMV) in Danzig in the autumn of 1925. It was Hilbert who asked him to deliver this report, and it was this report which gave rise to Hasse's famous report (Bericht) on class field theory which was to have an enormous impact on the later development (see [Ha-19261).

I would like to thank my friend Cornelius Greither for a linguistic improvement of the text.

References

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[Ar-1924al Artin, Emil, Quadratische Korper im Gebiete der hoheren Kon- gruenzen I, II, Math. Zeitschrift, 19 (1924), pp. 153-246.

[Ar-1924bl Artin, Emil, Ein mechanisches System mit quasiergodischen Bahnen, Abh. Math. Sem. Hamburg, (1924), pp. 170-175.

[Ar-19271 Artin, Emil, Beweis des allgemeinen Reziprozitatsgesetzes, Abh. Math. Sem. Hamburg, 5 (1927), pp. 353-363.

[AH-19251 Artin, Emil and Hasse, Helmut, ~ b e r den zweiten Ergiinzungs- satz zum Reziprozitatsgesetz der 1-ten Potenzreste im Korper

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Dedekind, Richard, Abr$ einer Theorie der hoheren Kongruen- zen in bezug auf einen reellen Primzahl-Modulus, J. reine angew. Math., 54 (1857), pp. 1-26.

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Furtwangler, Philipp, ~ b e r die Reziprozitatsgesetze zwischen I - ten Potenzresten in algebraischen Zahlkorpern, wenn 1 eine ungemde Primzahl bedeutet, Math. Ann., 58 (1904), pp. 1-50. Slightly changed reprint of an article in Gottinger Abhafld- lungen 1902.

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Furtwangler, Philipp, Die Reziprozitatsgesetze fur Potenzreste mit Primzahlexponenten in algebraischen Zahlkorpern, II, Math. Ann., 72 (1912), pp. 346-386.

Furtwangler, Philipp, Die Reziprozitatsgesetze fur Potenzreste mit Primzahlexponenten in algebraischen Zahlkorpern, III, Math. Ann., 74 (1913), pp. 413-429.

Hasse, Helmut, ~ b e r die Darstellbarkeit won Zahlen durch quadratische Formen im Korper der rationalen Zahlen, J. reine angew. Math., 152 (1923), pp. 129-148.

Hasse, Helmut, ~ b e r die ~ ~ u i v a l e n z quadratischer Forrnen im Korper der rationalen Zahlen, J. reine angew. Math., 152 (1923), pp. 205-224.

Hasse, Helmut, Symmetrische Matrizen im Korper der rationalen Zahlen, J. reine angew. Math., 153 (1924), pp. 12-43.

Hasse, Helmut, Zur Theorie des quadratischen Hilbertschen Nor- menrestsymbols in algebraischen Korpern, J. reine angew. Math., 153 (1924), pp. 76-93.

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Hasse, Helmut, A'quivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkiirper, J . reine angew. Math., 153 (1924), pp. 158-162.

Hasse, Helmut, Zur Theorie des Hilbertschen Normenrestsymbols in algebraischen Korpern, J . reine angew. Math., 153 (1924), pp. 184-191.

Hasse, Helmut, Das allgemeine Reziprozitatsgesetz und seine Erganzungssatze in beliebigen algebmischen Zahlkorpern fur gewisse, nicht-primare Zahlen, J . reine angew. Math., 153 (1924), pp. 192-207.

Hasse, Helmut, Direkter Beweis des Zerlegungs und Vertau- schungssatzes fur das Hilbertsche Normenrestsymbol in einem algebraischen Zahlkorper irn Falle eines Primteilers I des Re- lativgmdes 1, J . reine angew. Math., 154 (1925), pp. 20-35.

Hasse, Helmut, ~ b e r das allgerneine Reziprozitlitsgesetz der 1- ten Potenzreste im Korper kc der I-ten Einheitswurzeln und in Oberkorpern von kc, J. reine angew. Math., 154 (1925), pp. 96-109.

Hasse, Helmut, Zur Theorie des Hilbertschen Normenrestsymbols in algebraischen Korpern, J . reine angew. Math., 1 5 4 (1925), pp. 174-177.

Hasse, Helmut, Das allgemeine Reziprozitatsgesetz der 1-ten Potenzreste fur beliebige, zu 1 prime Zahlen in gewissen Oberkorpern des Korpers der I-ten Einheitswurzeln, J . reine angew. Math., 154 (1925), pp. 199-214.

Hasse, Helmut, Der zweite Erganzungssatz zum Reziprozitats- gesetz der 1-ten Potenzreste fur beliebige zu 1 prime Zahlen in gewissen Oberkorpern des Korpers der 1-ten Einheitswurzeln, J . reine angew. Math., 154 (1925), pp. 215-218.

Hasse, Helmut, Bericht iiber neuere Untersuchungen und Prob- leme aus der Theorie der algebmischen Zahlkorper, Teil I, Teil la, Jahresbericht der D.M.-V., 35 (1926), 36 (1927).

Hasse, Helmut, Neue Begriindung und Verallgemeinerung der Theorie des Normenrestsymbols, J . reine angew. Math., 162 (1930), pp. 134-144.

Hasse, Helmut, Die Struktur der R. Bmuerschen Algebrenklas- sengruppe uber einem algebraischen Zahlkorper. Insbesondere Begriindung der Theorie des Normenrestsymbols und die Her- leitung des Reziprozitatsgesetzes mit nichtkommutativen Hilfs- mitteln, Math. Annalen, 1 0 7 (1933), pp. 731-760.

Hasse, Helmut, "Zahlentheorie" , 3. Aufl., Akademie-Verlag, Ber- lin, 1969.

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Hasse, Helmut, "Mathematische Abhandlungen" , herausge- geben von Heinrich Wolfgang Leopoldt und Peter Roquette, Walter de Gruyter, Berlin, 1975.

Hasse, Helmut and Hensel, Kurt, ~ b e r die Normenreste eines relativ-zyklischen Korpers vom Primzahlgrad 1 nach einem Primteiler I von 1, Math. Ann., 90 (1923), pp. 262-278.

Hensel, Kurt, ~ b e r eine neue Begriindung der Theorie der algebraischen Zahlen, Jahresbericht der D.M.-V, 6 (1897), pp. 83- 88.

Hensel, Kurt, "Theorie der algebraischen Zahlen" , Teubner, Leipzig und Berlin, 1908.

Hensel, Kurt, "Zahlentheorie", Goschen, Leipzig und Berlin, 1913.

Hensel, Kurt, Die multiplikative Darstellung der algebraischen Zahlen fur den Bereich eines beliebigen Primteilers, J . reine angew. Math., 1 4 6 (1916), pp. 189-215.

Hensel, Kurt, ~ b e r die Normenreste in den allgemeinsten relativ- abelschen Zahlkorpern, Math. Ann., 85 (1922), pp. 1-10.

Hilbert, David, Die Theorie der algebraischen Zahlkorper, Jah- resbericht der D.M.-V, 4 (1897), pp. 175-546.

Hilbert, David, ~ b e r die Theorie des relativ-quadratischen Zahl- korpers, Math. Ann., 51 (1899), pp. 1-127.

Hilbert, David, ~ b e r die Theorie der relativ-Abelschen Zahlkor- a

per, Acta Math., 26 (1902), pp. 99-132. Slightly changed reprint of an article in Nachrichten der K. Ges. der Wiss. zu Gottingen 1898.

Kaplan, Pierre, "Teiji Takagi (1875-1960) et la dhcouverte de la thhorie du corps de classes", Typoscript, Tokyo, 1977.

Minkowski, Hermann, ~ b e r die Bedingungen, unter welchen zwei quadratische Formen mit rationalen Koefizienten ineinander rational transformiert werden konnen, (Auszug aus einem von Herrn H. Minkowski in Bonn an Herrn Adolf Hurwitz gerichteten Briefe). J. reine angew. Math., 106 (1890), 5-26; Ges. Abh., Bd. 1, pp. 219-239.

O'Meara, Timothy O., "Introduction to Quadratic Forms", Springer-Verlag, New York, 1963.

Scharlau, Winfried (editor), "Mathematische Institute in Deutschland 1800-1945" DMV, Vieweg, Braunschweig/Wies- baden, 1990.

Takagi, Teiji, ~ b e r eine Theorie des relativ Abel'schen Zahlkor- pers, Journ. of Coll. of Science, Univ. of Tokyo, 41, Art 9, 1920, pp. 1-133.

Takagi, Teiji, ~ b e r das Reciprocitiitsgesetz in einem beliebigen algebraischen Zahlkorper, Journ. of Coll. of Science, Univ. of Tokyo, 44, Art 5, 1922, pp. 1-50.

62 G. Frei

[Ta- 19731 Takagi, Teiji, "Collected Papers", Iwanami Shoten, Tokyo, 1973.

De'partement de mathe'matiques et de statistique Universite' Lava1 Ste-Foy, Que'bec, G1K 7P4 Canada E-mail address: gf reiQmat .ulaval . ca


Nonabelian Local Reciprocity Maps

Ivan Fesenko

There are several approaches to construct the reciprocity map, the essence of class field theory, which links the maximal abelian quotient (or sometimes the maximal abelian pro-pquotient) of the absolute Galois group of a particular field with an appropriate abelian object associated to the field such that certain functorial properties hold.

One of those approaches originates from works of Dwork, Serre, Hazewinkel [D, S, HI, H2], Iwasawa [Il, I21 and Neukirch [Nl, N2]. Recall it briefly.

Let F be a local field with finite residue field. Let Fur be the maximal unramified extension of F and let P be the completion of Fur . Let L be a finite Galois extension of F and LUr = LFUr, 2 = LF^.

For an element a of Gal(L/F) let 5 be any element of Gal(Lur/FUr) such that 51L = a and 5IFUr is a positive integer power of the Frobenius automorphism cp E Gal(Fur/F). Let C be the fixed field of 5; it is a finite extension of F .

Let G ~ ~ ( L / F ) " ~ be the maximal abelian quotient of Gal(L/F). Define the map [Nl, N2]

by (T -+ NCIF xC mod NLIF L* where nc is any prime element of C. Dur- ; ing the conference on class field theory in Tokyo, June 1998, Professor ; Tsuneo Tamagawa informed the author that similar constructions were

independently developed by Iwasawa. We call N the Neukirch-Iwasawa map.

Received August 29, 1998 Revised February 9, 1999

. This work was partially done during my visit to Japan in 1998. I would like to express my gratitude to the Royal Society, RIMS, JSPS and JAMS for support, to the organizers of the conference of class field theory, and to Professor

f Masato Kurihara for great hospitality during my stay in Tokyo Metropolitan University in June 1998. i

64 I. Fesenko

On the other hand, for a finite Galois totally ramified extension L / F of local fields there is a fundamental exact sequence [Se, (2.3)], [HI, (2.7)]

where V(L/F) is the subgroup of U; generated by elements uu-' with u E Uz, o E Gal(L/F) and c(o) = o(a) /n mod V(L/F) for a prime element a of L. Note that the same sequence for the maximal unramified extensions instead of their completion is exact.

Define the Hazewinkel homomorphism [HI, H2, 111

by H(u) = a where u = N- -(v) and c(a) = a(a)/a = v/cp(v) mod LIF V(L/F). This map can be extended to finite and infinite Galois extensions [HI, H2, 111.

The shortest way to deduce properties of N and H is to work with both maps simulteneously. For a finite Galois extension L / F the composition H o N coincides with the epimorphism Gal(L/F) + G ~ ~ ( L / F ) " ~ and the composition N 0 H is the identity map of UF/NLIFUL. Hence N is an epimomorphism with the kernel equal to the commutator group of the Galois group.

This approach with appropriate mofidications and generalizations works well in (p) class field theories of local fields with perfect residue field [F3, HI], higher local fields [Fl , F2, F4], complete discrete valuation fields with residue field of characteristic p [F5].

In this paper we shall define nonabelian reciprocity maps for arithmetically profinite Galois extensions of local fields extending the a p proach discussed above. For Fontaine-Wintenberger's theory of arithmetically profinite extension and fields of norms see [W], [FV, Ch. 111, sect.51. For simplicity we treat the case of totally ramified extensions, however, the constructions of this work can undoubtedly be defined for arbitrary Galois arithmetically profinite extensions, in particular, arbitrary finite Galois extensions of local fields. This paper contains a complete presention of the main construction; further applications is the subject of a forthcoming paper.

We shall use terminology "the field of norms" for finite extensions as well, meaning just the set of norm-compatible sequences in subexten-

Nonabelian Local

sions. In this case by UN(L/K) we

Reciprocit y Maps 65

mean the group of norm-compatible . . , sequences in the group of units of subextensions in LIK.

We shall work with maps NLIF, NLIF and HLIF. The map NLiF is a generalization of the map N. It injects the Galois group Gal(L/F) of a finite or infinite arithmetically profinite totally ramified extension L of a local field F into a certain subquotient Uo - /(IN(LIF) of the

N(L/F) . . - A h

group of units U - of the field of norms N(L/F) = N(L/F) of N(LIF)

h h

the arithmetically profinite extension L / F which is a natural Gal(L/F)- module.

The map NLIF is a 1-cocycle. It is compatible with the ramification filtration on the Galois group and the natural filtration on local fields.

We shall study the image of NLIF and show that there is a bijection

for a certain subgroup YLIF of UO- which contains UN(LIF). To N(L/F)

check the properties of NLIF we shall define a map

which acts in the reverse direction. The latter is a generalization of the fundamental exact sequence and the constructions of [HI, H2].

The set U0 - /YLIF with a new group structure given by N(LIF)

is isomorphic to Gal(L/F).

Recall that the field of norms N ( ~ F ) is isomorphic to IFFP((X)), so U - is isomorphic to lFFP[[X]]*. Thus, every Galois group of

N(LIF) a totally ramified arithmetically profinite extension L I F is isomorphic to a certain subquotient of IFFP[[X]]* which is endowed with the new (nonabelian in general) group structure on it.

The classical abelian reciprocity isomorphism is the P-component of theNLIF and HLIF. If R / F is the maximal abelian subextension of L/F ,

then the &-component of the NLIF and ELIF is in fact the metabelian reciprocity map introduced by Koch and de Shalit [K, KdSh].

66 I. Fesenko

Let F be a local field with finite residue field. Let cp in the absolute Galois group GF of F be an extension of the Frobenius automorphism of the maximal unramified extension Fur over F .

Let F p be the fixed field of cp. It is a totally ramified extension of F and its compositum with Fur coincides with the maximal separable extension of F . We shall work with Galois extensions of F inside FP. The reason why in the nonabelian theory one is deemed to work with extensions inside F p is explained in [KdSh, 0.21.

From abelian local class field theory and a compactness argument one deduces that there is a unique norm-compatible sequence of prime elements (aE) in finite subextensions of F'+'/F, see for instance [KdSh, Lemma 0.21.

Recall that a separable extension L of a local field K is called arithmetically profinite if the subgroup GLG& is of finite index in GK for every x (where G& is the upper ramification group of GK). Equivalently, L/K is arithmetically profinite if it has finite residue field extension and the Hasse-Herbrand function hLIK(x) = lim hEIK (x) takes real values for all real x 3 0 where E/K runs through all finite subextensions in L/K, see [W], [FV, Ch. 111, sect. 51. For an infinite arithmetically profinite extension L/K the field of norms N = N(L/K) is the set of all norm-compatible sequences

{(aE) : a~ E E*, E / K is a finite subextension of L/K)

and zero, such that the multiplication is componentwise and the addition (aE) + (bE) = (CE) is defined as c~ = limM NMIE (aM + bM) where hf runs through all finite subextension of E in L. An element of the field of norms has E-component for every finite subextension E / K of L/K. The field N is a local field of characteristic p with the residue field isomorphic to the residue field of L and a prime element t = ( r E ) which is a sequence of norm-compatible prime elements of finite subextensions of L/K. If the extension L/ F is totally ramified, then the discrete valuation VN(L/F) is given by ~ ~ ( ~ 1 ~ ) ((aE)) = vF(aF) = vE (aE). Every automorphism T of L over K induces an automorphism T of the field of norms: r((aE)) =

(TTE). If M is a separable extension of L, then one defines N(M, L/K) as the composit um of all N (F'I K ) where F' runs through finite extensions of L in M.

For FontaineWintenberger7s theory of fields of norms see [W], [FV, Ch. 111, sect. 51. For a finite subextension M / K of an arithmetically profinite extension L/K the extension LIM is arithmetically profinite; for every subextension M I K of an arithmetically profinite extension L/K the extension M / K is arithmetically profinite. One of the central

Nonabelian Local Reciprocity Maps 67

theorems of the theory of fields of norms tells that the absolute Galois group of N(L/K) coincides with G(N(LSeP, L/K)/N(L/K)) and the latter is isomorphic to G(LSeP/L), see [W, 3.2.21. Every abelian totally ramified extension is arithmetically profinite.

If L I K is finite, then denote by N(L/K) the set consisting of norm- compatible sequences in the multiplicative groups of finite subextensions in L I K and of 0. By UN(LIK) we mean the group of norm-compatible sequences in the group of units of subextensions in L/K.

Let L c Fq be a Galois (possibly infinite) totally ramified arithmetically profinite extension of F . The canonical sequence of norm- compatible prime elements (nE) in finite subextensions of P / F s u p plies the canonical sequence of norm-compatible prime elements ( r E ) in finite subextensions of L / F and therefore the canonical prime element X of the local field N(L/F). Denote by cp the automorphism of N(L/F)Ur - and N(L/F) corresponding to cp.

Using solvability of Galois extensions in the local situation fix a tower of subfields F = Eo - El - E2 - . . . , such that L = UEi , Ei/Ei-1 is cyclic of prime degree and Ei/F is normal. We can assume that the degree I Ei+1 : Ei ( = p for all i 3 io and I Ei, : Eo ( is relatively prime to P .

A h

Let N(L/Ei) be the field of norms of the arithmetically profinite h A

extension LIEi. It can be identified with the completion N ( ~ E ~ ) of the maximal unramified extension N(L/Ei)Ur of N (LIEi).

For a local field K the symbols UK, Ui,K denote, as usual, the group of units of the ring of integers and the higher groups of units.

Definition 1. Denote by U0 the subgroup of the group N(LIF)

UNEF) of those elements whose F-component belongs to Up.

Recall that every element of the group of units of a local field with separably closed residue field is (p - 1)-divisible, see for instance [I2,

L

Lemma 3.111. To motivate the next definition we interprete the map N for a finite

Galois totally ramified extension L / F in the following way. Since in this case both ax and a~ are prime elements of LUr, there is E E ULUr such that ax = aL&. We can take d = a c p . Then a:-' = Let 7 E Uz

be such that 7'p-l = E. Since (vup-l~-l)'+'-l = (7)("-1)p)p-1 , we deduce that E = 71u~-1rl( ' -o)~p with p E UL. Thus, for [ = 77"p-1

I. Fesenko

Definition 2. Define the map

NL/F: Gal(L/F) U ~ z F ) / U ~ ( ~ I ~ )

where U = (uEi) E U - satisfies the equation N(LIF)

Then, clearly, (us,) belongs to Uo and is defined modulo UN(LlF). N(LIF)

Note that U0 - /NU(LIF) is a direct product of the group of mul- N(L /F) , , ,

tiplicative representatives of F, a cyclic group Z/pa and a countable free topological Zp-module.

Remark 1. For a finite extension L / F the P-component of . In other NLIF (0) is equal to NzIp( mod NLIF UL where = a"-'

words, the F-component of NLI is the classical Neukirch-Iwasawa map N.

Lemma 1. Let M I F be a Galois subextension of L I F and EIF be a finite subextension of LIF . Then the following diagrams of maps are commutative:

Lemma 2. NLIF is injective and

N L / F ( ~ T ) = N ~ I ~ ( o ) O N ~ / ~ ( r ) .

Proof. If N L / F ( ~ ) = ( u ~ ) E uN(LIF), then (uE)v-' = 1; so o acts trivially on the prime elements T E , therefore a = 1. 0


Let U0 A be the filtration induced from the filtration U - n,N(LIF) n,N(LIF)

on the field of norms. For an infinite arithmetically profinite extension L / F with the HasseHerbrand function hLIF put Gal(L/F), =

G ~ ~ ( L / F )

Proof. Let T E Gal(L/F),. Then due to the properties of arithmetically profinite extensions [W, 3.3.2 and 3.3.41 there is a finite subextension Q/ F of L I F such that ~ 2 1 ; ~ E Un,E/ for every El > Q.

Choose a solution (up) of the equation (ug) '-9 = (TE)~- ' such that u p E U n , ~ for E' 3 Q. Then vG(((up) - 1 ) ~ ) 2 n for sufficiently

large El1 > Q [W, 2.3.2.2, 2.3.2.31. Hence (up) E U - n,N(L/F)'

If (TE)~- ' = ( U ~ ) ~ - P with (up) E U0 uN(LIF), then n+l,N(L/F) (TE)~- ' E uo - , and so a::' E for sufficiently large

n+l,N(LIF) El [W, 3.21. Thus, by [W, 3.3.2 and 3.3.41, T E Gal(L/F)n+l. 0

Remark 2. The set im(NLIF) isn't closed in general with respect to the multiplication in UNEF,/UN(LIF). However, Lemma 2 implies

. , , that being endowed with a new group structure given by

irn(NL/F) is a group isomorphic to Gal(L/F).

To study the image of NLIF we shall define after some preliminary considerations a map NLIF which takes values in Gal(L/ F) .

Recall that the norm map is surjective for finite extensions of local fields with separably closed residue field; see for instance [Se, 2.21.

Definition 3. Let oi be a generator of Gal(Ei/Ei-1). Let v p be

the discrete valuation of g i . Put si = v- - 1). Denote Xi = U 2 -' . Ei Ei

It is a Z,-submodule of U1,$. Xi is the direct sum of a cyclic torsion

group of order pni, ni 2 0, generated by, say, cyi (ai = 1 if ni = 0) and a free topological Zp-module Y,.

Note that if i < io, i.e. (Ei : EiVl( is relatively prime to p, then if a primitive pth root of unity C, were equal to u0%-' with u E Ei, then

70 I. Fesenko

A A

UP would belong to Ei-' and hence u E Ei-', a contradiction. Hence ai = 1, ni = 0 for i $ io.

Definition 3'. If ni = 0, set E U - to be equal to 1. If N(LIE2)

ni > 0, let A(') E UN(TEi) be a lifting of ai with the following restric-

tion: A!? isn't a root of unity of order a power of p (this condition can Et+ 1

be satisfied by multiplying the Ei+l-component of A('), if necessary, by yUi+l-' where y E U- sufficiently small).

Ei+l

pni

Lemma 3. ~f A(') # 1, then = A? belongs to Xi+' Ei+l

Proof. Clearly N2, = 1, so a+' = np-' Ei+l ~ ~ i t l - 1 with ~ + 1

p E Gal(Ei+i /Ei). We need to show that p = 1.

If E = NE- lg 6 E Ei , then N(p) belongs to NEi+l /Ei UE,+~ , and hence %+1

p = 1. If E # Ei, then EP = apw where a E UEz and w E UEz is a p-primary element (the extension Ei( fi)/Ei is unramified of degree p). If W E N E ~ + ~ ~ E ~ U E , + ~ , then N(p) = eP belongs to N E ~ + ~ / E ~ U E , + ~ , and hence p = 1. If w # NEi+l/EiUEi+l, then si+l = pe(Ei)/(p - 1).

Since A(:! E ~ 2 - ' is a primitive root of unity of order a power of p, we Ei Ei

deduce that 0 < si < e(Ei)/(p - l), hence (si,p) = 1. In this case it is well known and easy to see that si = si+l mod p, a contradiction. Thus p = 1. 0

From now on we assume that if F is of characteristic 0 and contains a primitive pth root of unity, then L / F is of infinite degree.

Definition 3". Let pilj, j E N be free generators of Y , which

include Pi whenever Pi is defined. Let ~ ( ~ l j ) E UN(TEl) be a lifting of

pi,, (i.e. B("j)si = &), such that if iO, = pi, then ~ ( " j j ) E k = B(") Ek =

A(y)~ni-l for lc 3 i.

E k

Definition 4. Define a map Xi + U - by sending a: nj Pi:;, N(LIEi)


C ' where 0 $ c $ ni - 1, cj E Zp, to n. '. We get a map 3

It depends on the choice of lifting.

Note that f i ( ~ ) ~ , = a .

Definition 5. Let

Lemma 4. The product oft(') in the definition of ZLIF converges. ZLIF is a subgroup of uo - . The subgroup YLjF contains uN(L/F).

N(L/F)

Proof. Let L I F be infinite. Let u p be the discrete valuation of $. Denote si = usi (nzi-' - 1). If hLIF is the HasseHerbrand function of

L I F , then ht jF(si) tends to +oo when i tends to infinity [FV, Ch. 111,

sect. 51. So, if q(EkIEi) is the minimal real number such that hEkIEi (x) =

x for x $ q(EklEi), then q(EklEi) tends to infinity when i tends to

infinity (see [FV, Ch. 111,(5.2)]). Note that vEi(zg: - 1) > si; so from

[FV, Ch. 111, (5.5)] we deduce that

v - ((z(') - I)$) 2 min((1 - p-') min q ( ~ k I E ~ ) , si). Ei k>i

Hence the product of di) converges. Due to the previous definitions the subgroup of U - generated

N(L/F) by fi (Us-') is equal to

t

Since = B(i+l)d , ZLIF is a subgroup of u - N(L/F)'

0

The following theorem is a generalization of the fundamental exact sequence.

72 I. Fesenko

Theorem 1. For every (u- ) E UO- there is a unique auto- Ei N(L/F)

morphism r in the group Gal(L/F) satisfying

(uA )I-9 = xr-l mod ZLIF. El

If (upi) E YLIF, then r = 1.

Proof. Assume that u - E UEJPl, and u - 6 UEj. Then Ej-1 E j

N- - u?? = 1, SO from the fundamental exact sequence u?? = E3IE3-1 Ei E3

"7j-lWuj-1 Ej

with rj E Gal(Ej/Ej-l), w E UE7. Both rj and wuj-' are

uniquely determined by (uEi). Let w(j) = f j (woj-I).

Now assume that for i > j we get

with uniquely determined 7,-1 E Gal(EiPl/F), w ( ~ ) E ~ ~ ( ~ 3 - l ) . Ek We

shall show that then the same is true for u k 9 . Ei

Let T,!-~ E Gal(Ei/F) be an extension of ri-1. Then

with E in the kernel of Nz ,- . F'rom the fundamental exact sequence 1- 1

ul"l 0,-1 weget E = rEi vi with vi E UE, 1 ( a < IEi : Ei-11. Write

Then

It remains to show that ~ i , w ( ~ ) are uniquely determined. If pi E

Gal(Ei/F), v(" E f i(u2- ' ) are such that I


since vt), w(') (10'-1 - , we deduce that the right hand side of the last Ei Et Ez

equation belongs to V(EilEi-l). Therefore, from the fundamental exact

sequence p = 1. Consequently v t ) = wk) and v(') = ~ ( 9 ) . Et Ei

Thus, by induction there is a unique automorphism r E Gal(L/F), T 1 E, = ri , satisfying

If (upi) E YLIF, then from the uniqueness we get r = 1. 0

Corollary. Thus, we get the map

defined by HLIF((uE)) = T. The composition of NLIF and HLIF is the

identity map of Gal(L/F).

Definition 6. Define the map

where r is the unique automorphism satisfying ( U ~ ) ~ - P = Xr-l mod ZLIF.

Note that U0 - /YLIF is a direct product of the group of multi- N(LIF)

plicative representatives of F , a cyclic group Z/pa and a countable free topological Z,-module.

Lemma 5. aLIF is injective.

Pm0f. If E ZLIF , then (up) E YLIF. 0

76 I. Fesenko

Denote R = Fab n Fv, M = R~~ n Fp. Since R / F and N(M, R /F) / N (R/ F) are arithmetically profinite, the extension M/ F is arithmetically profinite [W, 3.4.11.

Note that if T = H(uA') is the automorphism of Gal(R/F) corresponding via abelian class field theory to u-' , then the equation [(X)v-' = {u)(X)/X can be interpreted as (uE^)'-v = ( ? r ~ ) ~ - ' in the field of

norms of RIF . Hence g(F) can be identified with the set n(F) of all pairs (u, (up)) E UF x U0 which satisfy the equation (up)'-v =

N(RIF) (?rE)H(~-l)-l.

According to Corollary of Theorem 2 every coset of U0 - modulo N(M/F) . .

YMIF has a unique representative in im(NM/F). Send a coset with a

representative (u-) E Uo - satisfying (uQ )'-'+' = (TQ)~- ' with r E Q N(M/F)

Gal(M/F) to (ugl,(uP) E Uo- ). It belongs to n(F) by Remark 4. N(RIF)

Thus, we get a map

Now we construct an inverse map to g. For a pair (u, (up)) E UF x

Uo satisfying (uE^)'-v = ( T ~ ) ~ - ' fix a finite subextension EIF N(RIF)

of R/F. We claim that for every finite abelian subextension Q / E of M/E such that Q is normal over F there are unique u- E UG and

Q TQ E Gal(Q/F) satisfying

Writeus = N- -u with u E U-and observe that N- -(ul-v QIE Q QI E ? r ~ )

= 1 for a lifting T' E Gal(Q/F) of T. The group V(Q/E) is (9 -

1)-divisible, so @-' = wp-l ?rQ T ! - I ~ u - I with w E V(Q/E). From

the fundamental exact sequence for the abelian extension Q I E we get ul-cp 1-7' -

? r ~ - ?rG-'vv-' for some o E Gal(Q/E), u E V(Q/E). Hence

TQ-1

uQ^ = uvw satisfies N- nu- = u ~ , u?' =

QIE Q ?rQ , where TQ = 07'.

Q ~f u'^l-v = ?rQ 6 - l and N61zu& = up, then u~'u!-. belongs to

Q Q Q the kernel of N- so from the fundamental sequence we deduce that Q I ~ ' ?rTGlT6-' Q = ( u ~ ~ u L ) ' - v E V(Q/E). Since Q / E is abelian, T& = TQ,

Q Q "$ = "6.


Now let E1/F be a subextension of an abelian extension E2/F, let Q1 /El, Q2/E2 be abelian finite subextensions in M/F and let Q1 c Q2 be normal over F . Then N-- nu--, TQ, I Q l , where u--, T Q ~ constructed

QzIQI Qz Q2

for Q2/ E2 satisfies the conditions for Q1 / El, therefore the uniqueness

implies N-- Q , /Q~ --u-- Q~ = u- Q~ 7 TQZ I Q I r ~ ~ - Hence the pair (u, (uE)) E UF x UO- satisfying (us)'-' =

N(RIF) ( T E ) ~ ( " - ~ ) - ' uniquely determines TM E Gal(M/F) and ("6) E

U0 satisfying (u-)'+ Q = ( ?~Q)~M- ' . N(M/F)

Thus, we get the inverse map h: n(F) + U0 - /YMIF to g. N(MIF)

Now it is easy to show that the reciprocity map

of [KdSh] coincides with ('HMIF o h)-' and it associates TG' to (u, ( u ~ ) )

E n(F). The map (g o &IF) - ' is the inverse one. Thus, without using Coleman's homomorphism and the Lubin-Tate

theory (employed in the constructions of [KdSh]) one can deduce the metabelian reciprocity map as a partial case of ELIF. Note that the group structure on g(F) defined in [KdSh] corresponds to the group structure on im(NMIF) discussed in Remark 2.

Remark 6. Similarly one can deduce the reciprocity map constructed by Gurevich [GI for extensions L / F for which the n-th derived group of the Galois group is trivial.

References

B. Dwork, Norm residue symbol in local number fields, Abh. Math. Sem. Univ. Hamburg, 22 (1958), 180-190.

I. Fesenko, Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic, Algebra i Analiz, 3, no.3 (1991), 165-196; English transl. in St. Petersburg Math. J., 3, no.3 (1992), 649-678.

I. Fesenko, Multidimensional local class field theory. 11, Algebra i Analiz, 3, no.5 (1991), 168-190; English transl. in St. Petersburg Math. J., 3 (1992), 1103-1126.

I. Fesenko, Local class field theory: perfect residue field case, Izvestija Russ. Acad. Nauk. Ser. Mat., 57, no.4 (1993), 72-91; English transl. in Russ. Acad. Scienc. Izvest. Math., 43 (1994), 65-81.

I. Fesenko, Abelian local pclass field theory, Math. Ann., 301 (1995), 561-586.

I. Fesenko

I. Fesenko, On general local reciprocity maps, J. reine angew. Math., 473 (1996), 207-222.

I. B. Fesenko, S. V. Vostokov, "Local Fields and Their Extensions: A Constructive Approach", AMS, Providence, R.I., 1993.

A. Gurevich, On generalization of metabelian local class field theory, preprint (1997).

M. Hazewinkel, "Abelian extensions of local fields", Thesis, Amster- dam Univ., 1969.

M. Hazewinkel, Local class field theory is easy, Adv. Math., 18 (1975), 148-181.

K. Iwasawa, "Local Class Field Theory", Iwanami-Shoten, Tokyo, 1980, (Japanese).

K. Iwasawa, "Local Class Field Theory", Oxford Univ. Press and Clarendon Press, New York and Oxford, 1986.

H. Koch, Local class field theory for metabelian extensions, In Pro- ceed. of the 2nd Gauss Symposium. Conf. A: Mathematics and Theor. Physics (Munchen, l993), 287-300, de Gruyter, Berlin, 1995.

[KdSh] H. Koch, E. de Shalit, Metabelian local class field theory, J. reine angew. Math., 478 (1996), 85-106.

[Nl] J. Neukirch, Neubegrundung der Klassenkorpertheorie, Math. Z., 186 (1984), 557-574.

[N2] J. Neukirch, "Class Field Theory", Springer, Berlin etc., 1986.

[Se] J.-P. Serre, Sur les corps locaux B corps residue1 algkbriquement clos, Bull. Soc. Math. France, 89 (1961), 105-154.

[W] J.-P. Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; applications, Ann. Sci. E.N.S., 4 skrie, 16 (1983), 59-89.

Department of Mathematics University of Nottingham NG7 2RD Nottingham England E-mail address: i . f esenko0maths. nottingham. ac . uk


Embedding Problems with restricted Ramifications and the Class Number of Hilbert Class Fields

Akito Nomura

1 Introduction

Let k be an algebraic number field of finite degree, and 6 its absolute Galois group. Let L l k be a finite Galois extension with Galois group G,

and (E) : 1 + A + E 1) G + 1 a group extension with an abelian kernel A. Then an embedding problem (Llk, E) is defined by the diagram

where cp is the canonical surjection. When (E) is a central extension, we call (Llk, E) a central embedding problem. A solution of the embedding problem (Llk , e) is, by definition, a continuous homomorphism $ of 6 t o E satisfying the conditions j o $ = cp. We say the embedding problem ( L l k , ~ ) is solvable if it has a solution. The Galois extension over k corresponding t o the kernel of any solution is called a solution field. A solution $ is called a proper solution if it is surjective. The existence of a proper solution of (Llk, E) is equivalent to the existence of a Galois extension M I L I k such that the canonical sequence 1 --+ Gal(M/L) -t Gal(M/k) -, Gal(L/k) + 1 coincides with e.

Let S be a finite set of primes of L. An embedding problem with ramification conditions (Llk, E, S ) is defined by the diagram (*) , which is same to the case of (L lk , E ) . A solution $ is called a solution of (Llk, e, S) if MIL is unramified outside S, where M is the solution field corresponding to +. We remark that these definitions are a little different from those in [3] and [8], but essentially of the same nature.

Received August 28, 1998. Revised October 24, 1998.

80 A. Nomura

52. Central embedding problems

In this section, we quote some well-known results about central embedding problems without proofs. General studies on embedding problem are in Hoechsmann[5] and Neukirch[8].

Let k be an algebraic number field, and (Llk, e) a central embedding problem defined by the diagram (*) with a finite abelian group of odd order.

Fact 1. If L/k is unramified or (e) is split, then (Llk, e) is solvable.

Fact 2 (Ikeda[G]). If (Llk, e) is solvable, then (L/k,e) has a proper solution.

We remark that Fact 2 is always true in case A is abelian not necessary (E) is central.

For each prime q of k, we denote by kg (resp. Lq) the completion of k (resp. L) by q (resp. an extension of q to L). Then the local problem (Lq /kg , eq ) of (L/k, E) is defined by the diagram

where Gq is the Galois group of Lq/kq, which is isomorphic to the decomposition group of q in Llk, eq is the absolute Galois group of kq, and Eq is the inverse of Gq by j .

In the same manner as the case of (L/ k, E ) , solutions, solution fields etc. are defined for (Lq /kg , eq ).

Let p be an odd prime.

Fact 3. Let (E) : 1 + Z/pZ -+ E -t Gal(L/k) + 1 be a central extension. If (Lq/kq, eq) is solvable for every prime q, then (L/k, e) is solvable.

Fact 4 (Neukirch[8]). Let (e) : 1 + Z/pZ + E -t Gal(L/k) + 1 be a central extension, and assume that (Llk, e) has a solution. Let T be a finite set of primes of k, and M(q) be a solution field of (Lq/kq, E ~ )

for q of T. Then there exists a solution field M of (Llk, e) such that the completion of M by q is equal to M(q) for each q of T.

By using this fact, we can construct a good solution of (L/k, E).

Embedding Problems with restricted Ramifications 8 1

$3. Main theorem

Let LIK be a Galois extension of an algebraic number field K. We denote by Pl(L/K) (resp. P2(L/K)) the set of primes of L which is ramified in L I K and not lying above p (resp. lying above p). Let T be a finite set of primes of k, and denote by Bk(T) the set {a E k* l(a) = ap for some ideal a of k, and a E kgP for every prime q of T).

The following is a main theorem of this article.

Theorem. Let p be a n odd prime, and L/K/k a Galois extension such that L/K is a p-extension and that the degree [K : k] is prime to p. Let S be a finite set of primes of L, which contains the set Pl(L/K) and disjoint to P2(L/K), and (E) : 1 + Z/pZ + E -+ Gal(L/k) + 1 be a non-split central extension. Assume that the following conditions (Cl), (C2) and (C3) are satisfied.

(Cl) The embedding problem (Llk, e) has a solution. (C2) For every prime p of k lying above p, the local problem (Lp /kp,

cp) has a solution $j, such that Mp/Lp is unmmified, where Mp is a solution field corresponding to q p .

(C3) Bk(So) = k*', where So is the set of prime q of k such that q is the restriction of some prime contained i n S.

Then, (Llk, e, S) has a proper solution. That is to say, there exists a Galois extension Mlk such that

( i ) 1 + Gal(M/L) 4 Gal(M/k) -+ Gal(L/k) + 1 coincides with ( 4 , and

(ii) MIL is unmmijied outside S .

Remark. (1) There does not always exist a non-split central extension (E) : 1 + Z/pZ + E -+ Gal(L/k) + 1. The existence is equivalent to the non-vanishing of the cohomology group H2 (Gal(L/ k) , ZlpZ) .

(2) If k is the rational number field Q and LIK is unramified, then the conditions (Cl), (C2) and (C3) are satisfied.

As a simple case of the main theorem, we have the following. We treat the sketch of the proof of the following instead of the main theorem. For details, see [lo] and [13].

Proposition 1. Let p be a n odd prime, and L/K/Q a Galois extension such that LIK is a n unramified p-extension and that the degree [K : Q] is prime to p. Let (e) : 1 + Z/pZ + E -+ Gal(L/Q) + 1 be a non-split central extension.

Then there exists a Galois extension M/Q such that (i) 1 + Gal(M/L) -+ Gal(M/Q) --t Gal(L/Q) + 1 coincides with

(4, and

82 A. Nomura Embedding Problems with restricted Ramifications 83

(ii) MIL is unmmijied.

(sketch of the proof.) In this case, by using the general theory of embedding problems, we can easily see that the problem (L/Q,e) is solvable. By virtue of Fact 2 we can take a solution field Ml/L/Q such that every prime !Jl of L lying above p is unramified in Ml/L. Let 8 be a prime of L ramified in Ml/L, and q the prime number below 8. Then q r 1 mod p. Hence there exists a field F such that Q c F c Q(&) and that [F : Q] = p. Let M2 be the inertia field of 8 in M1 FIL, then M2 is also a solution field of (LIQ, e). And the number of primes ramified in M2/L is less than that of in Ml / L. By repeating this process, we can take a required extension.

$4. Applications

Let D be the group defined by

This is a non-abelian pgroup of order p3.

Proposition 2. Let K be a quadratic field, and assume that the p- rank of the ideal class group of K is greater than or equal to 2. Then there exists a Galois extension M / K such that the Galois group is isomorphic to D and that M / K is unramijied.

(sketch of the proof.) Let L I K be an unramified extension such that the Galois group Gal(L/K) is isomorphic to Z/pZ x ZlpZ. Then the Galois group Gal(L/Q) is isomorphic to

Let E =< x, y, z, t 1 xP = yP = zP = t2 = 1, Y-lxY = xz, zx =

xz, yz = zy,t-'xt = x-l, t-'yt = y-l , tz = zt >. Then 1 i< z >-+ E % Gal(L/Q) -+ 1 is a non-split central

extension, where j is defined by x -+ a , y -+ b, t -+ c. Since the Sylow subgroup of E is isomorphic to D l then by applying

Proposition 1, we can take a required extension.

Let K1 be the Hilbert pclass field of K, and K2 the central pclass field of K1/K. The following proposition is obtained by Miyake.

Proposition 3 (Miyake[7]). Let K be a quadratic field. Then the Galois group Gal(K2/K1) is isomorphic to Gal(K1/K) A Gal(K1 / K ) ,

where A denotes the exterior square. Further assume that the p-Sylow subgroup of the ideal class group of K is isomorphic to Z/pel Z x ZIPe2 Z x - . x Z/perZ (1 5 el 5 e2 < - . . 5 e,).

Then the Galois group Gal(K2/K) is isomorphic to

Let Q, be the prank of the unit group of k and Clk the ideal class group of k.

Proposition 4 (Nomura[lS]). Let p be an odd prime, and L/ K/k a Galois extension such that L /K is an unmmijied p-extension and that the degree [K : k] is prime to p. If p-mnk of the cohomology group H2(Gal(L/k), ZlpZ) is greater than ~ ,+p-rank Clk, then the class number of L is divisible by p.

(sketch of the proof.) There exists a finite set So of primes of k satisfying the conditions : (i) So does not contain any prime lying above p, (ii) Bk(So) = k*p, (iii) ISo 1 = ep + p-rank Clk.

Indeed, let F = k(*; a E Bk(0)). Then the Galois group Gal(F/k(Q)) is an abelian pgroup and isomorphic to ( z /~z ) m, where m = ep + p-rankClk. By Chevotarev's density theorem, there exist primes q l , q2 , - . - , qm such that the Frobenius automorphism of qi (i =

1,2, , m) generate Gal(F/k(Cp)). Then So = {ql, . . . , q,) is a required set.

Let S be the set of primes of L which is an extension of q E So. For each ( E ) : 1 -+ Z/pZ -t E -+ Gal(L/k) -+ 1, let ME be a Galois extension corresponding to a proper solution of (Llk, e, S) . Let M be the composite field of ME for all E. Then the Galois group Gal(M/L) is isomorphic to ( Z / P Z ) ~ , where m is equal to the p-rank of H2 (Gal(L/k) , Z/pZ) . For q E So, denote by M(q) the inertia field of in MIL, where is an extension of q to L. Since Gal(M/L) is contained in the center of Gal(M/ k), M (q)/ L/ k is a Galois extension. Then every prime of L lying above q is unramified in M(q)/L. Let M* be the intersection of M(q) for all q of So. If m > ISo], then M*/L is a non-trivial pextension. Hence the class number of L is divisible by p.

Proposition 5. Let p be an odd prime, and L the Hilbert p-class field of k. Assume that the p-rank of the ideal class group of k is greater than (1 + ,/-)/2, then the class number of L is divisible by p.

84 A. Nomura

(proof.) Since Gal(L/k) is abelian, the prank of H2(Gal(L/k), Z/pZ) is equal to n(n + 1)/2, where n is the prank of the ideal class group of k. By using Proposition 4;we have thus proved the proposition.

We investigate an application to the Boston's question, which is related to the Font aine-Mazur conjecture. For the Font aine-Mazur conjecture, see [I], [2] and [4].

Conjecture (Fontaine-Mazur) . Let Kur be the maximal unramified extension of an algebraic number field K. For any K, positive integer n and representation p : Gal(Kur/K) + GL,(Q,), the image of p is finite.

Let p be an odd prime. A pro-p group G is called powerful if G/GP is abelian, where the line denotes topological closure.

In [I] Boston introduced the following, which is equivalent to the above conjecture.

Conjecture (Fontaine-Mazur-Boston) . For any algebraic number field K , there does not exist an unramified pro-p extension K/K such that the degree [K : K] is infinite and that the Galois group is powerful.

Boston pointed out that this conjecture is closely related to the existence of unramified pextensions of a certain type, and introduced the following question.

Question (Boston). Let K be a number field, p an odd prime, and K(p) its pclass field. Suppose that the class number of K(p) is divisible by p. Then is there always an everywhere unramified extension M of degree p of K(p) such that M is Galois over K and exp(Gal(M/K)) = exp(Gal(K (p)/ K))? The " exp" stands for the exponent of the group.

Remark (Boston). (1) The truth of the Fontaine-Mazur conjecture implies an affirmative answer, when K has an infinite pclass field tower .

(2) Lemmermeyer noticed that the answer to this question is in the negative in general. He pointed out an example, due to Scholz and Taussky[l4]. The Galois group of the maximal unramified 3-extension of ~(d-4027) is isomorphic to < x, y 1 y("9y) = Y - ~ , x3 = y3 >. This group has a non-abelian subgroup of order 27 and exponent 9. Let K be the corresponding intermediate field, its 3-class field is an elementary abelian extension of degree 9 contained in no larger unramified extension with Galois group of exponent 3. Since the class field tower of K is finite, this is not a counter example of Fontaine-Mazur conjecture.

Embedding Problems with restricted Ramzfications

We produce some sufficient conditions for the answer

85

to Boston's question for K and p is affirmative. For detail and other results, see [ll] and [12].

Proposi t ion 6. (1) Let 1 and p be odd primes such that the order of p mod 1 is even. Assume that K / Q is an abelian 1-extension and the class number of K is divisible by p. Then there exists an unramified non-abelian p-extension M I K such that the exponent of Gal(M/K) is p, and therefore the answer to Boston's question for K and p is afirmative.

(2) Let p be an odd prime, and K a quadratic field. Then the answer to Boston's question for K and p is afirmative.

(sketch of the proof.) (1) There exists a Galois extension L/K/Q such that L /K is an unramified abelian pextension of exponent p. Un- der the assumption of p and 1, the cohomology group H ~ ( G ~ ~ ( L / Q ) , ZlpZ) is non-trivial. Hence there exists an non-split central extension (e) : 1 -+ Z/pZ --t E + Gal(L/Q) 4 1. By Proposition 1, there exists a Galois extension Ml/L/Q such that M1 gives a proper solution of (LIQ, E ) and that Ml/L is unramified. By group theoretical considerations, the pSylow subgroup E, of E is a non-abelian pgroup of exponent p, and the Galois group of M1/K is isomorphic to E,. Then Ml . K(p) gives an affirmative answer to Boston's question for K and p.

By using Proposition 2, we can easily prove (2).

References

N. Boston, Some cases of the Fontaine-Mazur conjecture, J. Number Theory, 42 (1992)) 285-291.

N. Boston, Some cases of the Fontaine-Mazur conjecture 11, J. Number Theory, 75 (1999), 161-169.

T. Crespo, Embedding problem with ramification conditions, Arch. Math., 53 (1989), 270-276.

F. Hajir, On the growth of pclass groups in pclass field towers, J. Algebra, 188 (1996), 256-271.

K. Hoechsmann, Zum Einbettungsproblem, J. reine angew. Math., 229 (1968), 81-106.

M. Ikeda, Zur Existenz eigentlicher galoisscher Korper beim Einbet- tungsproblem, Hamb. Abh., 24 (1960), 126-131.

K. Miyake, On the Ideal Class Groups of the pClass Fields of Quadratic Number Fields, Proc. Japan Acad., 68 Ser.A (1992), 62-67.

J. Neukirch, ~ b e r das Einbettungsproblem der algebraischen Zahlenthe- orie, Invent. Math., 21 (1973), 59-116.

86 A. Nomura

[9] A. Nomura, On the existence of unramified pextensions, Osaka J . Math., 28 (1991), 55-62.

[lo] A. Nomura, On the class number of certain Hilbert class fields, Manu- scripts Math., 79 (1993), 379-390.

[ll] A. Nomura, A Remark on Boston's Question Concerning the Existence of Unramified pextensions, J.Number Theory, 58 (1996), 66-70.

[12] A. Nomura, A Remark on Boston's Question Concerning the Existence of Unramified pextensions 11, Proc. Japan Acad., 73 Ser.A (1997), 10-11.

[13] A. Nomura, On embedding problems with restricted ramifications, Arch. Math., 73 (1999), 199-204.

[14] A. Scholz and 0 . Taussky, Die Hauptideale der kubischen Klassenkorper imaginarquadratischer Zahlkorper; ihre rechnerische Bestimmung und ihr Einfluss auf den Klassenkorperturm, J . Reine Angew. Math., 171 (1934), 19-41.

Department of Mathematics, Kanazawa University, Kanazawa 920-1 1 92, Japan E-mail address: anomuracDt . kanazawa-u. ac . jp


The History of the Theorem of Shafarevich in the Theory of Class Formations

Helmut Koch

The Theorem of Shafarevich or, as it is mostly called, the Theorem of Shafarevich-Weil always seemed to me to be the coronation of the cohomological approach to class field theory showing that the notion of the canonical class is much more than an auxiliary tool for proving the main theorems of class field theory. The history of the Theorem of Shafarevich and the development of the notion of the canonical class is quite interesting and this is the subject of my talk.

I begin with some remarks about the periodization of class field theory. Thereafter follows the formulation of the Theorem of Shafarevich and its background as it presents itself to the mathematician of today. Then I will speak about the local case which is the content of Shafare- vich's paper in Doklady 53 (1946). In the next section we consider the

I paper of Weil of 1951 "Sur la thborie du corps de classes" which had a great influence on the further development of class field theory. Then

: follows the development of the notion of the global canonical class and the formulation of the global theorem by Hochschild and Nakayama.

E I will conclude the talk with the consideration of the role played by

E Hasse and his school. I am very grateful to Jean-Pierre Serre, Sigrid Boge and Giinther

[ Frei who read a preliminary version of this paper and made suggestions

E which led to an improvement of this talk in content and form. i I am very much obliged to Wolfram Jehne who contributed to the 5

/ talk by means of many discussions of the subject with me over the last f years. I r: a

I 1 Some remarks about the periodization of class field theory L

1 The concepts of class field theory grew out of the work of Kro- necker [Kr1853, 18821 on cyclotomic fields and complex multiplication

[ and were formulated by Weber [We1891, 18971 and Hilbert [Hi1898].

Received September 2 , 1998.

88 H. Koch The History of the Theorem of Shafarevich 89

Ph. Furtwangler [Fu1907] in the unramified case and Takagi [Ta1920] in general proved the conjectures of Weber and Hilbert except for the Prin- cipal Ideal Theorem. Then Artin [Ar1924, 19271 added his reciprocity map to the theory and on its foundation the Principal Ideal Theorem was proved by Furtwangler in 1930.

With this result of Furtwangler the classical theory of class fields was established. The proofs were complete but looked very mysterious.

In the same year a new period of the theory began with three papers of Hasse [Ha1930a-c] creating local class field theory out of classical class field theory.

This was a period of reformulation and simplification of the theory which was completed by Tate [Tt1952] with the full establishment of the cohomological approach. Besides Hasse and Tate the main actors of this period were Chevalley, Herbrand, Nakayama, Hochschild and Weil, with Emmy Noether and her modern algebra in the background. It is this period 1930-1952 in which the Theorem of Shafarevich is placed. But before we are going into details I would like to give a brief presentation of the cohohomological approach to class field theory as it presents itself to the mathematician of today.

$2. The Theorem of Shafarevich in the Theory of Class For- mat ions

2.1. In our present understanding of class field theory, the Theorem of Shafarevich is a theorem about class formations. We have therefore to begin with a short consideration of this notion which was introduced by Artin-Tate [ArTt1952]. We will use a less abstract definition as is to be found in [Ko1992] which is adequate for our purpose.

Let F be a field and R / F a finite or infinite Galois extension. For our purpose F will be a local or global field and 52 the separable algebraic closure of F . We denote by fiF the category of finite extensions of F contained in 52. The morphisms of this category are the field homomorphisms which fix F elementwise.

A field formation is a functor A from fiF into the category Q of abelian groups such that the following properties are fulfilled.

(Ia) For any morphism cp of .RF the image A(cp) i s injective. If K, L are fields in fiF with K L, then we identify A(K) with its image in A(L).

(Ib) If L/K i s a normal extension i n fiF then

A ( L ) ~ ( ~ / ~ ) := { a E A(L) I ga = a for g E G(L/K)} = A(K).

The only interesting field formations for our purpose are AK = K X , the multiplicative group of K , if F is a local field, i.e. a complete field with discrete valuation and finite residue class field, called the local formation, and AK = CK, the idele class group of K , if F is a global field, i.e. F = Q the field of rational numbers or F = IF,(x), the rational function field over the field IF, with q elements.

2.2. Now we are going to define the notion of a class formation. We use the modified cohomological groups H (G, M) of Tate [Tt1952] and for any normal extension L /K in fiF we write for short

These cohomology groups have functorial properties. In particular we need the following:

(IIa) Let L/K be a finite normal extension i n fiF and let M be an intermediate field of L/K which i s normal over K . Then one has the inflation map

for all n 2 0. (IIb) Let LIK be a finite normal extension i n R / F and let M be an

arbitrary intermediate field of L/K. Then one has the restriction and the corestriction map

(IIc) Let L/K be a finite normal extension and s E G(fl/F). Then the compatible maps A(L) + A(sL), G(sL/sK) + G(L/K) given by a -+ sa for a E A(L) and t + sts-' for t E G(sL/sK) induce the map

A field formation K --+ A(K) is called a class formation if for any finite normal extension L /K in R / F the following axioms are fulfilled:

(IIIa) There is a canonical isomorphism

The preimage of 1 + [L : K]Z is called the canonical class and will be denoted by U L I K . In the following we write AL := A(L).

90 H. Koch The History of the Theorem of Shafarevich

( I I I ~ ) H ~ ( L / K ) = (0). (IIIc) For any finite extension M/K let L K + M be the map

induced by the compatible maps AL + ALM, G(LM/M) +

G(L/K). Then LL,MUL/K = [M : K]uLMIM.

From these axioms one derives the functorial properties of the canonical class: For L I K a normal extension and M an intermediate field of L /K one has

(IVa) h f M + L ~ M / K = [L : M ] u ~ , ~ i f M/K is normal. (IVb) R~SK-+MUL/K = UL/M, COTM-+KUL/M = [M : K]uL/K. (IVc) UsLlsK = s*uLIK with s defined as above.

2.3. In the cohomological setting the inverse of the Artin map, G -+ AK/NLIKAL, for an abelian extension L/K with Galois group G is given by

where fLIK(h,g) is a cocycle belonging to the canonical class UL/K. The map (2.1) was defined by Nakayama [Na1936] and is called the Nakayama map.

Tate [Tt1952] interpreted (2.1) as the map from H - 2 ( ~ ( ~ / ~ ) , Z) E

G(L/K) to H'(L/K) = AK/NLIKAL given by cup multiplication with the canonical class. More generally he proved that for any n E Z and any normal extension L / K in RIF the cup multiplication of B n ( ~ ( L/ K), Z) with the canonical class gives an isomorphism of H~(G(L/K) , Z) onto &+2 (LIK). (The case n = 0 is axiom (IIIa) and the case n = -1 is

axiom (IIIb)). For n = -2 this gives us an isomorphism of G/[G, G] onto AK/NLIKAL. One deduces the functorial properties of the Artin map from the functorial properties (IVa-c) of the canonical class. For a finite group G we denote by Gab the quotient group G/[G, GI = H - ~ (G, z). We keep the notation of (IVa-c) .

The following diagrams are commutative:

incl AK - AM

(Vb) 1 1 G(L/K)"~ -+ G(L/M)"~,

The bottom maps in (Va) and (Vc) are induced by projection and conjugation while the bottom maps in (Vb) are restriction and corestriction of H - ~ ( , Z). The restriction map is called transfer or Verlagerung. It was introduced by Artin in connection with the Principal Ideal Theorem [Ar1929]. But it already appears in Schur's paper [Su1902] as a nameless tool in the proofs. The corestriction map is induced by restriction to the subgroup.

For the proof of the fact that A(K) = K X for local fields K and A(K) = CK for global fields are class formations one uses apparently weaker axioms and proves the axioms above by the mechanism of group cohomology (see e.g. [Kol992]).

2.4. The Theorem of Shafarevich gives an answer to the following question: Let L/K be a normal extension in ffF and let MIL be an abelian extension given by the corresponding subgroup U of A(L). It follows from (Vc) that the extension M / K is normal if and only if sU = U for all s E G(L/K) and the conjugation of G(M/L) with an extension s' of s to G(M/K) corresponds to the action of s on A(L)/U.

Now assume that M I K is normal such that we have a group extension

We put G := G(M/K), H := G(M/L). Then if H is given as a GIH- module, where G/H acts on H by conjugation, the group extension (2.2) is determined by an element of H2(G/H, H) , which is defined by means of a 2-cocycle f (a, T) of G / H with values in H as follows. Take a set of representatives 7 in G for the elements T of G/H. Then

zf M / K normal, f (a, 7) = (T 5 m-I.

92 H. Koch

The question is the following: Which element of H2(G(L/K), G(M/L)) corresponds to this group extension. The answer is given by the following

Theorem 1 (Theorem of Shafarevich). The class in H~ (G(L/K), G(M/L)) corresponding to (2.2) is the image of the canonical class with respect to the map from H2(G(L/K), AL) to H2(G(L/K), G(M/L)) induced by the Artin map.

This theorem seems to me to be an example of pre-stabilized har- mony in the architecture of mathematics, independent of the human mind: The canonical class defined in the local case as an auxiliary notion in the theory of simple algebras and afterwards developed as a tool in class field theory, appears as the class being associated via the Artin map to any extension of Galois groups over the normal extension to which the canonical class belongs. Historically this picture is true for the local class field theory. But in the global case history is more complicated.

53. The Local Case

3.1. Hasse [Ha1930a-c] created local class field theory by means of global class field theory. Some assumptions he had to make were proved by F.K. Schmidt [Sc1930] to be always fulfilled. The corresponding local Artin map was called norm residue symbol, it is now sometimes called Hasse map.

Following almost immediately Hasse, Chevalley [Ch1930] and F.K. Schmidt [unpublished] built the theory on the theory of simple algebras, avoiding global class field theory. But a self contained definition of the local Artin map was only given later in the cyclic case by Hasse [Ha19331 and in general by Chevalley [Ch1933].

3.2. At the origin of the local canonical class is the notion of invariant of a simple central algebra over a local field as defined by Hasse [Ha1931]. In this paper Hasse shows that a central simple algebra A of dimension n2 over a local field K has a splitting field L of degree n which is unramified, hence L I K is cyclic.

Already Dickson [Di1914] considered simple algebras A defined by cyclic field extensions L/K in the form

with un = a for a fixed cr E K X and uJ = s(J)u for a generator s of G(L/K) and J E L. Such algebras are called cyclic or of Dickson type.

The History of the Theorem of Shafarevich 93

In the case of Hasse we can take for s the Frobenius automorphism of L/K and since a is determined by A only up to multiplication of cr by norms of L/K, we can assume cr = T", v = 0,1,. . . , n - 1, where T is a prime element of K . Hence A is determined by its dimension n2 and by its invariant X mod Z.

E. Noether [No19291 generalized the construction of Dickson for arbitrary Galois extensions LIK. Now we get a simple algebra A in the form

with us[ = g(<)ug for ( E L and

where a(g, h) is a 2-cocycle of G(L/K) with values in L X . This construction is determined by A and L /K only up to multiplication by a coboundary such that in fact d is determined by an element of H2 (G(L/ K ) , L '). These considerations show that the Brauer group of algebra classes which are central over K and split by L is isomorphic to H~ (G(L/ K) , L ) . If in particular K is a local field, then the algebra class is determined by its invariant and the canonical class of H2(G(L/K), L x ) corresponds to the algebra class with invariant & mod Z.

3.3. One of the most important steps towards the cohomological foundation of class field theory was a new interpretation of the local Artin map by Nakayama [Na1936]: He proved, using local class field theory, that the inverse map

v G(LIK) K X I N ~ I ~ ( L X )

: for a local abelian extension L/K is given by (2.1). More generally let L I K be a Galois extension with Galois group G

and let G' be the commutator subgroup of G. Then (2.1) induces by ' definition a homomorphism of GIG' into K INLIK (LX ). Y. Akizuki

[Ak1936] showed that this homomorphism is injective for arbitrary base fields K , if we use for the definition of (2.1) instead of UL/K an arbitrary element in H 2 (G, L X ) of order [GI. This implies NLIK (LX ) = NMIK(MX) for the maximal abelian subextension M I K of L/K if K is a local field.

3.4. The next important step in direction of the cohomological foundation of class field theory came in 1950, when Hochschild eliminated

94 H. Koch

the theory of simple algebras, working only with the corresponding factor systems [Ho1950]. This was in fact only a reformulation, but soon afterwards it became clear that the same procedure is possible in the global case.

3.5. In his paper [Sh1946] Shafarevich proved Theorem 1 in the local case. This paper of less than two pages is perhaps the shortest paper among the essential papers written about a mathematical subject. Besides some functorial properties of the canonical class which at that time appeared as functorial properties of the invariant of simple algebras he uses the Nakayama map (2.1) and a relation for factor systems by Witt [Wi1935]. In the latter paper, which consists only of one and a half pages, Witt proved two rules about classes of factor systems. Shafarevich used the second rule, which we formulate more generally as a property of 2-cocycles a(g, h) of a finite group G with values in a G-module A. Let H be a normal subgroup of G and let (519 E G) be a system of representatives of the classes of G I H in G.

Then

(3.1) f (g, h) := n xa(6, L)a(x, 5L)a-'(x, z), xEH

depends only on the classes of g and h in G / H and f (9, h) is a cocycle of G / H with values in AH. Furthermore, the class of f (9, h) in H2(G, A) is equal to the class of [H : {l)]a(g, h).

In the case of simple algebras we have A = L X . Witt shows that the algebra corresponding to [ H : {l)]a(g, h) with g, h E G, which a priori splits already over the fixed field L~ of H , is similar to the algebra corresponding to f (9, h) considered as factor system G I H with values in ( L ~ ) . Witt's motivation was to find an explicit expression for f (9, h) in terms of a(g, h).

Shafarevich's proof is so short that we reproduce it here: He formulates the theorem and the proof in terms of simple algebras over local fields and his theorem formulated in 1946 concerns only the local case. But if we pass from the simple central algebra with invariant & + Z to the corresponding canonical class, as we will do, then his proof goes through for class formations.

Proof of Theorem 1. We put G := G(M/K), H := G(M/L), hence G(L/K) = GIH. Let a(a, r ) be a cocycle of G / H belonging to the canonical class ULIK, let b(g, h) be a cocycle of G belonging to UM/K, and let c(g, h) be a cocycle of H belonging to UM/L. Furthermore, let a ( a , r ) be a cocycle of G / H belonging to the class of (2.2). Fix a representative 5 E G for a E G I H . Then a 7 = a ( a , r ) m . By means of

The History of the Theorem of Shafarevich

the Nakayama map (2.1) for MIL we can write

(3.2) a ( a , r) = a( n c(x, a(,, T))) with a, r E G/H, xE H

where a denotes the Artin map L X + H. We have R e ~ ~ + ~ u ~ ~ ~ = UM/L (IVb) and therefore

(3.3) a (o , r ) = a( n b(x, a(o, T))) with 0, r E GIH. xEH

On the other hand hfL--lM~L/K = [M : L]uM/K (Iva). Hence by the property (3.1) of Witt

a(gH, hH) - b(g, h)lMZL1 - 11 xb(5, L)b(x, $)b-l(x, 2)) XE H

where 6 := 8, which we can write in the form

(3.4) a(o, r) I-J x b ( ~ , f)b(x, 5 f ) b - I (x, m). xEH

Combining (3.3) and (3.4) we see that our assertion

4 0 , ~ ) - a(a(a ,7) )

, is equivalent to

(3.5) n b(x, a(o, T))U n xb(T,~)b(x, T ~ ) b - l ( x , m ) ~ . L

xEH xE H

I : Since U = NMILAM we have nxEH x b ( 5 , ~ ) E U. Furthermore, the cocycle property

xb(a(a, T), m)b(x, a (a , r ) m ) = b(xa(a, T), m)b(x, a(a, 7))

implies

N M / L ~ ( ~ ( o , r ) , m) n b(z, 5 7 ) = b(x, m) 11 b(x, a (o , 7)) xE H X E H xEH

This proves (3.5). 0

96 H. Koch The History of the Theorem of Shafarevich 97

$4. The Global Case

4.1. The idele group. Chevalley [Ch1936] considered the generalization of global class field theory to infinite abelian extensions. In terms of the old formulation of the theory this meant that one has to pass to ray class groups for bigger and bigger modules. He avoided this problem by passing from ray class groups to a new group which he baptized fundamental group and whose elements he called ideal elements. Later this group was called idele group and its elements ideles. In his fundamental group he introduced a topology such that the closure of the group of principal ideal elements, i.e. principal ideles, is the kernel of the natural map c p ~ from JK onto G$', where JK denotes the idele group of the number field K and G*he Galois group of the maximal abelian extension of K .

The map c p ~ is given by Hasse's norm residue symbol (L.o/KP )

where ap E Kp and Ly /Kp is an abelian extension of the completion KP of K for the place p (compare 3.1). Let {Kn In = 1,2, . . . ) be a sequence

00

of abelian extensions of K such that L = U K n is the maximal abelian n= 1

extension of K . Then the value of c p ~ at the idele nap is given by P

where the product runs over all places p of K and yn is a fixed place of Kn over p.

(4.1) shows that the introduction of ideles was prepared by Hasse [Ha1930a-c]. In particular the product formula for the norm residue symbol in [Ha1930a] means that the principal ideles are in the kernel of (PK. In [Ch1940] class field theory is presented with full proofs for the first time without using tools from complex function theory.

4.2. The further development of the global theory was very much influenced by Weil's paper [Wll95 11.

First of all Weil was fully aware of the fact that the fundamental object to be considered is the idele class group CK := JK /PK, where PK denotes the group of principal ideles, which one can identify with the multiplicative group K of K . He uses the topology in JK which is given by the product topology of the group UK of unit ideles: UK = n Up,

P where Up denotes the compact group of units in Kp if p is a prime ideal of K, and Up = K t if p is an archimedean valuation of K. Now PK is a discrete and therefore closed subgroup of JK.

Weil states without proof that the Grossencharacters of Hecke [He 19181 can be identified with the continuous characters of CK. The proof for this fact goes along the lines of the transition from ray class groups to the idele class groups as explained in [Ch1936]. But since Chevalley uses another topology for the idele group, he could not find this beautiful interpretation of Grossencharacters, which in fact shows that the introduction of infinite components of the idele group as the multiplicative groups KpX for archimedean places p was the "right" definition, while, if we are only interested in the interpretation of abelian extensions, the complex places, are useless and the real components could be reduced to {f 1). In contrast to [Ch1936], Weil considers not only number fields but also function fields of one variable over finite fields as base fields.

In his thesis of 1950 at Princeton University, J. Tate introduced Hecke's Grossencharacters in the same way as Weil in 1951. But this thesis was published only in 1967 ([Tt1950]).

The main problem stated and solved in [W11951] is related to the Theorem of Shafarevich: Let L I K be a finite Galois extension. Is it possible to find a natural group extension of G(L/K) with CL compatible with the natural action of G(L/K) on CL?

Let K first be a function field. Then the Artin map is an isomorphism of CL onto a dense subgroup of the Galois group Gtb of the maximal abelian extension L " ~ of L. Hence we have a natural group

: extension of G(L/K) with CL given by the group extension t

, , The case of an algebraic number field K is different. There we have b

: a non-trivial connected component ?DL of the unit element of CL and class field theory gives only a natural group extension of G(L/K) with

/ CL/DL. The problem of Weil is the question whether it is possible to lift this group extension of CL. He requires functorial properties for this

i lifting and he shows that there exists one and only one lifting GL, with ' these properties. Weil's motivation for the construction of this lifting to - the group extension

is its application to Lfunctions. By means of the group G L , ~ he defines a new kind of L-functions, now called Weil L-functions which combines the notions of Hecke and Artin (non-abelian) L-function. In his commentaries to his collected works Weil writes: Aussi aurais-je pu intituler mon mhmoire "le mariage d'Artin et de Hecke".

4.3. The group extension (4.2) defines an element in the group H2(G(L/K), C,), called later on the canonical class. But with respect

98 H. Koch

to the Theorem of Shafarevich in the global case Weil has nothing to add because he is not looking for an independent description of the class belonging to the group extension

but he takes this class for his purpose of defining the group extension (4.2).

Nevertheless, the paper [W11951] by Weil stimulated Nakayama to give an independent definition of the canonical class and to prove the Theorem of Shafarevich in the global case (together with Hochschild) [HoNa1952].

Weil writes in his commentaries to his collected works that Naka- yama got the manuscript of [W11951] before it was published and found an essential mistake in it. Weil was able to correct the mistake in time and Nakayama gave his independent definition of the canonical class already in his paper [Na1951] in the same volume of the Journal of the Mathematical Society of Japan in honour to Takagi that contains also Weil's paper.

Nakayama defines the canonical class in complete analogy with the local canonical class using the cohomological treatment of local class field theory given in [Ho1950]. The canonical class is defined by means of "Durchkreuzung" with a cyclic cyclotomic extension of K, a method which goes back to Chebotarev [Ce1926] and was used by Artin [Ar1927] to prove his reciprocity law. Cyclotomic fields play in the global case the same role as unramified extensions in the local case.

In [Na1952] the cohomological construction of class field theory is complete in so far as it is based only on index relations, the theory of cyclotomic fields and Hasse's sum relation for the invariants of Brauer algebra classes. He proves H1 (G(L/ K) , CK) = (0) for arbitrary finite Galois extensions of global fields K, while in [Na1951] this is only proved for cyclic extensions LIK. Furthermore, it is proved that Nakayama's canonical class is the same as Weil's canonical class.

Finally Hochschild-Nakayama [HoNa1952] proved that for any finite Galois extension L of a global field K the group H 2 ( G ( ~ I K ) , CL) is cyclic of degree [L:K] generated by the canonical class. This paper contains also a full treatment of the functorial properties of the Artin map including the transfer homomorphism (Verlagerung) .

Furthermore, the Theorem of Shafarevich is proved in the global case. This proof is identical with Shafarevich's proof (see 3.5) except for the fact that the second property of Witt is not used directly. The authors give a new and less elegant proof for it. They refer to Sha- farevich's Theorem and remark that if they apply their procedure to


the local case they "obtain a proof for Shafarevich's result". This is of course not surprising, since their procedure is identical to Shafarevich's.

4.4. The ideas of Weil, Hochschild and Nakayama were further developed in the seminar of Artin and Tate, 1951152, at Princeton University. There is an exposition of class field theory on the basis of group cohomology, the notion of class formation is introduced, and the groups G L , ~ of 4.2, called Wed groups, are defined for local and global fields in the scope of class formations. The Theorem of Shafarevich is treated as a theorem about class formations (exactly as in 2.4) and it is called "The- orem of Shafarevich-Weil". The notes of the seminar were published only in 1967 [ArTt1952].

An essential ingredient of the cohomological treatment of class field theory is still missing in the Artin-Tate notes: The modified cohomological groups which put together homology and cohomology groups for

, finite groups G and G-modules A to a sequence of groups { H ~ G , A) In E Z). The modified cohomology groups were introduced by Tate [Tt1952]. With these groups we get the picture of class field theory which we briefly described in 2.1-2.3.

4.5. With Tate's paper [Tt1952] the period of reformulation and simplification of class field theory was completed. The next period in

, the theory of algebraic number fields was distinguished by the study of infinite extensions: Iwasawa's theory of I?-extensions and the theory of

" maximal extensions with restricted ramification (Tate, Serre, Shafare- ' vich). But the description of this development lays outside the scope of 0, this talk.

1 From the time after 1952, I mention only two results which are re-

I lated to the Theorem of Shafarevich, both belonging to the local theory.

1 The first one is the Theorem of Sen and Tate [SnTt1963] which : clarifies the connection of the Theorem of Shafarevich with the filtration

given by the ramification groups in the upper numbering. I We keep the notation of section 2.4. Let F be a local field and let f

$ D be the division algebra with center K corresponding to the splitting field L and the canonical class ~ L / K . If a(o, T), a, T E G(L/K), is a

: cocycle in the class ULIK, then D is given as in 3.2 in the form

I +

; with u,c = a(J)u, for 5 E L and

100 H. Koch

Hence we have an imbedding of the local Weil group G L , ~ in D X given by taking u, as the representative of a E G(L/ K ) in G L , ~ . Then

We introduce a filtration {G;,K Iv E W+} in GLjK by means of

where L denotes the homomorphism of G L , ~ in G(L"~/K) given by the Theorem of Shafarevich.

Furthermore, let v be the exponential valuation of D, normalized such that V(T) = 1 for a prime element T of L, and let p(x) = pLIK (x) be the Herbrand function of LIK. Then one has the following

Theorem of Sen and Tate. Let G be an element of G L , ~ and

x E W, x > 0. Then g E GI!$ if and only if v(g - 1) 2 x.

The second result I want to mention here, belongs to the theory of Lubin-Tate extensions giving an explicit construction of the fully ramified extensions of a local field [LuTt1965]. If we apply this theory to a normal extension L of K in the notation of 2.4, then the transformation formula for the change of the prime element T of L for the formal multiplication by T can be interpreted as an explicit construction of the group extension 2.2. See [KodS1996] for details, in particular 2.2.

$5. The Contribution of Hasse and Jehne in the Time After the War

5.1. Hasse worked from 1946 to 1950 in Berlin, where he attracted a large group of talented students. In 1950 he went to Hamburg followed by almost all his students. His main research project during that time was the theory of field embeddings: Given a Galois extension L/K and a group extension

with abelian kernel A. Then one can look for a Galois extension F / K containing L/K such that there is an isomorphism cp of G(F/K) onto G such that the diagram

- G(F/K) 5 G(L/K)

1 II G G W K )


is commutative. In [Ha19471 the situation that F /L is given by class field theory is considered but the main interest of Hasse was concentrated on the case where F/L is given by Kummer theory.

The class field theoretic situation is considered in the language of ray class groups and the question of the corresponding two-classes is solved in some simple cases. In general this is called "Widerspiegelungsprob- lem" and Hasse writes at the end of the paper that one needs an essentially new idea to solve the problem.

5.2. One of Hasse's students, Wolfram Jehne, solved the Widerspie- gelungsproblem in his diploma in the spirit of the theory of simple algebras. In [Je1952] Jehne defined for this purpose the notion of an idele of algebras consisting of local algebras A; with center Kp for all places p of K such that A; is similar to Kp for almost all p. An idele of algebras is called embedding idele with respect to LIK of a Galois algebra AIR in the sense of Teichmiiller [Tel940] if the components A; are embeddings of Ap = A @K Kp (i.e. one can embed Ap in A; such that Ap is the centralizer of Kp in A;). TO such an embedding idele Jehne associated an element of H~(G(L/K), CL) and defined its invariant as the sum of the invariants of the local algebras A;. Finally the canonical class is the uniquely determined element in H2(G(L/K), EL) with in-

' variant & + Z. One finds a similar procedure in cohomological terms

I in [ArTt1952]. With this construction Jehne gave a proof of Artin's reciprocity law using only Hasse's sum relation for the invariants of algebra

I classes. 4

At the end of his paper Jehne proves the Theorem of Shafarevich in the global case in the same way as Shafarevich proved it in the local

: case by using the Nakayama map and Witt's second rule.

t 56. Concluding Remarks

6.1. Shafarevich proved his Theorem in 1945 in the local case. I Though his paper [Sh1946] was simultaneously published in English and

Russian it became known rather slowly in the West and in Japan. But it is reviewed in Math. Rev. 8 (1947), p. 250, by G. Whaples, and it is mentioned in the article of Chevalley [Ch1951]. It could have been known therefore also to other mathematicians who worked on class field theory around 1951. However, because of the isolation of the mathematicians in Germany at that time, it was unknown to Hasse's group in Berlin and Hamburg. Probably Weil did not know it when he wrote his paper [W11951], otherwise he would have mentioned it.

6.2. We have seen that the Theorem of Shafarevich in the global case was proved independently by Hochschild-Nakayama [HoNa1952]

102 H. Koch

and Jehne [Je1952] at the same time. A correct name for the Theo- rem would be therefore Theorem of Shafarevich-Hochschild-Nakayama- Jehne. Since this is too long and the proof of the Theorem in the global case is almost the same as in the local case after one has established the notion of a canonical class in the global case, it seems to me that "The- orem of Shafarevich" is a justified name. In this I follow Serre [Se1962] who called (p. 172) the Theorem in this way.

6.3. Hasse and Nakayama both could have proved the Theorem of Shafarevich in the local case in 1936 but it seems the time was not ripe for asking the question. Hasse finally published the problem in 1947 but only for algebraic number fields and in a, for that time, old fashioned form as a question about ray class groups. Nevertheless, his student Jehne solved the problem in terms of idele class groups. But Hasse failed to understand or accept the solution of Jehne, such that this result remained rather unknown.

6.4. Hasse [Ha19321 was the first to use factor system classes, i.e. elements of the group H2 (G(L/K), K ), in the context of algebraic number theory in connection with the theory of simple algebras. So he should be considered as one of the creators of the homological method in algebraic number theory. But he did not like it as a method of cohomology groups independent of the theory of algebras as can be seen from his talk [Ha19671 about the history of class field theory. Maybe, the theory of simple algebras was too important and dear to him since he was one of its creators in the thirties. He did not want to eliminate the simple algebras in the proofs of class field theory as was done in the local case by Hochschild [Ho1950]. This elimination paved the way for the cohomological approach to global class field theory. We see the cohomological method fully developed in the paper of Hochschild-Nakayama [HoNa1952].

References

Akizuki, Y., Eine homomorphe Zuordnung der Elemente der Ga- loisschen Gruppe zu den Elementen einer Untergruppe der Normklassengruppe, Math. Ann., 112 (1936), 566-571.

Artin, E., ~ b e r eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg, 1 (1924), 89-108.

Artin, E., Beweis des allgemeinen Reziprozitatsgesetzes, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 353-363.

Artin, E., Idealklassen in Oberkorpern und allgemeines Rezipro- zitatsgesetz, Abh. Math. Sem. Univ. Hamburg, 7 (1929), 46-51.

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Chebotarev, N.G., Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehoren, Math. Ann., 95 (1926), 191-228.

Chevalley, C., Sur la thdorie des restes normiques, Comptes rendus hdbdomadaires des sBances de 1'Academie des Sciences, Paris, 1 9 1 (1930), 426-42.

Chevalley, C., Sur la thdorie du corps de classes dans les corps de nombres algdbriques et dans les corps locaux, J . Fac. Sci. Univ. Tokyo Sect.,l-2 (1933), 363-476.

Chevalley, C., GBndralisation de la thhorie du corps de classes pour les extensions infinies, J. Math. Pures Appl., 1 5 (1936), 359-371.

Furtwangler, Ph., Allgemeiner Existenzbeweis fiir den Klassen- korper eines beliebigen algebraischen Zahlkorpers, Math. Ann., 6 3 (1907), 1-37.

Furtwangler, Ph., Beweis des Hauptidealsatzes fur Klassenkorper algebraischer Zahlkorper, Abh. Math. Sem. Univ. Hamburg, 7 (1930), 14-36.

Hasse, H., Neue Begrundung und Verallgemeinerung der Theorie des Normenrestsymbols, J . reine angew. Math., 162 (1930), 134-144.

Hasse, H., Die Normenresttheorie relativ-Abelscher Zahlkorper als Klassenkorpertheorie im Kleinen. J. reine angew. Math., 162 (1930), 145-154.

Hasse, H., Fuhrer, Diskriminante und Verzweigungskorper relativ- Abelscher Zahlkorper. J. reine angew. Math., 162 (l93O), 169-184.

Hasse, H., ~ b e r p-adische Schiefkorper und ihre Bedeutung fur die Arithmetik hyperkomplexer Zahlsysteme, Math. Ann., 104 (1931), 495-534.

Hasse, H., Theory of cyclic algebras over an algebraic number field, Trans. Amer. Math. Soc., 34 (1932), 171-214.

Hasse, H., Die Struktur der R. Brauerschen Algebrenklassen- gruppe iiber einem algebraischen Zahlkorper, Math. Ann., 107 (l933), 731-760.

Hasse, H., Invariante Kennzeichnung relativ-abelscher Zahlkorper mit vorgegebener Galoisgruppe uber einem Teilkorper des Grundkorpers, Abh. d. Deutschen Akad. d. Wissensch. Berlin, Math.-Nat. K1. 1947, Nr. 8, 5-56.

Hasse, H., History of class field theory, in Algebraic Number The- ory, edited by J.W.S. Cassels and A. Frohlich, Academic Press London 1967, 266-279.

104 H. Koch

[He19181 Hecke, E., Eine neue Art von Zetafunktion und ihre Beziehungen zur Verteilung der Primzahlen, I, 11, Math. Z., 1 (1918), 357- 376; 6 (1920), 11-51.

[Hi18981 Hilbert, D., ~ b e r die Theorie der relativ-Abelschen Zahlkijrper. Nachr. Ges. Wiss. Gottingen (l898), 377-399.

[Ho1950] Hochschild, G., Local class field theory, Ann. of Math., 51 (l95O), 331-347.

[HoNa1952] Hochschild, G., Nakayama, T., Cohomology in class field theory, Ann. of Math., 55 (1952), 348-366.

[Je1952] Jehne, W., Idealklassenfaktorensysteme und verallgemeinerte Theorie der verschrankten Produkte, Abh. Math. Sem. Univ. Hamb., 18 (1) (1952), 70-98.

[Ko1992] Koch, H., Algebraic Number Theory, Springer-Verlag Berlin 1997. [KodS1996] Koch, H., de Shalit, E., Metabelian local class field theory, J .

reine angew. Math., 478 (1996), 85-106. [Kr1853] Kronecker, L., ~ b e r die algebraisch auflosbaren Gleichungen I.

Sber. Preuss. Akad. Wiss. (1853), 365-374. [Kr1882] Kronecker, L., Grundzuge einer arithmetischen Theorie der alge-

braischen Grossen, J. reine angew. Math., 92 (1882), 1-122. [LuTt1965] Lubin, J., Tate, J., Formal complex multiplication in local fields,

Annals of Math., 81 (1965), 380-387. [Na1936] Nakayama, T., ~ b e r die Beziehungen zwischen den Faktoren-

systemen und der Normklassengruppe eines Galoisschen Er- weiterungskorpers, Math. Ann., 112 (l936), 85-91.

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[Na1952] Nakayama, T., Idele-class factor sets and class field theory, Ann. Math., 55 (1952), 73-84.

[No19291 Noether, E., Lectures a t the Gottingen University. [Sc1930] Schmidt, F.K., Zur Klassenkorpertheorie im Kleinen, J . reine

angew. Math., 162 (1930), 155-168. [Se1962] Serre, J.-P., Corps locaux, Hermann, Paris 1962. [Sh1946] Shafarevich, I.R., On the Galois group of padic field, Dokl. Akad.

Nauk SSSR, 53 (1946), 15-16. [SnTt1963] Sen, S., Tate, J., Ramification groups of local fields, J . Indian

Math. Soc., 27 (1963), 197-202. [Su1902] Schur, I., Neuer Beweis eines Satzes uber endliche Gruppen.

Sitzungsberichte der Preussischen Akademie der Wissen- schaften 1902, Physikalisch-Mathematische Klasse, 1013-1019.

[Ta1920] Takagi, T., ~ b e r eine Theorie des relativ-Abelschen Zehlkorpers, J. Coll. Sci. Tokyo, 41 (art. 9) (1920), 1-133.

[Tt1950] Tate, J., Fourier analysis in number fields and Hecke's zeta- functions. In Algebraic Number Theory, Thompson, Washing- ton, D.C. (1967), 305-347.


[Tt1952] Tate, J., The higher dimensional cohomology groups of class field theory, Ann. of Math., 56 (1952), 293-297.

[Te1940] Teichmuller, O., ~ b e r die sogenannte nicht-kommutative galoissche Theorie, Deutsche Math., 5 (1940), 138-149.

[We18911 Weber, H., "Elliptische Funktionen und algebraische Zahlen", Braunschweig 1891.

[We18971 Weber, H., ~ b e r Zahlengruppen in algebraischen Korpern I, 11, 111, Math. Ann., 48 (1897), 433-473; 49 (1897), 83-100; 50 (1898), 1-26.

[W11951] Weil, A., Sur la th6orie du corps de classes, J . Math. Soc. Japan, 3 (1951), 1-35.

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Humboldt-Universitat zu Berlin, Institut fur Mathematik, Rudower Chaussee 25, D-12 489 Berlin, Germany E-mail address: kochhathematik .hu-berlin. de


A Survey of pExtensions

Masakazu Yamagishi

This is a brief survey of what is known or unknown about the Galois group of the maximal pro-p-extension (p a fixed prime) of a number field which is unramified outside a given set of places. We are particularly interested in

presentation in terms of generators and relations cohomological dimension

of the Galois group. The contents are as follows. In Section 1 we recall basic facts on pro-pgroups. In Section 2 we review the structure of the Galois group of the maximal pro-p-extension of a local field. In Section 3 we state some known facts and unsolved conjectures about the structure of the Galois group of the the maximal pro-p-extension of a number field which is unramified outside a given finite set of places. In Section 4 we introduce some topics in Iwasawa theory. In Section 5 we state some

P known facts about the structure of the Galois group of the maximal

: pro-p-extension of a number field. Finally, as an application of Sections

; 3 and 4, we give some examples of free pro-pextensions of number fields ' in Section 6. i The author would like to thank the referee for valuable comments.

j Main references are Serre [54, I §3-§4] and Koch [26, 55-56]. Let G be a pro-pgroup.

1.1. Generators and relations

We put d(G) = dim H1 (G, ZlpZ) and r(G) = dim H ~ ( G , ZlpZ). d(G) is the minimal number of generators of G, which we also call the rank of G, and r(G) is the minimal number of relations of G.

Received September 1, 1998. Revised November 26, 1998.

M. Yamagishi

1.2. Cohomological dimension

The cohomological dimension and the strict cohomological dimension of G are defined by

cd(G) = inf{n; Hq(G, A) = 0 Vq > n,VA : discrete torsion G-module),

scd(G) = inf {n; H q (G, A) = 0 Vq > n, VA : discrete G-module),

respectively. We know the following facts:

cd(G) 5 n if and only if Hn+l (G, ZlpZ) = 0. cd(G) 5 scd(G) 5 cd(G) + 1. If H is a closed subgroup of G, then cd(H) 5 cd(G) and scd(H) 5 scd (G) . If G has non trivial torsion, then cd(G) = scd(G) = oo. Suppose cd(G) = n < oo, then scd(G) = n if and only if Hn(H, Qp/Zp) = 0 for all open subgroups H of G.

1.3. Euler-Poincard characteristic

If cd(G) is finite and Hi (G, ZlpZ) is finite for all i, we define the Euler-Poincar6 characteristic of G by

If X(G) is defined and H is an open subgroup of G, then x(H) is also defined and x(H) = [G : H]x(G).

1.4. Free pro-p-groups

G is called a free pro-p-group if and only if r(G) = 0, or equivalently, cd(G) 5 1. If G is a free pro-p-group and H is a closed subgroup of G, then H is also a free pro-pgroup since cd(H) 5 cd(G) 5 1. If, in addition, the rank of G is finite and H is open in G, then the rank of H is also finite and we have Schreier's formula:

d(H) - 1 = [ G : H](d(G) - I ) ,

which follows from Subsection 1.3.

1.5. DemuGkin groups

G is called a Demugkin group if it satisfies the following conditions:

(i) d(G) is finite. (ii) r(G) = 1.

A Survey of p-Extensions

(iii) The cupproduct

is a non-degenerate bilinear form.

The structure of Demuskin groups is known as follows. Suppose p > 2 for simplicity and let G be a DemuSkin group. Then we see by (iii) that d(G) = 2n is even and by (ii) that the maximal abelian quotient G " ~ is isomorphic to Z 2 - I x Zp/qZp, where q is either 0 or a power of

P .

Theorem 1.1 (DemuSkin [7]). Let p be an odd prime and G a Demuikin group with n and q as above. Then there exist generators X I , x2, . . . ,x2, of G such that the single relation for G has the form :

where [x, y] = ~ - ~ y - ~ x y .

See Serre [53] and Labute [34] for the case p = 2.

92. Local fields

Main reference is Serre [54, I1 $51. Let k be a finite extension of Q, k(p) the maximal pro-p-extension of k, and G = Gal(k(p)/k) the Galois group. The structure of G is determined. We use the following notation:

k : the algebraic closure of k. the group of pth roots of unity in &. PP .

Theorem 2.1. d(G) = N + 1 + 6 , r(G) = 6 .

Proof. By local class field theory H1 (G, ZlpZ) is dual to k /k '. The inflation homomorphism H~ (G, ZlpZ) -+ II2 ( ~ a l ( & / k) , ZlpZ) is an isomorphism and by the local duality theorem this last group is dual to

H0(~al(31-Ik), P,). 0

Corollary 2.2 ( ~ a f a r e v i ~ [47]). If 6 = 0, then G is a free pro-p- QTOUP.

M. Yamagishi A Survey of p-Extensions

Corollary 2.3. If 6 = 1, then G is a Demuikin group.

Proof. Since k 3 pp, we have H1(G,Z/pZ) G' k x / k x P and the cupproduct corresponds to the norm residue symbol, which is nondegenerate on k /k '. 0

Remark 2.4. If 6 = 1 a n d p > 2, then, with the notationofTheo- rem 1.1, the invariant q is the maximal power of p such that k contains the group of qth roots of unity.

Theorem 2.5. cd(G) 5 2, scd(G) = 2.

Proof. These follow from Corollaries 2.2 and 2.3. 0

Corollary 2.6. x(G) = - N

Remark 2.7. Let G be a pro-pgroup. It is known that G is a free pro-pgroup if and only if

d(H) - 1 = [ G : H](d(G) - 1)

for all open subgroups H of G. It is also known that G is a Demuskin group if and only if

for all open subgroups H of G (Dummit-Labute [8]). These characterization of free pro-pgroups and DemuSkin groups give alternative proofs of Corollaries 2.2 and 2.3.

It would be an interesting problem to consider a pro-pgroup G such that

for all open subgroups H of G, where c 2 3 is a fixed positive integer. A trivial example is G = Zp x Zp x . - - x Zp (C times). Are there any examples of such G which arise naturally in number theory? See Schmidt [50] for related topics.

$3. Global fields

Main references are Haberland [13] and Koch [26] (see also [28]). Let k be a finite extension of Q, S a finite set of places of k, ks (p) the maximal pro-pextension of k unramified outside S, and Gs = Gal(lcs(p)/k) the Galois group. Suppose that p is odd or that k is totally imaginary. Then since no archimedean place can ramify in a pro-p-extension of k, we may assume that S is disjoint from the set of the archimedean places of L. We use the following notation:

0 rl : the number of real places of k. 0 r2 : the number of imaginary places of k. 0 k, : the completion of k with respect to a place v of k.

the group of pth roots of unity in the algebraic closure k. PP -

0 Sp : the set of all places of k which are above p. Vs = {x E k X ; (x) = UP,x E k:p Vv E S)/kXP.

O = { 0 1 ( 6 = 1 , S = 0 ) (otherwise) '

Theorem 3.1 ( ~ a f a r e v i ~ [48] ) . d(Gs) = x 6 , - 6 - (rl +r2 - 1) + x [k, : Q,] +dimVs,

v E S v E S ~ S ,

r(Gs) 5 x 6, - 6 + dim Vs + 0. v E S

Two cases are of particular interest to us: one is the case where S is empty, the other is the case where S > Sp.

3.1. Case S = 0 It has been conjectured that every number field of finite degree can

be embedded in a number field with class number one (the class field tower ~roblem). In particular, G0 has been conjectured to be finite. Golod and ~ a f a r e v i ~ [ll] showed that if G is a finite pgroup then r(G) > (d(G) - 1 ) ~ / 4 holds (in fact r(G) > d(G)2/4 holds, see, for example, Roquette [46, Remark 141). Using this and Theorem 3.1, they gave examples of k (and p) with infinite Go.

Presentation of Ga in terms of generators and relations is not known in general; there seems no single example of infinite G0 whose minimal relations are completely known.

Suppose G0 # (1). It is known that scd(GO) 2 3 and conjectured that cd(GO) = oo (cf. Kawada [22, p.1111). Note that this conjecture is trivial if G0 is finite and # (1).

Fontaine and Mazur [9, Conjecture 5b] conjectured that G0 has no infinite padic analytic quotient. See Boston [3], [4], Hajir [14], Nomura [44], [45] for related topics.

If we allow the degree of the number field to be infinite, then interesting examples of unramified pro-pextensions are known. See Asada [2,

112 M. Yamagishi

Supplement] for a construction of an unramified SL2 (Zp)-extension (note that SL2(Z,) itself is not a pro-pgroup, but contains a pro-psubgroup with finite index), and Wingberg [64] for the case where the Galois group of the maximal unramified pro-pextension is a free pro-pgroup.

3.2. Case S > Sp

In this case, the inequality for r(Gs) in Theorem 3.1 is in fact an equality (Brumer [5]). For a proof by using the Poitou-Tate global duality theorem and a result of Neumann [39, Corollary 11, see Nguyen Quang Do [41, Proposition 111.

Example 3.2. k is called prational if Gsp is a free pro-pgroup. If k > pp and S > Sp, then

where Cls denotes the S-ideal class group of k (see, for example, Neu- kirch [38, 7.31). Hence if k > p,, then k is prational if and only if ISPI = 1 and p { IClsp I (see also [48, $41). A typical example is k = Q(pp) where p is a regular prime. See Movahhedi-Nguyen Quang Do [37], Movahhedi [36], Sauzet [49] for more examples of prational number fields and the arithmetic of such fields, and also G. Gras-Jaulent [12], Jaulent-Nguyen Quang Do [20] for related topics.

Wingberg [62] and [63] showed that in some cases Gs has a free pro- p product decomposition. Let 4, denote the decomposition subgroup of a place v in ks(p)/k (defined up to conjugate) and * the free pro-p product.

Theorem 3.3 ([62, Theorem A]). Suppose k > pp. Then

for some vo E Sp and for some free pro-p-group F if and only if vo does not split in ks(p)/k at all. If this is the case, then d(3) = [k,, :

Qp1+ 2 - IS1 - r2.

Remark 3.4. Wingberg showed more: if Gs does not have a free pro-p product decomposition of this form, then Gs is a pro-p duality group of dimension 2 which is not Poincark type. See also Schmidt [51].

A Survey of p- Extensions 113

If Gs has free pro-p product decomposition as in Theorem 3.3, then Q, coincides with Gal(k,(p)/k,) (Kuz'min [32]), which is a DemuSkin group. Therefore we know the relations of Gs; in particular, they all come from local relations.

Example 3.5 (essentially due to Kuz'min [32]). Let p = 3, k =

Q ( G , a). Then Gsp is a DemuSkin group of rank 4.

For free pro-p product decomposition of Gs in a different setting, see Neumann [40], Movahhedi-Nguyen Quang Do [37], Jaulent-Nguyen Quang Do [20] and Jaulent-Sauzet [21].

For the case where Gs is a Demuskin group, see Tsvetkov [58], Arrigoni [I] and Sauzet [49].

In general, presentation of Gs in terms of generators and relations is not known. In some cases, the class two quotient Gs/[Gs, [Gs, Gs]], where [ , ] denotes the topological commutator, can be described in terms of generators and relations. See Frohlich [lo], Koch [27], Ullom- Watt [59] and Movahhedi-Nguyen Quang Do [37]. Komatsu [29] treated the case where there is a global relation (i.e. not coming from local relations). See also Koch [26, $ 11.41.

The cohomological dimension of Gs is known:

Theorem 3.6. cd(Gs) 1 2.

I t For proofs, see Brumer [5], Kuz'min [30],[31], Neumann [39] and ! Haberland [13, Proposition 71.

,, Corollary 3.7. x(Gs) = -rz. d I j On the contrary, the strict cohomological dimension of Gs is not

known:

Conjecture 3.8. scd(Gs) = 2.

In the cases where the explicit structure of G s is known (i.e. Gs is a free pro-pgroup or a Demuskin group or Gs has a free pro-p product decomposition), this conjecture is true. See Corollary 4.3 for a relation with the Leopoldt conjecture.

The Galois group Gs is often compared to (the pro-p completion of) the fundamental group of a Riemann surface. For example, free pro- p product decomposition of Gs is an analogue of Riemann's existence theorem (Neumann [40]). See also [67].

114 M. Yamagishi

$4. Iwasawa theory

We introduce some topics in Iwasawa theory which are deeply connected with Gs. Main reference is Wingberg [61]. See also Washington [60] for Iwasawa Theory. We keep the notation of the previous section and suppose that S > S,.

4.1. The Leopoldt conjecture The following is Iwasawa's formulation [17, 2.31 of the Leopoldt con-

jecture.

Conjecture 4.1. k has exactly r2 + 1 independent Zp-extensions.

This conjecture has been verified in some cases; for example, k/Q is abelian (Ax-Brumer; see [60, 5.251).

Proposition 4.2. The Leopoldt conjecture is equivalent to

H2(Gs, QplZp) = 0.

For proofs, see, for example, Haberland [13, Proposition 181 and Nguyen Quang Do [41, Proposition 121. See also [67, $41 for related topics.

Corollary 4.3. scd(Gs) = 2 if and only if the Leopoldt conjecture is true for all finite subfields of ks (p)/k.

Proof. By Subsection 1.2, Theorem 3.6 and Proposition 4.2.

Let k, be the cyclotomic Zp-extension of k and Hs = Gal(ks(p) lk,) the Galois group. The following is called the weak Leopoldt conjecture for k,.

Proposition 4.4. H2(Hs, Qp/Zp) = 0.

See Schneider [52, Lemma 71 and Wingberg [61, 5.11 for proofs, and also Nguyen Quang Do [42, $21 for related topics.

4.2. Iwasawa invariants

In addition to k, and Hs as above, we use the following notation:

a r = Gal(k,/k) E Z,. a A = Z,[[r]]: completed group ring.

Xs = ~g~ = Gal(Ms/k,), where Ms is the maximal abelian pro-pextension of k, unramified outside S.

a X = Gal(L/k,), where L is the maximal unramified abelian pro-pextension of k, .

A Survey of p-Extensions 115

The Galois group r naturally acts on Xs and X by conjugation; therefore Xs and X are naturally A-modules. Concerning the A-module structure of Xs and X , we know the following facts:

0 Xs and X are Noetherian A-modules. 0 The A-rank of Xs is r 2 .

a The A-rank of X is 0, i.e. X is a torsion A-module. a The Iwasawa invariants p(X) and A(X) for the Noetherian A-

module X coincide with the usual Iwasawa invariants p(k) and A(k) of k,/k, respectively.

Proposition 4.5. The following two statements are equivalent:

(i) Hs is a free pro-p-group, (ii) p(Xs) = 0.

If k > p,, then these are equivalent to

(iii) p(X) = 0.

For proofs, see Iwasawa [19, Theorem 21 and Wingberg [6l, 5.3 and 7.91. It is conjectured that p(X) = 0 in general, and this has been verified in some cases; for example, k/Q is abelian (Ferrero and Washington; see [60, 7.151).

For a CM-field k, let k+ denote the maximal real subfield of k and A-(k) the minus part of A(k). The following is an analogue of the Riemann-Hurwitz formula.

Theorem 4.6 (Kida [23]). If k is a CM-field such that k > pp and p(k) = 0, and if K is a finite Galois p-extension of k which is also a CM-field, then we have p(K) = 0 and

2 ( X ( K ) - 1) = [K, : k,] 2 (A-(k) - 1) + E ( e , - I ) , W

where w ranges over all finite places of K, such that w p and w splits in K,/Kz, and e, denotes the ramification index of w in K,/k,.

Proof. (Cf. [61, $71.) Take S large enough so that ks(p) > K . It follows from 1.4 and Proposition 4.5 that p(K) = 0 (see also Iwasawa [18, Theorem 31). The Galois group Hs(k&) is a free pro-pgroup since p(k+) = 0, and is finitely generated since it has A-rank 0. Applying Schreier's formula to Hs(k&) > Hs(K&), we obtain a formula connecting A-invariants of Xs(k&) and Xs(KL). Then by duality, we obtain a formula connecting A--invariants of X (k, ) and X(K,). 0

For other proofs or generalization of this theorem, see, for example, Kuz'min [33], Iwasawa [19], Nguyen Quang Do [43] and Wingberg [65].

116 M. Yamagishi

55. The maximal pro-p-extension

Let the notation be as in Section 3 except that S is the set of all places of k (S was supposed to be a finite set in Section 3). We drop S in our notation. Hence k(p) is the maximal pro-pextension of k and G = Gal(k(p)/k).

Both d(G) and r(G) are countably infinite and a minimal presentation of G in terms of generators and relations is known (Koch [24, 531, [25] and Hoechsmann [15]; see also [26, $11.11 and [16]).

Theorem 5.1 (Serre [54, 11.4.41). cd(G) = 2.

Theorem 5.2 (Brumer [6, 6.21). scd(G) = 2.

See also Haberland [13, Section 61 for proofs of these theorems.

Corollary 5.3 (see Serre [55, Theorem 41). H2(G, Qp/Zp) = 0.

Theorem 5.4. Let Ic , be the cyclotomic Zp-extension of k. Then Gal(k(p)/k,) is a free pro-p-group of countably infinite rank.

For proofs, see Serre [54, 11,Propositions 2 and 91 and Miyake [35].

56. Free pro-pextensions

We consider the following problem: how large free pro-pgroups can be realized as Galois groups? To be precise, let k be a finite extension of 0, Fd a free pro-pgroup of rank d (unique up to isomorphism). A Galois extension is called an Fd-extension if the Galois group is isomorphic to Fd. We define the invariant

p = max{d; k has an Fd-extension),

which depends on k and p. Since k always has the cyclotomic Zp- extension, we always have p > 1.

Lemma 6.1 ([66, 2.11). An Fd-extension (d 2 1) of k is unmmified outside p.

Hence p is the maximal rank of free pro-p quotient of GSp. Consid- ering abelianization, we see that if the Leopoldt conjecture is true for k, then we have p 5 r 2 + 1. Some examples with p = r2 + 1 and p < r2 + 1 are known as follows.

Example 6.2. If Gsp itself is free (cf. Example 3.2), then p = ~(Gs , ) = rz + 1.


Proposition 6.3 ([66, 4.61). With the notation and assumption of Theorem 3.3, if GGsp has a free pro-p product decomposition as in the theorem, then we have

Proof. It suffices to know the maximal rank of free pro-p quotient of the DemuSkin group G,. Using a result of J. Sonn [56], which states that there exists a surjection from a DemuSkin group G to Fd if and only if d 5 d(G)/2, we obtain the desired formula.

In particular, if GSp is a DemuSkin group and if Ic is not totally real, then we have p < r 2 + 1.

Example 6.4 (cf. Example 3.5). Let p = 3 and k = Q ( G , a). We have p = 2 and 7-2 + 1 = 3.

See also [69] and Jaulent-Sauzet [21, 2.81 for related topics.

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Department of Intelligence 6 Computer Science Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan E-mail address: yamagisi0kyy .nitech.ac. jp


Galois Module Structure of p-Class Formations

Nguyen Quang Do Thong

0. Introduction

One of the main subjects in (classical) class field theory is to study the structure of the Galois groups AL of abelian extensions of a local or global field L. One natural next step would be to take a Galois extension L/K with given group G and to investigate the structure of AL as a G- module. This has been done by several authors (see e.g. [J2], [N2] ,. .., and the references therein), mainly from the padic point of view: they single out a prime number p and focus their investigation on the iZp[G]-module structure of the pSylow subgroup AL of .AL. In the most interesting cases, it happens that (for a fixed base field K), the modules AL constitute a so-called "pclass formation" (see $1); so the next natural step is to replace the AL by the modules XL belonging to any pclass formation. Adding noetherian conditions, we obtain (Thm. 3.2 (below)) that, up to projective summands, the Zp[G]-module XL is determined by its Zp-torsion tXL and a certain character XL of the group H2(G, tXL). This generalizes a former result of U. Jannsen on the "homotopy type" of AL ([J2], Thm. 4.5) and could probably be proved by extending the methods of [J2]. In order to throw some new light on the problem, we preferred instead to employ the technique of "envelopes" introduced by Gruenberg and Weiss ([GW], [W]) in their study of the Stark conjecture.

This paper (the first part of which is semi-expository) will be organized as follows: after recalling some known facts on p-class formations ($1) and the homotopy of modules ($2), we prove the main theorem in 53, essentially by giving a canonical description of the envelope of XL by means of a relative Weil group, and of the character XL by means of a "trace form". As an illustration, we study in 54 the arithmetic of

Received August 17, 1998 Revised February 18, 1999

124 T. Nguyen Quang Do

number fields which admit free pro-pextensions of maximal rank (joint work with A. Lannuzel).

1. On p-class formations

Let G be a profinite group. For all generalities concerning class formations X attached to G, we refer to IS], chap. XI. As a general con- vention, we will index the open subgroups of G by symbols K, L etc ... as if they were Galois groups. The submodule of X fixed by such an open subgroup GL will be denoted by XL. For any pair of open subgroups GL c GK such that GL is normal in GK (notation: GL aGK), the quotient GLIK := OK /GL acts on XL, and we will denote by H"L/ K, XL) and

by fii(L/K, XL) respectively the ordinary and the modified cohomology groups.

1.1. Definition. Let p be a fixed prime number. A p-class formation (G, X ) is a class formation with the additional property that all the modules XL of the formation are Z,-modules. Then Tate's theorem implies that, for each pair of open subgroups GL Q GK, the cup-product with the fundamental class ELIK E H2(L/K, XL) induces an isomor-

phism I?G~(L/K, Z,) @(L/K, XL) for every i E Z.

It is known that pclass formations "come essentially" from profinite groups G of strict pcohomological dimension two (notation: scd& = 2). Let us make this more precise:

For a profinite group G and an open subgroup GL, write AL for the pSylow subgroup of 4:b := G L / G i (the dash denotes the closed subgroup generated by commutators). If (G, X ) is a pclass formation, the reciprocity map induces a homomorphism WL : XL --, AL, and hence, for each pair of open subgroup GL aGK and for each i E Z, a homomorphism

I?'(L/K, XL) - I?'(L/K, AL). Besides, the class E H2(L/K, Gib ) associated with the extension 1 --, G L ~ -+ GK/GL - GK/GL -+ 1 induces

A .

by cupproduct a homomorphism H'-2(L/K, Z,) -+ fi"L/K, AL), and the theorem of ~a fa rev ie -~e i l gives a commutative triangle:


Kawada has defined G to be p-malleable if for each pair

125

of open sub-

groups QLoGK and each i E Z, the above homomorphism f k 2 ( L / K, Z,)

+ I?'(L/K, AL) is an isomorphism. In this context, the main result is Brumer 's theorem:

1.2 Theorem ([B], [K]). The following properties are equivalent: i) sc$G = 2; ii) G is p-malleable; iii) For every p-class formation (G, X), for each pair of open subgroups GL aGK and for each i E Z, the reciprocity map induces an isomorphism

Gi (L/K, xL) Ei (LIK, AL); iv) For each pair of open subgroups GL a GK, the transfer induces an

isomorphism AK 5 HO(L/K, AL); v) For GL running through the open subgroups of G, the modules AL constitute a p-class formation.

Proof. The equivalence between i), ii) and iii) is the content of Thm. 6.1 of [B]. The equivalence between ii) and iv) is the content of 52.3, Propos. 10 of [Ha]. The remaining equivalence is obvious. 0

We shall be mainly interested in the following examples:

1.3. Examples. 1) Let k be a padic local field (i.e. a finite extension of Qp) and G = Gk be the absolute Galois group of k. Then scd,G = 2. 2) Let k be a number field (supposed to be totally imaginary if p = 2), let S be a finite set of places containing the places above p and oo, and G = Gk(S) be the Galois group of the maximal S-ramified (i.e. unramified outside S) algebraic extension of k. Then cdPG 5 2, and scd,G = 2 if and only if all finite S-ramified extensions of k verify Leopoldt's conjecture for p (here, cd, and scd, denote of course the pcohomological dimensions). 3) If cd,G = 1, then scd,G = 2. This happens e.g. if G is some free profinite group 3.

Now take a finite group G and present it as a quotient G = Fm/R,, where 3, is free profinite on m generators. Let R$ be the pSylow of R$. The Lyndon resolution

is a 2-extension of Z,, the class of which is nothing but the fundamental class of H2 (G, R$) = EX^;,[^^ (z,, R e ) . Together with Schanuel's


lemma, it shows that, up to projective summands, the Zp[G]-module RZ does not depend on rn. By abuse of language, we will denote it by R$(G) or Rab(G), and call it "the" p-relation module of G. Considered as a given data of G, it will intervene in the Galois module structure of all pclass formations (see 2.4).

2. Homotopy of modules and envelopes

Let G be a finite group and A := Zp[G]. In this section, we recall some basic facts concerning the homotopy theory of A-modules of finite type. The main references are [J2] and [W].

2.1. Homotopy. Two A-modules M and N are homotopic (notation: M N N) if they differ by projective summands, i.e. if there exist two projective A-modules P and Q such that M @ P 21 N @ Q. The homotopy class of M is denoted by [MI. Classifying A-modules up to homotopy is almost classifying them up to isomorphism, because M 21 N if and only if M N N and M @I Qp 21 N @I Qp as Qp[G]-modules ([W], Chap. 6, Propos. 5).

A morphism of A-modules f : M + N is homotopic to zero (notation: f N 0) if f factors through a projective A-module. Two morphisms f and g are homotopic (f - 9) if f - g N 0. The morphisms homotopic to zero form a subgroup of HomA(M, N), and the quotient is denoted [M, N]. We write [f] for the class of f in [M, N]. One of our main tasks will be to determine [M, N] for suitable A-modules M and N. If D is a A-lattice (i.e. D is a A-module without Zp-torsion), [D, N] 21

I?'(G, Hom(D, N)) canonically ([GW] , 5.1). If D and D' are two A- lattices, we have an isomorphism [D, D'] 21 ID', D]* (Pontryagin dual)

induced by a pairing [D, Dl] @I [Dl, Dl % [D, Dl % Qp/Zp, wheie TD is

the so-called (algebraic) trace form so : [D, Dl + f i O ( ~ , Z p ) - Qp/Zp given by TD[g] = h t r a ~ e ( ~ ) ( m o d Z , ) ([GW], 5.8).

2.2. Envelopes. A (cohomologically trivial) envelope of a A- module M is an exact sequence of A-modules:

where C is a cohomologically trivial A-module and D is a A-lattice. It is not difficult to show that every A-module has an envelope, which is unique up to homotopy ([W] , Chap. 6, Lemma 9). By abuse of language,


we will call C "the" envelope of M. Taking Hom(D, -) and cohc we get a chain of isomorphisms

[D, Dl 2 do(^, Hom(D, D)) 2 H1(G, Hom(D, M)) ~ x t i

which sends [idD] to the extension class of (1).

The shift to cohomologically trivial modules makes things easier because of Jannsen's lemma: two cohomologically trivial A-modules are homotopic if and only if their Zp-torsion submodules are A-isomorphic ([J1], [W]). The problem is to recognize M starting from its envelope. There is a general "recognition theorem" due to Gruenberg and Weiss ([W], Chap. 6, Thm. 3), but we will be content with the following

2.3. Recognition lemma. Let M be a A-module, T its Zp- torsion. Fix a A-lattice D and suppose there exists an exact sequence

0 + M + C fi: D -+ 0, with C cohomologically trivial. Then the homotopy class [MI is determined by the homotopy class [fM] and the isomorphism class of T .

f Proof. Suppose we have two exact sequences 0 + M -+ C - f' D + 0 and 0 + M' + C' -+ D -+ 0 such that f N f', C and C'

are cohomologically trivial, and M and M' have the same Zp-torsion submodule T. Because D is a lattice, C and C' have the same torsion T; hence C C' by Jannsen's lemma. Write C @ P C1@P', with P and P'

f@O A-projective, and consider the surjective A-morphisms C @ P -H D and f '@O

C' @ P' -H D. Since f f', we are in the situation of the generalized Schanuel lemma ([J2], Lemma 1.3), which asserts that Ker(f $ 0) - Ker( f ' @ 0), i.e. M N MI. 0

In the situation of Lemma 2.3, let us write C = C/T. Then we have a commutative diagram

6 [D, C] - H' (G, Hom(D, T))

1 a [D, Dl - H' (G, Hom(D, M))

N

I


The lower horizontal isomorphism d has already appeared in 2.2. The upper horizontal isomorphism S is obtained analogously: from the ex- - act sequence 0 + T t C -+ C t 0, we get an exact sequence 0 +

Hom(D, T) + Hom(D, C) t Hom(D, C) + 0 (because C is a lat- A N

tice), and then by cohomology, an isomorphism HO(G, Hom(D, C)) -+

H1 (G, Hom(D, T)) (because Hom(D, C) is cohomologically trivial). The right vertical map is induced by the inclusion T L, M. The left vertical

map [D, C] --+ [D, Dl is given by [g] r [fM .g], by identifying Hom(C, D)

and ~ o m ( E , D) (because D is a lattice). The image of [fM] by the iso-

morphism [C, Dl = [C, Dl s [D, El* is the composite map

2.4. Special envelopes. In order to exploit Lemma 2.3, we must choose suitable lattices D. Let us say that a A-module is special, or admits a special envelope, if there exists an exact sequence:

where C is cohomologically trivial and IG is the augmentation ideal of A. The exact sequence (2) implies that fiG2 (G, Zp) : @(G, M ) for all i E Z. Moreover, the exact sequence 0 t Hom(Zp, .) -+ Hom(A, a ) t

Hom(IG, -) -+ 0 gives canonical isomorphisms H1 (G, Hom(IG, .)) - H 2 (G, .). We derive three consequences for a special A-module M: 1) The 2-extension 0 + M --t C t A --+ Zp -4 0 is described by the fundamental class (obvious definition) of H Z (G, M ) -- Ext? (q, M). 2) Comparing 1) with the Lyndon resolution (1.3), we get an alternative description of [fM] due to U. Jannsen ([J2], 4.5b). More precisely, consider the chain of isomorphisms:

- [C, IG] = [C, IG] ;do (G, H O ~ ( F , IG)) (by 2.1) - fiO (G, Hom(C, IG))

GH' (G, Hom(C, Rab))

(by the left part of the Lyndon resolution in 1.3)

Galois Module Structure of p-Class Formations 129

The image of [ f ~ ] in EX~;(C, Rab) is but the class of the pull-back extension:

3) The commutative diagram of 2.3 becomes

! This defines a character

: Taking into account the generalized Schanuel lemma, we find that 2.3 1. i becomes

I 2.5. Special recognition lemma. a) Two special A-modules are

homotopic if and only if their Zp-torsion submodules are A-isomorphic. 5

b) Let M be a special A-module and T its Zp-torsion. Then the homotopy ! class [MI is determined by the isomorphim class of T and the character . X T . In particular, XT = 0 ++ M - R"~(G) @ C, C being the special : envelope of M.

Remarks. a) Given T , there is a canonical construction of C due dJ to U. Jannsen ([Jl], [J2]). A free resolution An -+ Am + T* t 0 exists

if and only if there exists an exact sequence 0 -+ (A+)m f + (A+)n t C + 0 such that torzpC N T. Here T* is the Pontryagin dual of T and g5+ is the transpose map, obtained by applying the functor Horn(., A). "The" module C is the transposed module of T*, denoted by D(T*) .


b) However, the recognition lemmas remain unsatisfactory because we don't know to what extent the invariants [fM] and XT (in spite of the notation) depend on M. Fortunately, this difficulty can be solved for pclass formations (see 3.4).

3. Galois module structure of p-class formations

A pclass formation (G, X ) is called of finite type if all the Z,-modules XL belonging to the formation are of finite type (note that similar conditions are required in Mazur's theory of Galois deformations; see [Ma]). All examples in 1.3 are of finite type.

In order to apply the special recognition lemma, we must show

3.1. Proposition. Let (G, X ) be a p-class formation of finite type. For each pair of open subgroups GL a GK, the Zp[GK/GL]-module XL admits a special envelope.

Proof. We adapt the argument of [GW], 11.3. For simplicity, let us write G = GK/GL and A = Zp[G]. The augmentation ideal is defined by the exact sequence 0 -+ IG -+ A + Zp -+ 0. Taking Horn (. ,XL) and cohomology, we get an isomorphism H1(G, Hom(Ic, XL))

2 H2(G, Hom(Z,, XL)) = H2(G, XL). But H'(G, HOm(IG, XL)) - E X ~ ~ ( I ~ , XL) because IG is a lattice; hence we may consider the fundamental class ELIK as a class of E X ~ ~ ( I ~ , XL), corresponding to a certain extension:

The cohomology of this exact sequence and of the augmentation sequence gives a chain of connecting maps:

Standard augument of homological algebra shows that the composite map is but the cup-product by EL/K, which is an isomorphism because we have a class formation. Hence the map S is an isomorphism and fii-' (G, IG) 2 I?'(G,x~), Vi E Z. This shows that f i i (G,yL) = 0, Vi E Z.

The same argument works for every subgroup of G, because of the functorial properties of the fundamental class with respect to restriction. Hence YL is cohomologically trivial. 0


Remark. The extension of A-modules 0 + XL -+ YL -+ IG -+ 0 just constructed and the extension of groups 0 + XL -+ WLIK -+ G -+ 0 (the relative Weil group) defined by the fundamental class ELIK, are related by the so-called translation functor of Gruenberg and Weiss ([W], Chap. 3), which is an equivalence between two obvious categories.

Summarizing, we get

3.2. Theorem (compare with [J2], 4.5). Let (G, X) be a p-class formation of finite type. For each pair of open subgroups GL a GK, the Zp[GK/GL] -module XL is determined up to homotopy by: - the isomorphism class of its Zp-torsion tXL, and

- the character XL of H2(L/K,tXL) defined as the composite of the natural map H ~ ( L / K , tXL) -+ H2(L/K, XL) and the invariant map of

class formations invLIK : H ~ ( L / K , XL) I?'(L/K, ZP) -+ Qp/Zp. In particular the map H2(L/K, tXL) -+ H2(L/K, XL) is zero if and

only if XL R " ~ ( G ~ / G L ) @ D((tXL)*), where R " ~ ( . ) is the p-relation module and D(.) is the transposed module.

3.3. Special case. Let now G be a profinite group such that scdpG = 2, and for an open subgroup GL, let AL be the pSylow subgroup of G i b e By the theorem of Brumer-Kawada (1.2), (9, (AL)) is a pclass formation, and we will assume from now on that it is of finite type. For XL = AL, Thm. 3.2 then gives Thm. 4.5 a) and b) of [J2]. In this special case, the invariants which appear are of particular interest: 1) The special envelope of AL can be taken as BL := Ho(GL, IG), where IG is the augmentation ideal of the completed group algebra Zp[[G]] ([J2], [N2]). If moreover GK is a pro-pgroup on d generators and r relations, we

have a presentation 0 -+ AT -+ + BL -+ 0 (where A = Zp[GL/GK]) given by a matrix of Fox derivatives; in particular, BL (and hence tAL) is explicitly known if we know the relations of GK (see [N2]). 2) The module tAL is an important arithmetical invariant when G is a Galois group. If the base field lc is a padic local field, then tAL = p ~ ( ~ ) (the group of pprimary roots of unity in L), and the Galois module structure is entirely determined up to homotopy ([J1], [N1]).

If the base field lc is a number field and G = Gk (S) (EX. l.3.2), tAL is related by Iwasawa theory to padic L-functions (when they exist) and its structure has been studied intensively, e.g. in [N3]. 3) U. Jannsen ([J2], 4.5 c)) has given an alternative description of the character XL of H2(L/K, tAL) which shows that it does not depend on AL, but only on GL and tAL.


Let us summarize Jannsen's result: Let

be the p-dualizing module of 5;. It exists provided cdpG < 2, and is

characterized by the fact that H2 (G, M)* -- HomS (M, E?)) canonically for every ptorsion discrete G-module. Note that the dualizing module is the same for 4 and for every open subgroup GL. If scdPG = 2, then

tAL P H2(GL, Z,)* ([J1], [N1]) and E$') P l b L ~ A L with respect to the transfer. Thm. 4.5 c) of [J2] asserts that XL is the image of the identity

of E$') by the dual of the inflation map

This can be seen quickly as follows: The commutative diagram in 2.4.3 gives an isomorphism

which sends XL to [fL] (= the homotopy class associated with the exact sequence (3) in the proof of 3.1). This isomorphism is functorial and,

taking l%L, we get HompK(E$'), E?)) -- [IGK, IF,] by definition of

E?) and of the BL. But l iqL fL = id because l iqL AL = 0. We will now generalize this to get an analogous description of XL in

the general case. Let us add a few notations: for a class formation (G, X) , let tXL be the Zp-torsion of XL, X = 1 4 , XL and t X = limLtXL; similar notations hold for the AL. The reciprocity maps XL + AL induce

maps tXL - tAL and t X + tA E E?). Then:

3.4. Theorem. The notations are the same as in 3.2. Suppose moreover that sdcpG = 2. Then the character XL of H2(L/K, tXL) is the

image of the identity of E?) by the composite of the natural map

HornpK (E?), E?)) -+ HomGK(tX, E?)) and the dual of the inpation

map HornGK (tX, E?)) -. HZ (BK, tX)* + H2(L/K, tXL)*.

Proof. Because all our constructions are functorial, we have a commutative diagram

nat H 2 ( L / ~ , ~ A L ) * H~(L/K, tXL)*

I inf * I inf *

nat HornpK (E?), E?)) - HornPK (tX, E?)),


where the upper horizontal map sends the character XL corresponding to E AL to that corresponding to XL. By Jannsen's theorem, the conclusion 1 is obvious. 0 f

/

, Thm. 3.4 shows in particular that XL does not depend on XL, but only on tXL and GL.

4. On free pro-p-extensions

In this section, we fix a base field k, which is a padic local field or a number field, and we take G = Gr, or Gk(S) as in Examples 1.3.1 and 1.3.2. To simplify, we also suppose p # 2 and scd& = 2 (although a closer look at the proofs of Thm. 3.2 and 3.4 shows that weaker hypotheses could also work). Let G be a finite quotient of 4, corresponding to a Galois extension Klk. We stick to the notations of Thm. 3.2 and 3.4. Whereas the torsion module tAK is related to Iwasawa theory and p-adic L-functions, the character XK is related to the embedding problem

([JWI, [Nil). Because of the last statement of Thm. 3.2, we would like to study

the arithmetical implications of the nullity of XK. The approach in 2.3 is sharp enough, in principle, to allow us to kill X K . It is known ([JW], [N1]) that X K = 0 if and only if every embedding problem with pabelian kernel

admits a solution (= the existence of the dotted arrow). If we particularize further by supposing that G is a pgroup and

by replacing G by its maximal pro-pquotient G(p), then it is clear that every embedding problem:

admits a solution if the given epimorphism G(p) ++ G factors through a free prepgroup. This leads us to


4.1. Definition (see [N1]). A fme pro-p-extension, or Fd-extension, is a Galois extension Llk such that Gal(L/k) is isomorphic to a free pro- pgroup Fd on d generators.

Note that Fl 2 Zp is the only abelian free pro-pgroup. Define pk to be the maximal d such that k admits an Fd-extension.

In the local case, the interplay between the value of pk, the nullity of XK and the embedding problem is entirely known and can be summarized as follows:

4.2. Theorem ([JW], [N1]). Let k be a p-adic local field, and pk(p) the group of p-primary roots of unity in k. Suppose that G = Gal(K/k) is a p-group, and denote by d = d(G) := dim H1 (G, ZlpZ) the minimal number of generators of G. Then: A. If pk(p) = 1, the Galois group GK(p) is pro-p-free (~a fa rev i~ ) , pk =

1 + [ k : Qp], and AK 2 R ~ ~ ( G ) @ Z ~ [ G ] P * - ~ . B. If pk(p) # 1, the following four conditions are equivalent: i) XK = 0;

ii) AK R:b(G) @ D(PK (P)*); iii) Every embedding problem (4) admits a solution; iv) K l k is embeddable in an Fd-extension; If moreover G is abelian, the above conditions are equivalent to: v) Let ps be the minimum of the order of p ~ ( p ) and of the exponent of G. Then pk(p) C NKIk(K*), and for any h < s , the ZlphZ-module H1 (G, p,h ) is orthogonal to itself with respect to the Hilbert symbol of order ph.

As a consequence of v), pk = 1 + f [k : Q] (J. Sonn) .

The situation in the global case is much more complicated. Let k be a number field, let r2 = r2(k) be the number of complex places of k, S a finite set of primes of k containing the places above p and m, G = Gk(S) as in Ex. 1.3.2, A ~ ( S ) = G ~ ( s ) ~ ~ ( ~ ) , and ~ ( A ~ ( s ) ) = torkp(Ak(S)). ~t is known that every Fd-extension is Sp-ramified. Because of our assumption scdpGk(S) = 2, k verifies Leopoldt's conjecture; hence pk < 1 + r2. Examples are known for which pk < 1 + r 2 ([Y]). What would be a global analogue of Thm. 4.2? A. Because of Leopoldt's conjecture, it is obvious that

Number fields k for which Gk(SP)(~) is pro-pfree are called prutional and have been extensively studied in [MN].


B. If t(Ak (Sp)) # 1, we still have i) * ii) iii) + iv), but we don't know if iii) iv) (except in particular cases). The main difficulty is that we don't have any global analogue of v) (there is no "global Hilbert symbol"). One easy special case is the totally real case (r2 = 0):

4.3. Proposition. Suppose that the number field k is totally real, and let K l k be an S-ramified Galois extension such that G = Gal(K/k) is a cyclic p-group. The following three conditions are equivalent: i) XK = 0; ii) AK(S) II Zp @ ( ~ ( A K (S)), and t(AK(S)) is cohomologically trivial; iii) K lk is embeddable in a Zp-extension.

We omit the proof, which is obvious. The imaginary case (rz # 0) seems to be much deeper. To study the extreme situation, corresponding to pk = 1 +r2, we must appeal to hard (as yet unsolved) conjectures. The following results have been obtained in collaboration with A. Lannuzel. The proofs will appear elsewhere.

We start by recalling some conjectures of Greenberg, a "classical" one and a "generalized" one:

4.4. Greenberg's conjecture (GC). Let k be a totally real number field, k, = U n k, its cyclotomic Zp-extension, r = Gal(k,/k,), A', the modified p-class group of k, (=the p-class group divided by the classes of ideals above p), A', = 1% A',, X&, = A',, and A = Zp[[I']]

I the usual Iwasawa algebra. Then the following properties are equivalent: ; 1) The I?-module Ad, is zero. I

2) The A-module XS, is finite.

1 Greenberg's conjecture (GC) asserts that these equivalent properties hold for any totally real field (see [GI]) and has been verified for many / families of abelian fields, mainly quadratic fields (see e.g [IS] and the

! references therein). Note that Greenberg's original formulation concerns

f p-class groups instead of modified pclass groups, but is known to be i

equivalent to the formulation above when Leopoldt's conjecture always 1 holds along the cyclotomic tower - which is the case here.

In order to state a generalization of (GC) to any number field k, we - - must introduce: K,= the compositum of all Zp-extensions of k, r =

~ a l ( k , / k ) (2 Z;+'Z here), = z~[[F]] = the multivariable Iwasawa algebra,

Here L runs through all finite subextensions of k,/k, AL is defined as in 3.3 and AJL is as in 4.4.

T. Nguyen Quang Do

Let us recall the structure of X,:

4.5. Theorem (see [G2]). The I\-module X, has I\-rank r 2 . I t has no non zero pseudo-null submodule.

(Remember that "pseudo-null" means "killed by two co-prime ele-

ments of x". If I\ = A, "pseudo-null" is the same thing as "finite"). Thm. 4.5 is proved in [G2] by using induction and class field the-

ory. A simpler cohomological proof (by using the special envelopes BL), more in the spirit of this talk, can be found in [N2]. By means of Kum- mer's theory (after adding pp to k), Fitting's ideals, "Spiegelung" and - decomposition properties of primes above p in K,/k, one can show the following (not so easy) lemma.

4.6. Lemma. For a n imaginary number field k (r2 # O), the following are equivalent:

1) The F-module Ab, is zero.

2) The x-module Xk is pseudo-null. - 3) The A-torsion submodule of X, is zero (if k contains pp).

4.7. Greenberg's generalized conjecture (GGC) . The equivalent properties of 4.6 hold for every imaginary number field.

Property 4.6.2 has been verified for many families of quadratic number fields by J. Minardi, and a few biquadratic fields by D. Hubbard (two students of R. Greenberg). Note that the local analogue of 4.6.3 holds true ([N2], 4.3).

Remark. It is worth noting that, in spite of their common formulation in terms of class groups, the two conjectures (GC) and (GGC) are not of the same nature, because the property 4.6.3 definitely does not hold for totally real fields. Actually, for totally real fields, the torsion module torAX, is related by the "main conjecture" of Iwasawa's theory t o p-adic L-functions, hence is not trivial in general. The validity of 4.6.3 for imaginary fields, in this circle of ideas, would destroy any "naive" hope of constructing multivariable p-adic L-functions starting from torXX,.

We can now state the main theorem concerning the extreme case pk = 1 + 7-2. R. Greenberg told us a t the conference that his student Hubbard had obtained in his thesis [Hu] a similar, but somewhat weaker result (essentially because Lemma 4.6 is not available in [Hu]).

I


B I 4.8. Theorem ([L], [Hu]). Let k be an imaginary number field

I (r2 # 0) verifying (GGC). The following two properties are equivalent: , l ) p k = 1 + ~ 2 . F 2) k is p-rational, i. e. Gk (Sp) (p) is pro-p-free.

Note that , by the remark after 4.7, the extreme case pk = 1 + r 2

is radically different, according as lc is totally real (4.3) or is imaginary (4.8).

References

A. Brumer, Pseudo-compact algebras, profinite groups and class- formations, J. of Algebra, 4 (1966), 33-40.

R. Greenberg, On the Iwasawa invariants of totally real fields, Amer. J. Math., 98 (1976), 263-284.

R. Greenberg, On the structure of certain Galois groups, Invent. Math., 47 (1978), 85-99.

K. W. Gruenberg and A. Weiss, Galois invariants for units, Proc. Lon- don Math. Soc., 70 3 (1995), 264-284.

K. Haberland, "Galois cohomology of algebraic number fields", VEB Deutscher Verlag der Wissenschaften, Berlin (1978).

D. Hubbard, "The non-existence of certain free pro-pextensions and capitulation in a family of dihedral extensions of Q", Thesis, Uni- versity of Washington (1996).

H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields 11, Internat. Math. J., 7 6 (1996), 721-744.

U. Jannsen, On the structure of Galois groups as Galois modules, in "Number Theory, Noordwijkerhout 1983", Springer LNM, 1068 (1984), 109-125.

U. Jannsen, Iwasawa Modules up to isomorphism, in "Algebraic Num- ber Theory - in honor of K. Iwasawa", Advanced Studies in Pure Math., 17 (1989), 171-207.

U. Jannsen and K. Wingberg, Einbettungsprobleme und Galoisstruk- tur lokaler Korper, J. reine angew. Math., 319 (1980), 196-212.

Y. Kawada, Cohomology of group extensions, J. Fac. Sci. Univ. Tokyo, 9 (1963), 417-431.

A. Lannuzel, Sur les extensions pro-plibres d'un corps de nombres, prkpublication (1997).

B. Mazur, Deforming Galois representations, in "Galois groups over Q", MSRI Publ., 16 (1987), 385-437.

J. Minardi, Iwasawa modules for z:-extensions of number fields, Cana- dian Math. Soc. Conf. Proc., 7 (1987), 237-242.

A. Movahhedi and T. Nguyen Quang Do, Sur l'arithmktique des corps de nombres prationnels, Sdm. Thkorie des Nombres Paris 1987/88, Progr. Math. Birkhaiiser, Boston, 81 (1 990), 155-200.


[NI] T. Nguyen Quang Do, Sur la structure galoisienne des corps locaux et la thkorie d'Iwasawa, Compositio Math., 48 1 (1982), 85-1 19.

[Nz] T. Nguyen Quang Do, Formations de classes et modules d'Iwasawa, in "Number Theory, Noordwijkerhout 1983", Springer LNM, 1068 (1984), 167-185.

[N3] T. Nguyen Quang Do, Sur la Z,-torsion de certains modules galoisiens, Ann. Inst. Fourier, 36 (1986), 27-46.

[S] J.-P. Serre, "Corps locaux", Hermann, Paris (1962).

[W] A. Weiss, "Multiplicative Galois module structure", Fields Institute Monographs, 6 (1996).

[Y] M. Yamagishi, A note on free pro-pextensions of algebraic number fields, J. Thkor. Nombres Bordeaux, 5 (1993), 165-178.

U M R 6623 C N R S Laboratoire de Mathe'matiques Universite' de Franche-Comte' 16 route de Gmy F-25030 BESANCON CEDEX FRANCE


A Dieudonn6 Theory for p-Divisible Groups

Thomas Zink

1 Introduction

Let k be a perfect field of characteristic p > 0. We denote by W(k) the ring of Witt vectors. Let us denote by E -r F ~ , 5 E W(k) the Frobenius automorphism of the ring W(k). A Dieudonnk module over k is a finitely generated free W(k)-module M equipped with an F-

linear map F : M -+ M such that pM c F M . By a classical theorem of Dieudonnb (compare Grothendieck [GI) the category of pdivisible formal groups over k is equivalent to the category of Dieudonnk modules over k.

In this paper we will prove a totally similiar result for pdivisible groups over a complete noetherian local ring R with residue field k if either p > 2, or if pR = 0. For formal p-divisible groups (i.e. without &ale part) this is done in [Z2].

We will now give a description of our result. Let R be as above but assume firstly that R is artinian. The maximal ideal of R will be denoted by m. The most important point is that we do not work with the Witt ring W(R) but with a subring W(R) c W(R). This subring is characterized by the following properties: It is functorial in R. It is stable by the Frobenius endomorphism and by the Verschiebung

of W(R). We have ~ ( k ) = W (k). The canonical homomorphism W(R) -r W (k) is surjective, and its kernel consists exactly of the Witt vectors in W(m) with only finitely many non-zero components. The ring W(R) is a non-noetherian local ring with residue class field k. It is separated and complete as a local ring.

If R is an arbitrary complete local ring as above we set W(R) =

lim t- w ( R / ~ " ) .

Received September 24, 1998. Revised January 12, 1999.

140 T. Zink

Let us denote by fR c W(R) the ideal which consists of all Witt vectors whose first component is zero.

Definition 1. A Dieudonnk display over R is a quadruple (P, Q, F, V-') where P is a finitely generated free w(R)-module, Q C P is a submodule and F and V-l are F-linear maps F : P + P and V-l :

Q --+ P. The following properties are satisfied:

(i) fRp C Q C P and P /Q is a free R-module. (ii) V-l : Q ---+ P is an F-linear epimorphism.

(iii) For x E P and w E w (R), we have

In contrast with Cartier theory there is no operator V in our theory. The strange notation V-' is explained below by the relationship to Cartier's V. But there is a w(R)-1' inear map

which is uniquely determined by the relation V ~ ~ ( W V - ' ~ ) = w @ y for w E W(R) and y E Q (see 1221 Lemma 1.5 ).

If P is a Dieudonnk display over k , then the pair (P, F ) is a Dieudonnk module, and this defines an equivalence of categories.

Theorem: There is a functor ID from category of p-divisible groups over R to the category of Dieudonne' displays over R which is an equivalence of categories.

Let X be a pdivisible group over R and let P = ID(X) be the associated Dieudonnk display. Then heightX = rankm(R) (P) . Moreover the tangent space of X is canonically identified with the R-module P/Q.

I stated this theorem as a conjecture during the p-adic Semester in Paris 1997. Faltings told me that I should prove it using Proposition 19 below. We follow here his suggestion. In the proof we will restrict to an artinian ring R because the general case is then obtained by a standard limit argument.

Other generalizations of DieudonnB theory are Cartier theory, and the crystalline Dieudonnk theory, which was developed by Grothendieck, Messing, Berthelot, de Jong and others (compare de Jong [J]). Dieudonnk displays are explicitly related to both of these theories. More precisely we construct functors from the category of Dieudonnk displays to the category of crystals respectively to the category of Cartier modules. In

A Dieudonne' Theory for p-Divisible Groups 141

1 particular this explains the relationship between Cartier theory and crys- 1 talline theory completely. So far this relationship was only understood

in special cases (compare the introduction of Mazur and Messing [MM], and[Z3]).

We note that our theory works over rings with nilpotent elements, while the crystalline Dieudonnk functor is not fully faithful in this case. From our point of view the reason for this failure of crystalline Dieudonnk theory is that we can recover from the crystal associated to a Dieudonnk display P the data P, Q, F but not the operator V-l. On the other hand for a reduced ring the prime number p is a non-zero divisor in W(R) and therefore V-' may be recovered from the relation pV-l = F .

I Let us explain the relationship to Cartier theory. Like a Cartier module a Dieudonnk display may be defined by structural equations.

, Take any invertible matrix (a,) E G ~ ~ ( w ( R ) ) , and fix any number 0 5 d 5 h. We define a Dieudonnk display P = (P ,Q,F, V-l) as follows. We take for P the free w (R)-module with the basis el , . . . , eh . We set

The operators F and V-l are uniquely determined by ( I ) and by the following relations:

Fe j = x a i j e i , for j = l , ..., d

~ - 1 ~ . 3 - - C a i j e i for j = d + l , ..., h

Assume now for simplicity that R is an artinian ring. Let ER be the local Cartier ring with respect to p (see [Zl]). Then we may consider in the free ER-module with basis e l , . . . , eh the submodule generated by the elements

h

Fe j - aijei, for j = 1, . . . , d

ej - v ( E a g e i ) for j = d + l , ..., h

where F and V are now considered as elements of ER. The quotient by this submodule is the IER-module which Cartier associates to the

142 T. Zink

connected component of the pdivisible group X with the Dieudonnk display D(X) = P.

Finally we point out two questions, which we hope to answer in another paper.

Let X be a pdivisible group over R, and let P be the associated Dieudonnk display. Then we cannot verify in general that the crystal we associate to X coincides with the crystal Messing [MI associates to X . By [Z2] this is true, if X is connected.

The other probably easier question is, whether our functor respects duality. In [Z2] we proved the following: If X is a connected pdivisible group whose dual group Xt is also connected, the displays D(X) and D(Xt ) are dual to each other. The same is then automatically true for the Dieudonnk displays. It is not difficult to see that a positive answer to the first question gives also a positive answer to the second question.

52. Dieudonn6 Displays

Let R be an artinian local ring with perfect residue field k. There is a unique ring homomorphism W(k) -+ R which for an element a E k , maps the Teichmuller representative [a] of a in W ( k ) to the Teichmuller representative of a in R. Let m c R be the maximal ideal. Then we have the exact sequence

It admits a canonical section 6 : W ( k ) -t W(R) , which is a ring homomorphism commuting with F. It may be deduced from the Cartier morphism [Z2] (2.39), but it has also the following explicit Teichmuller description: Let x E W(k) . Then for any number n there is a unique solution of the equation Fn y, = x. Let Gn E W(R) be any lifting of y,. Then for big n the element Fn& is independent of n and the lifting chosen, and is the desired S(x).

Since m is a nilpotent algebra we have a subalgebra of W(m):

~ ( m ) = {(xo, sl, . . . ) E W(m) I xi = 0 for almost all i }

~ ( m ) is stable by and V . Moreover ~ ( m ) is an ideal in W(R) . Indeed, since every element in ~ ( m ) may be represented as a finite

N sum C "[ x i ] , it is enough to show that [xo]e E ~ ( m ) for xo E m and

i= 1 ( E W(R) . But this is obvious from the formula

A Dieudonne' Theory for p-Divisible Groups

We may now define a subring W ( R ) c W (R):

/ Again we have a split exact sequence i

? o - w(m) - W ( R ) w(k) - o 1

with a canonical section 6 of T.

Lemma 2. Assume that the characteristic p of k is not 2 , or that 2R = 0. Then the subring W ( R ) of W ( R ) is stable under and V .

Proof. Since 6 commutes with F, the stability under is obvious. For the stability under one has to show that

( 6 ) 6(Vx) -V6(x) E ~ ( m ) for x E W(k) .

If we write x = Fy and use that ~ ( m ) is an ideal in W ( R ) , we see that it suffices to verify ( 6 ) for x = 1. For the proof we may replace R by WN ( k ) for a big number N. In W ( W ( k ) ) we have the following formula using logarithmic coordinates (compare (7) below, and [Z2] 2.11) :

Our assertion is, that the Witt components of this Witt vector in W(W(k ) ) converge to zero in the padic topology of W(k) for p # 2 respectively that they become divisible by 2 in the case p = 2. We write

The ui are determined by the equations

An elementary induction shows ordp un = pn - pn-l - . - 1.

We remark that in the case pR = 0 the section 6 also commutes with V . Indeed, in this case taking the Teichmuller representative is a

144 T. Zink A Dieudonne' Theory for p-Divisible Groups 145

ring homomorphism k + R. We obtain 6, if we this homomorphism.

apply the functor W to

Since the ring W(R) has obviously all the properties mentioned in the introduction, i.e. the definition 1 has now a precise meaning.

We consider now a surjection S -+ R of artinian local rings with residue class field k as in the lemma. We assume that the kernel a of the surjection is equipped with divided powers yi : a -+ a. Then we have an exact sequence

and the divided Witt polynomials define an injective homomorphism:

If the divided powers are nilpotent in the sense that for a given element a E a the divided powers ypk(a) become zero for big k the homomorphism (7) becomes an isomorphism (compare [22] (3.4)). In this paper a pd-thickening is a triple (S, R, yi) which satisfies this nilpotence condition. We write an element from the right hand side of (7) as [ao, .. . , ai, ...I where ai E a are almost all zero. We call it a Witt vector in logarithmic coordinates.

The ideal a c W(S) is by definition the set of all elements of the form [a, 0, . . . ,0 , . . . ] where a E a. Let P be a Dieudonn6 display over S and 7 = PR be its reduction over R. Let us denote by Q the inverse image of Q by the homomorphism

Then V-' : Q -+ P extends uniquely to V-' : Q --t P such that V-'aP = 0.

Theorem 3. Let us consider a pd-thickening S --t R as above. Let Pi = (Pi, Qi, F, V-l) for i = 1 , 2 be Dieudonne' displays over S . Let - - - pi = (Pi, Qi, F, V-l) = Pi,R be the reductions over R. Assume we are given a morphism of Dieudonne' displays ii : P1 -t P 2 . Then there exists a unique morphism of quadruples

Proof. For the uniqueness it is enough to consider the case ii = 0. As in the proof of [22] Lemma 1.34 one obtains a commutative diagram

Since V-N[ao, a l , . . . ]x = [aN, aN+1 . . . ] F N x for [a0,. . . ] E &(a) and x E P2, any given element of ~ ( a ) p2 is annihilated by veN for big N. Since PI is finitely generated it follows that V - ~ U = 0 for big N. Then the diagram shows u = 0 which proves the uniqueness.

As in the proof of [Z2] Theorem 2.5 it is enough to consider the case where p1 = P2 = 7 and ?'i is the identity, if one wants to prove the existence of u. One simply repeats the proof of [22] Theorem 2.3 with w instead of W. The proof goes through without changing a word up to the last argument showing the nilpotency of the operator U defined by 1oc.cit. (2.16).

To complete the proof we have to show that for any ,-linear map S : L1 4 pN w(a)lpN+' ~ ( a ) @w(s) P2 there exists a number m such that Urn; = 0.

To see this we consider the following ~ ~ + l - l i n e a r map

By definition UmZ factors through the ,-linear map obtained from rm by partial linearization to an ,-linear map

But as in the proof of the uniqueness any given element of pNw(a) lpN+' w (a) @fi(S, P2 is annihilated by some power of V- ' . Since L1 is

. ,

a finitely generated w(s)-module, it follows that rm is zero for big m. This proves UmZ = 0 for big m. 0

Theorem 3 gives the possibility to associate a crystal to a Dieudonn6 display: Let P = (P, Q, F, V-l) be a Dieudonnk display over R. Let S -+ R be a pd-thickening. Then we define a functor on the category of pd-thickenings:

which lifts the morphism ?i.

146 T. Zink

where P = (P,Q,F,V-') is any lifting of P to S. The Theorem 3 assures that P is unique up to a canonical isomorphism. This functor is called the Witt crystal. We also define the Dieudonnk crystal

The filtration

is called the Hodge filtration. The following statement is similiar to a result of Grothendieck and Messing in crystalline Dieudonnk theory.

Theorem 4. Let C be the category of all pairs (P, Fil) where P i s a Dieudonne' display over R and Fil c Dp(S) is a direct summand which lifts the Hodge filtration of Vp(R). Then the category C is canonically isomorphic to the category of Dieudonne' displays over S .

This follows immediately from Theorem 3 (compare [Z2] 2.2). To a Dieudonnk display P = (P, Q, F, V-') we may associate a 3n-

display 3(P) = ( P I , Q', F, V-') where we set P = W(R) @k(R) P . The submodule Q' is defined to be the kernel of the natural map W ( R ) @ O ( ~ )

P + PIQ. The operators F and V-' for 3(P) are uniquely determined by the relations

We call a Dieudonnk display P over R V-nilpotent, if 3 ( P ) is a display in the sense of [Z2] 1.6. We recall that this is also equivalent to the following condition. Let Pk = (Pk , Qk, F, V-') be the Dieudonnk display obtained by base change to k. Then the operator V = p ~ - ' : Pk --+ Pk is topologically nilpotent for the padic topology.

If P is V-nilpotent a Dieudonnk crystal D3(p)(S) was defined in [Z2] 2.6. The trivial statement that the functor 3 respects liftings leads to a canonical isomorphism

Theorem 5. The functor 3 is an equivalence of the category of V-nilpotent Dieudonne' displays over R with the category of displays over R.


Proof. If R = k the functor 3 is the identical functor. By induction it suffices to prove the following. Let S + R be a pd-thickening and assume that the theorem holds for R. Then the theorem holds for S. But the category of Dieudonnk displays over S is decribed from the category of Dieudonnk displays over R and the Dieudonnh crystal. Since the same description holds for displays by [Z2] 2.7 , we can do by (9) the induction step. 0

Corollary 6. The category of p-divisible formal groups over R is equivalent to the category of V-nilpotent Dieudonne' displays over R.

This is clear because the corresponding theorem holds for displays by [Z2] Theorem 3.21. The equivalence of the corollary is given by the functor which associates to a V-nilpotent DieudonnB display the p divisible group BT(3(P)) . Let us describe this functor which will be simply denoted by BT(P).

Let X be a pdivisible group over R. It is an inductive limit of finite schemes over Spec R. Hence we have a fully faithful embedding of the category of pdivisible groups to the category of functors from the category of finite R-algebras to the category of abelian groups. We describe BT(P) by giving this functor.

Proposition 7. Let P be a V-nilpotent Dieudonne' display over R, and let S be a finite R-algebra. Let Ps = (Ps, Qs,F, V-') be the Dieudonne' display obtained by base change. Then we have an exact sequence:

Proof. First of all we note that S is a direct product of local artinian algebras satifying the same assumptions as R. Therefore the notion of a Dieudonnk display makes sense over S . Moreover we may assume that S is local with maximal ideal ms. In [Z2] we have considered BT(P) as a functor on the category NilR of nilpotent R-algebras. In this sense we have:

Let Pms = W (ms) B O ( ~ ) P c Ps. We set Qms = Pms n Qs. Then [Z2] Theorem 3.2 tells us that there is an exact sequence:

Let ks be the residue class field of S , and Pks the display obtained by base change. Then we have Ps/Pms = Pks and Qs/Qms = Qks . Hence

148 T. Zink

the proposition follows, if we show that the map V-' - id : Qk, + Pk, is bijective. Indeed, because V is topologically nilpotent on Pk, for the padic topology, the operator -V - V2 - V3 - . . . is an inverse.

53. The Multiplicative Part and the tale Part

For a pdivisible group G over an artinian ring there is an exact sequence:

Here Gc is a connected pdivisible group and Get is an &ale pdivisible group. The aim of this section is to show that the same result holds for Dieudonn6 displays.

Let us first recall a well-known lemma of Fitting (Lazard [L] VI 5.7):

Lemma 8. Let A be a commutative ring and I- : A -+ A a ring automorphism. Let M be an A-module of finite length and cp : M + M be a T-linear endomorphism. Then M admits a unique decomposition

such that cp leaves the submodules Mbij and Mnil stable, and such that cp is a bijection o n Mbij and operates nilpotently o n Mnil.

We omit the proof, but we remark that ~~~j and Mnil are given by the following formulas:

Here Image cpn is an A-module because T is surjective. In order to deal with a more general situation we add two complements to this lemma.

Let A be a commutative ring and a c A an ideal which consists of nilpotent elements. We set A. = Ala and more generally we denote for an A-module M the Ao-module MIaM by Mo. Let T : A + A be a ring homomorphism such that ~ ( a ) c a, and such that there exists a natural number r with r r (a) = 0. We denote by 1-0 : A. + AO the ring homomorphsim induced by I-.

Lemma 9. Let P be a finitely generated projective A-module and cp : P P be a I--linear endomorphism. Then cp induces a TO-linear endomorphism cpo : Po -+ Po of the Ao-module Po.


Let Eo be a direct summand of Po such that cpo induces a 1-0-linear isomorphism.

Then there exists a direct summand E c P which is uniquely determined by the following properties:

(i) v(E) c E . (ii) E lifts Eo.

(iii) cp : E -+ E is a I--linear isomorphism. (iv) Let C be an A-module which is equipped with a T-linear isomor-

phism $ : C t C. Let a : (C,$) + (P,cp) be an A-module homomorphism such that a o $ = cp o a . Let u s assume that ao(Co) c Eo. Then we have a (C) C E .

Proof. By our assumption on r we have an isomorphism

A %',A P = A @ T r , ~ O PO.

We define E to be the image of the A-module homomorphism

(11) (9')' : A @+,A, EO -+ P.

It follows immediately that cp(E) c E . Let us prove that E is a direct summand of P . We choose a AO-

submodule Fo c Po which is complementary to Eo:

Po = Eo @I Fo.

Then we lift Fo to a direct summand F of P. We consider the map induced by (1 1)

(12) (cpr)# : A @,r,a0 Eo - P/F.

By assumption the last map becomes an isomorphism when tensored with AOBA. Hence we conclude by the lemma of Nakayama that (12) is an isomorphism. We see that E is a direct summand:

P = E @ F

Applying Nakayama7s lemma to the projective and finitely generated module E, we obtain that

cp# : E - E 1

is an isomorphim.

150 T. Zink

Therefore we have checked the properties (i)-(iii). The last property follows from the commutative diagram

We have also a dual form of the last lemma.

Lemma 10. Let A, A. , T, TO be as before. Let P be a finitely generated projective A-module and

be a homomorphism of A-modules. Let Eo c Po be a direct summand of the Ao-module Po such that ipo induces an isomorphism

Then there exists a direct summand E C P of the A-module P , which is uniquely determined by the following properties:

(i) cp(E) c A @,,A E . (ii) E lifts Eo.

(iii) cp : P I E -+ A @,,A P I E is an isomorphism. (iv) Let C be any A-module, which is equipped with a n isomorphism

$ : C d A @T,A C . Let a : P d C be an A-module homomorphism such that Eo is i n the kernel of ao. Then E is i n the kernel of cr.

Proof. The proof is obtained by dualizing the last lemma with the functor HornA (- , A) except for the property (iv) . We omit the details, but we write down explicitly the definition of E. Let r be such that r T ( a ) = 0. From the isomorphism A P = A @ T r , ~ O Po we obtain a

map

Then E is the kernel of this map. 0

We will apply these lemmas in the situation where A = w(R), A. = W (k) and T is the F'robenius endomorphism of w(R), i.e. rw = F ~ . For this we have to convince ourself that the kernel ~ ( m ) of the


map W(R) i W(k) is nilpotent and that F r ~ ( m ) = 0 for a sufficiently big number r . By induction it is enough to prove that for a surjection of artinian rings S i R with kernel b such that pb = b2 = 0, we have ~ ( b ) ~ =F w (b ) = 0. This we know. Hence the lemmas are applicable and give the following:

1 Proposition 11. Let P be a finitely generated projective w(R)-

/ module and cp : P -+ P be an F-linear homomorphism. I Then there exists a uniquely determined direct summand prnUlt c P I

I with the following properties

(i) cp induces an F-linear isomorphism

pmult , pmult

I ' (ii) Let M be any w(R)-module and $ : M i M be an F-linear isomorphism. Let a : M i P be a homomorphism of w(R)- modules such that cr o $ = p o a. Then a factors through pmult.

Proof. Let us begin with the case R = k. For any natural number n the Frobenius induces an isomorphism : Wn(k) -+ Wn(k). There- fore Fitting's lemma is applicable to Pn = Wn(k) g W ( k ) P and cp, =

Wn ( I ; ) BW ( l c ) q. In the notation of that lemma we set PrnUlt = lim -- p,biJ . n

From the definition of P,biJ (see (10)) it follows that an : Mn + P, factors through p,bij. Hence a ( M ) c prnUlt + p n P for any n, which proves (ii) .

Let us now consider the general case. We set Po = W(k) @ c Y ( ~ ) P .

Then we have already proved the existence of We lift P T ~ ~ ~ by the lemma (9) to a direct summand prnUlt of P . Then that lemma states that prnUlt has the desired properties. 0

Proposition 12. Let R be as i n the last proposition. Let P be a finitely generated projective w(R)-module and let 9 : P -+ W ( R ) B F , f i ( R I P be a w(R)-module homomorphism. Then there exists a pro-

jective factor module Pet of P which is uniquely determined by the following properties.

(i) ip induces an isomorphism of w(R)-modules

152 T. Zink

(ii) Let M be a w(R)-module and $ : M -+ W(R) @ F , P ( ~ ) M be

an isomorphism. Let a : P -+ M be a homomorphism of w(R)- modules such that (id @ a) o ip = $ o a . Then a factors through pet .

Proof. Again we begin with the case R = k. Then : W(k) -+

W(k) is bijective. We denote its inverse by 7. Then we have a T-linear isomorphism

Hence we consider ip as a 7-linear map

To this map we apply Fitting's lemma as in the last proposition. We obtain the decomposition P = pbiJ @ pni1. Then we set Pet = pbiJ, and we obtain the lemma for W(k).

The general case is obtained, if we apply the lemma 10 to the situation A = W(R), A. = W(k) and TW = Fw for w E W (R). 0

We will now define the &ale part and the multiplicative part of a Dieudonn6 display over R.

Definition 13. Let P = (P, Q, F, V-') be a Dieudonnk display over R. We say that P is &ale if one of the following equivalent conditions is satisfied:

(i) P = Q. (ii) V# : P - W(R) @ F , ~ ( R ) P is an isomorphism.

Proof. Assume (i) is fulfilled. Then we have for any x E P the formula V#([V-'x) = [ @ x where [ E w (R). This implies that V# is surjective, and hence an isomorphism. Conversly if V# is surjective, we consider the composite of the following surjections:

Since the composite is by (2) induced by the inclusion Q c P, we conclude

But since P/Q is a projective R-module this implies P /Q = 0. Indeed F~~ =pa W(R) and p is not a unit in w(R).


Definition 14. Let P = (P, Q, F, V-') be a Dieudonn6 display over R. We say that P is of multiplicative type if one of the following equivalent conditions is satisfied:

(i) Q = IRP. (ii) F# : w (R) @F,P( R) P + P is an isomorphism.

Proof. The first condition implies that P is generated by elements of the form V-'("~I) = EFx, E E w(R), x E P . This implies the second condition.

Assume that the second condition holds. The image of a normal decomposition P = L @ T by F# gives a direct decomposition

Comparing this with the standard decomposition

we obtain p . W(R)V-' L = w (R)v-' L. Hence again since p is not a unit, we have W(R)V-'L = 0. This implies L = 0. 0

Let P = (P, Q, F, V-') be a Dieudonn6 display. Recall that P is called V-nilpotent if the following map is zero for big numbers N:

The Dieudonnk display is called F-nilpotent if the following map is zero for big numbers N:

Proposition 15. Let a : Pl = (PI, Q1, F, V-') + P2 = (P2,Q2, F, V-') be a homomorphism of Dieudonne' displays. Then a is zero, if one of the following conditions is satisfied.

(i) One of the Dieudonne' displays Pl and P2 is e'tale and the other is V-nilpotent.

(ii) One of the Dieudonne' displays Pl and P2 is of multiplicative type and the other is F-nilpotent.

Proof. By rigidity (i.e. the uniqueness assertion of Theorem 3) it is easy to reduce this proposition to the case where R = k is a perfect field. In this case the proposition is well known. 0

154 T. Zink

Proposition 16. Let P be a Dieudonne' display over R. Then there is a morphism P + Pet to an e'tale Dieudonne' display over R such that any other morphism to an e'tale Dieudonne' display P + Pl factors uniquely through Pet. Moreover Pet has the following properties:

1 ) The induced map P -+ Pt is suqective. 2 ) Let pnil be the kernel of P + Pet. Then (P"", P"" n Q , F, V - l )

is a V-nilpotent Dieudonne' display which we will denote by pnil.

Proof. The map V# : P + W ( R ) @F,&(R) P determines by Prope

sition 12 a projective factor module P 5 Pet such that V# induces an isomorphism Pet + W ( R ) @F,&(R) Pet. We consider the inverse map

It is induced by an F-linear map V-l : Pet -t Pt . We set Qet = Pet and F = pV-l : Pt + Pet. Then we obtain a Dieudonnd display Pet = (Pt , Qet , F, V - l ) . We will now check that the map a : P + Pet induces a homomorphism of displays P -+ Pet. To see that a commutes with F we consider the following diagram:

The right hand square is commutative by definition. Our assertion is that the left hand square is commutative. Since V# for Pet is an isomorphism it is enough to show that the diagram becomes commutative if we delete the vertical arrow in the middle. But this is trivial because V# o F# = p. Since we have trivially a ( Q ) c Pet it only remains to be checked that a commutes with V - l . For this we consider the diagram

By (2) the composition of the arrows in the first horizontal row is induced by the inclusion Q c P , while the composition in the lower horizontal row is the identity. We deduce the commutativity of the first square as before.

A Dieudonne' Theory for p-Divisible Groups 155 I 1

1 Hence we have a morphism of Dieudonnd displays a : P + Pet . the proposition is known for R = k and in fact easily deduced from Fitting's lemma. In this case V exists and Q = V P . It follows that the map

, Q -t Pt is surjective. In general we conclude the same by Nakayama's lemma. To show that (P"", P"" n Q , F, V - l ) defined in the proposition is a Dieudonnd display, it remains to be shown that P""/P"" n Q is a projective R-module. But because of the sujective map Q + Pet we have an isomorphism P"'~/P"" n Q 21 P / Q . The universality of P + Pet is an immediate consequence of Proposition 12. 0

Dually to the last proposition we have

Proposition 17. Let P be a Dieudonne' display over R. Then there is a morphism from a multiplicative Dieudonne' display pmUlt + P such that any other morphism P l + P from a multiplicative Dieudonne'

1 display Pl factors uniquely as P l + pmUlt + P . Moreover pmult has

i the following properties: !

1) The map prnUlt + P is injective and prnult n Q = I ~ P ~ ~ ~ ~ .

2 ) (p/prnult, Q / I R pmUlt, F, V - l ) is a n F-nilpotent Dieudonne' display.

I

Proof. We consider the map F : P + P , and we define the direct summand pmUlt c P according to Proposition 11. We define Q~~~~ = I R pmUlt, and obtain a Dieudonnk display pmult = (pmult, Qmult F, V - l ) which has the required universal property. To prove 1) we consider a normal decomposition P = L @ T. Let P = W ( k ) @k(R) P and let - -mult L and P be the images of L and pmUlt. Since the proposition is known (and easy to prove by Fitting's lemma) for R = k , it follows

-mult that @ P is a direct summand of P. By Nakayama's lemma one verifies that L @ pmult is a direct summand of P . Hence we may assume without loss of generality that pmUlt is a direct summand of T. From this 1) and 2) follow immediately, except for the F-nilpotence, which may be reduced to the case k = R. 0

54. The p-Divisible Group of a Dieudonn6 Display

In this section we will extend the functor BT of Proposition 7 to the category of all Dieudonnk displays, and show that this defines an equivalence of categories.

Let R be an artinian local ring with perfect residue class field k , satisfying the assumptions in the introduction. We will denote by the unramified extension of R such that is local and has residue class field

156 T. Zink

- k the algebraic closure of k. We will write I? = ~ a l ( E / k ) for the Galois group. Then I? acts continuously on the discrete module R.

Let H be a finitely generated free Zp-module. Assume we are given an action of I? on H , which is continuous with respect to the padic topology on H . The actions of I? on w(R) and H induce an action on w (R) @zP H. We set

One can show by reduction to the case R = k that P(H) is a finitely generated free w(R)-module and that the natural map

is an isomorphism. We define an ktale Dieudonnk display over R:

Here P (H) = Q(H) and V-l is induced by the map

Conversely if P is an &ale Dieudonn6 display over R we define H ( P ) to be the kernel of the homomorphism of Zp-modules

Hence the category of ktale Dieudonnk displays over R is equivalent to the category of continuous Zp[I']-modules which are free and finitely generated over Z,.

On the category of Dieudonnk displays over R we have the structure of an exact category: A morphism cp : Pl -+ P2 is called a strict monomorphism if cp : Pl -+ P2 is injective and Q1 = cp-l (Q2), and it is called a strict epimorphism if cp : PI -+ P2 is an epimorphism and cp(Q1) = Q2.

Proposition 18. Let P = (P, Q, F, V-l) be a V-nilpotent Dieu- donne' display over R. Let us denote by CR the cokernel of the map V-l - id : Qx --+ PE with its natural structure of a I'-module. Then we have a natural equivalence of categories


Proof. Let us start with a remark on Galois cohomology. Let P be any free and finitely generated w(R)-module with a semilinear r-action, which is continuous with respect to the topology induced by the ideals Vn w(R). Then we have

Indeed we reduce this to the assertion that for a finite dimensional vector space U over E with a semilinear continuous action of I', we have H1 ( r , U ) = 0, because this is an induced Galois module by usual descent theory. To make the reduction we consider first the case R = c. Then we have a filtration with graded pieces Hornzp (H, pnP/pn+'P). Since the cohomology of these graded pieces vanishes and our group is complete and separated for this filtration, we are done for R = z. In the general case we consider a surjection R -+ 3 with kernel a and argue by induction. We may assume that m . a = p . a = 0. It is enough to show that H1 (I?, Hornzp (H, w (a)P) = 0. Because w ( ~ ) P -

- en a @,,,,, ,, P ~ / I ~ ~ ~ the vanishing (14) follows.

We will use a bar to denote base change to R, i.e. QE = Q etc. Let us start with an extension from the right hand side of (13) :

It induces an exact sequence

of I?-modules. The same argument as above shows H1(r, Hornzp (H, - Q)) = 0, if we use a normal composition for Q. Hence the sequence (16) admits a I?-equivariant section over H :

We consider the function u : H -t P given by

Since we may change s exactly by a homomorphism of r-modules H -, Q, we obtain that the class of u in the cokernel of the map

is well-defined by the extension (15). Since the group cohomology vanishes this cokernel is exactly Homr(H, CE). This provides an injective

T. Zink

group homomorphism

Conversely it is easy to construct an extension of Dieudonnk displays over R:

from a homomorphism u E Homr (H, P ) by taking (17) as a definition for the operator V-' of P,. Then one has an action of r on P, for which the sequence (19) becomes r-equivariant. Taking the invariants by r we obtain an element in EX~'(P(H), P) whose image by (18) is u . Hence (18) is an isomorphism. 0

Remark. Our construction is functorial in the following sense. Let P' be a second V-nilpotent Dieudonnk display, and H' be a second ZP[r]-module which is free and finitely generated as a Zp-module. Let u1 E Homr(H, Ck) be a homomorphism to the cokernel of V-' - id : - Q' -+ P'. A morphism of data (P, H, ;) - (PI, HI, u' ) has the obvious meaning. Then it is clear that such a morphism induces a morphism of the corresponding extensions:

and conversely. Moreover, since there are no nontrivial homomorphims P - P(H1), we conclude

We associate to a continuous I?-module H, which is free and finitely generated as a Zp-module a Barsotti-Tate group as usual. The finite I?-module p-" H / H corresponds to a finite &ale group scheme G,. We set

n

The following analogue of Proposition 18 seems to be well-known.

Proposition 19. Let H be as above and let G be a formal p- divisible group over R. Then there is a canonical isomorphism of categories

Hornr (H, G(z)) -- ~ x t ' ( ~ F ( H ) , G)


Moreover this isomorphism is functorial i n the sense of the last remark.

Before we prove this, we remark that it implies the main theorem of this paper:

Theorem 20. There is a functor BT from the category of Dieu- donne' displays over R to the category of p-divisible groups over R which is an equivalence of categories. O n the subcategory of V-nilpotent Dieu- donne' displays this is the functor BT of Proposition 7.

Proof. By the last proposition the category of pdivisible groups over R is equivalent to the category of data (G, H, u : H - G(R)). But since we already know that the category of formal pdivisible groups is equivalent to the category of V-nilpotent Dieudonnk displays such that G(R) is identified with CE we conclude by the remark after Proposi- tion 18. 0

Proof of Proposition 19. Let us start with an extension

Let S be a local R-algebra such that the residue class field 1 of S is contained in a fixed algebraic closure of k. Then we obtain an exact sequence of I'l = ~al(%/l)-modules

In fact this sequence is exact because the flat ~ech-cohomology of a formal group vanishes (use [ZZ] 4.6 or more directly [Z3] 5.5).

Conversely, if we are given for any S an extension of rl-modules (21) which depends functorially on S we obtain an extension (20).

If we pull back the extension (20) by the morphism H @ Qp -+

H 8 Q,/Zp it splits uniquely as a sequence of abelian groups because G ( S ) is annihilated by some power of p. By the uniqueness it splits also as a sequence of rl-modules. Hence to give an extension (20) is the same thing as to give a homomorphism of rl-modules H - ~ ( 3 ) . The functoriality in S means in particular that we have a commutative diagram

which is equivariant with respect to rl c r k . Hence to give functorially extensions (21) is the same thing as a rk-equivariant homomorphism H + G@). 0

T. Zink

References

de Jong, A.J., Barsotti-Tate groups and crystals, in Proceedings of the ICM Berlin 1998, Vol.11, 259-265, Documenta Mathematica.

Grothendieck, A., Groupes de Barsotti-Tate et cristaux de Dieudonnk, Skm. Math. Sup., 45, Presses de 1'Univ. de Montreal, 1970.

Lazard, L., Commutative formal groups, LNM 443, Springer 1975. Mazur, B., and Messing, W. Universal extensions of and one-

dimensional crystalline cohomology, LNM 370, Springer 1974. Messing, W., The crystals associated to Barsotti-Tate groups, LNM

264, Springer 1972. Zink, Th., Cartiertheorie kommutativer formaler Gruppen, Teubner

Texte zur Mathematik, 68, Leipzig 1984. Zink, Th., The display of a formal pdivisible group, Preprint 98-017

SFB 343, Universitat Bielefeld 1998, to appear in Asterisque SMF. Zink, Th., Cartiertheorie uber perfekten Ringen, preprint, Karl-

WeierstrafkInstitut fur Mathematik, Berlin 1986. Zink, Th., Windows for displays of pdivisible groups, http://www. mathematik.uni-bielefeld.de/-zink, to appear in: Proceedings of a Conference on Texel Island 1999.

Universitiit Bielefeld Fakultat fur Mathematik POB 100131 33501 Bielefeld, Germany E-mail address: [email protected] eld. DE


Hilbert's 12th Problem, Complex Multiplication and Shimura Reciprocity

Peter Stevenhagen

Abstract.

We indicate the place of Shimura's reciprocity law in class field theory and give a formulation of the law that reduces the technical prerequisites to a minimum. We then illustrate its practical use by dealing with a number of classical problems from the theory of complex multiplication that have been the subject of recent research. Among them are the construction of class invariants and the explicit generation of ring class fields.

1 Hilbert's 12th problem

All variants of class field theory can be said to 'classify' in some i way the abelian extensions of a given field K. The classical examples are I those where K is a number field, a function field in one variable over a

I finite field, or a local field, but the second half of this century has seen

1 the birth of higher dimensional analogues as well [12]. I In the classical cases, the main theorem of class field theory provides

an anti-equivalence I t I t$ : AbK + Subx 1 ( between the category A ~ K of finite abelian extensions of K (inside some

: algebraic closure K of K) and the category Subx of open subgroups of ' a locally compact abelian group X = X ( K ) , which is entirely defined I 'in terms of K'. Here the morphisms in both categories are simply the I inclusions between fields and subgroups, respectively. In the three stan- I dard examples mentioned above, X ( K ) can be taken to be equal to the

idhle class group of K in the first two cases, which constitute the global case, and to the multiplicative group K* in the local case.

Acknowledgement: I thank N. Schappacher for drawing my attention to Sohngen's paper [15] during the conference.

Received October 7, 1998 Revised November 30, 1998

P. Stevenhagen

The definition of the anti-equivalence $I is entirely explicit: it maps a finite abelian extension L of K to the norm image NLIKX(L). The 'surjectivity on objects' of $I is the existence theorem of class field theory, which guarantees that every open subgroup H c XK is of the form NLIKX(L) for some finite abelian extension L of K , the class field of H . The problem of finding a 'direct description' of the extension L = [HI in terms of H is known as Hilbert 's 12th problem. Hilbert originally posed the problem for number fields, but it occurs in the other variants of class field theory as well.

Already for number fields, Hilbert's problem is not entirely well- posed, as one cannot say that the construction of class fields in the proof of the existence theorem is not 'explicit' or 'constructive'. However, the proof is not 'direct' in the sense that it does not generate the class fields over K itself, but over large auxiliary extensions of K . What Hilbert had in mind was an analogue for arbitrary number fields of the following theorem over Q.

1.1. Kronecker-Weber theorem. The abelian extensions of Q are generated by the values of the exponential function exp : T H e2'7riT at rational arguments T.

Even though the theorem exhibits the generators of the abelian extensions as values of a transcendental function, it is relatively easy to find the corresponding algebraic data, i. e., the irreducible polynomials in Z[X] corresponding to these generators. As exp[Q] S Q/Z is the subgroup of roots of unity in C*, these are the cyclotomic polynomials. Moreover, the action of ~ a l ( q / ~ ) on the roots of unity generating the maximal abelian extension Qab of Q yields an isomorphism

For local fields and for function fields over finite fields, there is an analogue of the statement in 1.1 that there is a module C* over the ring of integers Z of Q with the property that the torsion points of the Z-action on C* generate the abelian extensions of Q. In both cases, the abelian extensions are generated by the torsion points of a suitable module over a 'ring of integers' A C K . In the local case, A is the valuation ring and the module is provided by the Lubin-Tate theory of formal groups [lo]. In the function field case, there is some choice for A which has to be taken care of, and the modules one needs are rank-one Drinfeld A-modules [8].

As far as finding an analogue of the Kronecker-Weber theorem for number fields K # Q is concerned, Hilbert's problem is outstanding in

Complex Multiplication and Shimura Reciprocity 163

all but the special case of imaginary quadratic K . It is one of the main open problems in class field theory.

$ 2. Complex multiplication

The theory that generalizes 1.1 for imaginary quadratic fields goes under the name of complex multiplication. We let K further be imaginary quadratic, and use the unique infinite prime of K to view C as the archimedean completion of K . This enables us to evaluate complex analytic functions in the 'K-valued points7 of either C itself or the complex upper half plane H = {T E C : Im(r) > 0). It is our aim to generate the maximal abelian extension Kab of K using such values.

The maximal abelian extension Qab of Q , which contains K 7 is clearly a subfield of Kab. Weber tried to generate Kab over Qab using the values of the modular function j : H -, C . This is the unique holomorphic function on H that is invariant under the action of the modular group SL2(Z) and has a Fourier expansion of the form j(q) =

q-' + 744 + O(q) for q = eZTiT tending to 0. Weber thought incorrectly [18, $1691 that Kab is the compositum of Qab and the field Kj obtained by adjoining to K the values j ( r ) of the j-function at T E K n H. One does however come close.

2.1. Theorem. The maximal abelian extension Kab of K is an infinite abelian extension of KjQab with Galois group of exponent 2.

) Other functions are needed if one wants the full extension Kab rather f than the approximation 'up to quadratic extensions' from 1.2. There are

I two ways to proceed, and they appear to be rather different at first sight.

i The first method goes back to Fueter, Takagi and Hasse. Fueter,

[ who discovered the need of additional quadratic extensions, showed [4, 1 Hauptsatz, p. 2531 that Kab is contained in the extension of KJQab I generated by the division values ('Teilwerte') of the Weber function hK I associated to K . This function, which is not a modular but an elliptic 1

function, is the 'normalized' x-coordinate on a Weierstrass model of the i 1 elliptic curve E = C/OK associated to the ring of integers OK of K . It

can be viewed as a meromorphic function on C with period lattice OK. The precise definition, which depends on the number of roots of unity in K , can be found in 118, $1531, [9, Ch. 1, $51 or [3, $61.

Incomplete knowledge of the arithmetic nature of the division values of the Weber functions prevented Weber himself [18, $1551 from making extensive use of h~ in the theory of complex multiplication, and he uses Jacobi's elliptic function sn(z) as a substitute. Takagi, who devotes the final sections of his famous article on general class field theory to the

164 P. Stevenhagen

special case of imaginary quadratic K, follows this detour and provides explicit generators for Kab using Jacobi functions [17, Satz 371. A complete description of Kab using Weber functions is finally obtained by Hasse [7]. It reads as follows.

2.2. Theorem. Let K be imaginary quadratic with ring of integers OK = Z[rO]. Then Kab is generated over K(j(r0)) by the values hK(r) of the Weber function hK at T E K \ OK.

The second method, which plays a central role in Shimura's version of complex multiplication, sticks to modular functions, but uses infinitely many of them. More precisely, one needs modular functions of higher level as defined in [9, Ch. 6, $31. These functions form a field F, the modular function field over Q. The algebraic closure of Q in F is the maximal cyclotomic extension Qab of Q.

2.3. Theorem. Let K be imaginary quadratic, and pick T E K f l H . Then Kab is generated by the finite function values f (T), with f ranging over the modular function field F .

Theorems 2.2 and 2.3 are not as different as they may look. One can use Fricke functions to generate F over Q as in [9, Ch. 9, $31, and take T in 2.3 equal to the value TO from 2.2. Then the values of the various Fricke functions evaluated at TO coincide with the values of the Weber function h K O n K \ O K .

When comparing theorem 2.3, which fixes the argument but not the function, to the Kronecker-Weber theorem 1.1, one may wonder naively whether it is possible to replace the j-function in 2.1 by some other modular function f E F such that the simplicity of 1.1 is regained. Heinz Sohngen, a student of Emil Artin, showed in his thesis [15, Satz IV] that this is not possible.

2.4. Theorem. Let f E F be any modular function, and let Kf be the extension of K that is obtained by adjoining the finite function values f (T) for T E K n H to K . Then Kab has infinite degree over the compositum Kf Qab .

In order to be useful in practice, the theorems 2.2 and 2.3 need to be complemented by a description of the Galois theoretic properties of the generators of Kab. We will focus on Shimura's formulation [14], which has a reputation of being the most 'abstract' approach to complex multiplication. This is partly due to the heavy notation in which it is often couched. In addition, most expositions first go through a somewhat cum- bersome description of the multiplication of complex lattices by idkles.


In the next section, we furnish a concise description of Shimura's main results. It reduces notation to a minimum and avoids the usual 'componentwise' operations on idkles by a systematic use of profinite completions. The final three sections of the paper illustrate that this 'abstract' version is both an ideal instrument to obtain smooth concep tual proofs and a powerful algorithmic tool. In section 4, we prove a general result (4.4) that readily implies theorems 2.1 and 2.4. It encom- passes most of Sohngen's results [15] on ray class fields for orders in a rather painless way. Sections 5 and 6, which extend the recent work of Alice Gee and the author [5, 61 to arbitrary orders, deal with the construction of class invariants and the explicit generation of ring class fields. They show that Shimura reciprocity not only completely removes the mystery that long surrounded Weber's claims on class invariants, but also yields the Galois theoretic properties of such invariants that are needed for their use in computational settings.

53. Shimura reciprocity

Shimura's reciprocity law for K gives the action of the absolute abelian Galois group Gal(Kab/K) of K on the 'singular value' f (7) of a modular function f E F at T E K n H. It combines Artin7s reciprocity law from class field theory, which describes Gal(Kab/K) as a quotient of the idkle group of K , with the Galois theory of the field F of modular functions. It defines, for a jixed singular modulus T E K n H , an action of the idhle group of K on the modular function field F such that we have for every idkle x the innocuously looking identity

In this 'minimal notation version' of Shimura's reciprocity law the action of x on the value f (7) is via its Artin symbol, and the action of x on f E F is explained in this section. We avoid explicit multiplication of lattices by idkles by defining the action first for suitable subgroups.

A large subgroup of Aut(F) is obtained by considering F as an extension of the field Fl = Q ( j ) of modular functions of level 1 over Q. One has F = UN,l FN, where FN is the field of modular functions of level N over Q. one can view FN as the function field of the modular curve X(N) over the cyclotomic field Q(CN). Over the complex numbers, the curve X(N) is a Galois cover with group SL2(Z/NZ)/ f 1 of the j-line X ( l ) = P1. When working over Q, one has an isomorphism G a l ( F ~ / p ~ ) Z GL2(Z/NZ)/ f 1. It may be obtained by combining the 'geometric action' of the subgroup SL2 (Z/NZ)/ f 1 with the 'arithmetic

166 P. Stevenhagen

action' via the determinant map on the N-th roots of unity, cf. [9, Ch. 6, 531. The restriction maps between the fields FN correspond to the natural maps between the groups GL2(Z/NZ)/ f 1, and one finds the subgroup

Gal(F/Q(j)) = G L ~ ( ~ ) / k 1

of Aut(F) by taking the projective limit. We now pick an element T E K n H, and write AX2 + BX + C

with A E Z>o for the irreducible polynomial of T in Z[X]. Clearly, we have K = ~ ( m ) with D = B2 - 4AC. One easily checks that the lattice L, = Z . T + Z corresponding to T is an invertible 0-ideal for the quadratic order 0 = Z[AT] of discriminant D.

Corresponding to the subgroup Gal(F/Q(j)) c Aut(F), there is the subgroup Ga1(Kab/K(j(r)) C Gal(Kab/K). It is well-known that Ho =

K ( j ( r ) ) is the ring class field of K corresponding to the order 0 . It is a finite abelian extension of K whose Galois group over K is isomorphic to the class group of the order 0 . If T generates the ring of integers OK of K over Z, then K( j ( r ) ) is the Hilbert class field H = K(j( rO)) of K occurring in theorem 2.2.

It follows from class field theory that we may describe Gal(Kab/K) by an exact sequence

Here rk denotes the Artin map on the group of finite K-iddes K* =

(K @z 2)'. Note that I? = K @z 2 is the ring of finite adhles of K ,

and that I?* is the quotient of the full idl.le group of K obtained by 'forgetting' the infinite component C*. For imaginary quadratic K , this amounts to dividing out the connected component of the identity element. Inside I? we have the profinite completion

of the order 0 . Its unit group 6* c I?* maps under the Artin map unto Gal(Kab/Ho), so we have a diagram with exact rows

(3.2) 1 4 O* + 6* \k - Gal(Kab/HO) 1

l g r

1 - {&I} -- G L ~ ( ~ ) - Gal(F/Q(j)) 4 1.

The connecting homomorphism g, : 6* -+ G L ~ ( ~ ) sends the idkle x E

6* to the transpose of the matrix representing the multiplication by

Complex Mu2tiplication and Shimura Reciprocity 167

x on the free %module 2 . T + 2 with respect to the basis [T, 11. The defining identity for g, (x), which is often written as g,(x) (;) = ( y ) , may be expanded into the explicit formula

The map g = g, yields an action of 6* on F, and the Galois conjugate

(f (T))" of f (T) under the Artin symbol 8 ( x ) E Gal(Kab/Ho) of x E 6* can be computed from the reciprocity relation

(3.4) (f (r))" = (fg'x-")(r).

Whenever F is Galois over Q( f ) , we have the fundamental equivalence

Note that only the implication + is immediate from (3.4), the implication + requires an additional argument [14, prop. 6.331.

The content of Shimura's reciprocity law is that the natural Q-linear extension of the map g, in (3.3), which is a homomorphism

r connects the exact rows in the diagram

' extending (3.2) in such a way that (3.4) and (3.5) hold unchanged for i this map.

i The statement above is not complete without a description of the action of the group G L ~ ( Q ) of of invertible 2 x 2-matrices over the finite

adele ring Q = Q @z 2 of Q on F . It is obtained as in [9, Ch. 71 by

I writing the elements of this group in the form u . a, with u E G L ~ ( ~ ) in

I the subgroup for which we know already how it acts, and a E GL2(Q)+ a rational 2 x 2-matrix of positive determinant. Note that u and a are 1 not uniquely determined by the product u a, since we have

! G L ~ ( L ) n GL2 (Q)+ = SL2 (Z) c GL,(Q).

I

Nevertheless, the natural action of GL2(Q)+ on H via fractional linear 1 transformations induces a right action of GL2(Q)+ on F that can be

168 P. Stevenhagen

combined with the action of G L ~ ( % ) on F . A well-defined action of

G L ~ ( Q ) on F is obtained by putting

54. Ray class fields for orders

As our presentation in the previous section indicates, one can generate Kab over K in two steps. One first picks a quadratic order O C K, and considers the ring class field Ho of 0 . This is the finite abelian extension of K generated by the j-invariant j (O) of the order. The Galois group Gal(Ho/K) is isomorphic to the class group Cl(O) of the order 0, with the ideal class [a] E Cl(O) acting on j (0) by

The top row of (3.2) shows that the Galois group of Kab over Ho has a rather uncomplicated structure: as O* is a finite group consisting of the roots of unity in 0, the group Gal(Kab/Ho) is essentially the unit

group of the profinite completion 6 of 6 . This means that Kab can be obtained as the union of the finite extensions HN,o of HO corresponding to the finite quotients

of 6' for N E Z>l. We call HN,o the m y class field of conductor N for the order O. Its Galois group over Ho is isomorphic to (O/NO)* /im[O*] . If O is the maximal order of K , then HN,o is the ray class field of conductor N of K. We clearly have Hl ,o = Ho.

Let T E K n H be as in the previous section, and O the order corresponding to the lattice [T, 11. For any N E Z>l , - we obtain from (3.2) a diagram with exact rows

O* 4 (O/NO)* ---+ Gal(HN,o/Ho) - 1

(4.3) 19, {fl) - GL2(Z/NZ) 4 Gal(FN/Q(j)) 4 1

in which all groups are finite. Here g, is the natural reduction modulo N of the map g, in (3.2) and (3.3), and FN is the field of modular functions of level N.

It is clear from (3.4) that for every modular function f E FN of level N , the value f (T) is contained in the ray class field HN,o of conductor N for the order O corresponding to 7. In fact, a standard argument as


in [9, p. 1281 shows that the extension of Ho generated by the values f (T) for all f E FN is equal to HN,o. In fact, it suffices to adjoin the value of the Weber function for the elliptic curve C/L, at a generator of the cyclic O-module k L, / L,.

Let LN c Kab be the field obtained by adjoining to K all the finite function values f (T), with f ranging over FN and T ranging over K n H. As all orders O c K occur as the multiplyer ring of a lattice L,, we have

LN = limHN,o A C Kab,

where the injective limit is taken over all orders O c K. Writing Op = O @z Zp7 we have 6* = np 0; c I?*. The kernel of

the natural map 6* - (O/NO)* in (4.2) equals

6 ~ ~ ) = {z E 6* : x 1 mod* N ) = 0; x IIpIN(l + NOp)7 plN

so an inclusion of orders yields an inclusion of kernels and we find

For N = 1 this is the Anordnungssatz for ring class fields [3, $191. The field LN is the infinite extension of K corresponding to the

subgroup

I By class field theory, the Artin symbol of an idhle x E 6* acts trivially ' on the maximal cyclotomic extension Qab of Q if and only if its image 1 A

! under the norm map 6* + Z* h. Gal(Qab/Q) is trivial. As the norm

of an element x E %* c 6* is simply its square, we find the following Galois theoretic description of the compositum LNQab c Kab.

4.4. Theorem. Let LN c Kab be the field obtained by adjoining the finite values of the modular functions of level N at the points T E

K n H to K . Then the restriction of the Artin map I?* % Gal(Kab/K)

t o the subgroup %* c I?* induces a suvection

EiN)[2] = {X E z* : I m 1 mod* N and x2 = 1) 4 Gd(Kab/LNQab)

170 P. Stevenhagen

with kernel ziN) [2] fl {&I}. I n particular, Kab/LNQab is an infinite

abelian extension of exponent 2.

The map in 4.4 is an isomorphism for N > 3 and has a kernel of order 2 for N 5 2.

For N = 1 we have FN = Q(j ) , SO L1 = K j is the field occurring in theorem 2.1. This yields the following precise version of theorem 2.1.

4.5. Corollary. Let K j be as i n 2.1. Then there is a natural exact sequence

1 4 {&I} 4 @ {&I} + Gd(Kab/KjQab) 4 1. p prime

It follows from 4.4 that theorem 2.1 cannot be improved in a substan- tial way by replacing j by some other modular function f E F . In fact, by employing finitely many modular functions one always generates a subfield of the field LN for some N E and LNQab is a finite extension of KjQab. In particular, we see that SGhngen's theorem 2.4 is an immediate corollary of 4.4.

5 5. Class invariants

We have seen that the ring class field Ho corresponding to a quadratic order O C K is obtained by adjoining the value j(O) to K . The irreducible polynomial q5o of j(O) over K, which is known to be a polynomial in Z[X] with highest coefficient 1, is the class polynomial of the order 0 . The zeroes of are the j-values j(a) of the ideal classes [a] E Cl(O), and we can numerically determine +o from the complex approximations of its zeroes. This is much faster than the algebraic determination of as the divisor of some modular polynomial @,(X, X ) E Z[X] as in [2, p. 2971, which is only computationally feasible for a few very small 0 .

Weber noticed already that class polynomials have huge coefficients. In fact, they are so large that they are never useful in actually computing Hilbert class fields. For instance, for the quadratic field of discriminant

Complex Multiplication and Shimura Reciprocity

-95 the maximal order O has class polynomial

It was discovered by Weber that in some cases, 'small' elliptic functions f of level N > 1 can be used to generate Ho as well. In the example above,

one can take for f the Weber function a- of level 48 or the function

'((z+3)15) )2 of level 120. When evaluated at suitable T E KnH, these z( v(z) f yield elements f (T) E Ho with irreducible polynomials

over K . In such cases f (T) is said to be a class invariant of 0 . Weber used several modular functions of higher level in a rather

ad hoc manner to compute by hand a number of class invariants. His computations of class invariants in [18] are a mix of theorems, tricks, numerical observations, conjectures and open questions. Among them is the famous class number one problem, which already goes back to Gauss. Heegner's 1954 solution of the problem, which proved the completeness of Gauss's list of class number one discriminants, was not accepted because it relied heavily on the observations of Weber, which were not in all cases theorems. Only when Baker and Stark gave independent proofs in 1968 of the same result, it was realized that Heegner's proof was essentially correct [16]. The renewed interest in Weber's class invariants resulting from this led to new proofs and additional results [I, 161, but not to a systematic way to deal with such questions.

Shimura reciprocity enables us to determine in a rather mechanical way, for any given modular function f E F, the set of orders O = Z[T] for which the value f (7) lies in the ring class field Ho. As O determines T only up to an additive constant k E Z and the value f (T) may depend on k for functions f E F of higher level, we fix T to be the 'standard generator' of O c K having trace nKIQ(T) E {O,l). We will show that

172 P. Stevenhagen

with this normalization, the set of orders 0 for which f (T) is a class invariant for 0 can be described in terms of congruence conditions on the discriminant D = disc(0) modulo some integer n( f ). In fact, n( f ) divides 4N if f has level N.

Suppose we are given a modular function f in the field FN of modular functions of level N , together with the explicit GL2(Z/NZ)- action on f . In practice, this means that we know the action of the standard generators S, T E SL2(Z)/ f 1 on f and the action of the Galois group Gal(Q(CN)/Q) = (Z/NZ)* on the Fourier coefficients of f . We let 0 = Z[T] be the quadratic order of discriminant D, and X2 + B X + C E Z[X] the irreducible polynomial of T, and impose the mild restriction that Q( f ) c F be Galois.

From the top row of (4.3) we see that the value f (T), which a priori only is known to lie in the ray class field HN,o of conductor N for 0, is a class invariant for 0 if and only if the Artin symbols of all elements of (O/N0)* leave f (T) fixed. Shimura7s equivalence (3.5) shows that this is equivalent to the requirement that g,(x) fixes f for all x E (O/NO)* . Thus, we only need to compute a set of generators xi for the finite abelian group (DIN 0) * , compute their g,-images

using (3.3), and check whether these elements of GL2(Z/NZ) fix f E F N . If one finds that f is not left invariant by all g,(xi), a look at the

ij, [(O/NO)*]-orbit of f often suffices to see which modification of f does have this property. There are many examples in [5] where a small power of f , if necessary multiplied by a well chosen root of unity, turns out to have the desired property. We refer to 153 and [6] for a large number of examples.

The computation of generators xi of (O/NO)* and their g,-images in the group GL2(Z/NZ) only depends on the residue class modulo N of the coefficients of the irreducible polynomial X2+BX+C of T. Thus, if T

is the standard generator of 0 having B = -Tt.K/Q ( q ) E { O , l } , the pair (B mod N, C mod N) only depends on the residue class of D = B2 - 4C modulo 4N. This proves our claim for n( f ) made above.

There is a large supply of classical modular functions f of higher level that are, in a sense that can be made precise, 'smaller7 than the j-function, and to which the 'algorithm' above can be applied. The functions y~ = and 7 2 = ~ of level 2 and 3 are the simplest and most classical examples. The Weber functions f, f l , fa of level 48 ana- lyzed in [13] and, more generally, the normalized quotients of Dedekind Q-functions in [5,6], are other examples of small modular functions. They


give rise to integral class invariants for which the irreducible polynomials are much smaller than the class polynomials.

56. Computation of ring class fields

The method in the preceding section enables us to prove in a systematic way that certain singular values f (T) lie in the ring class field Ho corresponding to the order 0 = Z[T]. It does not tell us how to find the conjugates of f (T) over K. This is indispensable in computational class field theory, where one wants to compute the irreducible polynomial of f (T) over K in order to obtain an explicit generating polynomial for Ho. The need for explicit conjugates also arises in other situations, e.g. in primality proving [ll, p. 1191.

By class field theory, the Galois group Gal(Ho/K) is isomorphic to the class group Cl(O) of 0, and the elements of this group can conveniently be listed as reduced primitive binary quadratic forms [a, b, c] of discriminant D = disc(0). For our purposes, it suffices to know that these are triples [a, b, c] of integers satisfying gcd(a, b, c) = 1 and b2 - 4ac = D. They are reduced if they satisfy the inequalities I bl 5 a 5 c and, in case we have Ib( = a or a = c, also b 2 0. For any given discriminant D < 0, there are only finitely many such triples, and they are easily enumerated if D is not too large. The correspondence between reduced forms and elements of the class group is obtained by associat-

ing to [a, b, c] the class of the ideal with Z-basis [ , a]. Note that [a, b, c] and [a, -b, c] correspond to inverse ideal classes.

The classical formula (4.1) for the action of the class group of 0 = Z[T] on the canonical generator j ( ~ ) of Ho over K can be rewritten as

For a general modular function f E F with f (T) E Ho, Shimura reci-

procity enables us to determine the conjugate of f over Q( j ) for which we have

This is done by picking for every class [a] E Cl(0) an idhle x E I?* A

that generates the 6 idea l a @z Z. Such an element x exists since every invertible 0-ideal is locally principal. It is only determined up to multiplication by elements of 6'. As in the case of (3.3), this abstract description of x may be translated into a simple explicit recipe. If a is

174 P. Stevenhagen

the invertible 0-ideal with Z-basis [*,a] corresponding to [a, b, c], A

one has ii = x 6 for the idkle x = (xP), E K* with components

a i f p f a

(6.2) i fp I a a n d p f c

- b + a - a if p I a and p I c 2

for each rational prime p. The Artin symbol of the idkle x acts on Ho as the ideal class [a] E C1(0), so we have f ( T ) [ ~ . ~ + ] = f (T), for this x. Applying the reciprocity relation (3.4) for x-' and g = g,, we find

The element g,(z) E G L ~ ( Q ) is only determined by [a, -b, c] up to left

multiplication by elements u E g T [ 6 * ~ c GL2(2). However, the fact that f (T) is a class invariant exactly means that we have f" = f for such u, so (3.7) shows that the right hand side of (6.3) does not depend on the choice of the generator x of 2.

Let M E GL2(Q)+ c G L ~ ( Q ) be the transpose of the $-linear map

on I? = Q . T + Q . 1 that maps the basis [T, 11 to [v, a]. Then the

action of M on H satisfies M ( r ) = v. Putting u, = g, (x) . M-' €

G L ~ ( Q ) , we can rewrite (6.3) as

Comparing the defining identity M (1) = ((-b+y)12) for M to that for A

g,(x), we see that both elements are transposes of Q-linear maps on

I? = Q . T + Q 1 that map the %lattice 6 = 2 . r + 2 - 1 onto ii = 2 6 . It follows that u, = g,(x) . M-', being the transpose of an element that

stabilizes the 5-lattice 6 c I? spanned by the basis [r, 11, is actually in

GL? (2). This means that f "- = j is a conjugate of f over Q ( j ) . Thus

(6.4) tells us which conjugate j of f we have to take in (6.1). Computing the function f = fU- from f is another instance of the

problem considered in the previous section. Choosing x as in (6.2), it is straightforward to write down an explicit formula for the components of

u, E G L ? ( ~ ) at each Z, as in [ 5 ] . As before, all we really need is the image of u, in the finite group GL2(Z/NZ), with N the level of f .


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F. Morain, Primality proving using elliptic curves: an update, in "Al- gorithmic Number Theory", (J. P. Buhler, ed.), Springer LNCS 1423, 1998, pp. 111-127.

W. Raskind, Abelian class field theory of arithmetic schemes, in "K- theory and Algebraic Geometry": Connection with Quadratic Forms and Division Algebras, AMS Proc. of Symp. in Pure Math., 58 no.1 (1995), 85-187.

R. Schertz, Die singularen Werte der Weberschen Funktionen f, f l , f 2 ,

7 2 , 73, J. Reine Angew. Math., 286/287 (1976), 46-74. G. Shimura, "Introduction to the Arithmetic Theory of Automorphic

Functions", Iwanami Shoten and Princeton University Press, 1971. H. Sohngen, Zur komplexen Multiplikation, Math. Annalen, 111 (1935),

302-328. H. M. Stark, On a "gap" in a theorem of Heegner, J. Number Theory, 1 (1969), 16-27.

T. Takagi, ~ b e r eine Theorie des relativ Abel'schen Zahlkijrpers, J . Col- lege of Science, 41 no.9 (1920), 1-133, Imperial Univ. of Tokyo; in "Collected Works", Iwanami Shoten, 1973, pp. 73-167.

H. Weber, "Lehrbuch der Algebra, vol. 111", Chelsea reprint, original edition 1908.

P. Stevenhagen

Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden, The Netherlands E-mail address: psh@math . leidenuniv . nl


Hecke Module Structure of Quaternions

David R. Kohel

Abstract.

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shimura curves.

1 Introduction

The arithmetic of quaternion algebras underlies a number of areas of modern mathematics, from number theory and algebraic geometry to graph theory. A motivation for the present article, in particular, is the interplay with the arithmetic of Shimura curves. Thus in this article we present the arithmetic of quaternion algebras and describe the construction of the associated Hecke modules. Then we discuss the geometric relation with modular curves and Shimura curves and conclude with some calculations. In order to motivate what follows we begin here with an overview of the algebraic and geometric facets of the

' theory. The Hecke modules of this study are defined alternatively as divisor

groups on right ideal classes of a level m order O for a definite quaternion algebra of discriminant Dp, or as the monodromy group at p of a NQon

: model for the Jacobian of a Shimura curve X g (mp). The curve X g (mp) parameterizes abelian surfaces whose endomorphism rings admit a fixed embedding of a level mp order R in an indefinite quaternion algebra of discriminant D. The supersingular points of reduction modulo p correspond, by definition, to those abelian surfaces which split into a product of supersingular elliptic curves. By results of Buzzard [2], the singular points of the reduction modulo p are the supersingular points.

Received September 22, 1998. Revised December 11, 1998.

178 D. R. Kohel

The monodromy group, measuring winding numbers about these points, can be identified with the supersingular divisor group. The enhanced supersingular surfaces are functorially equivalent to the right ideals over the quaternion order 0. Moreover the Hecke operators, as with classical modular curves, are determined by the morphisms in the supersingular category. Thus the monodromy group can be formally replaced with a free abelian group on a basis of right ideals classes for 0, allowing one to work with a Hecke module of quaternions as a proxy for the supersingular divisor subgroup.

This paper is organized as follows. Section 2 contains the background machinery for quaternion algebras from a constructive point of view, with explicit examples given. In Section 3 we define Hecke modules in terms of supersingular elliptic curves as in Mestre [lo] and its generalization in terms of quaternions. Section 4 recalls the definition of Shimura curves as well as relevant theorems which establish the connection with the quaternion algebras. The final section is devoted to the analysis of specific Shimura curves using the corresponding Hecke modules. By demonstration, we show that the arithmetic of quaternion algebras is effective for computation, in contrast to the difficulty of computing Shimura curves and their Jacobians (see Elkies [6]). Moreover, by means of the geometric interpretation, Hecke modules of quaternions provide a means of elucidating the theory of Shimura curves.

52. Quaternion algebras

A finite dimensional algebra H over a field K is simple if it has no proper, nontrivial left or right ideals. If the center of H is K , then H is said to be central over K . With these prior definitions, we make the following definition of a quaternion algebra.

Definition 2.1. A quaternion algebra H over K is a central simple algebra of dimension four over K .

Since K is the center of H , it is clear that H must be a noncommutative ring. Throughout this work, the field K will be Q or one of its completions Q, or R.

Quaternion algebras are the simplest of noncommutative rings, and in this realm, are the analogues of quadratic field extensions. As in the case of number fields, the principle questions of arithmetic interest regard the orders in a quaternion algebra and the left and right ideal theory of such orders.

Hecke Module Structure of Quaternions 179

Example 2.2. The following three algebras are examples of quaternion algebras over Q.

1. The matrix algebra MI2(()) is the split quaternion algebra over Q. 2. The Qalgebra defined by generators i and j with relations

3. The Q-algebra defined by generators x and y with relations

2.1. Trace, norm, and involutions Let H be a quaternion algebra over K . Every x in H is contained

in a quadratic extension of K . Conversely every maximal commutative extension of K in H is quadratic. It follows that every x in H satisfies an equation

x2 - Tr(x)x + Nr(x) = 0,

where Tr(x) and Nr(x) are elements of K which we call the reduced trace and reduced norm respectively. The conjugate of x is defined to be the element

- x = Tr(x) - x,

and x H : is an involution of H . Now suppose that K = Q. We define an order in H to be a sub-

ring which is a Z-module of rank four. Let R be any such order, let {xl, x2, x3, x4) be a basis, and set

(x, y) = Nr(x + y) - Nr(x) - Nr(y).

Then the determinant of the matrix ((xi, xj)) is the square of an integer. We define the discriminant of R to the positive integer disc(R) such that

det ((xi, xj)) = disc(^)^.

The discriminant of H is defined to be the discriminant of a maximal order in H , which is well-defined by Theorem 2.6 below.

2.2. Completions, ramification, and splitting Let MQ denote the set of finite and infinite places of Q. For each

place v in MQ, we define

180 D. R. Kohel

where 27, is the unique central division algebra of dimension 4 over 0,. In the former case v is said to split in H and in the latter v is said to ramify. Quaternion algebras can be classified in terms of their ramification by means of the Brauer group.

The Brauer group Br(K) of a field K is an abelian group defined on a set of classes [A] of central simple K-algebras A. The equivalence relation is defined such that [A] = [B] if and only if

for positive integers r and s. The group operation in Br(K) is the tensor product, and for any central division algebra, the opposite algebra lies in the inverse class of the Brauer group.

By the Wedderburn theorem every central simple K-algebra is isomorphic to an algebra of the form M,(D) for a unique central division algebra D/K and positive integer n. It follows that the classes in Br(K) are in bijection with the isomorphism classes of central division algebras. From the description of the inverse operation, it follows that the 2-torsion subgroup Br(K)[2] is in bijection with the isomorphism classes of central division algebras with involution.

Theorem 2.3. The Hasse invariant defines canonical isomorphisms

1 (i) invp : Br(Qp) 4 Q/Z (ii) inv, : Br(R) 4 -Z/Z

2

for all finite places p and the infinite place, respectively, such that the sequence

defined by ([A]) = ([A @ Qv])v and inv( ([Av])v) = invv(Av), is exact.

In fact the 2-torsion group Br(Q)[2] has representatives among the quaternion algebras over Q, giving the following classification theorem (see VignQas [20] 11.3 Theorem 3.1).

Theorem 2.4. Every quaternion algebra over Q ramifies at an even number of places. Conversely for every finite set consisting of an even number of places of Q, there exists a unique quaternion algebra H/Q ramifying at exactly this set.


Example 2.5. As noted, the quaternion algebras lie in the 2- torsion subgroup of the Brauer group. Rom Theorem 2.3 and 2.4 we note that in particular that there exists a unique quaternion division algebra over IW. This algebra is the classical Hamilton quaternions

If a quaternion algebra ramifies at infinity, we say that it is positive definite, otherwise we say it is indefinite. It follows that a quaternion algebra is positive definite if and only if it embeds in the Hamilton quaternion algebra.

The following theorem, together with the definition of the discriminant, serves to determine the primes ramifying in a quaternion algebra H .

Theorem 2.6. Let H be a quaternion algebra over Q. The discriminant of an order R in H is mdisc(O), where O is any maximal order containing R and m is the index [O : R]. The discriminant of O is equal to the product of the finite primes ramifying i n H .

Proof. This is the content of VignQas [20], Chapitre I, Lemme 4.7 and Chapitre 111, Corollaire 5.3 and the discussion following. 0

Example 2.7. In Example 2.2 we see that by definition the split quaternion algebra (1) ramifies nowhere. The discriminant of the order R = Z(i, j ) of the quaternion algebra (2) is 4, so has index two in a maximal order in the algebra ramified at 2 and oo. The order generated by x and y in the algebra (3) is maximal. Indeed for the basis { ~ 1 , ~ 2 , ~ 3 , ~ 4 ) = { l , ~ , ~ , x ~ ) we have

which has determinant 372. Therefore R is maximal and the algebra ramifies at 37 and oo.

2.3. Orders and ideals

An order in a finite dimensional algebra H/Q is a subring containing a basis for H and which is finitely generated as a Zmodule. In number fields there exists a unique maximal order. For noncommutative algebras this uniqueness property fails. In the split quaternion algebra

182 D. R. Kohel

M2(Q) for example, the order M2(Z) is maximal, but the same is true for x-lM2(Z)x for all x in GL2(Q).

Let H be definite, and let R be a maximal order in H . Define 0 to be the finite adele ring of Q and 2 to be the subring of integral adeles, then set

h I?= H ~ ~ Q , f = ~ 8 ~ 2 .

As with number fields, we have a local-global principle for ideals. Under the inverse maps

I A

IE ~ n 2 ~ - 2 k

every nonzero right ideal I corresponds to a subset if of of I? of the form 2 E for an element 2 of I?'.

Every fractional right ideal of R is a locally free, rank one module over R, and conversely every such module embeds in H . Thus the isomorphism classes of locally free, right modules of rank one over R are in bijection with the set

H* \ I?* /~* .

Since H*\I?* is compact in the topology on the idele group I?*, and E* is open, the set of isomorphism classes is finite.

Definition 2.8. Let I be a right ideal for R. The left order of I is defined to be the subring S = {x E H I X I C I) of H .

It is clear that S is an order of H . Locally it is conjugate to R at all finite primes, so in fact S is also maximal.

Proposition 2.9. Every maximal order of H appears up to isomorphism as the left order of a right ideal for R.

Proof. Let S be another maximal order in H , and set

I = {x E H I Sx R).

Then I is a right ideal for R and a left ideal for S. 0

Corollary 2.10. The number of maximal orders of H , up to isomorphism, is bounded above by the number of right ideals of R, up to isomorphism.

Definition 2.11. The norm of a right R-ideal I, denoted N(I), is the positive integer generating the ideal {Nr(x) I x E I ) Z .


Lemma 2.12. Let R and S be maximal orders in H . Then the ideals I = {x E H I Sx R) and J = {x E H I Rx S) satisfy N(I ) = N ( J ) = m and JI = m.R and IJ = mS.

Proof. From the definition of I it is clear that I is a right ideal for R and a left ideal for S , and that moreover I C R. Set 7 = {T I x E I ) . Locally at each prime 1 we have Il = Slxl = ylRl for xl and yl in H;. Then

1171 = S ~ X ~ ? & S I = N ~ ( X ~ ) S ~ and 7111 = RljjlylRl =Nr(yl)R1.

But the norm of an ideal is locally defined, so we have Nr(xl)Zl =

Nr(yl)Z1 = mZ1. Thus globally we have 17 = m S and 71 = mR. It remains only to show that J = 7. Combining the equalities

17 = m S and 11 = mR with the inclusion I C R, we obtain mR c 7. Continuing similarly, we have m I C mS, so I C S. But then 7 C = S. Moreover 7 is a left R-ideal, so RT = 7 C S, and by the definition of J we have 7 C J . By symmetry we conclude that 3 C I, hence J C 7, so the desired equality J = 7 holds. 0

Proposition 2.13. The number of right ideal classes for a maximal order of H is independent of the maximal order and is the same for left ideals.

Proof. For two maximal orders R and S, set I = {x E H I Sx C R) and J = {x E H I Rx C S) as above, and let 11, . . . , Ih be a collection of representatives for the distinct right ideal classes of R. Then II J, . . . , Ih J is a collection of distinct right ideals for S. Since JI = mR and IJ =

mS, for m = N(I), the maps Ii H IiJ and Ji JiI compose to give Ii H mIi E Ii. Thus the ideals I and J determine bijections of the ideal classes for R and S . The second statement follows by taking conjugates.

0

Let I a right ideal for a maximal order R, with left order S, then R n S is the left order of the pair (R, I ) of right R-modules. This motivates the following definition.

Definition 2.14. An Eichler order R in H is defined to be the intersection of two maximal orders in H. The level m of R is the index of R in any maximal order containing it.

Example 2.15. Returning again to the quaternion algebras of Example 2.2, we have the following examples of maximal orders and ideals.

184 D. R. Kohel

1. Let H = M2(Z) be the split quaternion algebra, and set R = M2(Z). Then every right ideal is principle, and R is conjugate to every maximal order in H.

2. Let H = Q + Q i + Q j + Q i j , wherei2 = j2 = -1 and i j+ ji = 0, and set R = Z(i, j , w), where

Then R is a right principle ideal ring, and up to isomorphism, the unique maximal order in H.

It is also known that the order Z(i, j) is a principle ideal ring, and up to isomorphism, the unique order of level 2 in H. It is not, however, an Eichler order since locally there is a unique maximal order at a ramifying prime, so the intersection of two maximal orders is always maximal at the ramifying primes. Let R = Z(x, y) where

in the previously defined quaternion algebra ramified at 37 and oo. Then there are three ideal classes, with representatives:

Set z = 1 - x + y. We note that t = x - y, and the subring Z[z] is a maximal order of discriminant -23 in a quadratic imaginary extension of Q of class number 3. The ideals 11, 12, and I3 are generated by the ideal representatives (I), (2, z ) , and (2, t ) of this subring.

53. Hecke modules

Let ?? be the infinite dimensional Hecke algebra Z[. . . , Tn, . . . ] generated by commuting indeterminates Tn , indexed by the positive integers. We define a Hecke module to be a T-module M , which is free of finite rank over Z. We set T ( N ) equal to the subalgebra of T generated by Tn for all n relatively prime to N. We say that a homomorphism Ml 4 M2 is compatible with Hecke operators Tn relatively prime to N if it is a homomorphism of T(N)-modules.

In this section we describe two constructions of Hecke modules. The first construction, due to Mestre and Oesterlh [lo], is defined in terms


of supersingular points on classical modular curves. The second is a generalization in terms of quaternion algebras. We conclude the section by recalling a map of these modules to the standard Hecke module of classical modular forms.

3.1. Hecke modules on supersingular points Let k be an algebraic closure of a finite field of characteristic p,

let E l k be a supersingular elliptic curve over k, and let C be a cyclic subgroup of order m. We denoted by E the pair (E, C), which we will call an enhanced elliptic curve. Then the endomorphism ring Endk (E) of the pair is an Eichler order of level m in the quaternion algebra ramified at p and oo.

We define S to be a set of representatives of the isomorphism classes of enhanced elliptic curves of level m over k. Then an element E of S determines a point on Xo(mp)/k, in fact a double point of the reduction to k, so we can form the divisor group

and let X be the subgroup of degree zero divisors in M. The Hecke operators act on M and X by:

for all (n, mp) = 1, where the first sum is over the cyclic isogenies cp : E - F of degree n, up to isomorphism of the image curve F .

L

For two enhanced elliptic curves E and F let Isom(E, F) be the set I of isomorphisms from E to F . We define an inner product on M by 1

L

: extending ( , ) bilinearly to M x M. The Hecke operators Tn are Her- mitian with respect to the inner product:

I

([El,Tn([FI>> = (Tn([El), [FI). I

' T h e orthogonal complement to X in M is the rank one space generated over Q by the element

of M &, Q, which we call the Ezsenstein subspace of M.

186 D. R. Kohel

3.2. Hecke modules on quaternion ideals Let El1 be the category of enhanced supersingular elliptic curves

over k, let E be a fixed object, and set R = Endk(E). Then R is an Eichler order in the definite quaternion algebra H = R @z Q ramified at p and oo. Define ModR to be the category of right locally free rank one modules over R. Then Hornk@, -) : Ell 4 ModR is a functor and determines an equivalence of categories. The previous construction of Hecke modules was functorially defined hence carries over in terms of objects and maps in ModR.

We thus present the following construction as a generalization of the previous one. For a definite quaternion algebra H of discriminant D and an Eichler order R of level m, define S to be a set of representatives for the isomorphism classes of locally free, rank one right modules over R. Define the formal divisor group M to be the free abelian group

on the basis S. As before, we set X equal to the subgroup of formal degree zero elements. We say that a nonzero homomorphism cp : I --+ J is cyclic of degree n if

We note that with this definition J/cp(I) is a principle R-module. Based on the analysis of the torsion structure of supersingular elliptic curves by Lenstra [9], when R is the endomorphism ring of an enhanced supersingular elliptic curve E, this agrees, under the equivalence of categories with ModR, with the definition of a cyclic isogeny in Ell .

The Hecke operators Tn are defined as before as the operators

for (n, Dm) = 1 where the first sum is over cyclic R-module homomorphisms cp : I 4 J of degree n, up to isomorphism of J. In practice, I and J can be embedded in H as fractional right R-ideals such that the homomorphism cp is an inclusion, which gives

and we can equivalently sum over the inclusions of cyclic submodules J 4 n-l I.

Heclce Module Structure of Quaternions 187

As in the supersingular elliptic curve construction, we define an inner

extending ( , ) bilinearly to M x M. One verifies as before that the Hecke operators Tn satisfy:

and the orthogonal complement to X in M is the rank one subgroup E generated by a constant multiple of the element

of M @z Q, defining the Eisenstein subspace of M.

3.3. Hecke modules of classical modular forms The relation between Hecke modules of quaternions and modular

forms is given by the theory of the classical Brandt matrices developed by Eichler [5]. Further aspects of the theory and computation were developed by Pizer [l ll. The present formulation follows that of Kohel [B]. In this theory, there exists a Hecke-bilinear pairing with image in the space of modular forms:

where M is the Hecke module defined relative to an Eichler order of level m and discriminant D and N = Dm. In defining the pairing Q,

/ one first defines operators

such that A,, = AnA, for relatively prime n and s, and otherwise An+ = A,A,, - rAn for primes r not dividing N. Then Q is defined by

extended bilinearly to M x M. This pairing takes M x X and X x M ; to the space of cusp forms and takes E x M and M x E into the space

of Eisenstein series. For each pair of elements I and J in the basis S, the coefficients of

@([I]), [J]) are obtained as the representation numbers of the degree map

188 D. R. Kohel

on the rank four Z-module HomR(I, J ) , a quaternary quadratic form over Z. By standard results for theta functions (see for instance Chapter IX of Schoeneberg [17]), the series @([I], [J]) lies in M2 (ro (N), Z) .

In terms of the basis S, the matrices of the operators A, are the classical Brandt matrices, B,, and the matrices (@([I], [J]),) of n-th coefficients equal 2WBn, where W = (([I], [J])) is the diagonal Gram matrix of the inner product on M. Since the representation numbers are well-defined, we obtain operators A, for all n, and in particular define Hecke operators T, = A, also for p dividing N. By the recursions for A,, for all p not dividing N we have

where we denote also by Tp the p t h Hecke operator on M2(ro(N), Z).

54. Sh' lmura curves

Shimura curves provide a generalization of modular curves on which the previous construction has a natural interpretation. For an Eichler order R in an indefinite quaternion algebra, there is an associated Shimura curve which is the moduli space of abelian surfaces with a prescribed embedding of R in the endomorphism ring. For the present purpose we define a Shimura curve as a quotient of the upper half plane and discuss arithmetic constructions on the Jacobian varieties of these curves.

4.1. Construction of Shimura curves Let BIQ be an indefinite quaternion algebra over Q of discriminant

D, and let R be an Eichler order of level m in B. We fix once and for all an isomorphism

B 8 R M2(R).

Under this isomorphism there exists a well-defined left action of B* S GL(R) on the upper half plane 'FI. We set

and define the Shimura curve to be a model for the quotient

where the bar indicates the compactification. When D = 1, then I'A(m) = ro(m) , so this generalizes the standard definition of the modular curve Xo (m). When D is greater than one, then the quotient I't (m)\'FI is already compact.


4.2. Semistable reduction and monodromy groups For the construction of analogous Hecke modules on the Jacobians of

Shimura curves, we recall some general constructions of Grothendieck [7] on abelian varieties.

Let AlQ be an abelian variety with semistable reduction at a prime p, and let k be an algebraic closure of IF,. We define a finite subgroup @ = @(A, p) as the component group of the fiber at p of the Ndron model for A, and let T = T(A, p) be the toric part of the same fiber. We define the monodromy group of A at p to be

We note that Tlk is isomorphic to a finite product of copies of Gm, and thus X(A,p) is a free abelian group.

Let AV be the dual abelian variety of A, and suppose that there exists a canonical principal polarization c : A + A". Such is the case, for instance, if A is the Jacobian of a curve. There exists a canonical bilinear pairing,

u : X(A, P) x x ( A ~ , p) -+ Z,

such that (x, y ) = u(x, [(y)) defines a symmetric positive definite pairing on X(A, p). By the following result, Theorem 11.5 of Grothendieck [7], the monodromy pairing permits the determination of the component group @(A, p) of an abelian variety A.

Theorem 4.1. There exists a natural exact sequence

; taking x E X(A,p) to u(-, ((x)).

4.3. Hecke modules on Shimura curves We now apply the previous constructions to the Jacobians of Shi-

mura curves. As for classical modular curves, the Jacobian of a Shimura curve ~ : ( m ) is naturally equipped with Hecke operators T, for all n relatively prime to the level N = Dm. As defined by correspondences on divisor groups (see 57.4 of Shimura [16]), the Hecke operators embed naturally in the endomorphism ring of the Jacobian ~ g ( r n ) . Following the exposition of Takahashi [18], we summarize here results of Ribet [13], by which we can interpret the previous constructions of Hecke modules.

Theorem 4.2. Let D be a product of an even number of primes, and let p and q be distinct primes coprime to D. Then there exists a canonical exact sequence

190 D. R. Kohel

where

The homomorphisms are compatible with the Hecke operators T, for all n relatively prime to Dpqm. With respect to the monodromy pairings on X(Af,p) and X(A, q), the map L is an isometry with its image.

Proof. This exact sequence was proved by Ribet for D = 1 in [13], and the general case holds following the work of Buzzard [2].

By means of the following theorem we may interpret the construction of Hecke modules of quaternions in terms of the monodromy groups of the Jacobians of Shimura curves.

Theorem 4.3. Let H be a positive definite quaternion algebra of discriminant Dp, and let X(Dp, m) be the Hecke module for an Eichler order R of level m. Then there exists a canonical isomorphism

The isomorphism is compatible with Hecke operators T, for all n relatively prime to Dpm, and is an isometry with respect to the respective inner products. In particular, x ( JRqT (mp) , p), x ( J? (mq) , q) , and ~ ( ~ ~ ~ ' ( r n r ) , r ) are canonically isomorphic for distinct primes p, q, and r relatively prime to D.

Proof. This is a consequence of Theorem 4.7 and Theorem 4.10 of Buzzard [2], which prove the results analogous to Deligne and Rapoport [4]. The present formulation appears in Takahashi [18]. 0

$5. Examples and computations

Let X(Dp, m) denote the Hecke module constructed for an Eichler order of level m in the definite quaternion algebra of discriminant Dp. As in Theorem 7.14 of Shimura [16], the decomposition of the Hecke modules X(Dp, m) into its Hecke-stable subspaces give isogeny factors of the Jacobian J: (m). Of particular interest are the rank one factors, corresponding to elliptic curves covered by the curve x:(rn).

We consider in this section those Hecke modules X(Dp, m), for which Dpm divides 30. Under the isomorphism of Theorem 4.3, we


identify X(Dp, m) with X (J: (mp), p). Then from Theorem 4.2 we obtain the following six exact sequences:

The Hecke modules x ( J? (m), q)) can then be identified as the kernel of the corresponding projections, once the Hecke modules X(Dp, mq) and X(Dp, m) are determined.

By means of the quaternion algebra arithmetic described in Section 2 and the Hecke module construction in Section 3, it is possible to compute the Hecke modules X(Dp, m). Implementing this arithmetic in the computer algebra system Magma [I], the author determined bases for the modules X(Dp, m), together with the representations of the Hecke algebras on these modules.

Since the curves Xo(2), Xo(3), and Xo(5) have genus zero, the corresponding Hecke modules X(2, I), X(3, I ) , and X(5 , l ) are zero. Like- wise the curves Xo (6) and Xo (lo) have genus zero, so X (2,3), X(3,2), X(2,5), and X(5,2) are zero. Each of the remaining modules:

! are nontrivial, and the Hecke-invariant subspaces are all of rank one.

1. The table below summarizes the arithmetic data. The column de-

1 noted class refers to the isogeny class of corresponding isogeny factor in , Cremona [3]; the column (v, v) gives the self inner product of a gener- 1 ator v of the rank one eigenspace over Z, and a, is the eigenvalue of i the Hecke operator T,. We also note that the eigenspace generators

need not generate X(Dp, m). In the case of X(3,lO) and X(5,6) they generate a submodule of index two.

D. R. Kohel

Invariants of X(Dp, m)

From the previous exact sequences it is possible to identify kernel submodules of these Hecke modules. The modules X(J t5 (2),3) and X (Jd5 (2), 5) are isomorphic to X(5,6) and X(3, lo), respectively. The module X (J: (5), 2) can be identified as the rank one kernel of T2 + 1 in X(3, lo), and both X (J: (5), 3) and X ( JtO (3), 5) are canonically isomorphic to X(2,15). The module X (JiO (3), 2) can be identified with the rank one kernel of T2+1 in X(5,6). While the Hecke module X(30,l) does not enter into the exact sequences, by Theorem 4.3 it is canonically isomorphic to each of X ( Jt5 (2), 2), X ( J;' (3), 3), and x (J: (5), 5), by which we can identify again the common isogeny factor 30A of Jt5(2), JiO (3), and J: (5).

It is also possible to analyze the kernel of the maps of x (J: (mp) , q) to X(J: (m), q) induced by the quotients J: (mq) = J: (m) when q divides D. First, to have a concise representation of the above exacts sequences, we express a short exact sequence of the form 0 4 M' --+

M 4 M" x M" 4 0 in the nonstandard manner

where the double arrows are the projections to the factors. Note that in this notation M' is the intersection of the kernels of the two maps, and the condition of surjectivity is stronger than surjectivity on each of the components MI'. By the naturality of these exact sequences with respect to the two projections of ~ ? ( m ~ ) to J t ( m ) , we observe that

Hecke Module Structure of Quaternions

there exist exact sequences of sequences of the form:

in which W is isomorphic to X ( JtO (3), 2) and identified in X ( JJ5 (2), 3) %

X (5,6) as the kernel of the projection to X(5,3) x X(5,3), corresponding again to the isogeny factor 30A.

56. Further considerations

The above analysis gives a breakdown of the isogeny factors of the Jacobians of Shimura and classical modular curves, but ignores more subtle details omitted by a characterization only up to isogeny. For instance, the isogeny class of elliptic curves of conductor 30 consists of one isogeny class of eight curves, denoted 30A1 - 8 in the notation of Cremona [3], and which we denote by El through E8. The curves in this isogeny class are connected by isogenies over Q of degree 2 and 3 as indicated in the diagram below.

One defines an optimal quotient of an abelian variety to be a quotient with connected kernel, which is unique in its isogeny class, up to isomorphism. For instance, the optimal quotient of Jo(30) is the curve El.

194 D. R. Kohel

Roberts [15] finds the elliptic curves Jz(5) and Jt0(3) to be E6 and E2, respectively, and the optimal quotient of Jt5(2) in the isogeny class 30A to be E3.

Other arithmetic invariants of Shimura curves may be studied in the setting of quaternions. The component group of a special fiber of the N6ron model for ~ : ( r n ) ; congruence primes, as in Ribet [12]; and the degree of a parameterization of a modular elliptic curve may also be studied in this context. Theorem 4.1 provides the means for studying component groups. Congruence primes are defined as prime divisors of the index of the subgroup generated by eigenvectors in the Hecke module; it was noted in particular that 2 is a congruence prime for X(3,lO) and X (5,6). For classical modular curves the methods for computing degree of modular parameterizations are well-developed, and Cremona [3] has compiled extensive computations. The problem for Shimura curves has been studies by Ribet and Takahashi [14] and Takahashi [18], and this work can be applied in the analysis of the quaternion Hecke modules. Finally, in cases when the level is not square-free, it may be possible to extend the understanding of modular and Shimura curves by computation or proving results for Hecke modules of quaternions.

Acknowledgement. I express thanks for the insight provided by the thesis of Shuzo Takahashi, and for conversations with Ken Ribet on this subject. Diagrams and exact sequences were prepared using Paul Taylor's diagrams package [19] in I4w.

References

[ 1 ] W. Bosma, J . Cannon, and C. Playoust. The Magma algebra system I: The user language. J. Symb. Comp., 24 (1997), no. 3-4, 235-265.

[ 2 ] K. Buzzard. Integral models of Shimura curves. Duke Math Journal, 87 (1997), no. 3, 591-612.

[ 3 ] J . E. Cremona. Algorithms for modular elliptic curves. Cambridge Uni- versity Press, 2nd edition, 1997.

[ 4 ] P. Deligne and M. Rapoport. Les sch6mas de modules de courbes elliptiques. In Lecture Notes in Mathematics, 349, Springer-Verlag, 1973, 143-316.

[ 5 ] M. Eichler. The basis problem for modular forms and the traces of the Hecke operators. In W. Kuyk, editor, Modular Functions of One Vari- able I, Lecture Notes in Mathematics, 320, Springer-Verlag, 1973, 75- 152.

[ 6 ] N. Elkies. Shimura curve computations. In J . P. Buhler, editor, Algo- rithmic Number Theory, Lecture Notes in Computer Science, 1423, Springer, 1998, 1-47.


7 ] A. Grothendieck. SGA7 I, Expos6 IX. In Lecture Notes in Mathematics, 288, Springer-Verlag, 1972, 313-523.

8 ] D. Kohel. Computing modular curves via quaternions. Manuscript, 1998. 9 ] H. W. Lenstra, Jr. Complex multiplication structure of elliptic curves.

Journal of Number Theory, 56 (1996), no. 2, 227-241. LO] J.-F. Mestre. Sur la mkthode des graphes, Exemples et applications. In

Proceedings of the international conference on class numbers and fundamental units of algebraic number fields, Nagoya University, 1986, 217-242.

[ll] A. Pizer. An algorithm for computing modular forms on r o ( N ) . Journal of Algebra, 64 (1980), 340-390.

[12] K. Ribet. Mod p Hecke operators and congruences between modular forms. Invent. Math., 71 (1983), no. 1, 193-205.

[13] K. Ribet. On modular representations of ~ a l ( a / ~ ) arising from modular forms. Inventiones Math., 100 (1990), 431-476.

[14] K. Ribet and S. Takahashi. Parameterizations of elliptic curves by Shimura curves and by classical modular curves. Proc. Natl. Acad. Sci., 94 (1997), no. 21, 11110-11114.

[15] D. Roberts. Shimura curves analogous to X o ( N ) . Ph.D. thesis, Harvard University, 1989.

[16] G. Shimura. Introduction to the Arithmetic Theory of Automorphic Forms. Princeton University Press, 1971.

171 B. Schoeneberg. Elliptic Modular Functions. Grundlehren der mathematischen Wissenschaften, 203, Springer-Verlag, 1974.

181 S. Takahashi. Degrees of parameterizations of elliptic curves by modular curves and Shimura curves. PhD thesis, University of California, Berkeley, 1998.

191 P. Taylor. Commutative diagrams in w. CTAN archive a t ftp:// ftp.tex.ac.uk/ in tex-archive/macros/generic/diagrams/taylor/, diagrams.tex, ver. 3.86, 1998.

[20] M.-F. Vign6ras. Arithme'tique des Algkbres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, 1980.

Department of Mathematics, National University of Singapore E-mail address: k o h e l h a t h . nus. edu. sg


On Classification of Semisimple Algebraic Groups

Ichiro Satake

In this note we give a survey of the classification theory of semisimple algebraic groups over a number field. As is well known, for a given field F, the F-isomorphism class of such a group G defined over F is determined up to F-isogeny by the T-diagram" CF(G) and by the F - isomorphism class of the anisotropic kernel of G (see 52; [Sal], [TI). On the other hand, if G belongs to an inner type of an F-quasisplit group Go with center Z, then the F-equivalence class of an "inner F-form" (G, f ) of Go corresponds in a one-to-one way to a cohomology class in H1 (F, Go/Z), which in turn determines an element in H2(F, Z), denoted by Y F ( G , ~ ) (see 51; [Sa21).

For F = R (the field of real numbers), it is well known that the R- isogeny class of G is uniquely determined only by the I?-diagram Cw(G) (cf. [A], [Sa3], [TI), while for a padic field F, a fundamental result of Kneser [Kl] says that the F-equivalence class of an inner F-form (G, f ) of a simply connected Go is uniquely determined only by the cohomological invariant YF(G, f ) . In treating the case of a number field,

1 the key step is in the so-called local-global principle, or Hasse principle, which also plays an important role in the class field theory. The Hasse principle for (F, Go) (Go simply connected) had been established by Kneser and Harder ([K2], [K3], [HI]) except for the case of (Es), which was recently settled by Chernousov [Cher] (1989). On the other hand, for I?-diagrams, one can deduce the Hasse principle from a result in [H2] (see $4). Combining these results, one obtains a complete picture of the classification. We can formulate the main result in the following form.

MAIN THEOREM. Let F be an algebraic number field of finite degree and let V,,I denote the set of all real places of F . Let Go be an F-quasisplit simply connected semisimple algebraic group over F and let Z be the center of Go. Suppose there are given a collection of I?-diagrams {c (~ ) (v E Vm,l)) over R and c E H2(F, 2) such that, for each v E V&J,

Received September 21, 1998. Revised November 26, 1998.

t

198 I. Satake

there exists an inner Fv -form ( ~ ( ' 1 , f ( V ) ) of GO with y R ( ~ ( V ) , f ( V ) ) = cV and & ( G ( ~ ) ) = c("). Then there exists an inner F-form (G, f ) of Go (uniquely determined up to F-equivalence) such that yF(G, f ) = c and (G, f ) is Fv-equivalent to ( ~ ( " ' 1 , f (v ) ) (hence CFw (G) = ~ ( ~ 1 ) for all V E V & . (See, 55, Th. 7, 8.)

It should be noted that this result is quite analogous to the classical result of Minkowski [Mi] (1891) on the equivalence of quadratic forms with coefficients in Q. Here we see that the F-equivalence class of (G, f ) is uniquely determined by the cohomological invariant yF(G, f ) , which is an analogue of the "Hasse invariant", and a collection of r-diagrams {C(") (V E VmI1)) (or more precisely { ( G ( ~ ) , f(v))}) satisfying the above consistency condition, which is an analogue of the "signature(s)" of a quadratic form.

The above main theorem is essentially contained in a result of Sansuc ([San], Cor.4.5), which was generalized quite recently to the case of reductive groups by Borovoi ( [Bo~] , Th.5.11). In 55 of this note, we give a direct proof of it based on the Hasse principle. An explicit determination of the relevant invariants is given in 56.

1 Cohomological invariants ([Se], [Sa2]).

Let F be a field of characteristic zero and Go an algebraic group defined over F . Let Z denote the center of Go and set Go = Go/Z. Then - Go can naturally be identified with the group of inner automorphisms of Go, Inn(Go), by the correspondence g - I, (g E Go), g and Ig denoting the class of g mod Z and the inner automorphism I, : x H

gxg-' (x E Go), respectively. By an inner F-form of Go we mean a pair (G, f ) formed of an

algebraic group G defined over F and an F-isomorphism f : G + Go such that for all a E r = ~ a l ( p / F ) one has cp, = f u o f-l E Inn(Go), - F denoting the algebraic closure of F . Two inner F-forms (G, f ) and (GI, f') are said to be F-equivalent, if there exists an F-isomorphism cp : G + G' such that f ' o cp o f -' E Inn(Go). Sometimes, G alone is called an inner F-form of Go, or G and Go are said to be in the same inner type over F, if there exists an isomorphism f : G -+ Go such that (G, f ) is an inner F-form of Go in the above sense. In that case, two isomorphisms f , f ' of G onto Go satisfying this condition are said to be F-equivalent if (G, f ) and (G, f') are F-equivalent in the above sense.

Let (G, f ) be an inner F-form of Go. Then in the above notation it is clear that (9,) is a (continuous) 1-cocycle of r in Go E Inn(Go), i.e., it satisfies the condition cpgcp, = cp,, for all a, T E r. We denote the

On Classification of Semisimple Algebraic Groups 199

cohomology class of (9,) in H1 (F, Go) by c(G, f ) , or by cF(G, f ) if F is to be specified. Writing cp, = Ig, with g, E G,~(F), one has

and it is clear that (c,,,) is a (continuous) 2-cocycle of r in 2 . The cohomology class of (c,,,) in H 2 (F, 2) is denoted by y (G, f ) or yF(G, f ) . It is clear that these cohomology classes depend only on the F-equivalence class of the inner F-form (G, f ) .

From the exact sequence

one obtain an exact sequence

By the definition one has y (G, f ) = S(c(G, f )). Note that, since Z is abelian, H1 (F, Z) and H 2 (F, Z) have a structure of abelian group, while H1(F, Go) and H1(F,Go) are just a set with a distinguished element 1.

Now, conversely, suppose there is given an element J E H1 (F, -do). Let (9,) be a 1-cocycle representing J and let cp, = I,,. Then one can define a new action of on G~(F) by

(2) ,[ul - -1 - g , x g , for X E G ~ ( F ) ,

which defines an F-form of Go, denoted by (Go),. Then, writing f for the identity map (Go), + Go, one has an inner F-form ((Go)(, f ) of Go, whose F-equivalence class depends only on the cohomology class J, and one has c((Go),, f ) = J. Thus we see that the set of F-equivalence classes of inner F-forms of Go is in one-to-one correspondence with the

I cohomology set H1(F, Go). Clearly, one has c(G, f ) = 1 if and only if f is F-equivalent to an F-isomorphism.

I The following lemma ([Se], Ch.1, 5.7) will be useful later.

Lemma 1. Let (9,) and (I),) be 1-cocycles representing J, r] E H1 (F, Goo), respectively, and set G = (Go), and G = G/(center). Then

-- (cpglI),) is a 1-cocycle of T in G(F) and, denoting its cohomology class by J-lrl, one has (for a fixed J) a bijective map

The proof is straightforward. It is clear that, if (GI, f') is an inner F-form of Go corresponding to r ] , then (G', f-lf ') is an inner F-form of

200 I. Satake

G corresponding to t-lrl. If one identifies the center of G with Z by f , then one has

W 1 r l ) = W-'W in H2 (F, 2 ) . Since the sequence (1) (for G) is exact, it follows that

From now on, we assume that Go (and hence G, GI, etc.) is a (connected) simply connected semisimple algebraic group defined over F . Let T be a maximal torus in G defined over F and let X = X(T) denote the character module of T . Then one has

X r z', I = dimT = rank G.

Let @ = @(G, T) c X be a root system of G relative to T and let A be a basis of @; we call such a pair (T, A) a "coordinate" (defined over F ) in G. Let (TI, A') be another coordinate in G. Then, as is well known, there exists cp E Inn(G) such that one has cp(T) = T', cp* (A) = A', where cp* E t(cplT)-l; for simplicity, we write

cp : (T, A) + (TI, A').

The inner automorphism cp with this property is uniquely determined up to a right multiplication by Ig with g E T; hence cp I T and cp* are uniquely determined.

Now, let r = G a l ( F 1 ~ ) . Then for every a E I? there exists $, E

Inn(G) such that : (T, A) + (Ta, AO).

We set

(4) x[aI = for all x E X,

which is well defined and gives a new action of on X leaving A invariant (as a whole). Moreover, this Galois action, called a [I?]-action (or "*- action" in [TI), is defined intrinsically, independently of the choice of coordinates (defined over F); it is also inherited to all groups in the same inner type. In fact, let (GI, f') be another F-form of Go, (TI, A') a coordinates (defined over F) in GI, and let

$(, : (TI, A') -+ (T'O, A ' O ~

On Classification of Semisimple Algebraic Groups 20 1

with $(, E Inn(G1). Then there exists an F-isomorphism cp : G + G' such that one has cp o f-' o f ' E Inn(G1) and cp : (T, A) + (TI, A'). If (GI, f') is an inner F-form of G, then from cpa o cp-' E Inn(G1), one has $(, o cp = pa o $a on T, whence follows that

i.e., cp* is a [I?]-isomorphism of X onto X' = X(T1) (and the converse is also true).

We call a coordinate (T, A) in G F-admissible if the following two conditions are satisfied.

(i) T is defined over F and contains a maximal F-split torus A in G.

(ii) Let Xo denote the annihilater of A in X. Then the basis A is "adapted to Xo" in the sense that there exists a linear order in X for which all ai E A are positive and the following condition is satisfied:

X, E X, x > 0, x -- $ 0 (mod Xo) + > 0.

Let (T, A) be an F-admissible coordinate in G and set

n denoting the projection X + X = X/Xo = X(A). Then it is known (e.g. [Sa3]) that is a (closed) subsystem of @, of which A. is a basis, and that 5 is a system of F-roots of G relative to A (which becomes a root system in a wider sense) and n is a basis of 5. The closed (semisimple) subgroup of G corresponding to Ao, denoted by G(Ao), coincides with the semisimple part of Z(A) (centralizer of A) and is called the

/ (semisimple) "anisotropic kernel" of G over F (relative to (T, A)). More- over it is known that, for cp = Ig with g E N (T) (normalizer of T), the coordinate (T, cp* (A)) is F-admissible if and only if one has g E N(A)T and that, in particular, for cp = $, one has g E Z(A)T. It follows that A. is [I?]-invariant and the [r]-orbit decomposition of A - A. is given

by

Note that, if (TI, A') is another F-admissible coordinate in G with a maximal F-split torus A' and if cp E Inn(G) and cp : (T, A) -t (TI, A'),

202 I. Satake

then one has automatically cp(A) = A' (see Lem. 2 in $4). Thus Ao- part of A is also intrinsically determined, independently of the choice of F-admissible coordinate (T, A).

As usual, the basis A is expressed by a Dynkin diagram. The system C = (A, A,, [I?]) formed of a Dynkin diagram A, a [I?]-action on A, and A. will be called a I?-diagram (or "Tits index", or "Satake diagram") of G relative to (T, A). We express a E A. by a black vertex and a E A - A, by a white vertex. As noted above, the r-diagram of G is uniquely determined up to "congruence" (in an obvious sense) only by the F-structure of G. Hence we write C = C(G) or CF(G).

One has the following "isomorphism theorem" due to Tits and independently to the author (cf. [B-TI, [TI, [Sal], [Sa3]).

Theorem 1. Let G and G' be two simply connected semisimple algebraic groups over a field F of characteristic zero. Let (T, A) and (TI, A') be F-admissible coordinates in G and GI, respectively, and let

C = (A,Ao,[ r ] ) and C1=(A,Ab,[ r ] ' )

be the corresponding I?-diagrams. Then G and G' are F-isomorphic if and only if one has a congruence cp* : C + C' and an F-isomorphism cpo : G(Ao) + G1(Ab) such that cp* I A. coincides with 9:.

In the notation of the above theorem, suppose one has an F-isomorphism cp : G + GI. Then cp* is a congruence of C onto a I?- diagram (cp*(A), cp*(Ao), cp* [r]cp*-') of GI, which in turn is congruent to C'. Hence, combining these two congruence, one obtains a congruence C + C', which we call a congruence induced by cp.

For convenience, we recall here some well-known definitions. G is called "F-split" (or of Chevalley type), if there is an F-split maximal torus T = A in G. For such a T, the coordinate (T, A) (with any basis A) is F-admissible and the corresponding I?-diagram C has the property that A. = 0 and the [r]-action is trivial. Conversely, if C = CF(G) has this property, then G is F-split. G is called "F-quasisplit" (or of Steinberg type) if one has T = Z(A), or equivalently = 0. In this case, (T, A) is F-admissible if and only if A is I?-invariant (as a whole); and of course one then has A. = 0. Conversely, if A. = 0 in CF(G), then G is F-quasisplit. It should also be noted that G is "F-anisotropic" (i.e., F-rank G = 0) if and only if one has A = A. in CF(G).

$3. Classification over a local field.

It was shown by Chevalley [Ch1,2] that for any field F (of any characteristic) and for any Dynkin diagram A there exists uniquely (up to

O n Classification of Semisimple Algebraic Groups 203

F-isomorphism) an F-split semisimple algebraic group of adjoint type defined over F (the so-called Chevalley group). When F is algebraically closed, this gives a complete classification of (simply connected) sernisim- ple algebraic group over F . It follows also that for any field F, any Dynkin diagram A, and for any action of I? on A, there exists uniquely (up to F-isomorphism) an F-quasisplit simply connected semisimple algebraic group Go defined over F with CF(G) = (A, 0,r) (the uniqueness follows from Th.1). Therefore, for the classification theory over F of characteristic zero), it is enough to fix an F-quasisplit simply connected semisimple algebraic group Go over F and to determine all inner F-forms of Go.

For F = R, one has the following theorem.

Theorem 2. Let G and G' be simply connected semisimple algebraic groups defined over R. Then G and G' are R-isomorphic if and only if the I?-diagrams Cw(G) and Cw(G1) are congruent.

This follows from Theorem 1 and from the fact that a compact (i.e., R-anisotropic) Bform G is uniquely determined (up to R-isomorphism) only by its (unmarked) Dynkin diagram A (Weyl's theorem). A direct method of classifying r-diagrams over R was given by Araki. (See [A] or [Sa3], Appendix by Sugiura. For a more general method of classifying "Tits indices", see [TI. For the classification over R, cf. also [Mu], [Boll). For the determination of the invariant y over R, see $6.

For a p-adic field F (i.e., a finite extension of Q,) the following theorem of M. Kneser is fundamental. (For a uniform proof of it, see [Br-TI).

Theorem 3 ([Kl]). Let F be a p-adic field and G a simply connected semisimple algebraic group defined over F. Then H1 (F, G) = 1.

In view of the exact sequence (I) , this implies the following

Theorem 4 ([Kl]). Let F be a p-adic field. Let Go be a simply connected semisimple algebraic group defined over F and let Z be the center of Go. Then the map (G, f ) H y(G, f ) gives rise to a bijective correspondence between the set of F-equivalence classes of inner F-forms (G, f ) of Go and H2(F, 2 ) .

In fact, it is enough to show that the map S in the sequence (1) is bijective. It is known (Lem. 4 in $4) that when F is a p-adic field S is surjective. The injectivity follows from (3) and Theorem 3.

Theorem 4 shows that over a padic field F the simply connected semisimple algebraic groups are completely classified by the F-quasisplit

204 I. Satake

group Go (i.e., by the [I?]-action on A) and the cohomological invariant y E H~(F, 2 ) . From the result of classification, one sees that over a padic field F an absolutely simple "anisotropic" F-form G occurs only for the type ( 'A~) . Consequently, the cohomological invariant y (G, f ) reduces essentially to the classical Hasse invariant of central simple algebras (cf. [Kl] , [Sa3], and 56).

$4. Scalar extensions and Hasse principles.

Let G be a simply connected semisimple algebraic group defined over a field F of characteristic zero. We use the notation introduced in

$51, 2. Let F' be an extension of F and let I?' = G~~(F'/F'), F' being an

algebraic closure of F'. Identifying F with the algebraic closure of F in -I F , we denote the restriction of o' E I?' on F by o(F.

The scalar extension F1/F gives rise in a natural manner to canonical maps (homomorphisms) between cohomology sets (groups), which make the following diagram commutative:

For instance, for < E H1 (F, c) we denote by <F' the corresponding element in H1 (F', 6). Then, in the notation of 51, for < = cF(G, f ) H1 (F, Go) one has = CFI (G, f ).

Let (T, A) be an F-admissible coordinate in G and let

be the corresponding I'-diagram. Similarly, let (TI, A') be an F1-admissible coordinate in G with

Then there exists cp E ~ n n ( ~ o ) ( F ' ) such that cp : (T,A) 4 (T1,A'); then one has automatically cp(A) c A', where A and A' are maximal F-split resp. F1-split tori contained in T and T' (see Lemma 2 below). Therefore the induced isomorphism cp* has the following properties:

(7) cp* (A) = A', cp* (Ao) > Ato, and


cp*(X[u;l) = cp*(X)[u'I for all x E X, d E r l .

Note that the map cp* : C + C' is determined intrinsically, independently of the choice of coordinates (T, A), (TI, A'). The image by cp* of a [r]-orbit in C is a union of a finite number of [I"]-orbits in C'. In particular, the irnage of a white [r]-orbit is always a union of white [I"]-orbits.

Lemma 2. The notation being as above, let (T, A) (resp. (TI, A')) be an F- (resp. F1-)admissible coordinate in G and let p E Inn(G) be such that cp : (T, A) - (TI, A'). Then, for maximal F-split resp. F1-split tori A and A' contained in T and TI, one has cp(A) c A'.

Proof. First there exists cpl E Inn(G)(F1) such that cpl(A) c A'. Then there exists 9 2 = I,, , gz E Z(cp1 (A)) (P) such that cp2cpl (T) = TI. Then one has XA c &cp;(X0). Let Al be a basis of Q adapted to both (cp;cp?)-'(X&) and Xu; then 9jip;(Al) is a basis of Q1 adapted to X&. Therefore there exist

93 E N (A) n N (T) ( P ) and g, E N (A') n N (T I ) (F')

such that, for cp3 = Igg and cp4 = I,,, one has &A = Al and cp:cpacp;Al =A1. Then one has

By the uniqueness of such a map, one has cp = ( ~ 4 ( ~ 2 ( ~ 1 9 3 on T; hence, in particular, one has cp(A) c A', q.e.d.

Now let F be a number field (i.e., a finite extension of 0) and let V = vF denote the set of all places (i.e., equivalence classes of valuations) of F, and let VmT1 = V& denote the set of all real places. For v E V we denote by F, the completion of F with respect to the place v. In the above notation, we write <, for <F,, ; similarly, when C = CF(G) we

I write C, = CFV (G). For our purpose it is important to consider the canonical map

I

I (8) s : H~(F,G) - n H~(F,,G). I vEV I

[ Since, by Theorem 3, (F, , G) is trivial except for v E V,J, the map 8 can also be written as

206 I. Satake

Then the "Hasse principle" for H1, established by Kneser [KZ], [K3], Harder [HI], and Chernousov [Cher], can be stated as follows.

Theorem 5. Let G be a simply connected semisimple algebraic group defined over a number field F . Then the canonical map 9 in (8') is bijective.

For the proof, see [P-R] (Th. 6.6); the proof for the surjectivity of 9 (due to Kneser) is relatively easy. (It seems that no uniform proof for the injectivity of 8 is yet known.) For the Galois cohomology of the center Z, one has the following

Lemma 3. (i) The canonical map

is surjective. (ii) The canonical map

is injective. (Cf. [P-R], Prop. 7.8, Cor. 2 and Lemma 6.19.)

Lemma 4. If F is a p-adic field or a number field, then the map S : H1 (F, G) + H~(F, Z) in the sequence (1) is surjective.

(Cf. [P-R], Th. 6.20.)

In order to formulate another type of Hasse principle concerning the r-diagrams, let G be a connected semisimple algebraic group defined over F. (Note that here the simply connectedness is irrelevant.) Let (T, A) be an F-admissible coordinate in G and let B = B(A) be the corresponding Bore1 subgroup of G. For a subset Al of A we denote by G(Al) the corresponding (connected) semisimple closed subgroup of G and set P (Al ) = G(Al) B. Then it is known that P (Al ) is a parabolic subgroup of G and all parabolic subgroup of G is conjugate to a subgroup of this form. We denote by P (Al ) the conjugacy class of P(Al) , which can be identified with G/P(Al); thus P (Al ) has a natural structure of a projective variety.

Now, for a E I? one has Bu = B(Au) = $uB$;l and hence


It follows that P (Al ) is r-invariant if and only if Al is [I?]-invariant. Thus, in this case, P (Al ) is a variety defined over F .

We call a parabolic subgroup P of G F-pambolic if it is defined over F . From (1 1) it can be seen that, if A1 is [I?]-invariant and contains Ao, then P (Al ) is F-parabolic. It is known that all F-parabolic subgroup of G is conjugate (with respect to an element in G(F)) to a P(Al) with Al having this property. Thus one obtains

Lemma 5 ([TI). The notation being as above, suppose that Al is [I?]-invariant. Then the variety P(A1) is defined over F. It contains an F-rational point if and only if Al contains Ao.

Now, one has the following Hasse principle due to Harder ([HZ], Satz 4.3.3).

Theorem 6. Let G be a connected semisimple algebraic group defined over a number field F. Let Al be a subset of A invariant under [I?] and let P (Al ) denote the variety (defined over F ) of parabolic subgroup of G conjugate to P(Al) . Then P (Al ) has an F-rational point if and only if it has an F,-rational point for all v E VF.

By the above observation, one can rephrase this theorem in the following form.

Theorem 6' . Let G be a connected semisimple algebraic group defined over a number field F and let C = (A, A,, [r]) and Cv =

(A, at', [I'(V)]) (v E vF) be the r - resp. I?(v)-diagmms of G over F f and F,. Then A. is the smallest [r]-invariant subset of A containing I I all AP' (v E vF). i Otherwise expressed, one has the following Hasse principle for the i I?-diagrams: a [I?]-orbit in a r-diagram C is white if and only if it de-

1 composes in Cv into a union of white [I?(v)]-orbit for all v E v F .

I / 55. Classification over a number field. I

In this section, let F be a number field. We fix a simply connected 1 semisimple algebraic group Go defined over F . (In this section, the ! assumption for Go to be F-quasisplit is irrelevant.) The main results on I the classification of inner F-forms of Go can be formulated as follows.

Theorem 7. Let (G, f ) and (GI, f') be two inner F-forms of a simply connected semisimple algebraic group Go over a number field

208 I. Satake

F . Then (G, f ) and (GI, f') are F-equivalent (i.e., there exists an F - isomorphism cp : G -t G' such that cp o f -' o f ' E Inn(Gr)) if and only if the following two conditions are satisfied.

(i) One has y(G, f ) = y(G1, f'). (ii) (G, f ) and (GI, f') are Fv-equivalent for all v E V,,l.

Proof. The "only if" part is obvious. To prove the "if" part assume that the conditions (i),(ii) are satisfied. Then, by (i) the 1-cohomology classes J = c(G, f ) and J' = c(G1, f') are in the same fiber of the map 6 : H1 (F, Go) -+ H~ (F, 2 ) . Therefore, by the formula (3) there exists q E H1(F,G) such that P(q) = By the condition (ii) one has

= EL for all v E Vw,l, which implies that P(qv) = <;l[L = 1. Hence, for each v E V,,l, by the exactness of the sequence (1) (over F,), one has a([(")) = % for some <(") E H1(FV, 2 ) . By Lemma 3, (i), there exists [ E H1 (F, 2 ) such that Cv = [(") for all v E V,,l ; then one has a([), = a(Cv) = qv. Hence by Theorem 5 (injectivity of 8 ) one has a(<) = q, whence P(q) = 1 and so J = t', q.e.d.

It is clear that the condition (ii) in Theorem 7 can also be stated in the following form:

(ii') For v E v:,, let Cv = CFv(G), C: = CFV(G1). Then for each v one has a congruence Cv -+ CL induced by an Fv-isomorphism v(V) :

G -t G' such that cp(") o f-' o f ' E Inn(G1). An "existence theorem" for inner F-forms is given as follows:

Theorem 8. Let Go be a simply connected semisimple algebraic group defined over a number field F . Suppose there are given y E H2(F, 2) and, for each v E V&, an inner Fv-foms ( ~ ( " 1 , f (v ) ) of Go such that the following consistency condition (C) is satisfied:

(c) One has yv = YF,(G(~), f(v)) for a11 v E v,,~. Then there exists uniquely (up to an F-equivalence) an inner F-form (G, f ) of Go such that y(G, f ) = y and that (G, f ) is Fv-equivalent to ( ~ ( " 1 , f (V)) for all v E V,,l .

Proof. By Lemma 4 the map 6 : H ~ ( F , G ~ ) -+ H2(F, 2) in the sequence (1) is surjective. Hence there exists an inner F-form (G, f ) of Go such that yF(G, f ) = S(cF(G, f ) ) = y. Then by the condition (C) one has y ~ , (G, f ) = y ~ , (G("), f (")) for all v E V,,l ; this means that, if one puts E = cF(G, f ) , = CF,, ( ~ ( ~ 1 , f (v)), then Ev and [("I are in the same fiber of the map 6 in the sequence (I) over Fv. Hence by the formula (3) one has ~ ( q ( ~ ) ) = <;l<(v) for some q(v) E H1 ( F ~ , G). By Theorem 5 (surjectivity of 8) there exists q E H1(F, G ) such that one


has qv = q(") for all v E V,,J. Then, putting

one has

Thus (GI, f') is an inner F-form of Go satisfying all the requirements. The uniqueness follows from Theorem 7, q.e.d.

Remark 1. As will be shown in $6, one has H2(Fv, 2) = 1 for all v E VmI1, if Go is absolutely simple, F-quasisplit and of one the types (Al) (1 even), (E6), (E8), (F4), (G2). Hence in these cases, the above consistency condition (C) is automatically satisfied.

Remark 2. If F is totally imaginary, one has (analogously to Th.4) that the map 6 : H1 (F, Go) -+ H2(F, 2) is bijective. (For a similar result in the function field case, see [H3].)

Remark 3. The list of all possible I?-diagrams ("Tits indices") C(G) over a number field F was given in [TI. From our point of view, the same result can also be obtained by Theorems 6' and 8, using the classification over local fields. For groups of exceptional type, a method

: of explicit construction of F-forms was also given by Tits (see e.g. [Sc]) .

$6. Determination of the invariant.

In this section, Go is an F-quasisplit simply connected absolutely simple algebraic group over a number field F . We give an explicit determination of H~ (F, 2). At the end, we also give a list of y(G) for all R-forms G of Go. (Note that except for the case where Go is of type (Dl) (1 even) the invariant y(G, f ) is actually independent of f ; hence we omit f .) For convenience, we treat the case of groups of type (Dl) (1 even) separately.

f I) The case where Go is F-split (except the case (IDl), 1 even). I We denote by pn the group of n-th roots of unity in F viewed as a

/ group on which r is acting. Then, in the case of F-split Go (not of type ! ('Dl), 1 even), one has

(12) 2 ~ n ,

where n is given as follows:

210 I. Satake

It follows that

where Br(F) is the Brauer group of F and Br(F), denotes the subgroup of Br(F) consisting of those elements J with Jn = 1 (see [P-R], p.73, Lem. 2.6). Therefore over the local fields Fv (v E vF) one has

For the case n even, the invariant yw(G) for all inner R-forms G of Go is given in the list at the end of the section. For classical groups, the determination of this invariant is well known. For the case Go = E7, this can be done, e.g., by using the results in [Mu], [Sa2].

11) The case where Go in not F-split (except the case (2Dl), 1 even). There are three cases

Go = 2 ~ 1 , 2 ~ 1 ( I odd 2 3), 2 ~ 6 .

In these cases, there is a quadratic extension F1/F such that Go is split over F'. Then one has

and an exact sequence

(I5) 1 + F * / ( F * ) " N ~ I / ~ ( F ' * ) + H~(F, 2) -+

where N stands for N F t / ~ (see [P-R] , p.332, (6.31)). When n is odd (i.e., Go = 2A1 (I even), 2E6 ), one has


Therefore, if v E V(F) dose not decompose in F1/F (i.e., if v has a unique extension to F', denoted again by v, F' 8 Fv = FL), one has N : Br(F:), S Br(FV), and hence

If v decomposes in F1/F (i.e., if v has two extensions w, w' in F', F'@ Fv = FL @F;,), then one has

In either case, one has H2(Fv, 2 ) = 1 for v E VmIl. When n is even (i.e., Go = 2 ~ z (I odd), 2Dl (1 odd)), one has an

exact sequence

and Hz (F', Z) " Br(F1), .

Therefore, if v does not decompose in F1/F, then one has

(15"a) H2(FV, Z) F:/NCIF, (F':) S B T ( F ~ ) ~ .

If v decomposes in F1/F, then one has

(15"b) H2(&, 2) N B T ( F ~ ) ~ .

Thus in view of Lemma 3, (ii) one has actually (instead of (15"))

For the case n even, the invariant yw(G) for all inner R-forms of Go

I , is given in the list below.

111) The case where Go is of type (Dl) (1 even) Let F' be the smallest Galois extension of F such that Go is split

over F' and let [F' : F] = m; we write Go = mD1. Then there are the following four case:

I Go = ' ~ 1 , 2 ~ 1 ( I even > 4), 3 ~ 4 , 6 ~ 4 .

When Go = 'Dl, one has

I. Satake

H2(F, 2) 2 Br(F)2 x Br(F)2,

( 1 8 4 H2(&, 2) 2 Br(FV)2 x Br(FV)2.

When Go = 2Dl, one has

(I9) = RF1/F(p2),

(204 H2(fi , 2 ) Z Br(F;)2 for v not decomp. in F1/F,

(20b) H2(FV, 2) 2 Br(FV)2 x Br(FV)2 for v decomp. in FYF

When Go = 3D4, one has

(21) = R$?,~(M),

(224 H~ (F, , 2) = 1 for v not decomp. in Ff/F,

When Go = 6D4, we take an intermediate field Fl such that F C Fl c F' and [Fl : F] = 3. Then one has

If v does not decompose in Fl/F, then one has


If v decomposes in Fl/F but does not decompose completely in F1/F, then one has

If v decomposes completely in F1/F, one has

In all cases, one has H2(Fv, 2) = 1 for v E V& except for the case where v decomposes completely in F1/F. Hence for the determination of yw(G) it is enough to consider only inner R-forms of Go = 'Dl (1 even), which is given in the list below.

In the following list, one has 1 =rank G, r = R-rank G (which equals the number of white [I?]-orbits in Cw (G)), and the type of G over R is expressed by Cartan's symbol. An element in Br(R) is expressed by the corresponding Hasse invariant 0, 112 E (1/2)2/25. As remarked above, for all Go not included in this list, one has H2(R, 2) = 1. (For a complete list of r-diagrams over local fields, see [A], [Sa3], or [TI .)

I. Satake

*I1 I -.... * (1 odd)

1 - 2

DIII (1 even)

EVII I 7 I 0

EVI

compact 1 - '2

f - - - - - ' 0

DI ( 1 - r odd)

--Y--'

1 - r ( l 3 ) or0

1 1 ( ? , O ) or (0, 2)

7 1 - 2


DIII (1 odd)

References

S. Araki, On root systems and an infinitesimal classzfication of irreducible symmetric spaces, J . of Math., Osaka City Univ., 13 (1962), 1--34.

A. Bore1 and J. Tits, Groupes re'ductifs, Publ. Math., 27, IHES, 1965. M.V. Borovoi, Galois cohomology of real reductive groups and real

forms of simple Lie algebras, Funct. Anal. i evo Prilozhenia, 22 (1988), 63-64.

, "Abelian Galois Cohomology of Reductive Groups", Mem. A.M.S., 132, No. 626, 1998.

F. Bruhat and J. Tits, Groupes alge'briques simples sur un corps local: cohomologie galoisienne, decompositions d 'Iwasawa et de Cartan, C. R. Acad. Sci., 263 23A (1966)) 867-869.

V.I. Chernousov, On the Hasse principle for groups of type Es, Dokl. Akad. Nauk SSSR, 306 (1989), 1059-1063; = Soviet Math. Dokl., 3 9 (1989), 592-596.

C. Chevalley, Sur certains groupes simples, Tohoku Math. J., 7 (1955), 14-66.

, "Classification des Groupes de Lie Alg6briquesv, Vol. 1, 2, Skm. C. Chevalley, 1956-58, E.N.S., Paris, 1958.

M. Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen uber p-adischen Korpern I, Math. Z., 88 (1965)) 40-47; 11, ibid., 8 9 (1965), 250-272.

, Hasse principle for H' of simply connected groups, in "Al- gebraic Groups and Discontinuous Subgroups", Proc. Symp. Pure Math., 9 , A.M.S., 1966, pp. 159-163.

, "Lectures on Galois Cohomology of Classical Groups", TIFR, Bombay, 1969.

G. Harder, uber die Galoiskohomologie halbeinfacher Matrizengrup- pen I, Math. Z., 9 0 (1965), 404-428; 11, ibid., 92 (1966), 396-415.

, Bericht uber neuere Resultate der Galoiskohomologie halbeinfacher Gruppen, Jahresber. Deutschen Math. Vereinig., 70 (1968), 182-216.

, ~ b e r die Galoiskohomologie halbeinfacher algebraischer Gruppen, 111, J. fur reine u. angew. Math., 274/275 (1975), 125- 138.

H. Minkowski, ~ b e r die Bedingungen, unter welchen zwei quadratische Formen mit rationalen koefizienten ineinander rational transformiert werden konnen, Crelle J. fur reine u. angew. Math., 106 (1891), 5-26; = Ges. Abh., I, pp. 219-239.

S. Murakami, Sur la classzfication des alge'bres de Lie re'elles et simples, Osaka J. Math., 2 (1965), 291-307.

V. Platonov and A. Rapinchuk, "Algebraic Groups and Number The- ory", (Russian ed., 1991) English transl. by R. Rowen, Acad. Press, 1994.

-4 - - - - - 1 - 2

I. Satake

J.-J. Sansuc, Groupe de Brauer et arithme'tique des groupes a1ge'- briques line'aires sur u n corps de nombres, J. fiir reine u. angew. Math., 327 (1981), 12-80.

I. Satake, O n the theory of reductive algebraic groups over a perfect fields, J. Math. Soc. Japan, 15 (1963), 210-235.

-, O n a certain invariants of the groups of type Es and E7, J. Math. Soc. Japan, 20 (1968), 322-335.

-, "Classification Theory of Semi-simple Algebraic Groups (with an Appendix by M. Sugiura)", Marcel Dekker, New York, 1971.

R.D. Schafer, "An Introduction to Nonassociative Algebras", Acad. Press, 1966; Dover, New York, 1995.

J.-P. Serre, "Cohomologie Galoisienne", Lect. Notes in Math., 5, Springer Verlag, 1964; 5th ed., 1994.

J. Tits, Classijication of algebraic semisimple groups, in "Algebraic Groups and Discontinuous Subgroups", Proc. Symp. Pure Math., 9, A.M.S., 1966, pp. 33-62.

Department of Mathematics Chuo University 1-1 $27 Kasuga, Bunkyoku Tokyo 112-8551


The L-Group

Bill Casselman

It is an extremely useful thing to have knowledge of the true origins of memorable discoveries . . . It is not so much that thereby history may attribute to each man his own discoveries and that others should be encouraged to earn like commenda- tion, as that the art of making discoveries should be extended by considering noteworthy examples of it.

Leibniz (from the Historia et Origo Calculi Differentialis, translated by J. M. Child)

In the late 19601s, Robert Langlands introduced a number of ideas to the theory of automorphic forms and formulated a number of conjectures which gave the theory a new focus. I was a colleague of his at this time, and a good deal of my professional energy since then has been directed to problems posed by him. Thus it was not entirely inappropri- ate that when I was invited to this conference, Miyake suggested that I say something about those long gone years. I was rather reluctant to do this, and for several reasons. The most important one is that, unlike other mathematicians who have contributed to class field theory and whose work has been discussed at this conference-such as Weber, Takagi, Hasse, or Artin-Langlands himself is still very much alive, and can very well speak for himself. Indeed, in recent years he has shown himself quite willing to discuss his work on automorphic forms in an historical context. A second reason for hesitation on my part was that although my own professional life has practically coincided with that of Langlands' principal conjectures about automorphic forms, and although I have been both a professional and a personal friend of his for that period, my own contributions to the subject have been perhaps of too technical a nature to be of sufficiently general interest to talk about at this conference. A third reason was that if I really were to tell you something new and of historical interest, I would most of all want to


218 B. Casselman The L- Group 219

be able to refer to correspondence of Langlands during the late 1960's, which has up to now been available only to a few specialists, and details of which I could hardly include in a talk of my own.

However, last summer Langlands and I began a project which caused to me think again about Miyake's suggestion. With the assistance of many other people, we have begun to collaborate in publishing Langlands' collected works on the Internet. This is in many ways an ideal form of publication for something like this. For one thing, much of Langlands' work was first if not exclusively presented in unpublished correspondence and monographs hitherto not easily accessible. My original idea was simply to scan this material electronically for presentation in crude digital format. But Langlands was more ambitious. Currently several of the staff at the Institute for Advanced Study are retyping in T)$ not only the unpublished stuff, but in addition many of the published papers and books, for free distribution in electronic format. What we have done so far is now available at the Internet site

The site itself is one of several Internet sites partially sponsored by Sun Microsystems. The original idea for these sites was to make software easily and freely available to the public, but the proposal UBC made to Sun extends the concept of software to include a wide range of mathematical material.

At the moment I write this (July, 1998), what is on line includes

A letter to And& Weil from January, 1967 0 A letter to Roger Godement from May, 1967 0 A letter to J-P. Serre from December, 1967

Euler products, originally published as a booklet by Yale Univer- sity Press 'Problems in the theory of automorphic forms', contained in volume #I70 of the Lecture Notes in Mathematics

0 'A bit of number theory', notes from a lecture given in the early 1970's at the University of Toronto

Before the end of this summer of 1998, we will probably have also, among smaller items, the book Automorphic forms on GL(2) by Jacquet and Langlands, originally volume #I14 of the Lecture Notes in Mathemat- ics; the booklet Les d6buts de la formule de trace stable, originally published by the University of Paris; and the preprint distributed by Yale University on Artin L-functions and local L-factors, which has seen only extremely limited distribution.

One of the potential problems we expected to cause us some trouble was that of copyright ownership. But I am pleased to say that so far none of the original publishers has offered any obstacle at all to our project. Considering current controversies over these matters, I would like to say that in my opinion the only copyright policy regarding research publication which makes any sense from the overall perspective of the research community is one under which control automatically reverts to an author after, say, at most three or four years.

i The ultimate format of the collection has probably not yet been found, I but at the moment each item is accompanied by a few editorial remarks

as well as comments by Langlands himself. The papers themselves can be retrieved in any of several electronic formats produced from T)$

1 files. Nor is it entirely clear-at least to me-what the final fate of the I: collection will be, but the advantage of the way in which the project t

I is being carried out is that things will be made available as soon as possible, even if the first versions might be somewhat different from the final ones. My own contribution is essentially editorial, although I and one of my colleagues at UBC are also responsible for technical matters. I would like to point out that this manner of publication is the ideal one in many situations, and that if anyone would like to know exactly what sort of technical effort it involves, I will be happy to try to answer

c quest ions.

In the rest of this paper I will recall in modern terms the principal concepts introduced by Langlands in 1967 and shortly thereafter, and recount to some extent their origins. The crucial part of the story took place in January of 1967, when Langlands composed a letter to Weil in which the essential part of his program first saw light. Up to then, Langlands' own work on automorphic forms had certainly been impressive, but that single letter, which cost Langlands a great deal of effort, amounted to a definite turning point. What I have to say in the rest of this paper might be considered a kind of guide to reading both that letter and slightly later material. I will also include some informal remarks of an historical nature, and at the end a somewhat unorthodox collection of unsolved related problems. There are, of course, a number of surveys of this material, notably a few expositions by Langlands himself and that of Bore1 at the Corvallis conference, but it seems to me that there is still much room left for more of the same.

Incidentally, the letter to Weil was the first document posted on the UBC site.

220 B. Casselman The L- Group 22 1

Roughly speaking, there were two notable features to the letter. The first was that it incorporated in the theory of automorphic forms a radical use of adde groups and, implicitly, the representation theory of local reductive groups. The second was that it introduced what is now called the L-group. It was the first which attracted a lot of attention-and even perhaps controversy and resentment-at the beginning, but in the long run this was an inevitable step. Furthermore, the incorporation of adkle groups did not originate with Langlands, although in his hands they were to be more important than they had been. But it is the second feature which was really the more significant. In the intervening years, the L-group has come to play a central role in much of the theory of automorphic forms and related fields.

$1. Automorphic forms and adQle groups

Classically the automorphic forms considered in number theory are functions on the Poincark upper half plane H satisfying certain transformation properties with respect to a congruence group r in SL2(Z), some partial differential equation involving SL2(Q)-invariant differential operators on H, and some growth conditions near cusps. They include, for example, the 'non-analytic' automorphic forms defined first by Maass, which are simply functions on r \ H and eigenfunctions for the non-Euclidean Laplacian.

Tamagawa tells me it might have been Taniyama who first noticed that one could translate classical automorphic forms to certain functions on adkle quotients. More precisely, let I' be the principal congruence subgroup of level N in SL2(Z). Choose a compact open subgroup KN of npl GL2 (Zp) with these two properties: (1) r is the inverse image of KN with respect to the natural embedding of SL2(Z) into npl GL2 (4); (2) d e t ( K ~ ) = npl Zc . A common choice is

KN = {X I X [i 01 modulo N } .

For p { N, let Kp = GL2(Zp). The product Kf = KN nplN Kp is a compact open subgroup of GL2(Aj) and I' is the inverse image of Kf in SL2(Q). Since Z is a principal ideal domain, strong approximation tells us that the natural embedding

is a bijection. Here KR = S0(2), the elements of GL$'(R) fixing i in the usual action on H. Maass' functions on I?\% may therefore be

identified with certain functions on the adkle quotient GL2(Q)\GL2(W) fixed by KaKf, and holomorphic modular forms of weight other than 0 may be identified with functions transforming in a certain way under Kw.

If g is an element of G(AI) then we can express the double coset KlgKl as a disjoint union of right cosets gi Kf , and define the action of a kind of Hecke operator on the space of all functions on GL2(Q)\GL2 (A)/ Kn Kf according to the formula

It is not difficult to see that the Hecke operators Tp and T P , on I'\H correspond to right convolution by certain functions on padic groups GL2(Qp). More precisely, after a little fussing with weights to deal with the problem that classical Hecke operators involve a left action and the adllic operators a right one, we can make the classical operators Tp and Tp,, for p t N correspond to the double cosets

For classical automorphic forms, where one already has good tools at hand and where terminology is not bad, it might not be entirely clear why this translation to an adkle quotient is a good idea, but in other situations it makes life much simpler immediately. In particular, just as it does elsewhere in number theory, it allows one to separate global questions from local ones. Of course this always makes things clearer, but in this case especially so, and in fact just as with Tate's thesis it raises questions in local analysis which might never have otherwise appeared. For example, already even for classical forms one has to tailor the definition of Hecke operators to the level of the forms involved. In the adklic scheme, this fiddling takes place in the choice of KN, and the Hecke operators themselves become entirely local operators (depending only on the prime p).

As one has known since Iwasawa and Tate showed us how to look at I- functions, although adkles are a luxury for Q they are a virtual necessity for other number fields, where problems involving units and class groups, for example, otherwise confuse local and global questions enormously. For the moment, let W be the adhle ring of F. The exercise above for GL2(Q) thus suggests the following definition: An automorphic form for a reductive group G defined over a number field F is a function on the adklic quotient G(F)\G(A) with these properties:

222 B. Casselman

I t satisfies a condition of moderate growth on the addic analogue of Siege1 sets; it is smooth at the real primes, and contained in a finite dimensional space invariant under Z(g), the centre of the universal enveloping algebra of Gw, as well as Kw, a maximal compact subgroup of G(R); it is fixed with respect to the right action of some open subgroup Kf of the finite adde group G(Af).

Hecke operators are determined through convolution by functions on

Kf \G@f ) lKf .

The conditions on Gw determine an ideal I of finite codimension in Z(g) Z(k) , that of differential operators annihiliating the form.

One of the fundamental theorems in the subject is that for a fixed I, Kf , and Kw the dimension of automorphic forms annihilated by I is finite.

The group G will be unramified outside a finite set of prime DG, that is to say arises by base extension from a smooth reductive group over oF [1/N] for some positive integer N. For primes p not dividing N , the group G/Fp will therefore arise by base extension from a smooth reductive group scheme over op. One can express compact open subgroups Kf as a product Ks npes Kp, where S is a set of primes including Dc and for p # S we have Kp = G(op). The Hecke operators for Kf will include those defined by double cosets Kp \G(Fp)/Kp for p not in S .

In one of next sections I will recall why the algebra generated by the characteristic functions of these cosets is a commutative ring, the local Hecke algebra H p , whose structure one understands well. In this section I point out only that because of commutativity together with finite- dimensionality, it makes sense-and does no harm here-to impose on an automorphic form the condition that it be an eigenfunction for all but a finite number of Hecke algebras H p .

F'rom now on, let A(G(Q)\G(A)) be the space of automorphic forms on

G(Q)\G(A).

$2. The constant term of Maass' Eisenstein series

I will illustrate the convenience of adkle groups by calculating in two ways the constant term of Maass' Eisenstein series. In addition to illustrating adklic techniques, the calculation will be required later on.

The L - Group 223

Let = SL2(Z). For any complex number s with REAL(S) > 112 and any z = x + iy in H define

The point of this series is that for

we have

I M A G ( ~ ( Z ) ) = IMAG IMAG (z)

so that we actually looking at

where I?, is the stabilizer of im in r. It is generated by integral trans- lations and the scalar matrices f I, so the function IMAG(()Z) is r p - invariant. The series converges and defines a real analytic function on r \H invariant under r such that

AE, = (s2 - 1/4)E,,

where A is the non-Euclidean Laplacian. Simple spectral analysis will show that for REAL(S) > 112 the function E, is the unique eigenfunction of A on r \H asymptotic to yS+1/2 at m in the weak sense that the difference is square-integrable. A little more work will then show that it continues meromorphically in s and is asymptotic to a function of the form

y'/2+s + , ( , ) p s

as y -+ m for generic s, in the strong sense that the difference between E, and this asymptotic term is rapidly decreasing in y. (My current favourite explanation of the general theory is the lucid article by Jacquet at the Edinburgh conference, but of course in the particular case at hand one can follow the more elementary technique of Maass.) Of course the coefficient c(s) is a meromorphic function of s. It is easy to deduce that Es must therefore also satisfy the functional equation

224 B. Casselman

which implies that c(s) satisfies its own functional equation

It turns out also that y1 /2+s + C(S) y1 /2 - s is the constant term in the Fourier series of Es at oo, which is to say that

In this section we will calculate c(s) explicitly in both classical and adklic terms, to get a feel for the way things go in each case.

0 The classical calculation

The constant term of Es is

where

The L- Group 225

Here cp(c) is the number of integers mod c relatively prime to c. The terms in the sum are 'multiplicative' in the arithmetic sense, so the sum is also equal to

!

so that

E where t

t

1 0 The adklic calculation i'

Let G = SL2, and continue to let I' = SL2 (Z). For each finite prime p let Kp = G(Zp), and let Kf = n K,.

By strong approximation the natural inclusion of G(R) into G(A) induces a bijection

r \z c--+ G(Q) \G(A)/KwKf

or in other words G(A) = G(Q)G(R) Kf . To a function f (9) on r\G(R) therefore corresponds a unique function F on G(Q)\G(A) defined by the formula

226 B. Casselman The L- Group

if 90 lies in G(Q), g~ in G(R), gf in Kf . Let I, be the function on the adkle quotient corresponding to E, . Before I do this, let me explain simple generalizations of the classical Eisenstein series and constant term. Let P be the subgroup of G = SL2 of upper triangular matrices, N its unipotent radical. Let cp be a function on rpN(R)\G(R) which is a finite sum of eigenfunctions with respect to SO(2). Suppose also that cp satisfies the equation

where

is the modulus character of the group P. Then the series

will converge to an automorphic form on I'\G(R) if REAL(S) > 112, and continue meromorphically in s. If cp is invariant on the right by SO(2) it will be, up to a scalar multiple, the Eisenstein series E,.

If Q> is an automorphic form on r \G(R), define its constant term to be the function

r

on rp N(R)\G(R), where N(Z)\ N(R) is assigned measure 1

Thus if we apply the constant term to an Eisenstein series we get a map from certain functions on r p N (R)\G(R) to other functions on the same space. Rather than analyze this in detail, I will now explain what happens for adkle groups.

Because Q has class number one

Also because G = PK locally we have

and hence G(A) = P(A)N(A)A(R)Kf

where A is the group of diagonal matrices in G, or equivalently

Let Sp be the modulus character of P(A), taking h to the product of all the local factors bP(hp). Let p, be the unique function on P(Q)N(A)\G(A)/KwKf such that

cps(pg) = S;+1'2(~)cps(g) and 4 1 ) = 1.

0 The function E, is the rnerornorphic continuation of the series

If @ is an automorphic form on G(Q)\G(A), its constant term is defined to be the function

on P(Q)N (A)\G(A). This is compatible with the classical one in that this diagram is commutative (A denotes automorphic forms):

Therefore we calculate the constant term of E, to be the function

The point now is that we can apply the Bruhat decomposition

where

We can therefore express the constant term as

228 B. Casselman

The integral over N(A) is just the product of integrals over all the local groups N (Qp). We must therefore calculate the integrals

with ~ , , ~ ( h k ) = 6;+'l2(h) ( h E P(Qp)) .

In both real and p-adic cases we start with

We now must factor this as hk, In all cases we rely on the transitivity of the action of K on P' (Qp) . The group P is the stabilizer of the image in IP1 of image of the row vector [0 11, so in order to factor w-'n = pk we must find Ic in Kp taking [0 11 to [-1 -XI.

a The p-adic case

Let K = G(Zp) . If x lies in Zp then w-'n lies also in K , and there is nothing to be done. Else 1x1 = p-" with n > 0 and l / x lies in Zp. The row vector (-1 - X I is projectively equivalent to [x-' 11 so we may let

which gives us

For every integer n let

Calculate

a The real group

Here we normalize [- 1

where

Then

The L - Group

- X I to [(x2 + 1)-lI2 x(x2 + I ) - ' / ~ ] and let

0 -(x2 + 1)-'I2 x(x2 + I ) - ' /~ -(x2 + 1)lI2

The integral is therefore

These local calculations lead to exactly the same formula for c(s) as before, of course, but it seems fair to claim that we understand better why it has an Euler product.

$3. T h e Satake isomorphism

In this section and the next two I shall explain how the L-group is constructed. Suppose briefly that F is an algebraic number field, A its adkle ring. Recall that if cp is an automorphic form on G(F)\G(A) then cp is fixed by almost all the local compact groups G(Fp) . We know also from Hecke's analysis of classical automorphic forms that it's important to understand how certain p-adic Hecke operators act on 9.

For almost all primes p, the local group G(Fp ) is unramified in the sense that G splits over an unramified extension of Fp, which also means that G can be obtained by base extension from a smooth reductive group scheme over o = o p , the ring of integers of F. The Hecke algebra H = H(G(Fp) , G(o)) is defined to be the algebra of measures of compact support of G both right- and left-invariant under the maximal compact subgroup K = G(o), with convolution as multiplication. It is of course generated by the measures constant on single double cosets with respect

230 B. Casselman The L-Group 23 1

to K . Note that we can identify such measures with right K-invariant functions if we are given a Haar measure on G.

From now on in this section, let F be a p-adic field.

We are interested in homomorphisms of the Hecke algebra H(G(F) , G(o)) into C, and more generally in the structure of this algebra.

There is one simple way to obtain such homomorphisms. Let B be a Borel subgroup obtained by base extension from a Borel subgroup of G(o). Let S = SB be the modulus character of B, taking b to I detb(b) 1. We have an Iwasawa decomposition G = BK. Therefore, if x is a character of B trivial on B n K = B(o) (which is to say an unramified character of B), there is a unique function cp = cp, on G such that

cp,(bk) = ~ ( b ) 6 l / ~ ( b )

for all b in B, k in K . Up to a scalar multiple, it is unique with the

Right convolution by elements of the Hecke algebra H preserves this property, hence elements of H act simply as multiplication by scalars, and therefore from each x we obtain a homomorphism @, from H to @.

The normalizing factor S1l2 is there for several reasons, but among others to make notation easier in the result I am about to mention.

Suppose T to be a maximal torus in B , and w to be an element of K in the associated Weyl group W, and N the unipotent radical of B. If x satisfies some simple inequalities then the integral

will converge and satisfy the condition

The operator cp, H r,cp, also commutes with the Hecke algebra. The function rWpx will therefore be (generically non-zero) multiple of cp,,. As a consequence, the homomorphisms @, and @,, are the same.

This means that the map x H @, induces one from the W-orbits of unramified characters of T to a set of homomorphisms from the Hecke algebra H to @.

There is another way to set this up. Let T be a maximal torus in B and A a maximal split torus in T. The injection of A into T induces

an isomorphism of free groups A = A(F)/A(o) with T(F)/T(o). Re- striction from B to A therefore induces an isomorphism of the group of unramified characters of B (or T) with those of A. Let R be the group ring @[A] of A. Because G = B K, the R-module of all K-invariant

functions on G with values in R such that f (ntg) = t~5:'~(t) f (9) is free of rank one over R . Convolution by elements of the Hecke algebra are R-homomorphisms of this module, and therefore we have a ring homomorphism @ from H to R. Any unramified character x of T gives rise to a ring homomorphism from R to @, and the composition of this with @ will be the same as @,. The W-invariance we saw before now implies that the image of @ lies in R W . The following result is due in special cases to different people, but put in essentially definitive form by Satake:

Theorem. The canonical map constructed above from the Hecke algebra H to @ [ A ] ~ is a ring isomorphism.

In other words, all homomorphisms from H to (I: are of the form @, for some X. The point of Satake's proof is injectivity.

For example, let G be GL2(Qp), A the group of diagonal matrices in G. Suppose the character x takes the matrix a1 with diagonal (p, 1) to a1 and the matrix a 2 with diagonal (1, p) to a2. The ring @[A] is generated by the images of wl +a2 and (wlw2)* '. The Hecke operator Tp acts on cp, through multiplication by (al + a2) , and TP,p by a1 a2.

94. The dual group I. The split case

Suppose we are given a classical automorphic form of weight k for the congruence group I?, say an eigenform for the Hecke operator Tp with eigenvalue cp. Then by Deligne's version of the Weil conjectures c =

(ap+bp) with lap[ = Ibp( = p(k-1)/2, and also apbp = p"'. The diagonal matrix

where a, = ap/p(k-1)/2, pp = bp/p(k-1)/2, is therefore unitary. Of course the pair (ap, pp) is determined only up to a permutation. It was remarked by Sato and Tate in a slightly different context that the statis- tical distribution of the numbers cp as p varied seemed to be according to the SU(2)-invariant measure on the conjugacy classes determined by the pairs (ap, pp). This suggests that, more generally, an eigenfunction with respect to Hecke operators ought to be thought of as determining a conjugacy class in a complex group.


The simplest version of Langlands' construction of his dual group does exactly this. But there is a slight twist in the story.

Let G be any split reductive group defined over a p-adic field F. As before, let T B be a maximal split torus contained in the Borel subgroup B, and let W be the Weyl group of this pair (G, T). An eigenfunction with respect to the Hecke algebra of G(F) with respect to the maximal compact subgroup G(o), according to Satake's theorem, determines an element in the W-orbit of Hom(T (F) /T (0) , C ) . Following the suggestion of Sato-Take, we want to interpret this first as a W-orbit in a complex torus, then as a conjugacy class in some reductive group containing that torus.

Let ? be the torus we are looking for. We first pose an identification

which means that points on the torus T^ are the same as unramified characters of T(F) . Second, we fix a map

which identifies T(F)/T(o) with the lattice X,(T) of coweights of T . This map is characterized by the formula

for all t in T , x in X*(T). Equivalently, if p is a coweight of T then it is the image of p(w-') if a is generator of p. This allows to make the identification

?(c) = Hom(X, (T) , C )

Now if S is any complex torus then we have a canonical identification

since the coupling S(C) x X*(S) ---t C X

A

is certainly nondegenerate. If we set S = T we get an identification

which leads us also to pose

In other words, in some sense the tori T and ? must be dual to each another. In any event, this is a natural way to construct tori, since from almost any standpoint a torus is completely determined by its lattice of characters.

In summary:

0 Points on the torus F(c) may be identified with unramified complex characters o f T (F) .

0 Elements of T(F)/T(o) may be identified with rational characters o f F .

This kind of duality can be extended to one of reductive groups. Let C X* (T) be the set of roots of g with respect to T, and let CV be the associated set of coroots in X, (T) . The quadruple (X* (T), C, X, (T), CV) all together make up the root data of the pair (G, T). If, conversely, one is given a quadruple (L*, S*, L,, S,) where L, is a free abelian group of finite rank, L* is the dual of L, , S* is a root system in L* and S, a compatible coroot system in L, then we can find a reductive group defined and split over any field with these as associated root data. It is ~ unique up to inner automorphism.

i

I In our case, given the root data (X*(T), C, X,(T), CV) we get another set of root data by duality, namely the quadruple

(X*(T), CV,X*(T), C) = (x*(F), CV, x,(F), C).

Let be the reductive group defined over C associated to these data. For example, if G is simply connected and of type Cn then e is adjoint and of type B,. It is this involution of types that is at first a bit puzzling.

If we are given a system of positive roots in G, then the corresponding A

coroots determine also a positive system of roots in G, or in other words a Borel subgroup.

Here is Langlands' version of the Satake isomorphism in these circumstances:

Theorem. For a split group G over a p-adic field there is natural bijection between ring homomorphisms from the Hecke algebra to C and

A

W-orbits in ?(@) or equivalently semi-simple G(C)-conjugacy classes in F(C).

Or in yet another form:

Theorem. For a split group G over a p-adic field there is a natural ring isomorphism of the Heclce algebra H with the representation ring ~ e ~ ( d ) .

I


Example. For G = GL, this is all straightforward. An unramified character of the torus of diagonal elements is of the form

if [xi[ = qmi and ti = qs i . In fact the character is determined by the array of complex numbers t = (tl, . . . , t,). Permuted arrays will give rise to the same Hecke algebra homomorphism. But this means precisely that what really matters is the conjugacy class of the matrix

In GL,(C). This is compatible with what we have said just above, because the dual group of GL, is again just GL,.

Example. For SL, the dual group is PGL,(@). The torus ? is the quotient of the diagonal matrices by the scalars. This can be identified with the group of complex characters of the group of diagonal matrices in SL, simply by restriction of the corresponding identification for GL,. In particular, for n = 2 the diagonal matrix

corresponds to the character

Let me explain briefly what the dual group means for automorphic forms. Suppose now that G is a split reductive group defined over a number field F . Let cp be an automorphic form on G(F)\G(A) which is an eigenfunction for the Hecke algebra Hp for p not in a finite set of primes S. This means that for every f in a local Hecke algebra Hp there exists a constant cf such that Rf, = cfcp Then for each p not in S there

n

exists a unique conjugacy class i P p in G with the property that

trace, (Qp) = cf

whenever f is an element of the Hecke algebra Hp and .rr = '/rf is the n

corresponding virtual representation of G.

The connection between homomorphisms of the local Hecke algebra and conjugacy classes in 8 is rather straightforward. It may have been noticed by several mathematicians before Langlands called attention to it, but I can find no record of the observation. It is quite likely that, if it had been observed, it simply wasn't felt to be of great enough importance to be worth making explicit. In Langlands' hands, however, the dual group was to serve as an uncanny guide to understanding an enormously wide range of phenomena involving automorphic forms.

$5. The dual group 11. The unramified case

The first strong hint that dual group had nearly magical properties arose in Langlands' construction of the analogue of the dual group for arbitrary unramified p-adic groups. This, too, can be found in the original letter to Weil.

Now suppose only that G is an unramified reductive group defined over the p-adic field F. I recall that this means it is determined by base extension from a smooth reductive group scheme over op.

Let B be a Borel subgroup and T a maximal torus in B , containing a maximal split torus A. Let W be the restricted Weyl group. Satake7s theorem asserts that homomorphisms from the Hecke algebra Hp to C correspond naturally to W-orbits in A^(c) = Hom(A(F) /A(o), C ). The injection A -+ T gives us also an injection X,(A) -+ X,(T), hence a dual surjection

?(C) - X(C).

In these circumstances, when A # T it is not at all obvious how conjugacy classes in the dual group relate to W-orbits in A . It has always

I seemed to me that explaining this, although simple enough once seen, was one of Langlands' least obvious and most brilliant ideas. The trick is to incorporate the Galois group in the definition of the dual group.

The group G will split over an unramified extension E / F . let G be the Galois group of E/F, Frob the Frobenius automorphism. Because G contains a Borel subgroup defined over F, the Galois group permutes the positive roots of G over E, and this gives rise to a homomorphism from G to the automorphism group of 8. Langlands defined the full L- group L ~ E I F to be the semi-direct product 8 x 9. Here is his remarkable observation:

:i 4


Langlands' Lemma. Semi-simple e(@)-conjugacy A classes in 8(@) x

Frob correspond naturally to W-orbits in A.

If g lies in e then

g(go, ~rob)g- ' = (ggog-FrOb, Frob)

so that e-conjugacy in e x Frob is the same as twisted conjugacy. On the other hand: (1)

Frob = gOlgOgFrob 90

and (2) if a = Frobn then

Frob(go, ~ r o b ) ~ r o b - ' = (~robgo~rob- ' , Frob) = ( g P b , rob).

Therefore 6conjugacy in x Frob is the same as LG-conjugacy.

I outline here explicitly how the correspondence goes. First of all, every semi-simple E(c)-conjugacy class in E(c) x Frob contains of the form t x Frob with t in ?(c). Second, the W-orbit of image of t in A (@) depends only on the original conjugacy class. This at least gives us a map from these conjugacy classes to W-orbits in A^(@). Finally, this map is a bijection.

The simplest published proof of this Lemma can be found in Borel's Cor- vallis lecture. Like Langlands' original proof, it relies upon an old paper of Gantmacher's for a crucial point, but Kottwitz has pointed out to me that this point follows easily from a well known result of Steinberg's. This is explained briefly in a paper by Kottwitz and Shelstad, and I will sketch here without details a proof which combines the arguments of Borel and Kot twitz-Shelstad.

The restricted Weyl group is defined to be the image in Aut(A) of subgroup of the full Weyl group of the pair (G, T) which takes A into itself. In 56.1 of Borel's lecture it is shown that in the dual group the elements of W can be characterized as those elements of N&/? commuting with the F'robenius. Furthermore, 56.2 of Borel shows that every element of W can be represented by an element of Ni;(?)

commuting with the F'robenius. For s in ?(c), conjugation of t x Frob by s is equal to t(s/sFrob) x Frob. The kernel of the projection from ? to A is that spanned by elements s/sFrob. From this it is easy to see (56.4 of Borel) that the projection from ? to A induces a bijection of (?(@) x rob)/^^ (?) with A (@) / W. Every semi-simple conjugacy

class in 8 ( @ ) x Frob contains an element t x Frob with t in ?. This is

where Borel and Langlands quote Gantmacher, but I present here the argument of Kottwitz and Shelstad.

Let E be a Borel subgroup fixed by Frob containing ?. Given a semisimple element x x Frob in e(@), we want to find g in C(@) such that

g(x x ~rob)g- ' = gxg-FrOb x Frob = t x Frob

with t in ?(@) . Equivalently, we want to find g with the property that if we set

t = gxg-Fr0b

Let H = 6 or ?. Then tHt-' = H means that

or equivalently

since HFrob = H. In other words, since all Borel subgroups and tori are conjugate in e ( @ ) we are looking for a group H, fixed under conjugation by x x Frob. But a well known result of Steinberg guarantees that we can find some pair (B, , T, ) fixed by conjugation under x x Frob, so we are finished.

The map induced by inclusion from (?(@) x ~ r o b ) / N ~ ( ? ) into the

e-classes in e x Frob is an injection. This is proven in 56.5 of Borel (but note that there are quite a few simple typographical errors there).

This Lemma has as immediate consequence:

Theorem. There is a natural bijection between homomorphisms from the Hecke algebra H into @ and semi-simple E(c) or LG(@)-conjugacy classes in E(c) x Frob.

Example. The unramified special unitary group SU3.

Let F, be an unramified quadratic extension of F. Let

238 B. Casselman The L - Group

and let G be the unitary group of 3 x 3 matrices X with coefficients in F, corresponding to the Hermitian matrix w. This is an algebraic group over F-if R is any ring containing F then G(R) is made up of the matrices with coefficients in F, @ F R such that

where x H i? comes from conjugation in F,.

The group G(F) contains the torus T of diagonal matrices

with y in F,X.

Over F, this group becomes isomorphic to SL3, so e is PGL3(@). Let D = (1,a) be the Galois group of F,/F. The torus ? dual to T is the quotient of the group of complex matrices

by scalar matrices. The element a acts on e through the automorphism

X H U) t ~ - l ~ - l .

and, more explicitly, it acts on T by taking

tl 0 0 t,l 0 0

[ o O O t t2 0 3 ] - [ ; a;1 $1 The map

? H ~ ~ ( T ( E ~ ) / T ( ~ E , ) , CX )

takes the element diag(ti) to the character

The group A is generated by the element

and the map from ? to A takes diag(ti) to the character

If x is any unramified character of T, or equivalently of A, the element iX can thus be chosen as any element of !? such that tl/t3 = x(aV(w-l)) .

We shall need to know a bit more about the action of the Galois group on e. Let xi,j for i < j be the matrix with a single non-zero entry 1 in location (i, j ) . These form a basis for the Lie algebra ii. Then a interchanges xl,, and x2,3 and takes x1,3 to -21,s. This concludes my discussion of SU3.

Globally, an automorphic form is unramified at all but a finite number of primes. At an unramified prime p it gives rise to a homomorphism from the Hecke algebra Hp into @. It therefore also corresponds to a semi-simple G'(c)-conjugacy class Bp in e(@) x Frobp for all but a finite set of p. It is tempting to call this class the F'robenius class of the form at p, and I shall not resist the temptation.

This construction depends very weakly on the choice of splitting extension E/F, and one has a local L-group for every possible choice. In some ways the canonical choice is to let E be the maximal unramified extension of F .

One can also define an L-group attached to a global field F to be a semi-direct product

L~ = E x G(F/F)

and then one has also various embeddings of the local groups into this corresponding to local embeddings of Galois groups. Other variants of the L-group are also possible, with the Galois groups replaced by Weil groups or Weil-Deligne group. It was at any rate the introduction of Galois groups into the L-group which turned out to be incredibly fruitful. Incidentally, note that the definition of the extended L-group given in this section is compatible with that in the previous one, since when G is split the Galois group acts trivially on c.

240 B. Casselman The L- Group 24 1

Langlands himself has told me that the subtle point in his definition of the L-group was not the introduction of the Galois group, which he claims was more or less obviously necessary. Instead, he says, the point about which he worried was that the L-group should be a semi-direct product of D and rather than some non-trivial extension. I suppose he had the Weil group-a highly non-trivial extension--on the periphery of his mind. By now there is no doubt that his definition is correct, but it would be an interesting exercise to put together a simple argument to this effect.

56. The dual group 111. Why is the L-group important?

There were two questions which were answered, at least conjecturally, as soon as the L-group was defined.

How do we attach L-functions to automorphic forms?

In 1967 there had been already a long history of how to associate L- functions to automorphic forms in very special circumstances, but there was no systematic way to do this. In some cases it was not at all clear which was best among several choices. In terms of the L-group there was a natural guess. Suppose that cp is an automorphic form for a reductive group G defined over a number field F. Let <Pp be a representative of the corresponding Frobenius class of the L -group for p outside some finite set S of primes. Then for each irreducible finite-dimensional representation p of the L-group we define

L(s,p,cp) = n d e t ( I - %)-I.

~ 4 . 9

Of course there are a finite number of factors missing for primes in S, but these will not affect analytic properties seriously. Of course one conjectures this L-function to have all sorts of nice analytic properties- meromorphic continuation, functional equation, etc. The new feature here is the parameter p, and implicit in the construction of these functions was that p should play a role here analogous to that played by representations of the Galois group in the context of Artin's conjecture. This conjecture was made somewhat more reasonable, at least in Lang- lands' own mind and in lectures he gave very shortly after he wrote the letter to Weil, when he showed that the theory of Eisenstein series provided some weak evidence for it. It turned out that the constant term of series associated to cusp forms on maximal parabolic subgroups determined a new class of L-function of Langlands' form for which one could

at least prove analytic continuation. This was explained in Langlands' Yale notes in Euler products, and I shall say something about it further on. Later and more striking evidence that the L-functions suggested by Langlands were the natural ones was provided by several investigations which showed that the Hasse-Weil C functions of Shimura varieties were of Langlands' type. A result of this kind had been shown first by Eichler for classical modular varieties and later on by Shimura for more sophisti- cated modular varieties, but of course the relationship with the L-group was disguised there. What was really striking was that Deligne's formulation of Shimura's results on modular varieties and their canonical fields of definition fitted naturally with Langlands' L-group. This was first pointed out in Langlands' informal paper on Shimura varieties in the Canadian Journal of Mathematics, recently reprinted.

How are automorphic forms o n diflerent group related ?

There were many classical results, culminating in work of Eichler and Shimizu, that exhibited a strong relationship between automorphic forms for quaternion division algebras over Q and ones on GL2(Q). To Lang- lands this appeared as a special case of a remarkable principle he called functoriality. The functoriality principle conjectured that if G1 and G2 were two rational reductive groups, then whenever one had a group homomorphism from LG1 and LG2 compatible with projections onto the Galois group, one could expect a strong relationship between automorphic forms for the two groups.

The underlying idea here is perhaps even more remarkable. We know that an automorphic form gives rise to Frobenius classes in local L- groups for all but a finite number of primes. We know that L-functions can be attached to automorphic forms in terms of these classes. The functoriality principle asserts that the automorphic form is in some sense very strongly determined by those classes induced by L-group homomorphisms. Evidence for this idea was the theorem of Jacquet- Langlands in their book on GL2 which extended the work of Eichler- Shimizu to arbitrary global fields. This theorem was the first of many to come suggested by the functoriality principle, and its proof was the first and simplest of many in which the trace formula was combined with difficult local analysis.

The functoriality principle was especially interesting when the group G was trivial! In this case the L-group is just its Galois group component, and the functoriality principle asserts that finite dimensional representations of the Galois group should give rise to automorphic forms. This is because an n-dimensional representation of the Galois group amounts to


a homomorphism from the trivial L-group into that for GL,. Even more remarkable was the eventual proof by Langlands of certain non-trivial cases of Artin's conjecture, applying techniques from representation theory and automorphic forms. This was also strong evidence of the validity of the functoriality principle.

57. How much of this was in the letter to Weil?

Essentially all of it! At least the results. The proofs were crude or barely sketched, but better was perhaps not possible in view of incomplete technology. For example, even the work of Bruhat-Tits on the structure of local p-adic groups was not yet in definitive form. I have always found it astonishing that Langlands introduced the L-group full-grown right from the start. The scope and audacity of the conjectures in his letter to Weil were nearly incredible, especially because at that time the details of various technical things he needed hadn't been quite nailed down yet. The first time reasonably complete account appeared in Langlands' lecture in 1970 at a conference in Washington, the written version in the conference proceedings in the Springer Lecture Notes #170. It is instruc- tive for the timid among us to compare this account with the original letter, and with Godement's account in the Skminaire Bourbaki.

$8. Where does representat ion theory enter?

So far I haven't made any explicit reference to the representation theory of local reductive groups. I haven't needed it to formulate results, but without it the whole subject is practically incoherent. It already appears at least implicitly in the classical theory of automorphic forms, where one always knew that different automorphic forms were only trivially different from others. In current terminology this is because they occurred in the same local representation spaces. One place where local representation theory explains what is really going on is in the treatment of unramified automorphic forms above, where homomorphisms of Hecke algebras were attached to unramified characters X. Many things look rather bizarre unless we realize that we are looking there at the subspace of G(Op )-fixed vectors in the representation of G(Fp ) induced by x from the Bore1 subgroup. In fact, we really defining a class of local L-functions L(s, p, .rr) where now .rr is an unramified representation of a p-adic group. Satake's isomorphism asserts that there is a natural bijection between certain e ( ~ ) - c o n j u ~ a c ~ classes in local L-groups LG and irreducible unramified representations of the local group G(F). We can reformulate this result by saying that, given on G the structure of

a smooth reductive group over OF there is a natural bijection between irreducible unramified representations and splittings of a sequence

1 + E + L~ +< Frob >+ 1.

(using a suitable variant of L ~ ) . This is a special case of a local functoriality principle, which conjectures a strong relationship between homomorphisms from a local Galois group into LG and irreducible representations of the local group G. We know that at least for unramified representations T of G we have a whole family of L-functions L(s, p, T )

which vary with the finite dimensional representation p of e . This leads us to ask more generally how we might associate L-functions to representations other than the unramified ones. We know from Tate's thesis in the case of G, = GL1 that we should expect not only an L-function but in addition a local root number ~ ( s , .rr, p, +) as well which depends on a choice of local additive character + of the field. This idea was worked out in detail by Jacquet and Langlands for the case of GL2 through the theory of Whittaker models, and a bit later by Godement and Jacquet for all groups GL,, following Tate and Weil for division algebras. There are in fact several ways to attach both L-functions and root numbers to representations of a group G defined over a local field Fp, but the most natural and intriguing idea is this, which extends local class field theory in a remarkable way:

To each representation of a local reductive group G we should be able to associate a homomorphism from the Galois group or some variant (such as the Weil-Deligne group) into LG which i s compatible with the canonical projection from LG onto the Galois group. If p is a finite dimensional representation of L~ we can then expect the corresponding L-function and root number to be that obtained by Artin, Hasse, Dwork, and Langlands from the representation of the Galois group we get by composition.

There are a few mild but important modifications of this idea necessary, for example, to deal with certain poorly behaved 1-adic Galois representations, but although evidence for it is still somewhat indirect it seems very likely to be true. In particular if G = GL, we should expect irreducible cuspidal representations of G to correspond bijectively and naturally to irreducible n-dimensional representations of the Ga-

6 lois group. In my opinion the strongest evidence for the conjecture here r< r, * comes from work of Deligne, Langlands, and Carayol on the reduction of classical modular varieties in bad characteristic. Here G = GL2. Other convincing evidence comes from the remarkable results of Kazhdan and

I

244 B. Casselman The L - Group 245

Lusztig dealing with the best of the poorly behaved cases, covered by the Deligne-Langlands conjecture.

$9. Weil's reaction

Weil's first reaction to the letter Langlands had written to him was perhaps not quite what Langlands had hoped for. Langlands had written the letter by hand, and Weil apparently decided that the handwriting was unreadable! You can form your own opinion on this question, because at the UBC Sun SITE we have posted a copy of the hand-written letter in digital format (Weil's very own copy of the original was scanned by Mark Goresky in Princeton). At any rate, Langlands then sent to Weil a typed version. Copies of this were distributed to several mathematicians over the next few years, and this is how Langlands' idea became well known.

I do not believe that Weil ever made a written reply, but after all he worked only across Princeton from Langlands. Nonetheless, I think it is reasonable to guess that his first serious reaction was confusion. In spite of the fact that Weil had been one of the founders of the theory of algebraic groups, he may not have been familiar with the general theory of root systems, and this alone may have caused him technical difficulty. It also seems that although he had a hand in introducing representations into the theory of automorphic forms through his papers on Siegel's formulas, he was unfamiliar with the representation theory of Gelfand and Harish-Chandra, which was a major part of Langlands' own background. He says himself of his reaction to Langlands' letter (Collected Papers 111, page 45)

. . . pendant longtemps je n'y compms rien . . . At the time he received the letter, he was concerned with extending his 'converse theorem', which asserted that if an L-function and sufficiently many twists satisfied a certain type of functional equation, arose from a classical cusp form. He wanted to generalize this to automorphic forms for the groups GL2 associated to number fields other than Q, and troubles he was having with complex archimedean primes were almost immediately cleared up by Langlands' idea about the relation between representation theory and Galois representations. Also, after a while he worked out the case of GL, in some detail and gave a talk on it at Oberwolfach. In this case, as we have seen, the L-group is just GL, again, and many technical difficulties vanish. Finally, he wrote a short paper related to the local conjecture for GL2 over a p-adic field with residue characteristic two.

Weil also felt strongly, as he repeated often, that conjectures were to be evaluated according to the evidence behind them. There is much to be said for this attitude, since ideas often come cheaply and without support. Since Langlands' conjectures included Artin's conjecture about L-functions as a special case, and since it took a lot of work to verify even simple cases, or at least a lot of imagination to see how fruitful the conjectures would be in breaking up large problems into smaller ones, it could have been predicted that Weil would be skeptical. What he himself says is this (Collected Papers 111, page 457):

. . . je fus incapable de partager l'optimisme de Langlands & ce sujet; la suite a prouve' que j'avais tort. Je lui dis cependant, comme j'ai coutume de le faire en pared cas: "Theorems are proved by those who believe i n them."

Presumably a necessary, not a sufficient, condition.

1 0 L-functions associated to the constant term of Eisenstein series

Implicit in Langlands' conjectures is the idea that the L-functions he defines are precisely those of arithmetic interest. Not quite a conjecture, this should be taken rather as a working hypothesis. At the time he made the principal conjectures, the main evidence that he had for this idea came from the theory of Eisenstein series. In this section I will explain this evidence, and even a mild extension of what was known definitely to Langlands in 1967. Other explanations of this material can be found in Langlands' notes on Euler products and Godement's Bourbaki talk on the same topic. For technical reasons, both restricted themselves to the case of automorphic forms unramified at all primes of a split group. Developments in local representation theory that took place a few years later made it possible to extend the result somewhat beyond what can be found in the literature.

The basic idea is simple, but unfortunately it requires some technical preparation to introduce it. Let G be a semi-simple group defined over the number field F, P a rational parabolic subgroup with unipotent radical N and reductive quotient M. For the moment, let A be the adde ring of F. We can identify the induced representation

with a space of functions on P(F)Np(A)\G(A), which we can call without trouble the space d(P(F)Np(A)\G(A)) of automorphic forms on


the parabolic quotient P(F)Np(A)\G(A). The functions in this space can also be characterized directly.

Suppose that (n, V) is an irreducible representation of G(A) occurring in the subspace of induced cusp forms on P(F)Np(A)\G(A). Let Pi f ori = 1, - . . , n be the maximal proper rational parabolic subgroups containing P, for each i let bi be the modulus character of Pi, and for s in Cn let

For cp in V and s in Cn the function

also lies in the space of induced cusp forms. For REAL(s) sufficiently large the Eisenstein series

converges to an automorphic form on G(F)\G(A), and continues meromorphically in s to all of Cn. If @ is an automorphic form on G(F)\G(A) and Q is a rational parabolic subgroup, then the constant term of @ associated to Q is the function

0. Q (F)NQ (A) \G(A). Suppose, now that P and Q are two rational parabolic subgroups. Start with cp in the space of cusp forms in A(P(F)Np(A)\G(A)), and take the constant term of E[cp] with respect to Q. In effect, we are constructing a map from a subspace of

Formally, this is simple to describe. We calculate

Let T be a maximal split torus contained in both P and Q. The Bruhat decomposition tells us that P(F)\G(F)/Q(F) is a finite disjoint union P(F)wQ(F) as w ranges over an easily described subset of the Weyl

group of T in G. We can choose representatives of the Weyl group in G(F). Hence we can write

The constant term of E[cp] is then the sum

How to manipulate this expression in the most general case is a bit complicated . There is only one case that we are actually interested in, however-that when Q is an opposite of P. In this case, we may identify the reductive group M with the intersection P n P. Furthermore, there is only one term in the sum that we are interested in, that with w = 1. The term we are interested in is then

We know that n may be expressed as a restricted tensor product n = & r p and hence may assume that cp also is a restricted tensor product & p p . We may therefore express the adklic integral as a product

of local intertwining operators. We shall calculate some of these in moment. But whether they can be calculated explicitly or not it is known that all of them have a meromorphic continuation in s. For the finite primes this follows from a simple algebraic argument about the Jacquet module, while for the real primes it is somewhat more difficult. At any rate, this point now appears relatively straightforward, but in 1967 it was not known, and appeared difficult.


The representations 7rp will be unramified at all but a finite number of primes, and as we shall see in a moment in certain cases the constant term of the Eisenstein series can be written as a quotient of Langlands' L-functions for the inducing representation a and M. The Eisenstein series satisfies a functional equation

and from it we shall deduce that at least in favourable circumstances some of Langlands' Euler products possess a meromorphic continuation also. This argument does not allow us to deduce a functional equation for them, although it is compatible with one. Because of the technical problems with local intertwining operators, Langlands restricted himself in his writings to globally unramified automorphic forms. Presumably in order to simplify the argument for an untutored audience, he also restricted himself to split groups. The principal step in this discussion is to express the unramified terms in the product through the L-group. I will do this in detail for unramified rank one groups. The general case will follow easily.

For the moment, let F be an arbitrary p-adic field.

It suffices to look only at simply connected groups of rank one. There are then two types of unramified p-adic groups to be considered. The first is the restriction to F of a group SL2(E) where EIF is an unramified extension. The second is the restriction of a unitary group in three variables.

The group SL2(E)

Let q~ be the size of the residue field OE/pE, and let n be the degree of the unramified extension E I F . Let P be the group of upper triangular matrices in SL2(E), P that of lower triangular matrices. Let

be an unramified character of P (E) . Let T be the G(E)-covariant map

defined as the meromorphic continuation of

Let cp, be the function in Ind(x,IP(E),G(E)) fixed by G(OE) with vs (1) = 1, Fs the analogous function in Ind(x, [P(E), G(E)). We know that

79s = c(s)(P,

for some scalar c(s). From the calculation we made before for SL2(Qp) we can deduce that

where w is primitive n-th root of unity, since q~ = q;. On the other hand, the L-group associated to the restriction of SL2 from E to F is the semi-direct product of the cyclic Galois group G with the direct product of n copies of PGL2(C), G acting by cyclic permutation. Let iOpp = n-,v A root space of 0 corresponding to the dual root -aV, -&

A

the element of T corresponding to x,. We can write the formula for c(s) in the form

det(I - Adaopp (&))-' c(s) =

det(I - q,l~da.pp(ix))-l

The group SU3

Continue to let E be an unramified extension of F and E, an unramified quadratic extension of E . Let

and let G be the unitary group associated to the Hermitian matrix w, already introduced earlier in this paper. The upper triangular matrices in G form a Bore1 subgroup B, and its opposite can be taken to be the lower triangular matrices. The radical N of its opposite is the group of elements

250 B. Casselman

where x = g , z + z = x y .

The groups B and B intersect in the torus T of diagonal matrices

with y in E,X. We want to calculate the constant c(s) such that

then n will be in G(0) if and only if z lies in OE,. Otherwise we want to write it as hk with h E P, k E G(0) . We have

which if z @ 0 we can normalize to

We finally find

n =

We filter N by subgroups N, where z lies in p;. . The quotient No/N1 has size qi, the quotient N1/N2 has size q ~ . The groups N,,,, are all conjugate, as are the groups Nodd. Let x be the character

The L- Group 251

By expressing the integral for T over N as the sum of integrals over No, N-1 - No, etc. We find that

with

For the calculation, let X = (X6L12(av(w))). Then the integral is

Now let's interpret this in terms of the L-group, which I have already partly described in an earlier section. The L-group L ~ E o l E is the semi- direct product of PGL3 (@) and (1, o) , and the L-group eEo is the product of several copies of this and an induced action of the cyclic Galois group of E I F . Again let EoPP be the negative root space in c. I now claim that

To see this, we just have to calculate a detiiopp(& x rob). Here Fis chosen so

t l l t3 = x(aV(w-l)) , t3/tl = x(aV(w)).

But we can calculate

252 B. Casselman

I so that its matrix is

I I from which the claim can be verified. ~~~ *Representations induced from a Borel subgroup

Let now G be an arbitrary unramified reductive group defined over F, let B be a Borel subgroup. T a maximal torus in B, W the corresponding Weyl group. For each unramified character x of T and Borel subgroups P and Q containing T let

be the intertwining operator defined formally by

If x, y are elements of W with 1 (xy) = 1 (x) + 1 (y) , then we have a kind of functional equation

For any unramified character x and Borel subgroup P containing T there exists a unique function v,,p in Ind(xIP(F), G(F)) fixed by K with cp,,~=(l) = 1. F'rom the functional equation just above and the rank one calculations made earlier we can deduce easily that

11 and

*Representations induced from opposite parabolic subgroups

Suppose now that P is a parabolic subgroup of G, an opposite, M = P n P. If (a, U ) is an unramified representation of M(F) , then by the Satake isomorphism it corresponds to a conjugacy class io x Frob in

I the L-group of M. The same element represents in L~ the unramified representation Ind(alP(F), G(F)) of G(F). Suppose given in U

The L- Group 253

a particular vector cpu fixed by M ( 0 ) . In the induced representation Ind (a1 M (F) , G(F)) there will be a unique function cp, fixed by G ( 0 ) and such that cp,(l) = q q ~ . Define cp, similarly in the representation induced from p. Theorem. In these circumstances we have

where detiopp(I - (t*, x rob))^'

c(a) = detropp(I - q,l(i, x frob))-1 '

and P is the radical of a in 8. The proof of this formula follows almost immediately from the one in the previous section, because induction from parabolic subgroups is transitive.

The global constant term

Now we consider things globally. Let P be a maximal proper rational parabolic subgroup of the rational group G, a a cuspidal automorphic representation of M (A), a = Ind(a1 P(A) , G(A)). Let i$ be the element of ? representing the modulus character Sp. It lies in fact in the center of L ~ , since Sp is a character of M. It is also the image in T^ of the

I

; product n aV(m-I) .

~ E C ; i I. The vector space nOPP decomposes under i!b into eigenspaces with eigen- 1 values a,. Let pi be the representation of L~ on the eigenspace for I ai. P ?

The constant term of the Eisenstein series corresponding to the local S function p,

L(azs,pi,a) II 1 L(a,s + l ,pi , a ) .

In favourable cases (for example, when r = 1) this implies that the

1 L-function has a meromorphic continuation. More about exactly which L-functions arise is discussed in some detail in the Euler Products notes.

b I should add that although it was certainly impressive that Langlands was able to use the theory of Eisenstein series to prove in one stroke that several new families of Lfunctions possessed a meromorphic continuation, the technique was certainly limited. As observed by Langlands


himself, perhaps the most striking case was that where G = G2 and M = GL2.

A similar calculation for other terms in the Fourier expansion of Eisen- stein series, suggested by Langlands in the 1967 letter to Godement and carried out in detail much later by Shahidi, derives a functional equation for the L-function in the cases where a has a Whittaker model.

Langlands tells me that L-functions arising in the constant term of Eisen- stein series played a crucial role in his thinking, but exactly what role is not clear to me. In the notes on Euler products he credits Jacques Tits with the observation that they are of the form L(s, p, T ) where p is the representation on the nilpotent Lie algebra, but as far as I can see Tits could only have made this observation in Langlands' lectures at Yale in May, 1967, several months after the letter to Weil.

5 11. Some subsequent developments

Langlands realized the importance of the L-group much more clearly than any to whom he explained his conjectures. He immediately began to work out various ways in which it played a role. Already in May, 1967, we find him writing a letter to Godement conjecturing a formula relating Whittaker functions to the Weyl character formula applied to the L-group. (This was later to become the formula of Casselman and Shalika, who learned only after they had proven it that Langlands had conjectured it seven years before!)

Questions raised by his conjectures presumably motivated his exhaustive investigation of local L-factors and root numbers, later simplified to some extent by Deligne. The local functoriality principle received striking evidence from his work on the I-adic representations of modular varieties for presentation at the 1972 Antwerp conference, which I have already alluded to.

But perhaps most interesting was the appearance of phenomena related to L-indistinguishability. We have already seen that to some extent the functoriality principle asserted a kind of characterization of an automorphic form, or equivalently an irreducible representation of G(A), by its Frobenius classes. But what happens for GL2 turns out to be deceiving. For other groups, representations both global and local come in equivalence classes called L-indistinguishable, which means that as far as their L-functions are concerned they appear to be the same. For GL,, each equivalence class has just a single element in it. This notion of equivalence among representations turned out to be related to

a simple equivalence relation on conjugacy classes. Both these notions turned out to be necessary to understand the exact relationship between the trace formula and the Hasse-Weil zeta functions of Shimura varieties. Questions raised in this way were surprisingly subtle and complicated, and have occupied many first-rate mathematicians since they were brought to public attention in Langlands' lectures at the Univer- sity of Paris. Many of the most difficult, but presumably not impossibly difficult, open questions in the subject are concerned with these issues. (A succinct and admirable discussion of these matters was presented by Arthur at the Edinburgh conference.)

Another extremely interesting development was the extension of local functoriality to include Galois representations with a large unipotent component, for example those arising form elliptic curves with multiplicative reduction. Here arose the Deligne-Langlands conjecture, which predicted a complete classification of square-integrable representations of p-adic reductive groups occurring as subrepresentations of the unramified principal series. This conjecture was eventually proven by Kazhdan and Lusztig. Related matters were investigated in a long series of papers by Lusztig on the Hecke algebras associated to affine Weyl groups, where perhaps for the first time the L-group occurred as a geometrical object. In particular, L-indistinguishability appeared naturally in terms of local systems on the L-group.

12. Things t o look for

One can find elsewhere accounts of serious and outrageously difficult conjectures implicit in Langlands' construction of the L-group and Arthur's generalization of functoriality. I will not recall these conjectures, but instead I will pose here a number of more frivolous questions which are presumably more easily answered.

Even in the case of compact quotients, the role of L-indistinguishability in the Arthur-Selberg trace formula is not at all clear, as Arthur points out in his Edinburgh expos& This is presumably related to the rather formal aspect of proofs of the trace formula. Can one use ideas of Patterson, Bunke, and Olberich to elucidate the nature of L-indistinguishability? Even more formal are Arthur's arguments for non-compact quotients. What sort of analysis or geometry would make the trace formula seem natural? This is somewhat mysterious even for SL2 (Q).


Recently, following an extraordinary paper of Lusztig where intersection cohomology and the Weyl character formula appear together, Ginzburg and others have formulated the Satake isomorphism in terms of tensor categories of sheaves on a kind of Grassmannian associated to a group over fields of the form F ( ( T ) ) . In this context, the L-group is defined for the first time as a group rather than just formally. Is there a version of this valid for p-adic groups? Can one formulate and prove classical local class field theory in these terms? It is difficult to believe that one will ever understand the conjectured relationship between local Galois groups and representations of p-adic groups until one has a formulation of local class field theory along these lines. Geometry of the L-group first appeared, as I have already mentioned, in Lusztig's work on the conjecture of Deligne-Langlands. Lusztig showed in this work that Kazhdan-Lusztig cells in affine Weyl groups were strongly related to unipotent conjugacy classes in the L-group. Apparently still unproven remain conjectures of Lusztig such cells to cohomology of subvarieties in the flag man- ifold of relating the L-group.

a What replaces the L-group in analyzing Kazhdan-Lusztig cells in hyperbolic Coxeter groups? The phenomena to be explained can be found in work of Robert Bbdard, but not even the merest hint of what to do with them.

a Manin tells us that we should think of algebraic varieties at real primes as having the worst possible reduction. Is there any way one can use this idea to make better sense of representations of real groups? Can we use representation theory of either real or p-adic groups to explain Manin's formulas in Arakelov geometry? In his Ziirich talk, Rapoport mentioned a possible approach to local functoriality conjectured by Kottwitz and Drinfeld. The idea is highly conjectural, but any progress here would be interesting.

a Another approach to local functoriality was mentioned in Ginz- burg's talk at the ICM in Berkeley. This looks more interesting in light of the 'new' Satake isomorphism. Is there anything to it?

a One of the oddest puzzles in the theory of local L-functions in representation theory is the necessity of introducing an additive character to define the €-factors. There are two places in local representation theory where these arise naturally-in the theory of Godement-Jacquet for GL, and in the theory of Whittaker functions, which play a puzzling role. In a recent raper, Frenkel et al. interpret the explicit formula of Casselman-Shalika for

Whittaker functions in geometric terms. It would be interesting- illuminating both the meaning of local L-functions and L-group- if one could prove the formula in this context. It would also be interesting if one could similarly understand Mark Reeder's generalization of the Casselman-Shalika formula.

a I have proven above a formula for the effect of intertwining operators on unramified functions on a p-adic group, which has a striking formulation in terms of the L-group. The proof is entirely computational, however. Can one explain this formula directly in terms of the L-group? Extend it to ramified representations? Similarly deduce Macdonald's formula for unramified matrix coefficients?

a There has been a lot of work on the classification of irreducible representations of local reductive groups in the past several years, but the Galois group plays no apparent role in these investigations. Is there any way to introduce it there? Is there any way to generalize Kazhdan-Lusztig's work on the Deligne-Langlands conjecture to ramified representations?

I refrain from commmenting on overlap among these problems.

References

J. Arthur, Stability and endoscopy: informal motivation, Proc. Symp. Pure Math., 61(1997), A. M. S., pp. 433-442.

A. Borel, Automorphic L-functions, Proc. Symp. Pure Math., XXXIII(1979), edited by A. Borel and W. Casselman, A. M. S.

P. Cartier, Representations of p-adic groups, Proc. Symp. Pure Math., XXXIII(1979), edited by A. Borel and W. Casselman, A. M. S.

I. Fkenkel, D. Gaitsgory, D. Kazhdan, and K. Vilonen, Geometric realization of Whittaker functions and the Langlands conjecture, preprint, 1997.

S. Gelbart and F. Shahidi, Analytic Properties of Automorphic L-functions, Academic Press, 1988.

R. Godement, Formes automorphes et produits Euleriens, S6minaire Bourbaki, Expos6 349, Paris, 1968.

B. H. Gross, On the Satake isomorphism, to appear in a volume of the London Mathematical Society Lecture Notes, edited by Taylor and Scholl, later in 1998.

H. Jacquet, Notes on the analytic continuation of Eisenstein series, Proc. Symp. Pure Math., 61(1997), A. M. S., pp. 407-412.

R. E. Kottwitz and D. Shelstad, Twisted endoscopy I, to appear in AstBrisque.

258 B. Casselman

[lo] R. P. Langlands, introductory comments on the reprint of the first CJM paper on Shimura varieties, Can. Math. Soc. Selecta, volume 11, 1996.

[ll] R. P. Langlands, Where stands functoriality today?, Proc. Symp. Pure Math., 61(1997), A. M. S., pp. 443-450.

[12] G. Lusztig, Some examples of square-integrable representations of semisimple padic groups, T. A. M. s . , 227(1983), pp. 623-653.

[13] Yu. I. Manin, Three-dimensional geometry hyperbolic geometry as GO-

adic Arakelov geometry, Invent. Math., 104(1991), pp. 223-244. [14] I. Mirkovic and K. Vilonen, Perverse sheaves on loop Grassmannians and

Langlands duality, preprint, 1997. [15] M. Rapoport, Non-archimedean period domains, Proceedings of the ICM

at Zurich, Birkhauser, pp. 423-434. [16] M. Reeder, padic Whittaker functions and vector bundles on flag mani-

folds, Comp. Math., 85(1993), pp. 9-36. [17] I. Satake, Theory of spherical functions on reductive algebraic groups over

p-adic fields, Publ. I. H. E. S., 18(1963), pp. 1-69. [18] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. A. M. S.,

BO(l968). [19] Andrk Weil, volume I11 of Collected Papers, Springer-Verlag, 1979.

Department of Mathematics, the University of British Columbia Vancouver, Canada E-mail address: [email protected] . ca


Groupe des Obstructions pour les Dhformations de Repr6sentations Galoisiennes

Roland Gillard

Un des thkmes les plus importants en thkorie des nombres est l'ktude du groupe de Galois absolu, GQ, de la clhture algkbrique de Q. Une des faqons possibles d'aborder ce problitme est de se concentrer sur les reprksentations & coefficients dans une Z,-algkbre noethkrienne R :

GQ + GLN(R). Ces reprksentations se factorisent en gknkral par un quotient Gs de GQ par des groupes d'inertie en dehors d'un ensemble fini S : on a alors de hypothkses de finitude de prksentation. Pour N = 1, on retrouve ainsi la thkorie du corps de classes puisque c'est en fait le quotient abklien maximal qui intervient. La thkorie des dkformations de reprksentations peut donc Stre vue comme une tentative pour gknkraliser la thkorie du corps de classes en sortant du cadre abklien.

Depuis l'article fondateur de Mazur [18], on sait que les problitmes de dkformation des reprksentations galoisiennes sont (pro)-reprbsentables sous des conditions raisonnables; les anneaux sont de la forme R =

O[[Tl, . . . , Td]]/I, avec 0 un anneau de Witt. On sait aussi, que pour que l'ideal I, idkal des relations soit nu1 (le problkme alors appelk sans obstruction), il suffit qu'un certain groupe de cohomologie H 2 soit nul.

Le but de ce travail est de prkciser ce rksultat en construisant et en ktudiant un groupe d'obstructions, sous groupe du H 2 ci-dessus, et en montrant que le rang de ce groupe est exactement le nombre minimal de g6nQateurs de I. On montre ensuite que ce groupe, ou une variante concernant la caractkistique p, est fabriquk & l'aide de cupproduits de 1-classes et aussi de produits supQieurs analogues aux produits de Massey.

1

La premikre partie reprend la mkthode de Vistoli [31] ktudiant les relations et obstructions dans un cadre gkomktrique.

Une question qui se pose donc est de savoir si on peut trouver des

! exemples oh le sous groupe serait nu1 sans que le groupe de cohomologie


260 R. Gillard Groupe des Obstructions 261

le soit. Uiie autre serait l'expressiori directt du sous-groupe en terines (111 prohliime initial (par exeinple le module galoisieri de d4part).

Le paragraphe 1 cibcrit les conditions de Sclilessinger d'existence d'linc algi.bre ~inivcrsellc. Le paragraphe 2 axiomatise la 1n4thode de Vistolil en supposant que le foncteur posskde une thCorie linkaire d'obstruction. Ccci pcrmct dc reprendre plus explicitement la construction tlc R, d'introduire le groupe d'obstruction OF ct de montrer que sa dimension et egale au noinbre de relations dans R. Discliter les conditions cl'existeiice d'une telle th6oric nous aurait entrain4 trop loin, surtout que sur cliaque exerilple classique de foncteur de dkformations, cette question lie prbseiite guCre de difficult4. Le paragraphe 3 s'intkresse au cas iiitroduit par Mazur des repr4sentations d'un groupe, cependarit en adoptant le cadre plus g4n4ral des sch4inas en groupes lisses comme dans Tilouine [30].

La deuxitme partie se concentre sur le cas des dkforrnations de rcpr6scntations de groupes. Elle fait le lien entre la premiiire partie et la question de Maiilir portant slir la dimension relative de R, puis <<calclilc>> lc groupe des obstrlictions en caractCristiquc p en montrant q11'011 pclit l'cngcndrcr avcc lc cup-produit et des produits supkrieurs. Le \5 montrc l'intervcntion des opbrations de Bockstein.

La deri1il.1-e partie developpe quelques remarques arithm6tiques: re- tour sur la conjecture de Leopoldt dans le 56.1, nullit6 du groupe des obstructions dam 1e 56.2 et contr6le de l'idkal correspondant dans le 5 6.3.

Premihre partie

Groupe des obstructions Cette partie se place dam un cadre trks g4nkral aiialogue tl celui de

Schlessi~iger .

51. Foncteurs pour les anneaux artiniens

Or1 fixe un riomhre preniier p et line extension finie K de Q,, d'an- iioau de valuation 0, d'uniformisante T . de valuation T-adique vo et de corps rbsiduel k. Soit C la catkgorie des 0-alghbres locales artiniennes coniplktes de corps rksiduel k. Pour A dans C, on note r n ~ son id6al maximal. Pour rr, entier, on appelle C,, la sous catkgorie pleinc des algkhres A

'je rernercie A. hl6zard qui rn'a transmis l'article [31]

annul6es par m:". On appelle 1-extenszon une surjection A' -+ A dans C dont le noyau I est annul6 par m ~ l . Un morphisme de 1-extensions est un diagramme cornmutatif dkfini par des morphismes A' -, B' et A - B. Une 1-extension est dite petite si en plus I est un idkal principal. La catkgorie C est une sous-catkgorie pleine de la catkgorie C des 0-algkbres locales noethkriennes, compli3tes de corps rksiduel k.

1.2. Foncteurs

On fixe un foncteur covariant F de C dans Ens la catkgorie des ensembles. L'espace tangent de F est par dkfinition tF = F(k[c]). Pour R un anneau de C, on note hR le foncteur Hom(R, 0 ) . On considi3re des couples (A, a) avec A dans C ou C, et a dans F(A) ; un morphisme de couples (A', a') - (A, a ) est un morphisme u : A' + A tel que a = F(u)(al) . Si R est dans C, un couple (R,() est R complktk par une suite cohkrentes de 6, E F(R,), avec Rn = ~/m;l+ ' . Un morphisme de couples (R, J) + (A, a ) est un morphisme (R,, (,) -+ (A, a ) pour n assez grand. On dit qu'un couple (R, J) repre'sente (ou est universe1 pour) F si le morphisme de foncteurs hR + F : Hom(R,A) = Hom(R,, A) +

F(A) , p -+ F(p)((,) est un isomorphisme pour tout n et pour tout A dans C,,; Si un tel couple existe on dit que F est pro-reprksentable. On dit que (R, e ) est versel si ce rnorphisme est lisse: ceci revient 8. demander que pour tout morphisme de couples comme ci-dessus ou u est 1-extension, tout morphisme u : (R, () -+ (A, a ) se relkve en u' : (R, J) + (A', a') rendant commutatif le triangle

On dit que (R, J) est une enveloppe si ce couple est versel et si de plus hR -+ F induit un isomorphisme sur les espaces tangents: (mR/(.rr) + m;)' - F(k[c]). Enfin on dit que (R, () est n-verse1 (resp. est une n-enveloppe) si cet anneau est dans C, et y est versel (resp. est une enveloppe) pour la restriction de F . Si (R, () est versel (resp. m-versel) pour R dans C (resp. C,), alors (R,, (,) est versel (resp. (R,, J,) est verse1 oh J, est l'image de J,, pour n < m). Cette remarque vaut aussi pour les couples universels ou les enveloppes.

1.3. Conditions de Schlessinger

Soit F un foncteur de C vers Ens avec F(k) = {e), l'ensemble 8. un point. Soit p : A' -+ A et p' : A" -+ A deux morphismes de C. On a une

262 R. Gillard Groupe des Obstructions

application canonique: 1 A.2 Changement de foncteur

(1) Q, : F(A' X A A") + F(A') X F ( A ) F(A").

Rappelons le thkorkme fondamental de [28].

Th6orkme 1.1. Le foncteur F a une enveloppe si et seulement si les 3 conditions suivantes sont ve'rifie'es: HI . Q, est une surjection pour tout p et toute petite surjection p'. HZ. Q, est une bijection si A = k et A" = k [ ~ ] . H3. tF est u n k-espace vectoriel de dimension finie.

De plus, ces conditions e'tant suppose'es satisfaites, F est pro-repr8 sentable si et seulement si H4. @ est u n isomorphisme si p = p' et est une petite surjection.

On sait en effet que H2 implique que t~ est un k-espace vectoriel. Si V est un k espace vectoriel de dimension finie, k[V] dksigne l'anneau k @ V oii la restriction de la multiplication & V x V est nulle. Par recurrence avec H2, on obtient un isomorphisme canonique: F(k[V]) 1 t~ @k V. Pour toute 1-extension de noyau I, p : A' --+ A, l'isomorphisme A' x k k[I] 2 A' X A A', inverse de [28](2.16) induit compte tenu de H1 et H2 une surjection:

Lemme 1.2. La surjection pre'ce'dente munit F(A1) d'une action de t F @k I a : a' t a a', respectant les fibres de F(A1) -' F(A) et y induisant une action transitive.

Dans toute la suite on suppose que F vkrifie les conditions Hl,H2,H3.

1.4. Fonctorialit6s

1.4.1 Changement de 1-extension

Un morphisme de 1-extensions (A' -+ A) 4 (B' -' B), de noyaux respectifs I et J induit un diagramme commutatif

ainsi qu'un morphisme tF €3 I -' tF @ J et F(A1) -+ F(B1) est kquivariant pour les actions dkcrites dans la section 1.3.

Un morphisme de fonteurs cp : F -t F' vQifiant tous les dewc les conditions de Schlessinger induit un morphisme d'anneaux cp* : R' + R, de sorte que le diagramme nature1 induit

soit commutatif.

$2. Obstructions

2.1. Th6orie lin6aire des obstructions On dit qu'un foncteur F posskde une the'orie line'aire des obstruc-

tions si

- i) il existe un k-espace vectoriel TF

- ii) il existe pour chaque 1-extension p : A' t A de noyau I et chaque klkment a E F(A) un klkment Op,, E T(I) := TF @k I dkpendant fonctoriellement des donnkes (A' -+ A, a).

- iii) o,,, est nu1 si et seulement si a est dans l'image de l'application F(A1) -+ F(A); on dit alors que a se relkve & A'.

2.2. Relkvement maximal Soit A' 4 A une 1-extension de noyau I et a E F(A).

Lemme 2.1. I1 existe u n plus petit sous-ide'al I, de I tel que a se relive r i A, := A1/Ia.

Ce lemme est d&j& dans [28], bas de la page 213: il suffit de considgrer S l'ensemble fini non vide (il contient I ) des idkaux K c I tel que a se relkve 8. AK = A1/K et d'observer que S est stable par intersection. En effet si K1 et K2 sont deux tels idkaux, quitte & agrandir K1 sans changer K1 n K2, on peut supposer que K1 + K2 = I ; de sorte que

et on conclut avec la surjection de (1)

fournie par H1 et une rgcurrence pour &tendre la propriktk des petites extensions aux 1-extensions.

264 R. Gillard Groupe des Obstructions

2.3. Groupe des obstructions 2.4. Construction d'une 1-enveloppe

I1 peut arriver que TF soit trop gros; on va 6tudier ce qui intervient rkellement , le groupe des obstructions qui est son sous-k-espace vectoriel OF engendrk par tous les 616ments (Id @ cp)(op,,) fournis par les p et a avec p petite extension ainsi que cp : kerp + k induisant

Ceci nous permet d'knoncer l'analogue de [31], rksultat principal de cette partie:

ThBorhme 2.2. Soit F un foncteur ve'rifiant les conditions de Schlessinger et ayant une the'orie line'aire des obstructions, cf. 2.1. Soient r et s les dimensions respectives de tF et OF sur k: l'anneau (uni-)verse1 de F peut 6tre e'crit comme quotient d'un anneau de se'ries formelles sur O ci r inde'termine'es par un ide'al de relations dont s est le nombre minimal de ge'ne'rateurs.

Ce th6orkme ne sera demontr6 qu7en 2.7. Commenqons par quelques lemmes sur les o,,, .

Lemme 2.3. Soit p : A' -+ A une 1-extension de noyau I : o,,, est dans le sous-groupe OF @k I de TF @k I.

De'monstmtion: Comme dans [31], on prend une base ei de I pour 6crire Op,, = C oi @ei. Pour voir que les oi sont tous dans OF, on utilise la fonctorialit6 des o,,, en considQant l'id6al Ii engendr6 par tous les ej sauf ei ainsi que la petite extension A1/Ii + A.

Ceci permet de pr6ciser le lemme 2.1 en choisissant une base wi, i = 1, . . . , s de OF; en reprenant les mGmes notations, on peut alors kcrire o,,, = C:I; wi B xi avec xi E I.

Lemme 2.4. L 'ide'al I, est engendre' par les xi.

De'monstration: Soit J l'id6al engendr6 par les xi. L'image de o,,, dans OF @ I/J est nulle donc a se relkve B Aj; par minimalitk, I, C J . Comme a se relkve B AII,, l'image de o,,, dam OF@I/Ia est nulle, ce qui impose la nullit6 des images de chaque xi dans III,, d'oh J C I,. Ceci se voit comme plus haut en utilisant la petite extension correspondant B un suppl6mentaire de la k-droite engendr6e par l'image de xi dans I/Ia; on forme ainsi un quotient de A, oh a n'a pas de relkvement, ce qui est absurde.

On part de Ro = k[t>]. On prend une base ei,i = l;.. , r de tF et on appelle e: la base duale, on considkre.17anneau A = A, des s6ries formelles B r variables XI , . . . , X, sur 0, Xi correspondant 8. ef . On a canoniquement tA 21 tF. On munit A de v ~ , la valuation mA-adique. On peut compl6ter & par to de faqon B obtenir un couple universel dans la catdgorie Cb des O-algkbres A annukes par (rr) + m:. En effect Ro -+ k posskde une section donc l'el6ment de F(k) se relkve dans Ro. H2 implique que l'application (1) est une bijection ce qui montre que F(&) est en bijection avec tF @k t> 2. End(tF). I1 est alors facile de voir que C ei @ e: est universel puisqu7il correspond B 17identit6 de End(tF).

Partant de (Ro, to) comme plus haut, considkrons la 1-extension R1 = A/m; + Ro: le noyau est l'id6al engendrk - par 17image de rr. On a une obstruction au relkvement de to B Rl, 01 = w @ x, oh x peut &re pris 6gal & 0 ou B l'image de rr et on obtient un idkal qu70n note - Il dans A/mi d7image r6ciproque nothe Il dans A: II est nu1 ou peut &re engendr6 par rr. Ce dernier cas correspond & des dkformations limit6es B la caractkristique p. On choisit un relkvement quelconque t1 B R1 : = All l , cf. 2.4 pour l'existence. L'anneau R1 admet toujours tF comme espace tangent.

Lemme 2.5. Le couple (R1,J1) est une 1-enveloppe.

De'monstration: I1 suffit de prouver que ce couple est 1-versel. On se place dans la situation de 1.2 avec A' -+ A une petite extension de C1. La premikre &ape consiste B simplement compl6ter le diagramme de 0-algkbres:

avec une diagonale montante R1 + A'. Utilisant que & est verse1 dans Ch pour obtenir

avec A0 = A/(rr) et Ab = At/(rr), d'oh un relkvement de R1 + A en U; : R1 + A X A ~ Ab N A'II XA,, Af/(r) 2 A'lI n (T).


Soit J = I n (T); il reste B relever ub : R1 -+ A'/ J & A'. Considkrons alors le diagramme:

iil -+ A'

1 1 R1 -t A'/J

obtenu en relevant ub B A et observant que ce relkvement se factorise par Rl. De plus, ub induit la deuxikme ligne du diagramme suivant:

Par fonctorialitk des obstructions aux relkvements, l'image de 01 E TF 8 T R ~ dans TF 8 TA' est nulle; ainsi l'irnage de Il est nulle dans TA', si bien que l~homomorphisme R1 -+ A' se factorise par R1 fournissant le relkvement u' : R1 -+ A' de u recherchk dans cette premikre &ape.

I1 reste maintenant & modifier u' pour que le diagramme de couples

soit commutatif. Pour cela, on considkre le diagramme:

Hom(R1, A') --t F(A1)

1 1 Hom(R1,A) F(A)

les flkches horizontales &ant dkfinies par v -+ F(v)(J1). Sur la fibre de u dans la premikre colonne, cf. lemme 1.2, on a

une action homogkne de tRl @ I, sur celle de a dans la deuxikme, une action homogkne de tF 8 I ; les deux klkments de F(A1) sont reliks par un a E t F @ I : a' = aaF(u')(Jl). Si on identifie tRl & tF il suffit de prendre v = a l U' pour assurer a' = F(v)(Jl) et achever la dkmonstration.

2.5. Construction d'une n-enveloppe On suppose construit un couple (R, , J,) , qui soit une n-enveloppe

pour F; nous construisons maintenant une (n+l)-enveloppe (%+l, Les raisonnements ressemblent & ceux dkveloppks pour passer de Ro B R1. On suppose que R, est de la forme A/I, avec I, idkal contenant m:+'. On considkre la 1-extension = A l m ~ l , + Rn et <,; on applique le lemme 2.4 en kcrivant l'obstruction au relhvement o,+l = xtIS wi 8 ui, en utilisant la meme base de OF et des 6lkments

ui de In/mAIn qu'on relhve par des klkments de meme nom dans I,. L'idkal In+1 qu'ils engendrent avec mAIn dans A dkfinitle quotient maximal R,+l de R,+~ oh J, se relkve. On choisit un relkvement

Lemme 2.6. Le couple est une (n t- 1)-enveloppe.

Par rkcurrence, on dkduit que In+1 c (T) + m i si bien que l'espace tangent de est bien kgal & celui de F. I1 suffit donc de vkrifier la (n + 1)-versalitk en prockdant en gros comme dans le lemme 2.5 avec une petite extension maintenant dans C,+l et des morphisme de couples

(Rn+l? Jn+l) -+ (A'a) et (A'ya') -+ (A,a). La premihre &ape consiste B complkter le diagramme de 0-algkbres

avec un morphisme R,+l -+ A'. En utilisant la propriktk de lissitk incluse dans l'hypothkse de versalitk du couple (R,, J,), on commence par dkduire l'existence d'un relkvement au produit fibrk A x A',, oh A, (resp. A',) dBsigne le quotient de cet anneau par la puissance (n+ 1) ikme de l'idkal maximal. Cet anneau s'identifie & A'/Jn+l avec Jn+l = I n mn+l . On relkve alors le morphisme R,+l 4 A'/ J,+l en un morphisme A -, A'. Comme le noyau de A' -t A est annul6 par m ~ t , et que le noyau de A + est In+1, on voit que mA In+1 est dans le noyau de ce relkvement: on a une factorisation en R,,+~ -+ A'. On considkre alors le diagramme

- pour conclure que le noyau de R,+l -+ A' contient I,+l par le meme argument de fonctorialitk des obstructions. Dans la deuxikme ktape, on ajuste alors le relkvement trouvk griice au lemme 1.2 et & l'isomorphisme tRn+, @ I 7 tF 8 I comme dans la section 2.4.

I

1 2.6. Sur les gdndrateurs de In

1 Dans cette section, on reprend la mkthode de [31] pour ktudier I,.

1 Lemme 2.7. La famille des ide'aux I, ve'rijie la relation de re'cur- t rence:

In+i + mi+' = In.


De'monstration: L'inclusion du membre de gauche dans In est claire. La surjection naturelle qui en rksulte est un isomorphisme par unicit6 de l'enveloppe de F dans Cn cf. [28] prop. 2.9.

Considkrons des 6lkments de A, ui (i = 1, , s), comme dans 2.5 dont les images engendrent In+l/mAIn; soit Jn+1 l'idkal de A engendrk par ces ui: on a donc In+1 = m~ . In + Jn+l.

Lemme 2.8. On a les kgalitks d'ide'aux de A i) In = mxfl + Jn+l, ii) In+l = mi+2 + Jn+1.

Dkmonstmtion: L'inclusion m;+' c In se voit par rkcurrence puisque mAIn C In+1. De plus les ui sont dans In+1 c In.

On a aussi In = mi+' + In+1 = mn+' A + m~ . In + Jn+l et on en dkduit i). En reportant dans la relation In+1 = m~ . In + Jn+17 on en tire ii) griice au lemme de Nakayama.

Rappelons le lemme klkmentaire [31] (7.11).

Lemme 2.9. Soit M un module de type fini sur un anneau local A; Deux systkrnes de ge'nkrateurs xi et yi (i = 1, . . . , s) ayant m6me nombre d'klkments sont relib par une matrice carrke inversible 8 = (OiTj) d coeficients dans A : yi = Cr Oi, jxj.

Lemme 2.10. Les idkaux In peuvent &!re engendrks par m;+'

et s polyn6mes fin) de degrk < n de fagon d avoir: fjn) = fin+') mod m:+'.

De'monstration: Fabriquons les fin) par rkcurrence en prenant comme

fil) les ui trouv6s en 2.4. On peut les choisir de degrk _< 1. Appliquons

le lemme 2.9 aux ui de 2.5 et aux fjn) en prenant comme module le quotient in/m:+' et en tenant compte du lemme 2.8 i): on touve des

BiJ E A tels que fjn) = B i , j ~ j mod m:"; on peut donc engendrer

In+, par ml+2 et des fin+') tels que fjn+') = C; Bi,juj mod (n+l) (n+l) Pour cela on peut choisir fjn+') de forme f!") + gi , g, somme

de mon6mes ax:' . . XFr avec v~ (ax:' - - . X,".) = n + 1.

Ainsi chaque suite fjn) est convergente vers une limite fi dans A. Si on note I l'idkal engendrh par les fi, on a In = m:" +I et il est clair que, en posant R = AII et en appelant J la suite des J,, le couple (R, J) est une enveloppe pour F.

On prockde comme dans [31]. Notons f n l'image de I (ou de In) dans An = ~/m;+l : Le lemme d'Artin-Rees implique que la surjection

canonique I/mA I -+ fn/mAfn est un isomorphisme pour n assez gros. Considkrons la situation de 2.5 avec la 1-extension Rn+1 = A/mAIn -+

Rn = Alln et le couple (Rn,Jn); l'obstruction & relever Jn a dkj& kt6 kcrite sous la forme = cir; wi @ ui, en utilisant notre base wi de OF et des 6lkments ui de In/mAIn. Pour i fixk, choisissons une petite extension pi : A; -+ Ai, un 616mknt ai E F(Ai) et 9 : kerpi 2 k de f a ~ o n que wi soit &gal B cp(o,,,,) E OF. On construit un diagramme commutatif:

En fait puisque A: est supposke 6tre dans C,, fi se factorise par le quotient de &+l par la puissance n + 1-ikme de son idkal maximal, soit encore le quotient Rn de A par mAI + m y f ' , nous donnant le nouveau diagramme commutatif

L'obstruction B relever En B & est l'image Zn+1 = Ciz; Wi @ iii de

on+l = ~ i z ; wi@ui dans 0 ~ ~ f ~ l m ~ f ~ . Par restriction aux noyaux, f? induit un homomorphisme f ,/mAfn I + m:" /mA I + m",' -+ ker pi d'oh

envoyant par fonctorialitk on+l sur wi. L7616ment Sn+1 = wi @ Tii - fournit une application (T,/~AI,)* -+ OF : g -+ (Id @ g)(&+l). Le raisonnement prkckdent montre que pour n assez gros pour que Cn contienne toutes les algkbres A!,, i = 1 , . . . , s , cette application est surjective; ceci montre bien que s est le nombre minimal de gknkrateurs de 7,) donc de I . Ceci achkve la dkmonstration du thkorkme 2.2.

On a un diagramme commutatif:


avec 5, et sn surjectives et in injective. Pour n assez gros 5, est bijective. La dkmonstration du lemme 2.10 montre que Im in, l'image de in, est engendrke par les ui. On dkduit du prkckdent un nouveau diagramme

De plus 17application canonique

induit un isomorphisme : Im in+' + Im in pour n assez grand. Notons

maintenant uin) 1786mknt de A not6 ui dans 2.6. On voit ainsi que dans

le raisonnement du lemme 2.10, on peut prendre fin) F ujn) mod m~",

on a alors fin+ ') F ujn) mod mz+2. On remarque que, puisque on+2

s'envoit sur on+', 8,(uin+')) = uin), si bien que les congruences ci- dessus sont vQifikes par rhcurrence pour tout entier m > n : On peut donc assurer l'kgalitk

Ceci prouve la conskquence suivante de 2.1 ii):

Proposition 2.11. On suppose que les s ge'ne'rateurs fl , . . . , f, de I sont choisis comme on vient de le faire. Etant donne' une petite surjection p : A' + A et un e'le'me'nt a de F(A), correspondant 6 un - homomorphisme 8 : Rn - A pour n assez gros; pour tout : A - A' relkvement de a, on a

DeuxiAme part ie

Deformat ions de represent at ions

$3. Relations pour les dbformations de reprbsentations

3.1. Thborie de Mazur Fixons un groupe I2 vQifiant les conditions de finitude de [18]: on

demande que la pro-pcomplktion de chaque sous-groupe ouvert ait un

nombre fini de gknkrateurs topologiques. Comme dans [30], considQons un schkma en groupes G lisse sur 0, linhaire, rkductif et dkployk. On note ZG son centre; pour un tel groupe, on note G le groupe alg6brique sur k dkfini par sa fibre spkciale ainsi que g l'algkbre de Lie de G.

On fixe une reprksentation p : II - G(k). Pour chaque 0-algkbre A artinienne on considkre VA l'ensemble des reprksentations p : ll+ G(A) qui par composition avec G(A) + G(k) donnent p. On forme le quotient

P; F(A) de Va par l'action par conjugaison de G(A) = ker(G(A) - G(k)).

ls,. On obtient ainsi un foncteur F . On sait, par [30] (prolongeant [18]), que Yi > F est reprksentable moyennant des conditions sur le centre: il suffit i

qu7au niveau des composantes connexes, le centralisateur de l'image de *, p dans G soit kgal au centre de et que le centre de G soit formellement lisse sur 0 : en effet, dans ces conditions les hypothkses Hi de 1.3 sont vkrifikes. Pour t oute 1-extension de noyau I, l'application exponent ielle induit une suit exacte

8. I' & si bien, cf. [15], que la question des relkvements d'une reprksentations 4 II - G(A) & G(A1) s'ktudie & l'aide de la cohomologie de I2 B valeurs

dans g @k I. Ce k-espace vectoriel est considQk comme un II-module g r k e B l'action adjointe de G(k) sur g composke avec 7. On en dkduit

I1 est aussi classique, cf. [18] et [30], qu7en prenant

TF = H2(n, 01,

t on obtient une thkorie linkaire des obstructions.

3.2. Description de l'anneau universe1

Le foncteur posskde donc toutes les propriktks dkcrites plus haut.

Thborkme 3.1. Soit F le foncteur des de'formations de p et OF le sous lc-espace vectoriel de TF = H2(II, g) engendre' par les obstructions des petites extensions. De'signons par r et s les dimensions respectives de / H' (II, g) et OF sur k: l'anneau universe1 X de F peut 6tre 4crit comme

F quotient d 'un anneau de series fomnelles sur 0 & r inde'temnine'es par

1 un ide'al de relations dont s est le nombre minimal de ge'ne'rateurs.

1 En commentaire, la question cf. [18] et [30] sur l'kgalitk de la di-

1 mension relative de R sur 0 (dans un contexte de corps de nombres) se dkcompose en deux parties:


Question 3.2. i ) L'anneau R est-il une intersection complite relative sur 0 ?

i i ) Le k-espace vectoriel H 2 ( n , g) est-il engendre' par les obstructions difinies par toutes les petites extensions?

I1 est inthressant de remarquer que la dkmonstration classique, cf. [18] et [30], de l'inkgalith dim H 2 (H, g) 2 dimk (IA/mA I ) se fait en prouvant que 17application canonique dkfinie par J, (n assez gros)

est injective; cette application se factorise 6videmment par OF et la m6thode suivie ici montre en fait (In/mAIn)* + OF est surjective; elle est aussi injective puisqu'on a vu que s = dimk OF majore le nombre minimal de g6n6rateurs de I .

Remarque: La dimension relative est en fait remplac6e dans [18] et [30] par la dimension de la fibre sp6ciale i.e. celle du quotient RITR. On a alors un 6nonch analogue au pr6c6dent en remplaqant OF par son sous k-espace vectoriel OF) engendr6 par les obstructions des petites extensions annule'es par p. Le but des 5 suivants est de calculer ce sous k-espace vectoriel, cf. 4.4.

54. Obstructions et produits de Lie-Massey

Pour H et p comme en 3.1, on cherche & calculer les obstructions fournies par les petites extensions. On va faire le lien avec le produit de Lie-Massey dont on r6sume la d6finition. La mdthode reprend les idkes de Laudal [16] et [1712. Elle utilise une r6currence bas6e sur le calcul prkliminaire suivant .

4.1. Calcul de l'obstruction pour une 1-extension On fixe une 1-extension p : A' + A de noyau I . Soit 3 : n + G(A)

une representation. On relkve 6 en une application not6e p : ll + G(A1) on dkforme p en p' dkfini par pl(g) = [1+ S(g)] p(g) oh 6 : II -+ g 8 m ~ l et pas seulement g 8 I . Soient a le 2-cocycle associ6 B p et a' celui associk & p'. Le premier calcul exprime a' B 17aide de a et 6.

Lemme 4.1. 1) La composition de p' et de G(A1) -+ G(A) est u n homomorphisme si et seulement si o n a: d6 - S U 6 + a E g @I I.

2) S i cette condition est ve'rifie'e, alors a' = d6 - 6 U 6 + a.

2je remercie J. Bertin qui m'a signal6 17existence de [16] et m'a incit6 B regarder les produits de Massey

De'monstmtion: On commence par le calcul oil les produits sont calcul6s & partir de ceux de Mn(k):

Comme p est un homomorphisme, a est & valeurs dans g @I I et annul6 par m ~ l donc par les valeurs de 6; le membre de droite est donc 6gal &

En regardant 17image dans G(A) on voit & quelle condition 176galit6 pr6c6dente modulo I est vkrifi6e avec a' = 0 mod I d'oh la partie 1). Pour la partie 2), griice B l'hypothkse, a' est & valeurs dans g @I I et est aussi annul6 par ~ A I donc par les valeurs de 6: le membre de gauche se simplifie en 1 + al(g, h) + S(gh). Ainsi:

ce qui prouve la formule.

4.2. Produits de Lie-Massey On r6sume les dbfinitions, d6jB anciennes de Retakh [25,26]. On

notera que celles ci ont kt6 aussi introduites pour des considkrations topologiques par Tan& [29].

On travaille avec une algkbre de Lie gradu6e L = $,L, munie d7un crochet [. , a ] : L x L + L, homoghne: [L,, L,] c L,+, antisymhtrique (au sens gradu6) :

x E L,, y E L,, et v6rifiant l'identitk de Jacobi

si x E L,, y E L,, z E L,. Pour un Bl6ment de L,, on introduit sa variante Z = (-l)n+lx; on pose 1x1 = n, on a donc

2 74 R. Gillard

On suppose L munie aussi d'une structure diffkrentielle d : Lp + LP+l telle que d(x u y) = dx U y + (-1)1"1x U dy et d2 = 0.

Etant donnk n klkments € 1 , . . . , E , de la cohomologie H(L) de L, on appelle donne'e de dkfinition du produit [el, . . . , E,], une famille d'61k- ments SZ de L, indicke par les sous-ensembles propres (ni vides, ni tout) de (1,. . . , n), telle que 61,. . . , 6, reprksentent respectivement €1,. . . , E ,

et soumise B la condition (13) suivante. Pour chaque sous-ensemble I , on considkre les partitions rkgulikres I = J U K en sous-ensembles propres disjoints avec inf (J) < inf (K)

oh (J, K ) parcourt l'ensemble partitions rkgulihes de I et E(J, K ) dksigne un signe habilement choisi (c'est +1, si les ~i sont de degrk impair). La cochaine

oh (J, K ) parcourt maintenant l'ensemble partitions rkgulikres de 1,. . ., n, est en fait un cocycle [26], prop. 1.6. Le produit de Lie-Massey ([25], [26]) ou de Whitehead ([29]) [el, . . . , E,] est alors dkfini comme la classe du cocycle de (14).

On sait construire une telle structure L en partant d'une algkbre graduke (associative): il suffit de dkfinir un nouveau produit en posant [x, y] = xy - (-l)lXl'lylyx. Un exemple (B un d6tail ennuyeux prks) de telle L est fourni par le complexe de cochaines d'un groupe B valeurs dans une sous-algkbre de matrices g c MN(k), en prenant le produit (associatif) Cp(Il, g) x C4 (II, g) + CP+q (II, g) : x, y --' z = x U y d6fini

Par

Notons m la multiplication des matrices MN (k) 8 MN (k ) 4 MN (lc) et L l'involution A 8 B + B 8 A dans MN (k) 8 MN (k): Le cup produit U

est induit par fonctorialit6 par m. Sur la cohomologie Hpf Q(II, g 8 g) un produit d'une chaine de degr6 p et d'une chaine de degrk q est tranformk par L par multiplication par le signe (-1)pQ.

On notera x U y le crochet x U y - (-1)pQy U x obtenu plus haut. I1 provient du produit dans MN (lc) 8 MN (lc) --' MN (k) en appliquant le morphisme dkduit par fonctorialitk m - m o L, c'est B dire le crochet de Lie: Comme g est une sous-algkbre de Lie, on tombe en fait

Groupe des Obstructions 275

bien dans Hp+q(II, g). Pour L'(H, g) = C' (II, g), on parlera de l'algkbre diffkrentielle graduke (ADG) pour le produit U et d'algkbre de Lie diffkrentielle gradu6e (ALDG) pour le produit U. Ainsi la relation de

i commutativitk (10) n'est vQifike que sur la cohomologie, ce qui ne change pas le rksultat clef de [26]; c'est le dktail 6voquk plus haut. On rajoutera quasi (QALDG, QADG) pour marquer que la relation de commutativitk n'a lieu qu'B homotopie prks.

Pour des classes de cohomologie S et de degrk 1:

4.3. Obstructions et produits de Lie-Massey On se place dans la situation pure et on va voir que l'obstruction de

4.1 est donnke par un produit de Lie-Massey. Considkrons l'anneau

ainsi que son quotient U(m) par l'image du monhme t 1 . . . t,. On s u p pose donnke une reprksentation B valeurs dans G(U(m)) et on cherche l'obstruction B la remonter dans G(U(m)). Comme U(m) + k a une section on peut prendre p = p dans le lemme 4.1. Une base de l'dkal maximal de U(m) est donnke par les monhmes XI = xi, . . . xi,, 1 5 t < m. En enlevant X I . . . x,, on obtient une base de U(rn). Ainsi avec les notations du lemme, la dkformation gknQique de p B U(m) est donnke par pl(g) = [I + S(g)] p(g) ; on peut dkcomposer 6 E g 8 mu(,) en C 61 8 XI,

somme sur les sous-ensembles non vides de (1, . . . , m). La p&rtie 1) du lemme dit B quelle condition on a une dkformation

de p Q U(m) : dS = 6 U 6 mod (xl . . . x,). On dkcompose sur la base de monhmes: en regardant les termes de degrk 1, il faut dkjB que les 61,. . . ,6, soient des 1-cocycles. On note €1,. . . , em leurs classes. En degrk supkrieur, on obtient dbZ = 6 J U bK, somme sur toutes les partitions de I en rkunion disjointe de sous-ensemble non vides J et K . Mais J et K jouent un rhle symktrique donc en restreignant la somme aux partitions rkgulikres (J, K):

en remarquant que les classes de d6part e; sont de degrk 1; ainsi la reprksentation II + ~ ( v ( m ) ) kquivaut B une donnke de dkfinition pour le produit de Lie-Massey [e l , . . . , c,] en prenant L(Il, g) comme ALDG: celui-ci (formule (14)) est alors don& par l'inverse de la classe de a' (notations du leknme 4.1).

En consid6rant le cas m = 2 et en faisant varier e l et € 2 , on obtient

276 R. Gillard

Proposition 4.2. Le groupe des obstructions contient l'image H;(II, g) du cup-produit U : H1(II,g) x H1(II, g) + H2(II, g).

4.4. Produits de Laudal en caracthristique p

On reprend la construction de l'anneau universe1 de 2.5: la thkorie ressemble B celle de la prop. 4.2, & la complication prks que les monbmes ne sont plus linkairement indkpendants sur k et que les carr6s ne sont plus nuls; on va expliciter les obstructions on+l, ou plut6t leur image en caracthistique p. On rappelle le diagramme issu de 2.5:

Rn+l - Rn (Pn

Soit S, resp.Sn, resp.G, la rkduction de R, resp.&, resp.Rn modulo T:

- (Pn

- Sn+l - S n

1%+1 \P, 1% Sn+l - S n

(Pn

On choisit une base de mon6mes pour Sn et une base Bn+1 de monbmes pour le quotient ker& de mnc1/mnc2. On note la

rkunion de En et Bn+13. On peut assurer par rkcurrence que B:+, - contienne

Lemme 4.3. Tout mon6me s'exprime mod fj d l'aide de EL+,:

(19) xu = A,2, mod (T) + I + mn+2

Soit pn un relkvement de p B Sn : pn peut s'kcrire sous la forme

somme sur En. On relkve pn B Sn+l en pn+,(g) = [l + C6,(g) 8 z,]P(g) somme sur Ti',+, et on cherche l'obstruction b ce que pn+l soit - un homomorphisme. Comme Sn+1 + Sn est une 1-extension, le lemme 4.1 donne la condition sur les 6@, p E Bn:

--

3 B n et g+l correspondent respectivement & des bases des grad~& GT(S,) et GT(%+I)


oh dans le deuxikme membre vl et v2 parcourent En. Leur somme n'est plus dans En, mais le lemme 4.3 permet de dkcomposer sur En les deux membres de l'kgalitb de la partie 2) du lemme 4.1; en identifiant alors le coefficient de x,:

On voit alors en appliquant la deuxikme partie du lemme 4.1 et la famille d'kgalitks prkckdentes que l'obstruction on+l mod p est kgale B la somme sur Bn+1

Par dkfinition, pour p E Bn+l, la variante & la Laudal du produit de Lie-Massey des classes ~i des Si (coefficient du gknQateur xi, i = I, . . . , r ) est la classe ~ ( p ) du coefficient a, = Cul, ~ul+,,(6ul u S.,)

de x, dans l'expression ci-dessus. Elle nkcessite la donnke des 6,' v E vkrifiant (20) avec les coefficients Pup comme dans le lemme 4.3.

Remarque: On fera attention au fait que les 6, ne sont pas des COCYCLES. On ne peut donc dkcomposer leurs classes.

En s'appuyant sur 52.7, et en notant B, la rkunion de des Bn+1, n >_ 1, on obtient

Theorhme 4.4. Le groupe OF) des obstructions des d&foonations de p en caracte'ristique p est le sous-groupe HgL (11, g) de H2 (II, g) engendre' par les produits de Laudal ~ ( p ) avec p E B, . I1 contient le groupe

Hi (K 0).

Question 4.5. Peut-on faire un lien entre la filtration de ll et celle de H 2 ?

Remarque: R peut &re construit par rkcurrence & l'aide de L(II, 0). On trouve 1es Pup, En et donc 6, pour v dans En, on en tire alors a,; ceci permet de complbter le sous-groupe de H2. En en prenant une base wi et en dkcomposant on+l dessus, on obient Sn+1 comme dans 2.5. I1 serait intkressant d' associer directement & R un L(R) et de vQifier qu'il est homotope & L(II, g).

La construction exhibe en fait un groupe d'obstruction de niveau 5 n + 1, H:+l (II, g) et on a la suite d'inclusions:

278 R. Gillard

$5. Ophrateurs de Bockstein

Groupe des Obstructions

On cherche B quitter la caractkistique p. Commenqons par discuter l'obstruction initiale dkfinie par le passage

de Ro = k[rl, . . . , r,] B R I = OIXl, . . . , x,]/m2, cf. 2.4. Une premikre obstruction se prksente: peut-on relever ;Zi B G(O/(r2)) : si non, r est dans l'idkal I des relations et toutes les dkformations restent en caractkistique p. Dans la suite de ce §, on suppose qu'un tel relkvement p existe, ce qui est le cas dans les situations provenant de la gkomktrie arithmktique (courbes elliptiques, formes modulaires).

L'ktape suivante ktudie l'obstruction like au monbme rXi, i fixk; pour cela on prend la 1-extension avec A' = C3[Xi]/(r2, X,") et A = OIXi]/(r,X,") = k[ri]. Le noyau de l'homomorphisme p : A' + A est engendrk comme k-espace vectoriel par r et rXi . Le noyau de l'homomorphisme de groupes G(A1) + G(A) est g@ [Or + O r x i ] . Mais Xi correspond B un morphisme II -+ G(k [ri]) : pi (9) = [I + Si (g)Xi]p(g) avec Si cocycle B valeurs dans k. On le remonte dans 0 / ( r 2 ) : ce n'est plus un cocycle mais son bord dkfinit un 2-cocycle B valeurs dans le noyau g@OrXi. Avec l'aide du lernrne 4.1, on constate donc que cette obstruction dans H2(II, g @OrXi):H2(II, 0) est l'image de la classe de Si dans H1 (II, g) par l'opkration de Bockstein, donnke par le cobord de la suite exacte de cohomologie dkfinie par g + g2 + g oh g2 est la sous-algkbre dkfinie par G dans Mn (O/(r2)) . On dkduit alors facilemant:

Proposition 5.1. L'ide'al des relations contient l'uniformisante .rr si et seulement si ;Zi n'a pas de rel6vement a 0 / ( r 2 ) . S i de tels rel6vement existent, l'image de I modulo l'ide'al (r2, XiXj, i et j = 1, . . . , r ) contient la classe de rXi si et seulement si l'image du 1- cobord Si, correspondant par dualite' la variable Xi, par l'op4ration de Bockstein H1(II,g) -+ H2(II,g) est non nulle. Le sous-groupe des obstructions a relever Zj 6 OIX1, . . . , X,]/(r2, XiXj, i et j = 1, . . . , r ) est engendre' par l'image de l'ope'ration de Bockstein.

Remarque: Cette proposition est intkressante pour discuter sur les exemples la question 3.2 puiqu'elle explicite des klkments du H2 qui sont dans OF.

Question 5.2. on peut continuer en e'crivant des suites de Bock- stein, mais assez vite on bute sur les problPlmes de dknominateurs en caracte'ristique p (cf. formule de Campbell-Hausdorfl): peut-on remonter a 0 en utilisant des vecteurs de Wit t?

56. Quest ions arit hm6t iques

On se place dans la situation des extensions de corps de nombres. On part de p une reprksentation du groupe de Galois GF = Gal(o/F), pour F une extension de Q de degrk fini. Soit K = F(p) l'extension de F dkfinie par le noyau de p et G = Gal(K/F). Soit S un ensemble de places de F contenant les places de ramification de p ainsi que les places au dessus de p et a. Dksignons par Ks (resp. Fs) la plus grande p extension de K (resp. de F) non ramifike en dehors des places au dessus de S et posons II = Gal(Ks/F), P = Gal(Ks/K) et Gs = Gal(Fs/F); soit G , O, k comme avant. On considkre p comme une reprksentation de II.

* - ., 6.1. Sur Gs

On rksume des rksultats de [13], cf. aussi [21], et on explicite [18] 1.10 lemme 4 en termes de cup-produits dans la cohomologie de Gs. Les invariants principaux de Gs sont son nombre de gknhateurs:

!

d(S) = 7-2 + 1 + r ( S ) ,

et son nombre de relations:

oh ps dksigne la dimension de Es = Hom(Vs, ZlpZ) et Vs le groupe quotient par F* du groupe des x E F* qui sont des puissances locales pikmes dans S et qui engendrent un idkal qui est aussi une puissance pihme.

Si la reprksentation p est B valeurs dans le groupe linkaire GLN(k) et si N = 1, R est simplement z~[[G:~]]. Soit A le dkfaut de la conjecture de Leopoldt (cf. [32]); il intervient dans une dkcomposition q b 1 zz+l+A

P @ ~ i ~ l Zp/pni Zp, avec ni > 0: alors H1 (G:~, Z/pZ) =

H1 (Gs, ZlpZ) est de dimension

On peut remarquer que H2(Ggb, Z/pZ) est la somme directe de 2 sous espaces vectoriels de dimension r (engendrk par des klkments de Bock- stein) et ( d ( 2 S ) ) (engendrk par des cup-produits). On a donc

280 R. Gillard

Le nombre de gknkrateurs de R sur Z, est d(S). Le nombre de relations

est r = dimk o$), & comparer & dim H~ (ll, Z/pZ) = dim H~ (GS, Z/pZ) = r(S) = d(S) - (r2 + 1) = r + A. On voit que dans ce cas R est une intersection complkte. Avec les notations du 54, on peut donc reformuler l'knonck du lemme 4 de [18]:

Proposition 6.1. Le de'faut de la conjecture de Leopoldt est &gal ci ci l'excks de la dimension de Kmll duns le cas N = 1. C'est la dighence des dimensions entre H2 (ll, ZlpZ) et H:L (ll, ZlpZ).

6.2. Annulation des obstructions par r6duction l'action t riviale

Le but de ce 5 est d'expliquer la mkthode gknQalement suivie pour l'ktude des dkformations. On peut noter que chaque obstruction like & une petite extension p correspond & un problkme (faible) de plongement & ramification restreinte ([20], [22]). En effet, partant d'un homomorphisme ll -+ G(A) (son noyau correspondant & une extension L de F) et de la donnke d'une surjection G(p) : G(A1) + G(A) dkfinie par p on cherche & le relever en ll --t G(A1) avec un diagramme commutatif; c'est exactement le problkme de plongement de L I F dkfinie par G(p) avec la condition de restriction de la ramification & S. On sait que la mkthode locale-globale est classique dans le problkme de plongement; c'est celle qui est suivie en g4nQal pour vQifier la nullitk du groupe des obstructions, cf. aussi [3] pour une variante dans le cas << Bore1 >. On montre en gknQal la nullit6 du H2 global en s'appuyant sur celle des H~ locaux. La mkthode s'achkve en gdnQal en vkrifiant la nullitk d'un groupe de classes qui assure qu'il n'y pas de nouvelles relations (purement globales). On relira [24] pour la prise en considkration des nouvelles obstructions dans le problkme de plongement.

Dktaillons la mkthode 8. l'aide de quelques lemmes. La premikre ktape est d'utiliser la dualit6 globale de Poitou-Tate. Pour un module galoisien M pour l'action de Gal(Ks/F), annul4 par p, on note LLIi(Ks/F, M) le noyau de l'application obtenue en prenant le produit des applications dans les groupes de cohomologie locaux:

Lemme 6.2. Le groupes U12(Ks/F, M) et UI1 (Ks/F, Hom(M, p,)) sont en dualite' exacte.

Le lemme suivant donne une condition d'annulation assez souvent vQifike. Posons K' = K(pp). Distinguons les conditions:


- C1) K ne contient pas le groupe p, : le groupe A := Gal(K1/K) n'a que 1 comme point fixe dans pp et est d'ordre premier & p : Hi(A, Hom(g, p,)) = 0 si i > 0, d'oh H1(K'/F, Hom(g, p,)) =

H1 (K/F, Hom(g, pp)A) = H1 (K/F, Hom(g, 4)) = 0.

- C2) F contient dkjB p,, G = GL2, p dkfinit une surjection sur GL2(IFp) et p 2 5. On a alors un isomorphisme de modules

galoisiens Hom(g, p,) 2 Hom(g, IF,) et on peut reprendre le raisonnement de [8], lemme 1.2.

- C3) On suppose ici que p provient d'une courbe elliptique dkfinie sur F = Q avec une condition d'irrkductibilitk absolue pour la restriction & Gal(Ks, k) avec k = Q( J(-1)Plp), c'est le sous- corps quadratique de Q(p,), et on s'appuie sur le lemme 19 de [7]; go est le sous-module des matrices de trace nulle.

On peut alors knoncer:

Lemme 6.3. Si une des conditions Ci pre'ce'dentes est ve'rifie'e, le groupe H1 (Kt / F, Hom(M, p,)) est nu1 pour M = g pour C1 ou C2 et M = go pour C3.

Notons Cls(K1) le quotient du groupe des classes du corps K' par le sous-groupe engendrk par les classes des idkaux premiers au-dessus de S. On note Cls(K1), son plus grand quotient annul6 par p.

Lemme 6.4. Avec M comme duns le lemme pre'ce'dent, on a un isomorphisme de Gal(K1/F)-modules: LLI1 (Ks/K1, Hom(M, p , ) ) S ~ o m

( M €4 C1s(K1)p, PP).

De'monstration: En effet l'action de Gal(Ks/K1) sur Hom(M, p,) &ant triviale, on peut sortir Hom(M, p,) dans H1 (Ks/K1, Hom(M, p,)) de faqon Gal(K1/F)-kquivariante pour se ramener au calcul classique de LLI1 (Ks/Kt , p,) par la thkorie de Kummer.

On a donc obtenu:

Proposition 6.5. En supposant une des conditions Ci, ainsi que l'absence d'obstmction locale en S, la nullit6 du module HomGaqKtIF) (M €4 Cls(K1),, p,) entraine la nullite' du groupe H2(ll, M) et donc l'absence d'obstmctions duns le problkme de de'formation de p (ci de'ter- minant fix6 duns le cas C3).

Remarque: On a donc un critkre extrement brutal pour pouvoir avancer une rbponse positive B la question de Mazur. On aimerait,

282 R. Gillard

comme pour la conjecture de Leopoldt, pouvoir disposer d'un raffine- ment en termes de diviseurs dans un module d'Iwasawa associk. On pourrait aussi rkver d7une variante analytique en termes de 0 ou de pile d7une fonction L padique.

6.3. Etude du contrble de l'idkal des relations C'est une question naturelle d'ktudier le contrde de 17idkal des rela-

tions relativement aux extensions. I1 est & noter que pour les anneaux de dkformations eux-meme, Hida ([lo], [I 11) a montrk 17existence d'un tel contrde moyennant des hypothkses raisonnables. Ce paragraphe rksume sa th6orie et montre que la conjonction d'un contrGle des relations et de la nullitk d'un invariant p d'Iwasawa implique la nullite' des relations. On considkre d7abord une extension galoisienne F1/F incluse dans Ks. On fera ensuite varier F' parmi les ktages de la Zp-extension cyclotomique F,/F. Pour simplifier on suppose F' et K linkairement disjointes sur F . On note II' et A les groupes de Galois de Ks sur F' et de F' sur F . On suppose que 17action de II et donc de II' sur le sous-module des matrices de trace nulle go de g est sans point fixe. On se restreint au cas oh le groupe G est simplement GLN. Considkrons les hypothkses ci-dessous oh x d6signe un relkvement de det(p) B valeurs dans 0.

- HH1) le dkterminant de la dkformation est fix&: det(p) = x

- HH2) l'action adjointe de II' est sans point fixe dans go

- HH3) p ne divise pas la dimension de l'espace de la representation p

- HH4) Le centralisateur de l'image de p', la restriction de p & II', est rkduit au centre (HH2 est la condition d6rivke).

Hida, cf. [lo] prop. A 2.3, et [I 11 cor. 3.2 dhmontre, sous HHl), HH3) et HH4), que l'anneau des dkformations (& dkterminant fix6) R relatif B II et p s7identifie B l'anneau des coinvariants de l'anneau de dkformations R' de p' pour 17action par fonctorialitk de A. I1 en d6duit que l'espace cotan- gent t* de R s7identifie aux coinvariants de celui tl* de R'. ConsidQons la prksentation de R' B 17aide de l'algkbre de sQies formelles compl6tke de l'algkbre symktrique sur tl*: si cette prksentation est e'quivariante, on dkduit que l'id6al des relations pour ll s'identifie aux coinvariants dans 17id6al correspondant pour II': c7est cette condition qu'on appelle contr6le de l'ide'al des relations duns l'extension F1/F; par dualitk on obtient une injection sur les groupes d70bstructions respectifs 0 - 0' . Supposons maintenant que ce phknomkne existe pour tous les ktages F, IF de la Zp-extension cyclotomique F,/ F , supposke linkairement


disjointe de K . Notons IIL le groupe de Galois de Ks sur F,: on a ainsi un diagramme:

On d6duit un diagramme analogue en passant B la limite sur n oh on pose ll', = Gal(Ks/F,)

r e s - ~ 2 ( r I ' , , go) .

Notons L I F la sous-extension de K dkfinie par le psous-groupe de Sylow de Gal(K/F). Iwasawa, cf. [12] th. 3, a remarquk que les nul- litks des invariants p des Zp-extensions cyclotomiques K, et L, de K et L sont kquivalentes. De cette nullitk, on dkduit celle des groupes H 2 (KSIK,, go) ou H 2 (Ks/L,, en interprktant comme Kuzmin, cf. [14] $5, la nullitk de p en termes de libertk de pgroupes pro-finis. On peut alors dkduire la nullitk de H2(II',, go) de la nullitk de l'invariant p pour la Zp-extension K,/K par un argument de restriction et corestriction, cf. [5] I11 th. (10.3). On a donc:

Proposition 6.6. E n supposant K / F et F, line'airement disjointes, si les idkaux de relations sont contr6le's dans Ees extensions F,/F, alors la nullite' de 17invariant p d'Iwasawa pour K,/K, ou L,/L si on pre'fire,implique celle de l'ide'al des relations dans 17anneau R des dt5formations de p h de'temninant 6x6.

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[18] B. MAZUR, Deforming Galois Representations, In K. Ribet et J.-P.Serre Y. IHARA, editors, Galois Groups over Q, number 16 in MSRI, pages 385-437, 1987.

[19] B. MAZUR, An introduction to the deformation theory of Galois representations, IN G. CORNELL ET J. H. SILVERMAN E T G. STE- VANS, editors, Modular forms and Fermat's last theorem, pages 243- 312, Springer Verlag, 1997.

[20] J . NEUKIRCH, ~ b e r das Einbettungsproblem der algebraischen Zahlen- theorie, Inv. Math., 21(1973), 59-116.

(211 0 . NEUMAN, On pclosed number fields and an anlogue of Riemann's existence theorem, In Frohlich, editor, Algebraic number fields, pages 625-647, Acad. Press, 1977.

[22] A. NOMURA, Embedding Problems with Restricted Ramifications and the Class Number of Hilbert Class Fields, Expos15 B cette confkrence (CFT 1998).

[23] B. PERRIN-RIOU, Fonctions L p-adiques des r4pre'sentations p-adiques. Astbrisque., 229. Paris: 200 p. (1995).


1241 G. POITOU, Conditions globales pour les problkmes de plongement B noyau abdien, Ann. Inst. Fourier, 29(1979), 1-14.

[25] V. S. RETAKH, Massey operations in the cohomology of Lie super- algebras and deformations of complex analytical spaces, Funct. Anal Appl., 11(1977), 88-89.

[26] V. S. RETAKH, Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras, JPA Alg, 89(1993), 217-229.

[27] J. P. SERRE, Corps Locaux. Hermann, Paris, 1996. [28] M. SCHLESSINGER, Functor of Artin rings, Trans. Am. Math. Soc.,

130(1968), 218-22. [29] D. TANRE, Homotopie rationelle: Moddes de Chen, Quillen, Sullivan.

Lecture Notes in Math., 1025. Berlin-New York: Springer, 211 p. (1983).

[30] J. TILOUINE, Deformations of Galois representations and Hecke algebras. Publ. Mehta Res. Inst., Narosa Publ. Dehli, 1996.

[31] A. VISTOLI, The deformation theory of local complete intersections, pr6publication klectronique: alg-geom/9703008, 1997.

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Institut Fourier, UMR 5582 Universite' Joseph FOURIER et C.N.R.S. BP 74 F-384 02 Saint Martin d'H2res E-mail address: roland. gil lardkanadoo . f r


Abelian Varieties over Q(&) with Good Reduction Everywhere

RenQ Schoof

Abstract.

The elliptic curve with Weierstrass equation y2 + &XY - Y = x3 - (2 + &)x2 has good reduction modulo every prime of the ring of integers of Q(&). We show that every abelian variety over Q(&) that has good reduction everywhere is isogenous to a power of this elliptic curve.

1 . Introduction

In [12] B. Setzer shows that the elliptic curve £ given over Q(&) by , the equation

1 has good reduction at all primes of the ring of integers z[&]. This can be seen from the fact that the discriminant of E is equal to the

I unit (5 + 2&)3. Let F = Q(-, G) denote the unique unramified

quadratic extension of Q(&). In this paper we prove the following two

[ theorems. Below we give the easy proof of the fact that the two results directly imply one another.

I

Theorem 1.1. Every abelian variety over Q(&) that has good

reduction at all primes of z[&] is isogenous over ~ ( 4 ) to a power of the elliptic curve £.

Theorem 1.2. Every abelian variety over F that has good reduc- t ion at all primes of the r;lng of integers OF is isogenous over F to a power of the elliptic curve E .

Received December 7, 1998 Revised February 12, 1999

288 R. Schoof

For several number fields K it is known that there do not exist any non-zero abelian varieties over K at all with good reduction everywhere [I, 5, 111. In contrast, Theorems 1.1 and 1.2 say that over the

number fields $ ( A ) and Q(-, G), there exists, up to isogeny, precisely one simple abelian variety with good reduction everywhere. At present Q(&) and its unramified extension Q ( G , G) are the only number fields for which I know how to prove a statement like The- orem 1.1 or 1.2. Under the assumption of the Generalized Riemann Hypothesis however, one can prove similar results for a few more number fields.

We briefly sketch the proof of Theorem 1.2. An abelian variety with good reduction everywhere over F = Q ( G , G) is by definition the generic fiber of an abelian scheme A over OF. Let g = dim(A). Let E denote an abelian scheme of dimension 1 over OF with generic fiber isomorphic to Setzer's curve E. We show that for every n 2 1 the finite flat subgroup schemes A[2n] and Eg[2"] of Zn-torsion points are isomorphic over OF. Faltings' isogeny Theorem [4] implies then that A and E g are isogenous.

In order to show the statement about the torsion points, we give a rather complete description of the commutative finite flat group schemes over OF of rank a power of 2. In Section 2 we first determine all simple such group schemes; we invoke A.M. Odlyzko's discriminant bounds [9] and the theorems of J.-M. Fontaine [5] and V. AbraSkin [I] on the ramification of the action of the Galois group on the points of these group schemes. Since we use the "unconditional" Odlyzko bounds, it is crucial that we consider group schemes whose rank is a power of 2 rather than a power of some prime p > 2. The next step is the determination of several extensions of the simple group schemes by one another. This is the content of Section 3. In this section we make use of a local result of C. Greither's [6]. From this and an application of Weil's Riemann Hypothesis for abelian varieties over finite fields, we obtain severe re- strictions on the structure of the group schemes A[2n]. In Section 4 we prove that A[2n] S Eg[2n] for all n 2 1 and we derive Theorem 1.2 from this.

In the remainder of the introduction we collect some information concerning the elliptic curve E . The j-invariant of E is 8000. Therefore E acquires complex multiplication by Z [ G ] over the extension F =

Q(-, G) of Q(&). The endomorphism 9 E End(E) induces a 2-isogeny from E to El that is defined over Q(&) . Here El denotes the quadratic twist of E that is associated to the extension Q(-, G).

From this we easily deduce that Theorem 1.2 implies Theorem 1.1.

Abelian Varieties over Q(&) 289

Suppose that A is an abelian variety over Q(&) that has good reduction everywhere. Then A also has good reduction everywhere over the extension F and hence it is isogenous to E g over F. Taking Weil re- strictions, it follows that A x A' is isogenous to E g x Elg over Q(&). Here A' denotes the twist of A associated to the quadratic extension Q(&) c F. Since E is isogenous to E', Theorem 1.1 follows.

Conversely, to see that Theorem 1.1 implies Theorem 1.2, we consider an abelian variety A over F that has good reduction everywhere and we take its Weil restriction to $(&). Since the extension Q(&) c F is only ramified at the infinite primes, the Weil restriction has good reduction everywhere over Q(&). Therefore it is isogenous to E 2 g ,

where g = dim A. Theorem 1.2 now follows easily by extending the base field to F.

We make some final remarks concerning the curve E . The three points of order 2 of E have their x-coordinates equal to x = -1 and

l+fi*(G+a)i respectively. Their y-coordinates are given by y = 2

. The point with x = -1 is the only 2-torsion point that is

rational over Q ( G , G ) . The curve E has exactly six torsion points defined over Q(&). They are (0,O) and its multiples (2+ &, -5 - 2&),

( 4 7 i& ), (2 + fi, O), (0 , l ) and co. Over F the curve E has exactly 18 rational torsion points, nine of which are the 3-torsion points. The curve E admits two Q(&)-rational isogenies of degree 3. The kernel of one consists of the Q(&)-rational points of order 3. The other has the

points (- 1, (*G+l)(*m+l) 2 ) and co in its kernel. Dividing E by either of its rational subgroups of order 3, we obtain two more elliptic curves over Q(&) that have good reduction everywhere. The j-invariants of

&

these curves are equal to 8OOO(49 f 1 2 f i ) ~ (5 f 2fi)2. The curves admit complex multiplication by the non-maximal order Z[3-1. See [8] for a description of these curves and their Q(&)-rational isogenies. The existence of these curves shows that we cannot replace "isogenous" by "isomorphic" in Theorems 1.1 and 1.2.

92. Simple 2-group schemes

In this section we determine all simple finite flat commutative group schemes over the ring of integers of F = Q ( G , m) of rank a power of 2. The main result is Theorem 2.3. In this section we denote by 5, a primitive n-the root of unity. As usual we let i = C4.

1

290 R. Schoof

Let p be a prime and let R be a commutative domain with 1. In this section and the next we study finite flat commutative group schemes of ppower order over various rings R. We call such group schemes p-group schemes. Examples are provided by the constant group schemes Z/pnZ and their Cartier duals ppn. For later use we recall a construction, due to N. Katz and B. Mazur, of certain pgroup schemes of rank p2. See [7, Interlude 8.71 or [Ill for more details. For a unit E E R* we consider the R-algebra

The scheme G, = Spec(A) is a finite flat group scheme over R with multiplication of two points (t , i) and (s, j) (with tp = ci, sp = ~j and 0 5 i , j < p) given by

(ts, i + j); if i + j < p ,

( ~ s / E , i + j - p); if i + j 2 p.

The group scheme G, is an extension of Z/pZ by pp. It is killed by p and its K-valued points generate the extension K(&, fi) of the quotient field K of R. Two group schemes G, and G,, are isomorphic if and only if E/E' is a p t h power.

A pgroup scheme is called simple if it does not admit any closed flat subgroup schemes other than 0 and itself. Any pgroup scheme of rank p is simple. Every pgroup scheme admits a filtration with closed flat subgroup schemes whose successive quotients are simple.

In our main application we take p = 2 and R = OF, the ring of integers of F = Q ( G , G). Let q = -+ G. The unit group 0: is generated by C6 and q. There lies a unique prime over 2 in OF. It is generated by and its residue field has 4 elements. We'll see below that F does not admit any non-trivial everywhere unramified abelian extensions.

We already mentioned the fact that the constant group scheme 2 / 2 2 and its Cartier dual p2 are simple 2-group schemes over OF. Another simple 2-group scheme is provided by the kernel E[T] of the endomorphism .rr = f End(E) of Setzer's elliptic curve E that has been described in the introduction. Since E has good reduction everywhere, E[T] is finite and flat over OF. Since E has supersingular reduction at the unique prime over 2, the group scheme E[T] is local and has a local Cartier dual. The main result of this section is Theorem 2.3. It says that these three group schemes are the only simple 2-group schemes over OF.

The proof of Theorem 2.3 involves a rather detailed knowledge of

Abelian Varieties over Q(&) 29 1

certain extensions of the number field F that are only ramified at the unique prime over 2. We isolate the facts we use in two lemmas.

Lemma 2.1. Let q = + and let K be a number field satisfying

F c K c F(<167 &).

Then there is only one prime of K lying over 2 and K does not admit any non-trivial abelian extension that is at most tamely ramified at this prime.

Proof. The degree [F (Cl6, fi) : F] is equal to 8. Note that F (i) =

Q(C24). We let K1 = F(i, fi) = Q((24r fi) and K2 = F ( C l 6 fi) = Q(&, fi). The fields K satisfying F C K c K2 fit in the following diagram:

The techniques to prove this lemma are standard. We first compute the discriminants A and the root discriminants aK = 1 A 1 l/[K:QI Of

all subfields K. All fields K are abelian extensions of F that are unramified out-

side 2. We compute the conductors of the Dirichlet characters of Gal(K2/F). By [3, Lemma 61, the root discriminant of Q(<24) is equal

1 to 4fi. Applying the conductor discriminant formula to the extension F c Q(C24), we see that the conductor of the corresponding quadratic

L 6 character is equal to (a)2 = (2). On the other hand, since q r I 1

1 + a (mod 2), it follows from class field theory that the 2-part of the ray class group of F of conductor (2) has order at most 2. It follows / that 4(C4) is the full ray class field of conductor (2) of F and that the

292 R. Schoof

character corresponding to Q(CZ4) is the unique character of conductor (2). Since the discriminants of the polynomials T2 f q are equal to (4), the quadratic characters corresponding to the extensions F(fi) and F ( f i ) each have conductor (4), possibly divided by an even power of ( G ) . Since neither character can have conductor (2), both have conductor equal to (4). Finally, since the root discriminant of Q(C8) is equal to 8 f i (see [3, Lemma 6]), it follows easily from an application of the conductor discriminant formula over F that both characters of order 4 of Gal(Q(C48)/F) have conductor ( G ) 5 . This implies that all four characters of order 4 of Gal(K2/F) have conductor (J_-2)5.

An application of the conductor discriminant formula over F gives that the root discriminants of the intermediate fields are given by

It follows that the prime over 2 in K2 is totally ramified over F. Hence there is in every intermediate field K precisely one prime lying over 2.

Next we apply Odlyzko's discriminant bounds [9, p.1871 to show that none of the subfields K admit a non-trivial everywhere unramified abelian extension. For every subfield K, let hK denote the degree of the maximal abelian everywhere unramified extension H K of K . We must show that hK = 1 for each K . We start with F itself. The root discriminants of F and HF are equal to < 4.899. Odlyzko's discriminant bounds imply that [HF : Q] < 6 and hence [HF : F] = 1. In a similar way, Odlyzko's discriminant bounds imply at once that h~ = 1 for K = Q(C24) and K1. The bounds imply that hK < 2 for the two subfields K = F( f i ) . Since only the prime over 2 ramifies in these quadratic extensions of F, it follows from [14, Thm. 10.41 that hK = 1 for K = F( f i ) . Odlyzko's bounds imply that hK < 3 for K = Q(C48) and K = Ki . Both fields are cyclic extensions of degree 4 of F . Since the common quadratic subfield Q(C24) admits no non-trivial abelian everywhere unramified extension, it follows from [14, Thm. 10.81 that hK cannot be equal to 3. Since only the unique prime over 2 is ramified, hK cannot be equal to 2 either. Therefore hK = 1 for both fields K . Finally Odlyzko's bounds show that hK2 < 5. Since the Galois group of K2 over Q(C24) is isomorphic to the Klein four group, the odd


part of hK, is equal to the product of the odd parts of the numbers hK corresponding to the three quadratic subfields K . Therefore the odd part of hK, is trivial. By [14, Thm. 10.41, the degree hK2 cannot be 2 or 4 either. Therefore it is 1.

Finally we apply class field theory to show that none of the subfields K admits an abelian extension that is at most tamely ramified at the unique prime over 2. Since hK = 1 for each subfield K , the ray class group of conductor the unique prime over 2 is isomorphic to the multiplicative group F: modulo the images of the global units. Since C3 E 0; for each subfield K , this ray class group is trivial for each K . This proves the Lemma.

Lemma 2.2. Let L be a Galois extension of Q that contains F and for which the following hold:

- F c L is unramijied except at the unique prime over 2; - i and fi are contained i n L;

- the root discriminant dL of L satisfies dL < 8&. Then [L : F] is a power of 2.

Proof. Since dL < 8& < 19.596, Odlyzko's bounds [9] imply that [L : $1 < 380. This implies that [L : F(i , fi)] < 380116. Therefore we have the following inclusions of fields (the superscripts indicate the relative degrees) :

Since the degree [L : F(fi, i)] is less than 60, the Galois group Gal(L/F(fi, i ) ) and hence the Galois group 7r = Gal(L/Q) are solvable groups. We will show that 7r is a 2-group.

The largest abelian extension F' of Q inside L contains F(i) =

Q(C24). Since L is only ramified at the prime over 2 of F and since OL < 8&, there are only two possibilities for F'. Either F' = Q(C24) or F' = Q(C48). This shows that 7r/7r1 = Gal(F1/Q) is a 2-group. Here 7r'

denotes the commutator subgroup of 7r. We distinguish the two cases.

Case 1. F' = Q(&). Since [L : Q] < 380, we have the following diagram of extensions (the superscripts indicate the relative degrees) :

By Lemma 2.1, the field Q(C48) does not admit an abelian extension that is at most tamely ramified at its unique prime over 2. This implies that 7r'/7r1' is a finite 2-group.

294 R. Schoof

The order of d/dl is at least 2. If it is actually equal to 2, the fixed field of dl is Q (C48, &) . By Lemma 2.1, this field does not admit an abelian extension that is only tamely ramified at the unique prime over 2. Therefore dl/dl' is a 2-group. This implies that n-'/d" is a 2-group which is cyclic modulo its commutator subgroup. Therefore d/dl' itself is cyclic and hence dl/dl' and hence d' are trivial. It follows that T is a 2-group.

If the order of d/d l is at least 4, we have that #TI' 5 5, so that d' is abelian. If #dl has order 2 or 4, the group 7r is a 2-group and we are done. If not, then d' is cyclic of odd order and the exact sequence

is split. Since Aut(x1') is abelian, d is in the kernel of the homomorphism .rr ---t Aut(dl) which is induced by conjugation. This implies that 7r' is isomorphic to the direct product of d/dl and TI'. This shows that .nl' is trivial. This shows that .rr is a 2-group as required.

Case 2. F' = Q(C24). Since [L : Q] < 380 we have the following diagram of extensions.

By Lemma 2.1, the field Q(C24) does not admit an abelian extension that is at most tamely ramified a t the unique prime over 2. This implies that d/dl is a finite 2-group.

The order of .rrl/d' is at least 2. If it is actually equal to 2, the fixed field of TI' is Q (c24, Jii) . By Lemma 2.1, this field does not admit an abelian extension that is only ramified at the unique prime over 2. Therefore 7r1'/?t-"' is a 2-group. Since d/d1 is cyclic, the group .rr1/d" is itself cyclic. Therefore T" is trivial and T is a 2-group.

Therefore we assume that #(d/dl) 2 4 and hence #TI' 5 11.

Claim. The odd part of dl/d"' is cyclic.

Proof of the Claim. If the odd part of d l / P is is not cyclic we have that d'/~"' Z C(3) x C(3) where C(3) denotes a cyclic group of order 3. It follows that dl' is trivial, that #(.rr1/.n") = 4 and that [L : Q] = 8 . 4 . 9 = 288. Let K denote the fixed field of TI'. We have the following inclusions

The field K is an abelian extension of Q(C24) of degree 4 which is only ramified at the prime over 2. By Lemma 2.1, the subfield Q(c24, fi)


does not admit a non-trivial unramified abelian extension. Therefore the field K is a ramified quadratic extension. This implies that the prime over 2 in Q(Q4) is totally ramified in K and hence its residue field is again F4. Since C3 E K , it follows from class field theory that K does not admit any cyclic extension of degree 3 which is tamely ramified at the unique prime over 2. Therefore, if d'/.rrl" Z C(3) x C(3), the extension L is everywhere unramified over K . This implies that OL = d K .

Let x denote the quadratic character corresponding to the extension Q(G4) c Q(C24, Jii). We saw in the proof of Lemma 2.1 that the root

discriminant of Q(C24, Jii) is equal to 25/4&%. Since C8 - 1 generates the unique prime over 2 in Q(C24), an application of the conductor discriminant formula over Q(C24) then implies that the conductor of x is equal to (& - I ) ~ . Suppose (C8 - l)a is the conductor of one of the other two non-trivial characters of Gal(K/Q(C24)). If a > 6, both these characters have conductor (& - l)a and applying the conductor discriminant formula to K gives that

In other words a < 9 and hence a 5 8. Therefore aL = aK < 227/831/2 < - t i I

17.9697. This inequality obviously also holds when a < 6. Odlyzko's I bounds imply then that [L : Q] < 170. This contradicts the fact that

I [L : Q] = 288. Therefore it cannot happen that iil'/?i'" E C(3) x C(3) and our claim follows. I Let P denote the minimal subgroup T"' c P c d' of odd index in dl. By the claim, #/P is cyclic so that Aut(d l /P) is abelian. There- fore d is in the kernel of the natural map 7r ---t Aut(dl /P) and the action by conjugation of d/d l on dl/P is trivial. Since the exact se-

1 I is split, we conclude that a"/P is trivial. In other words, 7r"/?r1" is a

b 2-group. i: i: If a"/?rl" is trivial, it follows from the solvability of a that ?rl' is

trivial and hence that 7r is a 2-group. If ~ r " / d " has order 2, and d/d l 1 has order 4, the group a/rrl" would be a group of order 64. But this

1 is impossible since all groups of order 64 have an abelian commutator

t subgroup. In all other cases either #(d1/d'') > 4 or #(d/dl) 2 8 so 1 that the order of T is divisible by 128. Since #n 5 380, it follows that

.rr is a 2-group as required.

Theorem 2.3. The simple 2-group schemes over OF are 2 /22 , pz and E [TI.

296 R. Schoof

Proof. Suppose that G is a simple 2-group scheme over OF. Then G is annihilated by 2. Let GI be the product of all Gal(F/Q)-conjugates of G and of the Katz-Mazur group schemes G, where e E 0: runs through a set of representatives of 0g / (0>)2 . Then G' is a finite flat group scheme over OF that is annihilated by 2. Let L be the extension that we obtain by adjoining the points of GI to F . This is an extension of F that is unramified except possibly at the unique prime of F lying over 2. By construction, L is a Galois extension of Q and it contains & for every E E 0:. Since GI is killed by 2, the results of Fontaine [5] and AbraSkin [I] imply that the root discriminant dL of the number field L satisfies

dL < dF21f * = 4 6 5 19.596.

Therefore all conditions of Lemma 2.2 are satisfied and we conclude that Gal(L/F) is a 2-group.

Since all points of our simple group scheme G become rational over L, the Galois group Gal(L/F) acts on the group G(F). Since both these groups are 2-groups, there is a non-trivial fixed point. Such a point generates a G~~(F/F)-submodule of G(F) of order 2. In other words, the point generates a closed subgroup scheme of rank 2 of the base change of G to F. The Zariski closure of this subgroup scheme inside the OF-group scheme G is a finite flat closed subgroup scheme of G over OF. By simplicity it must be equal to G. This shows that G has rank 2.

To complete the proof we observe as in [lo, p.11 or [13, Example (3.2)], that for any ring R, any finite flat R-group scheme of rank 2 is isomorphic to Galb = Spec(R[X] /(X2 + a x ) ) with comultiplication given by X H u +v + buv E R[u, v]/(u2 +au, v2 + av) for certain a, b E R satisfying ab = 2. Two group schemes Ga,b and Gatjb/ are isomorphic when a = ua' and b = u-'b' for some u E R*. In order to apply this

2 to the ring OF, we observe that 2 = -- is a factorization of 2 into prime factors and hence that there are, up to isomorphism, three group schemes of order 2 over OF. They correspond to a = 1, G, 2 respectively. We recover 2 /22 and p2 by taking a = 1 and a = 2 respectively. The group scheme corresponding to a = is local and self dual. It is isomorphic to the group scheme E[T].

This proves the theorem.

53. Extensions

As in the previous sections we let F = Q(-, fl) and write OF for the ring of integers of F. In this section we compute various

Abelian Varieties over ~(fi) 297

extensions of the three simple 2-group schemes p2, E[T] and 2/22 by one another over OF. We do this by determining the extensions locally as well as generically. The global result over OF follows from an application of an exact Mayer-Vietoris sequence.

The following result is in [ l l ] .

Proposition 3.1. ( "Mayer- Vietoris") Let K be a number field, let p be a prime and let G and H be two p-group schemes over OK. Then there is a natural exact sequence

Here G,, GI P and G1 denote the base changes of G to OK €4 Z,, OK[l/p]

and K @ Q, respectively. Similarly for Hp, . . . etc.

Most maps in the sequence of the proposition are the obvious ones. The only one that needs explanation is

We use the fpqc covering {Spec(OK €4 Z,) , Spec(OK [1 /p]) ) of Spec(OK) to give a simple description of the category of pgroup schemes over OK and to define 6. See [2, Thm. 2.61 for this description and [ll] for the definition of the map S and for a proof of the fact that the sequence is exact.

Proposition 3.2. We have the following:

(i) ExtL, (p2, 2/22), Extb, (E[T], 2/22) and ExtL, (p2, E[T]) are all trivial.

(ii) EX~;,(Z/~Z, 2/22) is cyclic of order 2; it is generated by the extension 2/42.

(iii) Ext&,(p2, p2) is cyclic of order 2; it is generated by the extension pq.

Proof. As we have seen in the previous section, the field F does not admit any unrarnified quadratic extensions. This is the main arithmetical ingredient in the proof.

298 R. Schoof

(i) Suppose G is an extension of p2 by 2/22. In other words, there is an exact sequence of 2-group schemes over OF

Over the completion of OF at the unique prime over 2, the connected component gives a section and the sequence is split. Therefore the action of G ~ z ( F / F ) on G(F) is everywhere unramified. It follows that the Ga- lois action is actually trivial. So, the extension is locally as well as generically trivial.To finish the proof we observe that Homo, (p2, 2/22) =

HomoFNz2(p2, 2/22) = 0. Moreover, since there is only one prime of OF lying over 2, both groups HornF (p2, 2/22) and HornoFBg2 (p2, 2/22) have order 2. Therefore it follows from the Mayer-Vietoris sequence of Prop. 3.1 that the extension is split over OF. This proves (i).

Since E[n] is a local group scheme, exactly the same proof shows that Ext;, (E[n], 2/22) vanishes. By Cartier duality we then also have

that Ext;, (p2, E[a]) = 0. This proves (i).

(ii) Any extension G of 2 /22 by 2 /22 is Btale. Therefore, adjoining its points to F gives rise to an everywhere unramified extension of degree at most 2. It follows that the Galois action on G(F) is actually trivial. By Galois theory, the &ale group schemes over OF are determined by their generic fibers. It follows that G Z 2 /42 or that G 3 2 /22 x 2 /22 as required. (iii) This part follows from part (ii) by Cartier duality.

Let 0 2 denote the completion of OF at the unique prime over 2. We have that 0 2 2 Z2[[3] [ m ] . Let F2 denote the quotient field of 0 2 . The following result is a special case of a result of Greither's (61.

3 Lemma 3.3. Let U = { u E 0; : u 1 (mod )}. Then

Moreover, the quadratic extension of F2 generated bg the points 0.f the extension of E[n] by E[T] corresponding to u E U/(U n o ; ~ ) is the field F2 (6).

The first statement is a special case of [6, Cor. 3.6 (a)]. If one carefully goes through Greither's proof, one finds that the field extension corresponding to the unit u E U is F 2 ( 6 ) .


Proposition 3.4. The extension group Ext;, (E[n], E[nl) is cyclic of order 2; it is generated by the extension E[2].

Proof. Let G be an extension of E[n] by E[n] over OF. We recall that F2 denotes the quotient field of the completion of OF at the unique prime over 2. By Lemma 3.3, the field obtained by adjoining the points of G to F2 is F 2 ( 6 ) , where u is some unit that is congruent

3 to 1 (mod fl ). Since -u -- 1 (mod f12), the discriminant and conductor of this quadratic extension of F2 divide the discriminant of

the polynomial T2 - OT - (1 + u)/2 which is 4 / m 2 = ( G ) 2 .

Since the action of Gal(F/F) is unramified outside 2, this implies that the field obtained by adjoining the points of G(F) to the number field F is contained in the ray class field of F of conductor (2) = (@)2. In the previous section we have already seen that the ray class field of F of conductor (G) is F itself. Since -1 - 1 (mod 2) and q =

I p+ = 1 + (mod 2), we see that the ray class group of F of conductor (2), which is isomorphic to (OF/(2)) */ (- 1, q, &) , has order at most 2. On the other hand, the extension F c F(i) is easily seen to have conductor (2). This shows that the ray class field of F of conductor (2)

I 1

is equal to F(i) . It follows from this and the Mayer-Vietoris exact sequence of

Prop. 3.1 that Ext;, ( ~ [ n ] , E[T]) has order 2. As we remarked at the end of the introduction, the elliptic curve E admits only one rational point of order 2. Therefore E[2] is a non-trivial extension of E[n] by E[T] and hence this extension represents the non-trivial element of Extk, (E[a] , El*]) and we are done.

j I The extensions of 2/22, p2 and E[n] by one another that are not

listed in Propositions 3.2 or 3.4, are in general not trivial. For instance, i there is an exact sequence I

! The Katz-Mazur group schemes G, represent non-trivial classes in

~ x t b , ( z / ~ z , ~ 2 ) .

54. Abelian varieties

Let F = Q(a, G). In this section we derive a structure theorem concerning 2-group schemes over OF and prove Theorem 1.2. As is explained in the introduction, the endomorphism ring End(E) of

300 R. Schoof

the Setzer curve E is isomorphic to Z [ n ] . We write n for the endomorphism -. As usual, we denote the kernel of multiplication ?rm : E - E by E[?rm]. It is a finite flat OF-group scheme of rank 2m. For every m, n 2 0 we have an exact sequence

Lemma 4.1. The natural homomorphisms

are both isomorphisms. Here the leftmost group is a subgroup of a quotient of End(E); the group i n the middle is the group of OF-group scheme homomorphisms f : E[nn] -+ E[?rm] and HomF(E[nn], E[?rm]) denotes the group of homomorphisms E [ ~ ~ ] ( F ) --+ E [ P ] ( F ) of G ~ ~ ( F / F ) - modules.

Proof. It is a standard fact that the map $9 is injective. It follows that cp is injective. To see that $ is injective, let f : E[?rn] -t E[.rrm] be an OF-morphism of groupschemes for which $( f ) = 0. This means that f * (IB @ F) = F @ f * ( I B ) is zero. Here IB denotes the augmentation ideal of the Hopf algebra B of E[?rm]. By flatness it follows that f *(IB) = 0, so that f = 0. Since both cp and $ are injective, it suffices to show that #HornF (E[nn] , E[?rm]) is at most #(End(E)/(?rn)) [?rm] = 2min(n9m).

n = 1. We proceed by induction with respect to m. When m = 1, the statement is obvious, When m = 2, we observe that the only Galois submodule of order 2 of E[?r2] = E[2] is E[?r]. Therefore the image of any homomorphism E[T] ---t E[2] is contained in E[?r] and hence HomF(E[.;], E[?r]) Z HomF(E[?r], E[2]) has order 2.

When m > 2, we consider the exact sequence

where O(f) = 2 - f = f . 2. It follows that 0 = 0 and hence that #HomF(E[.rr], E[2"]) = #HomF(E[n], E[2]) = 2.

n > 1, m < n. We proceed by induction with respect to n. Consider the exact sequence


Since n > m, no f E HornF (E[nn] , E [?rm]) is injective. Since E [?r] is the unique minimal Galois submodule of E[nn], this means that O( f ) is zero and it follows that the first two groups in the exact sequence are isomorphic. Therefore #HomF(E [?rn], E[?rm]) = #HornF (E[?rn-'1, E[?rm]) = 2" as required.

n > 1, m 2 n. The same exact sequence implies that

and hence, inductively, that #HornF (E[?rn], E[nm]) 5 2n-1 - 2 = 2n. This completes the proof of the lemma.

Lemma 4.2. For each m 2 1,

; is a group of order 2. Moreover, the group ~ [ ? r ~ + ' ] is a non-trivial extension of E[?r] by E[.rrm].

?

Proof. For m = 1, this is Prop. 3.4. For m > 1, we apply the func- : tor Hom(E[?r], -) to the exact sequence 0 ---t E[n] -+ E[?rm+'] -+

E[am] -r 0. It follows from Lemma 4.1 and the fact, established in ' Prop.3.4,that~xt~,(E[n],E[?r])hasorder2,thatthereisaninjective

homomorphism

1 It follows by induction that # ~ x t b , (E [TI, E[nm]) 5 2. The exact se-

I quence 0 -t E[?r] -t E[?rm+'] - E[?rm] + 0 is not split, because it is not even split over F. Therefore the extension group has exactly 2 elements, a s required.

p

Theorem 4.3. The objects of the form @L=lE[?rna] form a full obeiian subcategory C of the category of fppf sheaves over OF. t,

Proof. We need to show that kernels exist in C. Since E[?r] is self- 1 I dual, cokernels then exist by duality. The category C obviously has all j the other properties of an abelian category. Let therefore

302 R. Schoof

be a homomorphism of fppf sheaves or, equivalently, of group schemes. By Lemma 4.1, there are endomorphisms fij E End(E) that induce g. Therefore the kernel K of g is isomorphic to the kernel of

Consider the commutative diagram

0 0

where (7rnt ) and A denote the homomorphisms

i fl 1

f l s 7rn1

0

respectively. The diagram has exact rows and columns and it easily implies that

A K 2 K1 = ker(ET - E'+").

Since Z [ q is a principal ideal domain, there exist an invertible r x r- matrix B and an invertible (r + s) x (r + s)-matrix B' with entries in End(E) % z[-] so that

Abelian Varieties over Q(&)

This shows that K is isomorphic to the kernel of the map

Since the ideal generated by the determinants of the r x r submatrices of A contains a power of 7r, all entries gi divide some power of 7r, so that g 2 - - f 7rni for some ni. Therefore K has the required form. This proves the theorem.

Corollary 4.4. Let G be a finite flat OF-group scheme that admits a filtration by closed group schemes isomorphic to E[T]. Then

Proof. We proceed by induction with respect to the rank of G. This means that we may assume that we have an exact sequence of 2-group schemes over OF of the form

By Lemma 4.2, the group ~xt;, (E[T], @L1E[7rna]) is a vector space of dimension r over Fa. It is generated by the extensions ej for j = 1, . . . , r , where ej is the extension

By Theorem 4.3, the group of extensions of E[T] by @:=lE[ana] in the category C form a subgroup of ~ x t b , ( ~ [ r r ] , $:=1E[7rni]). Since all extensions ej are extensions in C, the two extension groups are actually equal. This implies that G is isomorphic to an object in C and the corollary follows.

Corollary 4.5. Every 2-group scheme G over OF admits a filtration with closed flat subgroup schemes

for which

304 R. Schoof Abelian Varieties

- G2 is diagonalizable; i.e. isomorphic to a product of group schemes of the form p2k;

- G1/G2 is a product of group schemes isomorphic to ~ [ r r ~ ] ; - GIG1 is constant; i.e. isomorphic to a product of group schemes

of the form ~ 1 2 ~ 2 .

Moreover, the mnks of the subgroup schemes G1 and G2 are invariants of the group scheme G and do not depend on the filtration.

Proof. We filter G with closed flat subgroup schemes Gi in such a way that the subquotients are simple. By Theorem 2.3, the simple 2-group schemes are isomorphic to 2/22, p2 or E[rr]. By Proposi- tion 3.3 (i), we can modify the filtration as follows. If for some index i there are successive steps GiPl -+ Gi -+ Gi+1 in the filtration with Gi/Gi-l N 2 /22 and Gi+1/Gi r p2, then we can replace Gi by another subgroup scheme GI with Gi- 1 ct GI ct Gi+1 so that Gi/Gi- 1 N p2 and Gi+1/Gi N 2/22. Similarly, if for some index i there are successive steps Gi-1 -+ Gi --t Gi+1 with Gi/GiPl N 2 /22 and Gi+1/Gi Z E[rr], we can replace Gi by another subgroup scheme G!, with Gi-1 -+ G!, - Gi+1 so that GilGi-1 N E[T] and Gi+1/Gi N 2/22. Finally, if for some index i there are successive steps Gi-1 Gi -+ Gi+1 with Gi/Gi-1 N E[T] and Gi+1/Gi N p2, we can replace Gi by another subgroup scheme Gi with GiMl --t GI --t Gi+1 so that Gi/GiPl E p2 and Gi+l/Gi E E[rr].

This implies that G admits a filtration 0 c G2 c G1 c G with the property that G2 is filtered by copies of p2 only, G1/G2 is filtered by copies of E[T] only and GIGl is filtered by Z/2Z's only. It follows from Prop. 3.3 (ii) and Galois theory over OF that GIG1 is constant. By Prop. 3.3 (iii) and Cartier duality Gq is diagonalizable. Finally, by Cor. 4.4, G1/G2 has the required form. This proves the first statement of the corollary.

The second statement follows from the fact that the rank of GIG1 is equal to the order of the group G(&) and that the rank of G2 is equal to the order of G~"" '(K). These ranks are additive in exact sequences.

Proof of Theorem 1.2. Let A be an abelian variety of dimension g > 0 over F = Q ( G , J--I?) with good reduction everywhere. As before, we write A for an abelian scheme over OF with generic fiber isomorphic to A. By Cor. 4.5, the 2-group scheme A[2] admits a filtration

Suppose that G1 # A[2]. In other words A[2] admits a constant quotient of rank at least 2. By the second statement of Cor. 4.5, the 2-group scheme A[2"] admits, for each n, a constant quotient Cn of rank at

least 2". In other words, there is an over OF

over Q(&) 305

exact sequence of 2-group schemes

where H, is a closed flat subgroup scheme of A[2"] and Cn is constant of rank at least 2". Consider the abelian variety A/Hn. It admits the constant group scheme Cn as a closed subgroup scheme. Reducing A/H, modulo some prime ideal p of OF with residue field isomorphic to F, we find that

C" (Fd - (AIH") (F,)

and hence, by Weil's Theorem

This cannot hold for every n 2 1. Therefore A[2]/G1 = 0. By Cartier duality it follows that G2 = 0 as

well. This shows that A[2] and hence every A[2"] admits a filtration with all simple subquotients isomorphic to E[rr]. It follows from Corollary 4.4 that -

Inspection of the geometric points easily shows that r = g and ni = 2n for each i. Therefore

It follows that the 2-adic Tate modules of A and E g are isomorphic Gal(F1~)-modules. By Faltings' theorem [4, Cor. 2 of Thm. 41 this implies that A and E g areisogenous.

This completes the proof of Theorem 1.2.

References

[ 1 ] AbraSkin V.A., Galois moduli of period p group schemes over a ring of Witt vectors, Izv. Ak. Nauk CCCP, Ser. Matem., 51 (1987); English translation in Math. USSR Izvestiya, 31 (1988), 1-46.

[ 2 ] Artin M., Algebraization of formal moduli, 11. Existence of modifications, Annals Math., 91 (1971), 88-135.

[ 3 ] Birch B. J., Cyclotomic fields and Kummer extensions, in "Algebraic Number Theory", (Cassels J. W .S. and F'rohlich A., eds.), Academic Press, London, 1967.

[ 4 ] Faltings G. , Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Invent. Math., 73 (l983), 349-366.

306 R. Schoof

Fontaine J.-M., I1 n'y a pas de variktk abklienne sur Z, Invent. Math., 81 (1985), 515-538.

Greither C., Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Zeitschrift, 210 (1992), 37-67.

Katz N. and Mazur B., "Arithmetic moduli of elliptic curves", Annals of Math. Studies 108, Princeton University Press, Princeton, 1985.

Kida M., Reduction of elliptic curves over certain real quadratic number fields, Math. Comp., 68 (1999), 1679-1685.

Martinet J., Petits discriminants des corps de nombres, in "Journbes Arithmktiques 198OV, (J.V. Armitage, ed.), CUP Lecture Notes Series 56, Cambridge University Press, Cambridge, 1981.

Tate J. and Oort F., Group schemes of prime order, Ann. Scient. ~ c o l e Norm. Sup., 3 (1970), 1-21.

Schoof R., Abelian varieties over cyclotomic fields with good reduction everywhere, in preparation.

Setzer B., Elliptic curves with good reduction everywhere over quadratic fields and having rational j-invariants, Illinois J. Math., 25 (1981), 233-245.

Tate J., Finite flat group schemes, in "Modular Forms and Fermat's Last Theorem", (Cornell G., Silverman J. and Stevens G., eds.), Springer- Verlag, New York, 1997.

Washington L. C., "Introduction to cyclotomic fields", Graduate Texts in Math. 83, Springer-Verlag, Berlin Heidelberg New York, 1982.

Dipartimento di Matematica Universitci di Roma "Tor Vergata" 1-001 33 Roma ITALY E-mail address : schoof Qwins . uva . nl


Hodge Cycles and Unramified Class Fields

Hiromichi Yanai

In the theory of complex multiplication, we obtain ramified class fields by the torsion points of a CM abelian variety and unramified class fields as certain fields of moduli (cf. [S-TI). Several authors studied the class fields obtained by complex multiplication ([K], [R], [S], [Maul, [O]).

When the abelian variety has "many" Hodge cycles (hence the Hodge group is "small"), it is known that the ramified class fields are "small" (cf. [R]). On the contrary, we know no clear relation between Hodge cycles and unramified class fields. In certain cases, however, the existence of exceptional Hodge cycles helps us to say something about the unramified class fields.

In the previous paper [DCM], we have shown a method of constructing CM abelian varieties with exceptional Hodge cycles and we have given several examples in which we can determine the degrees of the unramified class fields obtained as fields of moduli.

In the present note, we shall generalize the result of [DCM] and explain how the exceptional Hodge cycles influence the unramified class fields.

1 Except ional H o d g e Cycles

Let A be a CM abelian variety of type (K, S) defined over a subfield of C with dimA = d, where K is a CM-field of degree 2d and S is a CM-type of K (cf. [S-TI). Let H g = Hg(A) be the Hodge (or the special Mumford-Tate) group of A, that is a sub algebraic torus of GL(H1 (A, Q)). It is known that dim H g 5 d. When A is simple and dim H g < d, there exist certain Hodge cycles on a product A'" of several copies of A that are not generated by the divisor classes; we call such Hodge cycles exceptional (cf. [DCM]) .

Received August 31, 1998. Revised April 12, 1999.

308 H. Yanai

In this section, we give a method of constructing CM abelian varieties with exceptional Hodge cycles. This generalizes [DCM Theorem 2.11.

Consider a chain of three distinct CM-fields

which we regard as subfields of C . Let 2d, 2dl, 2d2 be the degrees of K, K1, K2 over Q , respectively. Let r, rl, r2 be the sets of the embeddings of these fields into C and let : r -+ rl, 7rl2 : rl + r2 be the canonical surjections.

Take a CM-type S of K satisfying the following condition:

For each p E r2, there exists a nonnegative integer a, such that for each T E I'l with rl2(T) = p, # {o E S 1 (0) = T) = a,.

d Under this condition, we have a, + a,, = [K : K1] = -, where p

d 1 denotes the complex conjugation.

Our theorem is

Theorem 1.1. Take a CM-type (K,S) satisfying the condition (*). Let A be a CM abelian variety of type (K, S) and let H g be its Hodge group. Then we have

Remark 1.2. Our theorem gives an upper bound for the dimension of the Hodge group under the condition (*). We should note that L. Mai [Mail has discussed certain lower bounds.

If the abelian variety A considered in Theorem 1.1 is simple, we have exceptional Hodge cycles on A itself (see Section 2). This does not necessarily hold in general degenerate (i.e., dimHg < d) cases (cf. [W]).

Proof of Theorem. Since our argument is similar to that of [DCM], we give only an outline.

For a CM-field F, TF = ResFlq(G,) denotes the algebraic torus corresponding to the multiplicative group FX and T> denotes the kernel of the norm map NFIF+ : TF + TF+ , where F+ is the maximal real subfield of F.

Since K acts on the cohomology group H1 (A, Q), we can regard TK as a subgroup of G L ( H ~ (A, 4)). It is known that H g c T: C TK (cf.

PCMI

Hodge Cycles and Unramified Class Fields 309

We consider several algebraic tori and morphisms between them:

where NKIK1 is the norm map and j is the closed immersion. Put

T = N&, W i 2 ) ) n ~ , f .

Then we can show

Hg C T and d im(T$ /~) = d l -d2

(see the proof of [DCM Theorem 2.11). Since d i m ~ ; = d, these imply our Theorem. 0

Example 1.3. Let K be the cyclotomic field of 37-th roots of unity. We naturally identify the Galois group r = Gal(K/Q) with the group (Z/37Z) '. Let K1, K2 be the subfields of K of degree 12 and 4 over Q , and H1, H2 be the corresponding subgroups of r, respectively. Then H1 = {1,26,10) and H2 = H1 U 16H1 U 34H1. The coset decomposition of r is

Put p = H2, p' = 2H2 E r2, so that pp = 36H2, pp' = 35H2. Taking up = 2, a,/ = 3 , a,,, = 1, aPpt = 0, we choose two elements from each of H1, 16H1, 34H1 and choose all elements of 2H2. From 36H2 and 35H2, we must choose the elements that are not complex conjugate to any elements selected before. In this way, we get a simple CM-type S satisfying the condition (*) ; for example,

In this case, we have dim H g = 18 - (6 - 2) = 14. Moreover, we can construct an exceptional Hodge cycle by the elements of (for example) H1 U 21H1 = {1,10,21,25,26,28) (cf. Remark 2.2 below).

$2. Unrarnified Class Fields

In this section, we explain how the exceptional Hodge cycles obtained in Theorem 1.1 restrict the degree of the field of moduli.

From now on, we assume that the CM-field K is abelian over Q. Let A be a simple, principal (in the sense of [S-TI) CM abelian variety of type (K, S) and let H g be its Hodge group. Let XK = X(TK) be the

310 H. Yanai

character group of TK, that is naturally identified with the group ring Z [TI; each element of XK can act on the ideal classes of K .

Let HgL c XK be the group of the annihilators of Hg in XK. Take an embedding 0 : K + End(A) @ Q inducing the CM-type S and take a polarization C of A. Then the field of moduli M of the triple (A, 0, C) is an unramified class field over K . We denote the ideal class group of K by CK and denote the subgroup of CK corresponding to M by CK(S).

The next proposition is none other than [DCM Proposition 3.11.

Proposition 2.1. For A E CK and z E HgL , we have Ax E

CK (S).

In the rest of this section, we assume that K, K1, K2 and S are as in Theorem 1.1. We can describe certain elements of HgL explicitly:

Proposition 2.2. For p E r2 and r, r' E rl satisfying r # r' and ~ ~ ~ ( 7 ) = r12(7') = p, put

T h e n we have

(i) Y E HgL and Y @ (W, (ii) for each A E CK, AY E C1,

where C1 i s the image of the natural map CKl -+ CK.

Remark 2.3. The element y is corresponding to a (complex valued) exceptional Hodge cycle on A. See [HI, [PI.

Proof of Proposition 2.2. The following characterization of the elements of HgL can be found in the proof of [DCM Proposition 1.11.

For x = C n , y E XK, ,Er

We rewrite y so that y = m7y. Then, for each g E T, 7 E r

Hodge Cycles and Unramified Class Fields 311

Hence y E HgL. The assumption r # r' implies yp # y; this proves y $t! ( ~ 2 ) ' . We have thus obtained (i).

We can easily see AY = (NKIK1 ( A))~+I 'P E Cl . This proves (ii). 0

Let D be the subgroup of CK generated by the ideal classes of the form AY(~~'T"), where A E CK and p, r, T' are as in Proposition 2.2. Then, by Proposition 2.1 and 2.2, we have

In some cases, it may happen that D = C1 C CK(S). If this is the case then we can say that a sub CM-field K1 with "large" class number causes a "small" unramified class field.

Finally, we recall the examples in [DCM]. In [DCM Example 3.21, K is the cyclotomic field of 31-st roots of

unity; K1 and K2 are the sub fields of degree 6 and 2 over Q , respectively. Here we have hK = hk = hKl = hGl = 9 and #CK(S) = 3.

In [DCM Example 3.41, K is the cyclotomic field of 61-st roots of unity; K1 and K2 are the sub fields of degree 20 and 4 over Q, respectively. Here we have hK = h ; = 41.1861, hK, = hkl = 41 and #CK (S) = 41.

In both examples, we have D = C1 = CK(S).

References

F . Hazama, Hodge cycles on Abelian varieties of CM-type, Res. Act. Fac. Sci. Engrg. Tokyo Denki Univ., 5 (1983), 31-33.

T . Kubota, On the field extension by complex multiplication, Trans. AMS, 118 , No.6 (1965), 113-122.

L. Mai, Lower bounds for the rank of a CM-type, J. Number Theory, 32, No.2 (1989), 192-202.

E. Maus, On the field of moduli of an abelian variety with complex multiplication, Symp. Math., X V (1973), 551-555.

A. I . Ovseevic, Abelian extensions of fields of CM type, Func. Anal. Appl., 8, No.1 (1974), 14-20.

H. Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann. o f Math., (2) 8 8 (1968), 161-180.

K. A. Ribet, Division fields of abelian varieties with complex multiplication, M6m. Soc. Math. France, No.2 (1980), 75-94.

G. Shimura, O n the class fields obtained by complex multiplication of abelian varieties, Osaka Math. J., 14 (1962), 33-44.

312 H. Yanai

[S-TI G. Shimura and Y. Taniyama, Complex Multiplication of Abelian Vari- eties and Its Application to Number Theory, Publ. Math. Soc. Japan (1961).

[W] S. P. White, Sporadic cycles on CM abelian varieties, Compositio Math., 88 (1993), 123-142.

[DCM] H. Yanai, O n Degenerate CM-types, J. Number Theory, 49, No.3 (1994), 295-303.

Aichi Institute of Technology Yakusa, Toyota 4 70-0392, Japan E-mail address: yanai(0ge. aitech. ac. jp


Recent Progress on the Finiteness of Torsion Algebraic Cycles

Noriyuki Otsubo

In this article we review recent results on the finiteness of torsion algebraic cycles on certain surfaces over number fields.

1. Let X be an algebraic variety over a field k. For an integer i > 0, let X 9 e the set of schematic points of codimension i (equivalently, the set of integral closed subvarieties of codimension i of X) . The Chow group of algebraic cycles of codimension d modulo rational equivalence is defined by

where ~ ( x ) denotes the residue field at x. Chow groups are natural generalization of Picard group, but little is known on their structure in general.

Bloch was the first to study the close relation between algebraic cycles and algebraic K-theory. Let K,(X) be the algebraic K-group defined by Quillen. Let Kn be the Zariski sheaf on X which is associated to the presheaf U H Kn(I'(U, Ox)). Define similarly 'Fln(Z/lm(r)) by the sheafification of the &ale cohomology functor U H Hz(U, CLg) for a prime number 1 # ch(k). Then, if X is smooth we have the following isomorphisms called Bloch7s formula:

Received September 1, 1998. Revised April 13, 1999.

314 N. Otsubo

On the other hand we have the Riemann-Roch theorem:

As to the structure of higher K-groups we have

Conjecture 1.1 (Bass). Let X be a regular scheme of finite type over Z. Then Ki(X) is a finitely generated abelian group for i 2 0.

As a corollary of higher dimensional class field theory we have

Theorem 1.2 (Bloch[B13], Kato-Saito[K-S]). For a scheme X of finite type over Z, the Chow group of zero-cycles CHo(X) is a finitely generated abelian group.

From now on, we mainly consider a projective smooth variety X over a number field k. Let X be a proper smooth model of X over Ok[l /N] for some N. Then the natural maps

are surjective, and we expect CHd (x) to be finitely generated.

Conjecture 1.3 (Tate[T] , Beilinson[Be] , Bloch[B14], Bloch-Kato

P-Kl ) (i) For d 2 0, c H d ( X ) is a finitely generated abelian group.

(ii) rank(CHd (x)/c Hd (x) horn) = dim(Hz; (fl, QP (d)) ) = -~rd,,d+~ L ( H ~ ~ (x), s) .

(iii) rank(CHd (x)~,,) = dim(H) (k, HZ:-' (fl, Qp(d))) - - o ~ ~ , , ~ L ( H ~ ~ - ' ( x ) , s) .

Here, CHd ( x ) h O m is the kernel of the cycle map CHd(X) -+ H:$(X, Qp(d)) , and H) (k, -) c H1 (k, -) is a vector-space analog of the Selmer group defined by Bloch-Kato[B-K] (see 54).

Remark 1.4. If d = 1, tfien CH1 (X) = Pic(X) and (i) follows from the Nkron-Severi theorem and the Mordell- Weil theorem.

If X is an elliptic curve and d = 1, (iii) is a part of the Birch- Swinnerton-Dyer conjecture.

2. On the finiteness of torsion part of Chow group of codimension two, there has been considerable progress for varieties over various fields not only number fields. In particular, for varieties with Hgar(X, Ox) = 0,

Torsion Algebraic Cycles 315

we have many general results (see [CT2] and [Crr3] for much more). The first example was the following:

Theorem 2.1 (Bloch[B12]). Let X be a rational surface over a

number field which is a conic bundle over P1. Then, Ker(CH2(X) 2 Z) is finite.

This was generalized:

Theorem 2.2 ([Colliot-Thkl&ne[CTl]]). The same holds for every rational surface.

While Bloch used the theory of quadratic forms, Colliot-Thklkne proved rather simply using the following significant theorem of Merkur- jev and Suslin called Hilbert 90 for K2.

Theorem 2.3 ([Merkurjev-Suslin[M-S]). For a field k and a prime number 1 # ch(k), we have an isomorphism K2 (k)/ln P H 2 (k, p$?).

With this theorem and the Bloch-Ogus theory[B-01, Bloch gave the following exact sequence [Bll] (cf. [CT2]) :

NHZ~ (x, Qp/%(2))

:= K ~ H , ~ , ( x , Qp/Zp(2)) -+ ~ , 3 , ( k ( ~ ) , QpIZp(2))).

! Therefore, to show the finiteness of torsion of CH2(X) , we are to I show that the first group Hiar (X, K2) is sufficiently large. The most 1 general result known so far is the following: I I Theorem 2.4 (Colliot-Thklkne-Raskind[CT-R] , Salberger [Sa] ) . Let

X be a projective smooth vdriety over a number field. I f H;,,(X, Ox) / = 0, then CH2(X)to, is finite. I / 3. One of the crucial difficulties in the situation H;,,(X, Ox) # 0 is that

1 we need essentially new elements in Hi,,(X, K2) called indecomposable

: in the sense that they are not contained in the image of the product map

316 N. Otsubo Torsion Algebraic Cycles 317

(even after any finite extension of the base field). Importance of the group HAar(X, K2) can also be seen from the

following localization sequence in K-theory. Let X be a proper smooth model of X over Qk[l /N] and X, be its closed fiber over a prime v. Then we have an exact sequence:

(3.1)

-- H&,(x, K2) @ ~ i c ( ~ , ) - C H ~ ( X ) - C H ~ ( X ) -+ 0.

vtN

It is known by [CT-R] that the pprimary torsion subgroup C H 2 (X){p) is a cofinitely generated Zp-module (i.e. a direct sum of finite copies of Qp/Z, and a finite pgroup). Therefore, if Ker(CH2(X) -t CH2(X) ) =

Coker(d) is torsion then C H ~ ( X ) { ~ ) is also cofinitely generated as a Zp-module and hence the n-torsion of CH2(X) is finite for any n.

For the self-product X = E x E of a modular elliptic curve over Q, we have the following elements in Hk,(X, K2) constructed by Flach[F12] and Mildenhall[M] using the theory of modular curves and modular units. For a prime p where E has good reduction there is an element of Hiar(X, K2) whose image by the boundary map d of (3.1) is trivial at 1 # p, and a non-zero constant multiple of the class of the graph of the Frobenius endomorphism of E (mod p) at 1 = p. Mildenhall used them to show the torsionness of K ~ ~ ( C H ~ ( X ) -t CH2(X)) , and Flach used them to detect the Selmer group associated to the Galois representation

s y m 2 ( ~ p ( ~ ) ) . Based on their results Langer and Saito proved

Theorem 3.1 ([Langer-Saito[L-S]). Let E be a semi-stable elliptic curve over Q with conductor N. Then CH2 (E x E) {p) is finite for p 6N, and trivial for almost all primes.

When E has complex multiplication we have

Theorem 3.2 ([Langer[Ll], Langer-Raskind [L-R], [Ol]). Let E be an elliptic curve over Q with complex multiplication by the ring of integers in an imaginary quadratic field K . Let N be its conductor and p I / 6N be a prime number. Then, under some assumption which is satisfied if N is a power of a prime, C H ~ (E x E ) {p) and C H ~ (EK x

EK){p) are finite, where EK := E @Q K . Moreover, the same holds for the associated Kummer surfaces K I ~ ( E x E) and Km(EK x EK).

Remark 3.3. For an abelian surface A over a field with characteristic # 2, the Kummer surface Km(A) associated to A is a K3-surface obtained by blowing up sixteen singularities of the quotient A/{& 1) corresponding to the points of order 2 on A. Also in the semi-stable case, we can show in the same manner the finiteness of CH2(Km(E x E)){p) for E and p as in Theorem 3.1.

Finally we introduce

Theorem 3.4 ([Ol]). Let F be the Fermat quartic surface over Q defined by

Put K = Q(-), FK = F @p K , and let p 1/ 6 be a prime number. Then, C H ~ ( F ) { ~ ) and CH2(FK){p) are finite.

Remark 3.5. Let Y be a projective smooth variety over a padic field k' (i.e. [k' : Qp] < m) which has a projective smooth model 2J over the integer ring. Then, if Hiar(Y, Oy) = 0, CH2(Y)tor is known to be finite (cf. [CT2])

If K ~ ~ ( c H ~ ( ? ) ) -t CH2(Y)) is torsion, then this is finite, and the prime-to-p part of CH2(Y)tOr is finite because CH2(2J)tor has the property [Ra]. For X and p as in Theorems 3.1, 3.2 and 3.4, the first step of their proofs show the finiteness of K e r ( c ~ ~ ( 2 J ) -, CH2(Y)) for Y = X @Q k'.

Other examples over padic fields are the product of two (possibly different) elliptic curves [Sp] and a class of Hilbert-Blumenthal surfaces

[L21.

4. Now let us recall the outline of the method of Langer-Saito[L-S] which is also used in [Ll] and [Ol]. Let X be one of the surfaces of Theorems 3.1, 3.2 or 3.4 over Q, and p be a prime number satisfying the assumption. The proof for XK is parallel. By (2. I ) , since CH2 (X) is cofinitely generated by the above result of Mildenhall and the corresponding results for the Kummer surface [Ll] [Ol], and for the Fermat quartic surface [Ol], it is enough to show that H;,,(X, K2) @ Qp/Zp is the maximal divisible subgroup of NH;(X, Q,/Zp(2)). We modify

318 N. Otsubo

N H$ (X, Qp/Z, (2)) by the subgroup

and reduce to prove

Then these groups are embedded by using Hochschild-Serre spectral sequence into the Galois cohomology group H1(Q, A) where A :=

H$ (x, Qp/Zp(2)). Taking further the localizations with local conditions we obtain

where H;(Ql, A) is the unramified part for 1 # p, and defined by using the Fontaine ring Bcri, for 1 = p (see [B-K]). The Selmer group of A is defined by

S(Q, A) := Ker(cr),

and its analogue

H; (Q7 v) := Ker (HI (Q. V) + @ H; (al, v) all 1 Hi(Q7V))

is similarly defined. Then we are to prove:

(i) In (4.1), the image of the first group in the final direct sum coincides with that of the second one;

(ii) The Selmer group S(Q, A) is finite.

The key to show (i) is the following commutative diagram with the vertical isomorphism for I { N. Define V := ~2~ (X, Qp(2)), and let the subspace Hi(Ql, V) c H1(Q1, V) be the whole space for 1 # p and the one defined in [B-K] using the Fontaine ring BdR for 1 = p. Then we

Torsion Algebraic Cycles

have

where Dl is the 1-part of the boundary map of (3.1) tensored with Q,. The proof of its 1 = p part requires recent results in padic Hodge theory. Since we know that dl is surjective for 1 { N, the composition of (4.1) is "almost" surjective modulo many delicate arguments such as the difference between Qp-coefficients and Qp/Zp-coefficients or the bad reduction primes.

The part (ii) is more arithmetic in nature. When X = E x E for a semi-stable E, the Selmer group is studied in [F12]. When E has CM, there are results [Fll], [W], [Dl, all of which are based on the two-variable Iwasawa main conjecture proved by Rubin[Ru].

5. Once the result for E x E (or EK x EK) is obtained, it is not so difficult to prove the statement for the associated Kummer surface. One should notice, however, that the proof will be more complicated if the 2-torsion points of E is not defined over Q (or the CM field K) , in which case we have possibly infinite Selmer group and we need some tricks as in [Ol].

Finally, we consider the Fermat quartic surface F. The key is a geometric construction connecting F to a Kummer surface which enables us to use the results on E x E . It is known [Ka-Sh] that F is constructed from the product of two copies of the Fermat quartic curve C, by taking blowing-up, quotient by a finite group and blowing-down. We can find a finite morphism C -+ E where E is an elliptic curve, such that it induces a finite morphism --, Km(E x E) of degree 2 where F is the blowing-up of F at certain eight points. This E has complex multiplication by Z[-].

Pull-backs of this morphism induce a commutative diagram:

where the vertical maps are the cycle maps. We can define explicitly eight divisors on P whose divisor (resp. cohomology) classes generate

320 N. Otsubo

the cokernel of the upper (resp. lower) map. This means that both the Picard group and the second cohomology group of (then, of F) are described by those of Km(E x E) and the classes of the explicit divisors.

The crucial difference from the other cases is the fact that the Selmer group for F or FK is not finite.

Theorem 5.1 ([Ol]). Let A = H,:(F, QP/Zp(2)). Then we have

corankap (S(Q, A)) = 2, corankzp (S(K, A)) = 4.

This breaks the part (ii), but we can separate from A = H&(F, Qp/Zp(2)) a part which causes the infinite Selmer groups and treat this directly without taking localizations. This part is controlled by

the classes of the eight divisors mentioned above and the multiplicative group of Q(C8) because the divisors are defined only over Q(C8). In view of the exact sequence

for a number field k, this explains why our Selmer groups are infinite. By a conjecture of Bloch-Kato[B-K], the Z,-corank of the Selmer

group of a general motive should coincide with the rank of a certain motivic cohomology group defined by K-theory, and then with the order of vanishing at an integer point of the L-function by Beilinson's conjecture [Be]. These are wide generalization of Conjecture 1.3. Note that for a (),-representation V of ~ a l ( E / k ) of geometric origin, its Ga- lois stable Z,-lattice T and A = V/T, we have corankzp (S(k, A)) =

dimClp(H;(k> V)). In our situation the desired equalities are

Remark 5.2. We have H&(-, Q(2)) -. Hi,,(-, Kz) 8 Q. The subscript Z means the integral part, that is, the elements extending to an integral model over the whole integer ring.

We have in fact

Theorem 5.3 ([Ol]). (i) ords=1L(H2(F), s) = 2, ~ r d , , ~ h ( H ~ ( F ~ ) , S) = 4.

(ii) There exist two (msp. four) elements in HL(F,Q(2))z (resp.

Torsion Algebraic Cycles 32 1

))z) whose image by the Chern class map generate the m u - imal divisible subgroup of the Selmer group.

Since we have explicit description of the Picard group, (i) follows from the functional equation and the Tate conjecture ([Fall using [Ka-Sh]). The elements of (ii) are constructed using the eight specific divisors mentioned above and certain units of Q(C8). These are decom- posable over Q(C8), but not over Q nor K .

Remark 5.4. This method is generalized in [02] to construct elements in HZ" (x, Q(m + r))z (r 2 I) , for any projective smooth variety X over a number field. The image of these elements under the Chern class map generate the Selmer group of V1(r) c H , ~ , ~ ( X , Q , ( ~ + r ) ) where V' is the sub-representation of H,~,"(X, Qp(m)) generated by the classes of cycles on X of codimension m.

References

A. A. BEILINSON, Higher regulators and values of L-functions, Itogi Nauki i Techniki, Sovremennyje Problemy Matematiki, 24, VINITI, Moskva, 1984, 181-238; English tranl. in: J. Soviet Math., 30 (1985), 2036-2070.

S. BLOCH, Lectures on Algebraic Cycles, Duke Univ. Math. Series, Durham, 1980.

, On the Chow groups of ceratin rational surfaces, Ann. Sc. Ec. Norm. Sup., 14, (1981) 41-59.

, Algebraic K-theory and class field theory for arithmetic surfaces, Ann. of Math., 114 (1981), 229-266.

, Algebraic cycles and values of L-functions I, J. Reine. Angew. Math., 350 (1984), 94-108.

S. BLOCH and K. KATO, L-functions and Tamagawa numbers of motives, in: P. Cartier et a1 (eds.), The Grothendieck festschrzfl, vol.1, Progr. Math., 86, Boston Basel Stuttgart, Birkhauser, 1990, 333-400.

S. BLOCH and A. OGUS, Gersten's conjecture and the homology of schemes, Ann. Scient. EC. Norm. Sup. 4e sdrie, t.7 (1974), 181-202.

J.-L. COLLIOT-TH~LENE, Hilbert's theorem 90 for K2, with application to the Chow group of rational surfaces, Invent. Math., 71 (1983), 1-20.

N. Otsubo

, Cycles alghbriques de torsion et K-th6orie alghbrique, in: Arithmetic of Algebraic Geometry, Lecture Notes in Math., 1553, Springer-Verlag, 1993, 1-49.

, L'arithmhtique du groups de Chow des zhro-cycles, J . Thhor. Nombres Bordeaux, 7 (1995), no. 1, 51-73.

J.-L. COLLIOT-THELENE and W. RASKIND, Groupe de Chow de codimension deux des vari6ths dhfinies sur un corps de nombres: Un thhorkme de finitude pour la torsion, Invent. Math., 105 (1991), 221-245.

J. DEE, Selmer groups for Hecke characters a t supersingular primes, preprint.

G. FALTINGS, Endlichkeitssatze fiir abelsche varietaten iiber zahlkorpern, Invent. Math., 7 3 (1983), 349-366.

M. FLACH, Selmer groups for the symmetric square for an elliptic curve, Ph. D. Thesis, University of Cambridge, 1990.

, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math., 109 (1992), 307-327.

K. KATO and S. SAITO, Global class field theory of arithmetic schemes, Contemp. Math., 55, vol.1, 255-331.

T . KATSURA and T . SHIODA, On Fermat varieties, T6hoku Math. J., 31 (1979), 97-115.

A. LANGER, 0-cycles on the elliptic modular surface of level 4, T6hoku Math. J., (2) 5 0 (1998), no. 2, 291-302.

, Zero cycles on Hilbert-Blumenthal surfaces, Duke Math. J., 1 0 3 (2000), no. 1, 131-163.

, Finiteness of torsion in the codimension-two Chow group: an axiomatic approach, in: B. B. Gordon et a1 (eds.), The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000, 277-284.

A. LANGER and W. RASKIND, Torsion 0-cycles on the self-product of a CM elliptic curve, J . Reine. Angew. Math., 516 (1999), 1-26.

A. LANGER and S. SAITO, Torsion zero-cycles on the self-product of a modular elliptic curve, Duke Math. J., 85 (1996), 315-357.

A. S. MERKUR'EV and A. A. SUSLIN, K-cohomology of Severi- Brauer variaties and the norm residue homomorphism, Izv. Akad. Nauk. SSSR Ser. Mat., 46, No. 5 (1982), 1011-1046; English transl. in: Math. USSR Izv., 21, No.2 (1983), 307-340.

S. MILDENHALL, Cycles in a product of elliptic curves, and a group analogous to the class group, Duke Math. J., 6 7 (1992), 387-406.

Torsion Algebraic Cycles

N. OTSUBO, Selmer groups and zero-cycles

323

on the Fermat quartic surface, J . Reine Angew. Math., 525 (2000), 113-146.

, Note on conjectures of Beilinson-Bloch-Kato for cycle classes, manuscripta math., 101 (2000), 115-124.

W . RASKIND, Torsion algebraic cycles on varieties over local fields, in: J. F. Jardine and V. P. Snaith (eds.), Algebraic K-thery: Con- nection with Geometry and Topology, Lake Louise 1987, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., 279, Kluwer Acad. Publ., Dordrecht, 1989, 343-388.

K. RUBIN, The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math., 1 0 3 (1991), 25-68.

P . SALBERGER, Torsion cycles of codimension 2 and 1-adic realization of motivic cohomology, in: S. David (eds.), Se'minaire de the'orie de nombres, Paris 1991-92, Birkhauser, Boston, 1993, 247-277.

M. SPIESS, On indecomposable elements of K1 of a product of elliptic curves, K-Theory, 1 7 (1999), no. 4, 363-383.

J . TATE, Algebraic cycles and poles of zeta functions, in: Arithmeti- cal algebraic geometry, Harper and Row, New York, 1965.

A. WILES, Modular elliptic curves and Fermat's last theorem, Ann. of Math., 1 4 1 (1995), 443-551.

Added in Proof. After the first manuscript was written, Langer [L3] axiomatized the method for the finiteness of CH2(X) t , , including the case where the Selmer group is not finite, in which case the generalization of Theorem 5.3 (see Remark 5.4) should be useful.

Department of Mathematics, Chiba Unibersity, Yayoi-cho 1-33, Chiba, 263-8522, Japan E-mail address: otsubohath. s . chiba-u . ac . jp


Finiteness of a certain Motivic Cohomology Group of Varieties over Local and Global Fields

Kanet omo Sat o

INTRODUCTION

In this paper, I would like to survey my recent research [22]. I would like to express gratitude to the organizers for giving me this opportunity to write this manuscript.

Let k be a global field, i.e., an algebraic number field (case (N)) or a function field in one variable over a finite field (case (F)). Let X be a projective smooth geometrically connected Ic-variety. Let 1 be a prime number invertible in k. The l-adic regulator map of Soul6 [24]

is a central topic in the arithmetic geometry. Here H h (X, Q(n)) denotes the motivic cohomology and is defined by the n-th Adams eigenspace of the algebraic K-group K2n-i(X)Q (1171 and [25]), and the right hand side is the continuous etale cohomology group (cf. Jannsen [9]). The coefficient Ql (n) in the right hand side means the n-th Tate twist of Ql. In the case i = 2n, it is known that this map coincides with the cycle map for the Chow group of algebraic cycles of codimension n modulo rational equivalence ([9] 6.14) :

We write F* for the Hochschild-Serre filtration on the continuous etale cohomology group w.r.t. the covering X @ k kSeP + X. For instance, F2 of Htont(X,Ql(n)) is defined by the image of the Hochschild-Serre mapping

Received August 31, 1998. Revised December 21, 1998. The research for this article was supported by JSPS Research Fellowships

for Young Scientists.

326 K. Sato

and it is trivial if i < 2. In this manuscript, we start from the following conjecture, which is based on the philosophy of mixed motives of Beilinson [2] and Bloch [3] (cf. Jannsen [ll] 11.6) and the Beilinson- Deligne-Jannsen conjecture (cf. [l 11 11.4, 12.18).

Conjecture 1. For arbitrary integers i and n satisfying 0 5 n < d + 1 and 0 j i j 2n (d := dimX), the following map induced by r;'",

is injective.

The cases (n, i) = (0,O) and (d + 1,2d + 2) are trivial. It is also the case, when (n, i) = (1,2), by the fact that the Picard group of X is a finitely generated Z-module and by the Kummer theory for the Picard variety (cf. Reskind [19] Appendix). As for the conjecture on the image of F)", see [ll], 12.18, and Bloch 141, 55. Conjecture 1 at least implies the following:

Conjecture 2. For integers i and n satisfying 1 < n 5 d + 1 and 2 < i < 2n, the image of the 1-adic regulator map r;'" intersects with F2 trivially:

1m(ri7") n F~H:,, (x, Ql (n)) = 0.

This is clearly true in the case (n, i) = (1 ,2 ) by the above remark. In this manuscript, we are concerned with Conjecture 2. The main result

Theorem 3 ([22] Corollary 5.4). Let k be the case (F), and X be a proper smooth variety over k. Then Conjecture 2 is true in the case i = n + 1 with n at least 2.

In the proof, the finiteness result stated below ($2, Theorem 6) will play an important role (see 52). By the Merkur'ev-Suslin theorem ([15], SIB), a result of Soul6 ([25], Th6orkme 4 (iv)), and Theorem 3, we can show the following:

Corollary 4 ([22] Theorem 0.2). Let k, X , and I be as in The- orem 3. Then we have

Here the superscript (2) means the second Adams eigenspace, and the subscript I-div means the maximal 1-divisible subgroup.

Finiteness of a certain Motivic Cohomology Group of Varieties 327

We will not prove this corollary here (cf. [22], 56). Accoding to Bass' general conjecture [I], 59 (predicting that algebraic K-groups of a regular scheme of finite type over Spec Z should be finitely generated Z-modules), the right hand side in Corollary 4 would be trivial. In other words, the injectivity problem of is reduced to the Bass conjecture by this corollary.

If k is a function field, Conjecture 2 is true in several cases. We will review them in the first section. On the other hand, if k is a number field, there are only a few known cases (cf. Langer and Raskind [14], Theorem 0.2). One of the difficulties lies in the point that one needs, in a step of proofs, some local-global principle (cf. (1.2) below), which is known to hold in the function field case, but have not been proven yet in general in the number field case (cf. [14] Theorem 5.5).

1 Review of known results

Throughout this section, k, X and 1 are as in Theorem 3. We write d for the dimension of X. Then Conjecture 2 is known to be true in the following cases (Figure 1).

(0) (n, i) = (1, 2)- (1) X has potentially good reduction everywhere. (2) i < n. (3) (n, i) = (d + 1,2d + 1). (4) (n, i) = (d, 2d). (5) i = 2n, 2 j n < d - 1 (with an additional geometrical assump-

tion).

Theorem 3 corresponds to the line (6) in Figure 1. We shall review the local-global argument of Raskind briefly ([I91

Proposition 3.6; see also [22] Theorem 5.1), which is a key step in the proof of the cases (1)-(5). We will also use this argument in our proof of Theorem 3 (cf. 52). For a place p of k, we write kp for the completion of k at p, and write r;;; for the regulator map for Xa, . We fix a finite set S of places of k containing all the places where X does not have good reduction.

First, in the cases (1)-(5), we have

for any place p of k (Case (1): Deligne [6] Corollaire 3.3.9, and Nekov6.i. [16] Theorem D (i). Case (2): Remark 5 below. Case (3): Saito [20] p.64, Theorem 4.1. Case (4): Raskind [I91 the earlier part of the proof

K. Sato Finiteness of a certain Motivic Cohomology Group of Varieties 329

Fig. 1. Table of the known cases

of Proposition 3.2. Case (5): [16] Theorem D (ii). See also Corollary 7 below). Then by a diagram chase which is not so difficult, we can see that 1m(ri9") n F2H:,,,(X, Ql(n)) is contained in the image of the following 01-vector space:

Here Gs := Gal(ks/k), and ks denotes the maximal galois extension of k unramified outside of S. Finally, the map a;" is injective by results of Jannsen since i 5 2n ([lo] $6 Theorem 4, [19] Theorem 4.1).

Remark 5. In the case i 5 n, one can show that for every place

p of k,

Therefore F2H:,(Xkp, Ql(n)) = 0 for any place p of k. If X has potentially good reduction a t p, (1.3) immediately follows from Deligne's proof of the Weil conjecture [6] 3.3.9. If X does not have potentially good reduction at p, (1.3) follows from the alteration theorem of de Jong [12], the Rapoport-Zink theorems [18] Satz 2.21, 2.23, and the Weil conjecture.

$2. Finieness theorem

In this section, we will prove the vanishing (1.1) for the case i = n+l . We call the completion of a global field at a non-archimedean place a local field. The essential result is the following:

Theorem 6 ([22] Theorem 2.1). Let K be a local field, and X a proper smooth variety over K. Let 1 be a prime number diflerent from the characteristic of K , and n an arbitrary integer at least 2. Then the

group N'H:;' (x, Q/G (n)) n F2H,",e' (x, 0 1 1 ~ ~ (n))

is finite. Here Ql/Zl(n) := 1&, (pp)@", and p p denotes the etale sheaf of 1"-th roots of unity. N' denotes the coniveau filtration and F' denotes the Hochschild-Serre filtration (see the map ap below).

In the case n = 2, this finiteness was originally proved by Salberger [21]. We will give a rough proof of Theorem 6 later. Admitting Theo- rem 6, we prove

Corollary 7. Let k be a global field, and X be a proper smooth variety over k. Let 1 be a prime number which is diflerent from the characteristic of k, and n be an arbitrary integer at least 3. Then for every non-archimedean place p of k, we have

1m(r;I:17") n F~H:~ ' ( x k P , Q1 (n)) = 0.

Here r;Ill'" denotes the regulator map for Xkp .

330 K. Sato

This corollary and the argument in 51 imply Theorem 3. The condition that "k is a function field" in Theorem 3 was used to control the Ql -vector space (1.2).

Proof of "Theorem 6 Corollary 7". Let p be an arbitrary place of k. Consider the image of the group

I := 1m(r;:17") n F2H:;' ( X k , , Ql (n))

under the canonical map

Note that I is a divisible group. By a result of Soul6 [25] 2.1 Thkorkme 1, the image of I is contained in the subgroup

which is finite by Theorem 6. Therefore I has trivial image in this group and is contained in ker(7r). On the other hand, ker(7r) is finitely generated as a Z1-module by the exact sequence

and the fact that H:;l(XkP, Zl(n)) is a finitely-generated Z1-module. Hence ker(.rr) contains no non-trivial divisible subgroup, and I is trivial.

Q.E.D.

Finally, we state the outline of a proof of Theorem 6. In the following, cohomology groups of a scheme are taken over the etale topology. Cohomology groups of a field mean etale cohomology groups of the spec- trum, or equivalently, Galois cohomology groups of the absolute Galois group. We consider the following composite map:

Here the first arrow is the Hochschild-Serre mapping, and the subgroup F2 of HE;' (x, Ql /Z1 (n)) is defined by the image. On the other hand, the subgroup N1 is defined by the kernel of the s a n d map. Therefore, our task is to prove that a? has finite kernel.

We write OK for the ring of integers of K, and write IF for the residue field of K. Thanks to the alteration theorem of de Jong [12], the problem is reduced to the case X has a regular model proper flat over OK with strict semi-stable reduction (cf. [22] (2.1)). In the following, we

Finiteness of a certain Motivic Cohomology Group of Varieties 331

assume that X has regular model X/OK as above, and that 1 # ch(F) (see [22] 54 for the ch(IF)-primary case). We write R*W&/Z1 for the sheaf of vanishing cycles, and Jn (resp. 7) for the set of the generic points of the intersections of n irreducible components of Y := X@o, F (resp. Y @p IF).

Intuitively, we compute the quotient of weight -2 of Hn-I (X, Ql /Zl (n)) by the Rapoport-Zink theorems [18] Satz 2.21, 2.23 and the Weil conjecture [6], and prove the finiteness of ker(aY). Precisely, we can construct the following commutative diagram (cf. [22] (2.2)) :

and prove that the map ,By is injective (loc. cit. (2.3)-(2.4)), and that the map yf has finite kernel (loc. cit. Lemma 2.6). Moreover, we can show that y? is injective for almost all primes (# ch(IF)) by a theorem of Gabber [7], and hence that the group in Theorem 6 is trivial for almost all 1 ([22], Lemma 3.2).

Remark 8. The local-global principle of Jannsen ([lo], Theorem 3) and Theorem 6 imply that for a proper smooth variety over a number field and for an arbitrary integer n > 2, the group

is finite [22], 54. i

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(ed.) K-Theory, Arithmetic, and Geometry. (LNM Vol. 1289 pp. 1-26) Berlin Heidelberg New York: Springer-Verlag 1987.

[3] Bloch, S., Algebraic K-theory, motives, and algebraic cycles, Proc. of I.C.M. Kyoto, 1990, 43-54.

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[4] Bloch, S., Kato, K., L-functions and Tamagawa numbers of motives, Cartier, P., Illusie, L., Katz, N. M. et al. (eds.) The Grothendieck Festschrift I. pp. 333-400: Birkauser 1990.

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[8] Hyodo, O., Kato, K., Semi-stable reduction and crystalline cohomology with logarithmic poles, Pkriodes padiques. Skminaire de Bures, 1988. (Astkrisque 223, pp. 221-268): Sociktk Mathkmatique de France 1994.

[9] Jannsen, U., Continuous e'tale cohomology, Math. Ann., 280(1988), 207- 245.

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[12] de Jong, A. J., Smoothness, semi-stability, and altemtions, Publ. Math., Inst. Hautes ~ t u d . Sci., 83(1996), 51-93.

[13] Kato, K., Semi-stable reduction and padic etale cohomology. Phriodes padiques, Skminaire de Bures, 1988. (Astkrisque 223, pp. 269-293): Sociktk Mathkmatique de France 1994.

1141 Langer, A., Raskind, W., 0-cycles on the self-product of a CM elliptic curve over Q, preprint.

1151 Merkur'ev, A. S., Suslin, A. A., K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izv., 21(1983), 307-34 1.

1161 NekovAE, J., Syntomic cohomology and p-adic regulators, preprint. 1171 Quillen, D., Higher algebraic K-theory I, Bass, H. (ed.) Algebraic K -

theory I. (LNM vol. 341 pp. 85-147) Berlin Heidelberg New York: Springer-Verlag 1973.

[18] Rapoport, M., Zink, T., ~ e b e r die lokale Zetafunktion von Shimurava- rietaten. Monodromiefiltmtion und verschwindende Zyklen in ungle- icher Charakteristik, Invent. Math., 68(1982), 21-101.

[19] Raskind, W., Higher 1 -adic Abel- Jacobi mappings and filtmions on Chow groups, Duke Math. J., 78(1995), 33-57.

1201 Saito, S., Class field theory for curves over local fields, J . Number Theory, 21(1985), 44-80.

[21] Salberger, P., Torsion cycles of codimension two and 1-adic realizations of motivic cohomology, David, S. (ed.) Skminaire de Thkorie des Nombres 1991/92 Boston: Birkhauser .l993.

[22] Sato, K., Abel- Jacobi mappings and finiteness of motivic cohomology groups, Duke Math. J., lO4(2OOO), 75-112.

[23] Serre, J.-P., Cohomologie Galoisienne, (Lecture Notes in Math. 5) Berlin, Heidelberg, New York: Springer-Verlag 1965.

[24] Soulk, C., Operations on &tale K-theory. Applications, Dennis, R. K. (ed.) Algebraic K-theory, Oberwolfach 1980, Part I. (LNM vol. 966 pp. 271- 303) Berlin Heidelberg New York: Springer-Verlag 1982.

PSI ---- , Ope'rations en K-the'orie alge'brique, Canad. J . Math., 37(1985), 488-550.

1261 Tate, J., Relations between Kz and Galois cohomology, Invent. Math., 36(1976), 257-274.

[27] Tsuji, T., p-adic e'tale cohomology and crystalline cohomology in the semi- stable reduction case, preprint.

Gmduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 4 64-8602, Japan E-mail address: kanetomo0math. nagoya-u. ac . jp


Iwasawa Theory - Past and Present

Ralph Greenberg

Dedicated to the memory of Kenkichi Iwasawa

Let F be a finite extension of Q. Let p be a prime number. S u p pose that F, is a Galois extension of F and that I' = Gal (F,/F) is isomorphic to Z,, the additive group of padic integers. The nontrivial closed subgroups of I' are of the form r, = rpn for n 2 0. They form a descending sequence and r/r, is cyclic of order pn. If we let F, = F$, then we obtain a tower of number fields

such that F,/F is a cyclic extension of degree pn and F, = U, F,. In 1956, at the summer meeting of the American Mathematical Society in Seattle, Iwasawa gave an invited address entitled A theorem on Abelian groups and its application to algebraic number theory. The application which he discussed is the following now famous theorem.

Theorem. Let pen be the highest power of p dividing the class number of F,. Then there exist integers A, p, and v such that en = An + ppn + v for all suficiently large n.

Iwasawa's proof of this theorem is based on studying the Galois group X = Gal (L,/F,), where L, = U, L, and L, is the pHilbert class field of F,. (That is, L, is the maximal abelian pextension of F, which is unramified at all primes of F,. By class field theory, L, is a finite extension of F, and [L, : F,] = pen .) The extension L,/F is Galoisian, and one has an exact sequence

Since X is a projective limit of finite abelian pgroups, we can regard X as a compact Z,-module. (Z, denotes the ring of padic integers.) But

Received November 16, 1998. Revised May 13, 1999.

336 R. Greenberg

there is also a natural action of I? on X. If y E r and x E X , one defines y(x) = yxy-', where 7 E Gal (Lw/F) is such that TIFm = y. All of this structure allows Iwasawa to study the growth of [L, : F,] which, as we mentioned, is equal to pen.

The details of the proof of the above theorem were published in 1959 in [ Iw~] . Iwasawa has written more than twenty papers about the theory of Zp-extensions. (He referred to extensions F,/F as described above as I?-extensions until the late 1960's, when he switched to calling them Zp-extensions.) These papers introduced many new ideas which have really blossomed over the years. Several hundred papers have been written pursuing various aspects of these ideas, which have turned out to be fruitful in a number of different ways. In this article we will give a somewhat sketchy and personal account of these ideas and how they have developed. Along the way we will mention some of the many open questions which this topic has provided.

1. The relationship between the structure of X (together with the action of I?) and the groups Gal (L,/F,) is rather easy to establish if we assume that F has just one prime p lying over p and that this prime is totally ramified in F,/F. The prime p would then be the only prime of F which is ramified in F,/F. For if q is any prime of F not lying over p, then q could be at most tamely ramified in the abelian extension Fn/F for any n. As is well-known, the ramification index for q in Fn/F must then divide N(q) - 1, where N(q) denotes the cardinality of the residue field for q. It follows that the inertia subgroup of I? for q would be finite, which implies that it must be trivial since I? is torsion-free. This argument shows in general that only primes of F lying over p can be ramified in a Zp-extension F,/F.

Let L: denote the maximal abelian extension of F, contained in L,. Obviously, F, C L: and L, C L:. Let p, denote the unique prime of F, lying over p, which is the only prime of F, ramified in L:/Fn. Clearly L, = (L:)'", where I, denotes the inertia subgroup of Gal (LEIF,) for p,. Now I, n Gal (L:/F,) = 0 since L:/F, is unramified. Therefore L: = L, F, and, since L, n F, = F,, we have Gal (Ln/Fn) 2 Gal (L:/F,). On the other hand, Gal (L,/ L:) is precisely the derived subgroup of G, = Gal(L,/F,). We have an exact sequence

Let yo be a fixed topological generator of r. (This means that the subgroup generated by yo is dense in r. It suffices to choose yo E such

that yOJFl is nontrivial.) Then y, = y~~ is a topological generator of I?,.

Iwasawa Theory - Past and Present 337

Since I?, acts on X by inner automorphisms, one can see that y,(x)x-' is a commutator in G, for each x E X . It is not hard to show that the derived subgroup of G, is precisely { ~ , ( x ) x - ~ ( x E X). Changing to an additive notation for X , we write this as w,X, where w, = yn - 1. Therefore, Gal (L:/Fw) E X/w,X, giving the result that

(1) Gal (LJF,) 2 X/w,X

for all n 2 0. This isomorphism is induced by the restriction map from X to Gal (Ln/Fn).

Let A be a discrete, pprimary, abelian group on which I? acts continuously (as automorphisms). Assume that Arn = {a I a E A, y,(a) = a ) is finite for all n 2 0. (Iwasawa uses the term "strictly l?-finite" for such an A.) The structure theory which Iwasawa develops in [Iw3] then allows him to prove that ( A ~ " I = p X n + ~ ~ n + " for all sufficiently large n, where the integers X and p are described in terms of the structure of A and where v E Z. He applies this to A = HomCont(X, Qp/Zp). The action of r on this group is induced by the action of I? on X . Note that X/w,X is the maximal quotient of X on which I?, acts trivially. Hence ~~n = Hom(X/w,X, Qp/Zp) is finite and has the same order as X/w,X. Iwasawa's theorem would then follow (in the special case where F has just one prime above p, totally ramified in F,/F).

Serre gave a Seminaire Bourbaki lecture on Iwasawa's results in 1959. There he introduced a somewhat different approach which Iwa- sawa soon adopted. The idea is to view X as a module over the ring A = Zp[[T]] by letting T act on the Zp-module X as wo = yo - 1. This makes X into a Zp[T]-module. One can easily show that the action of T on X is "topologically nilpotent", i.e., any open subgroup of X contains T n X for n >> 0. Then X does become a A-module. It turns out to be a finitely generated, torsion A-module. (This is true without any assumption about the primes of F over p. In the special case that we have been considering, it follows easily from the fact that X/TX 2 Gal (Lo/Fo) is finite, together with a version of Nakayama's Lemma for compact A-modules.) Serre then derives Iwasawa's structure theorem from a classification theorem for such A-modules.

This classification theorem is quite easy to state:

Theorem. If X is any finitely generated, torsion A-module, then there exists a A-module homomorphism

338 R. Greenberg Iwasawa Theory - Past and Present

with finite kernel and cokernel, where each fi(T) is an irreducible element of A and each ai is a positive integer for 1 < i < t . The value of t, the prime ideals (fi(T)), and the corresponding ai's are uniquely determined by X , up to their order.

A A-module homomorphism with finite kernel and cokernel is often called a pseudo-isomorphism. The ring A is a UFD, but not a PID. Furthermore, A is a complete, Noetherian local ring (with maximal ideal m = (p, T)) and is regular of dimension 2. The prime ideals of height 1 are principal. One of them is (p) = PA. The others have a unique generator of the form f (T) = T' + U ~ - ~ T ' - ' + . . . + ao, where 1 > 1, ao, a1 , . . . , a'-1 E pZp, and f (T) is irreducible as an element of Qp [TI. (A polynomial of this form, irreducible or not, is called a "distinguished" polynomial.) We will assume that each fi(T) is either p or an irreducible, distinguished polynomial. Then we define

One refers to fx(T) as the characteristic polynomial of X . The invariants X and p which occur in Iwasawa's theorem can be described just in terms of fx(T). (No hypothesis on the primes of F lying over p is needed.) It turns out that X = deg(fx (T)) and that p is just the largest integer such that pp divides fx(T) in A (or Zp[T]). One can also describe X and p in terms of the A-module X. We have X/Xzp-tors 2 Zi . This determines X just in terms of the structure of X as a Zp-module. As for p, let Y = Xzp-tor,. Since A is Noetherian, Y is finitely generated as a A-module. It therefore has finite exponent pC as a group. For i 2 0, p iY/p i+ '~ is a module over the ring h = AlpA, which is simply IFp[[T]], with IFP = Z/pZ. Then p is just the sum of the x-ranks of the modules piY/p"lY, where 0 5 i < c - 1. It is often better to think of A in a more intrinsic way as Zp([r]], which by definition is Lim Zp[Gal (Fn/F)]. This inverse limit is defined by the Zp-algebra ho- t

momorphisms Zp [Gal (F,/ F)] -t Zp [Gal (Fn/ F)] (for m > n) induced by the restriction maps c -t clFn for o E Gal (Fm/F). The identification of Zp [[I?]] with Zp [[TI] depends on the choice of topological generator "/o for r. One identifies yo with 1 + T.

We continue with the special case where only one prime p of F lies over p and F,/F is totally ramified at p. Then pen = IX/wnXI for n > 0. To study how these orders grow, one reduces to the case of a A-module of the form Y = A/(g(T)), where g(T) is one of the fi (T)ai 's.

Then Y/w,Y will also be finite and we have

where we write a ;;: b to mean that two nonzero padic integers a and b satisfy ab-' E Z,X. This is not hard to verify. The quotient ring A/w,A can be identified with the group ring Zp [Gal (Fn/F)] . (Thinking of A more intrinsically as A = LimZp[Gal (Fn/F)], one has a surjective

e

homomorphism A -t Zp[Gal (Fn/F)] defined by sending T = yo - 1 to the element yolFn - 1. The kernel of this homomorphism is generated by w,.) Thus, A/wnA is a free Zp-module of rank pn. Multiplication by T on A/w,A is Zp-linear and has eigenvalues C - 1, where [pn = 1. Now Y/wnY is the cokernel of multiplication by g(T) on A/wnA. This map is Zp-linear and has determinant n< g([ - I) , where Cpn = 1. This

implies (2). If g(T) = pm, then one gets IY/wnYI = pmpn. If g(T) =

T' + a l - l ~ ' - ' + + ao, where plai for 0 < i < I, then the valuation of g(C - 1) is the same as that of (C - 1)' when [ has sufficiently large order. One then finds that IY/wnYI = p'n+" when n >> 0, where v is a constant. Putting all of this together, and taking into account the finite kernel and cokernel, one obtains that JX/wnXI = pXn+ppn+" for n >> 0, where v is a constant.

A more detailed account of the proof of Iwasawa's theorem (including the general case where one must keep careful track of the inertia groups for primes above p in Gal (L;/F,)) can be found in [ I w ~ ] , [Sell, or perhaps more conveniently in Washington's book [Wa]. In the general case, it sometimes happens that X/w,X is infinite. This happens precisely when fx([ - 1) = 0 for some pn-th root of unity [.

2. We know very little about the Iwasawa invariants X and p associated to an arbitrary Zp-extension F,/F. We will just mention two rather special results.

Proposition (2.1). Assume that the class number of F is not divisible by p and that F has only one prime lying over p. Let F,/F be any Zp-extension. Then X = p = v = 0.

Proposition (2.2). Assume that p splits completely in F/Q. Let F,/F be a Zp-extension in which every prime of F lying over p is ramified. Then X(F,/F) > r z , where r a denotes the number of complex primes of F .

340 R. Greenberg

Proposition 2.1 is stated in [ I w ~ ] , and follows easily from a result proved in his earlier paper [Iwl]. We can prove it as follows. First note that the unique prime p of F lying over p must be ramified in F l /F . Otherwise Fl would be contained in the pHilbert class field Lo of F = Fo, contradicting the assumption that p doesn't divide the class number of F. This implies that p is totally ramified in F,/F. Using the notation described before, we have X/TX E Gal (Lo/ Fo) = 0. Hence TX = X and therefore X = 0 (because the action of T on X is topologically nilpotent). But then Gal (Ln/Fn) = X/wnX = 0 for all n, which clearly means that X = p = v = 0, as stated.

To prove Proposit ion 2.2, we need the following existence theorem for Zp-extensions.

Theorem. Let F denote the cornpositurn of all Zp-extensions of F . Then

Gal E z:,

where r 2 + 1 5 d 5 [F : Q]

One consequence of this theorem is that F will have infinitely many distinct Zp-extensions when F is not totally real. The proof of the theorem is a straightforward application of the idelic form of class field theory. Let

where UF denotes the group of principal units in the completion Fp of F at p. Then U0 can be considered as a Zp-module and

rankzp (U') = [Fp : Qp] = [F : Q]. P I P

The Artin map defines a homomorphism from U0 to G a l ( F 1 ~ ) with finite cokernel (isomorphic to Gal (Lo n F/F)). To describe the kernel of this homomorphism, let E denote the group of units in F and let E0 denote the subgroup of units e = 1 (mod p) for all plp, which has finite index in E . We can consider E0 as a subgroup of U0 by using the natural injection F -+ n Fp. Then the topological closure @ of E0 in

P I P U0 is a Zp-submodule; it is the image of E0 @z Zp and thus has Zp-rank bounded above by rankz(~O). Let H denote the kernel of the Artin map U0 -+ Gal (FIF). Then H can be characterized as the smallest

Iwasawa Theory - Past and Present 34 1

Zp-submodule of U0 containing @ and such that UO/H is torsion-free. Clearly [H:@] < co. The theorem follows from this because UOIH has Zp-rank equal to [F:Q] - rank%(@) and rankzp(@) 5 rankZ(E) = rl + 7-2 - 1, where r l = [F : Q] - 2r2 is the number of real primes of F.

It is extremely likely that we have the equality d = 7-2 + 1 in the above theorem. This is known as Leopoldt's Conjecture. It is clearly equivalent to the assertion that

the second quantity being r = rl + r 2 - 1. More concretely, it can be stated as follows. Let 01,. . . ,an denote the distinct embeddings of F into 8, where n = IF: Q]. Suppose that el, . . . , e, are generators of E0 (modulo the subgroup of roots of unity). Then the conjecture asserts that the n x r matrix [ l 0 g ~ ( 0 ~ ( ~ ~ ) ) ] ~ ~ ~ ~ ~ , 1 < t ~ ~ has rank r . Leopoldt considered just the case where F is totally real. Then he conjectured the nonvanishing of the so-called padic regulator for F, the determinant of the r x r submatrix obtained by omitting any row (well-defined up to f 1). The above formulation in terms of rankzp(Gal (FIF)) is due to Iwasawa and has been proven when F is an abelian extension of Q or of an imaginary quadratic field. In these cases, it follows from Brumer's padic version of a famous theorem of Baker concerning linear forms in logarithms of algebraic numbers.

Obviously F, C F. But it is also true that F C L, under the stated assumption that Fp = Qp for all plp. This is because the inertia subgroup Ip of Gal ( F p ) for each such p is precisely the image of U: (contained

in U0 as a direct factor) under the Artin map U0 + Gal ( F ~ F ) . Thus Ip E Zp. But the image of Ip in Gal (F,/F) under the restriction homomorphism is also isomorphic to Zp because p is ramified in F,/F. Therefore, Ip n Gal (FIF,) = 0, which implies that the primes of F, lying over any such p are unramified in FIF,. Since primes not dividing p are unramified in every Zp-extension of F , and hence in FIF, it follows

that F C L,. Thus X = Gal (L,/F,) has ~ a l (F/F,) - zZ-' as a quotient. This implies that indeed X = rankzp (X) > d - 1 2 r2. In fact,

I? acts trivially on the quotient Gal (FIF,) and consequently fx(T) is divisible by TT2. If r2 > 0, we obtain examples where X/woX is infinite, because wo = T.

Several of Iwasawa's papers discuss the values of X and p in the important case where F = Q(G), Fl = Q(Cp2), . . . , Fn = Q(Gn+l), . . . , where, for any m 2 1, we let C, denote a primitive m-th root of unity. Then F, = Un Fn is the so-called cyclotomic Zp-extension of F . If p is

342 R. Greenberg

a regular prime (i.e., a prime such that the class number of F = Q(Cp) is not divisible by p), then Proposition 2.1 immediately implies that X = p = v = 0 for this (and any) Z,-extension of F . We will denote the Iwasawa invariants for F,/F by A,, p,, and v,. How can one compute them for irregular primes p?

It is customary to factor the class number hp of F as hp = hih;, where h: denotes the class number of the maximal real subfield Ff =

Q(Cp + C i l ) of F. There is, of course, a similar factorization of the class number of Fn. The maximal real subfields form a Z,-extension F& = Un F$ of F + . The Iwasawa invariants can then be written as Xp = Xp + A:, pp = p; + p:, vP = v; + v:, where A:, p: , v: are the invariants for F&/F+. If X = Gal (L,/F,) as before, then Gal (F,/F&) r 2/22 acts on X (by inner automorphisms). Assuming that p is odd, we then get a decomposition X = X- @ X+. We have fx (T) = fx- (T) fx+ (T). The invariants A:, p;, and v: can be recovered from fx+ (T) (or from X + ) just as described earlier, and similarly for A;, p i , and v;. For example, A; = deg( fx- (T)) = rankzp ( X - ) . Let So denote the pprimary subgroup of the ideal class group of F = Fo. Then, by class field theory, So 2 Gal (Lo/Fo). Now Gal (F/F+) acts on So in an obvious way and on Gal (Lo/Fo) by inner automorphisms. The isomorphism is compatible with these actions (and even for the actions of A = Gal (F/Q) on both groups, which we will consider later). Corre- spondingly, we can write So = Si @ S: and the power of p dividing h i (respectively, h:) is just the order of Sg (respectively, s:). Our earlier arguments show that

A well-known conjecture of Vandiver states that p f h:. That is, S: = 0. . s we This would imply that X + = 0, and hence A: = p: = v: = 0 A

will mention later, there is considerable numerical support for Vandiver's conjecture.

Iwasawa's paper On some invariants of cyclotomic fields, published in 1958, is devoted to finding criteria for the nonvanishing of p,. One such criterion is an infinite sequence of congruences involving Bernoulli numbers. We will state the first two of these congruences below. It is in fact sufficient to consider p; TO justify this, Iwasawa refers to a theorem of Takagi [TI which states that

for all n 2 0. Here we let Sn denote the pprimary subgroup of the ideal class group of F,, which can be decomposed as Sn = S; @ S: by the


action of Gal (Fn/F$). The proof of (3) is based on the Spiegelungsatz (the Reflection Principle). It then follows that:

p: > 0 o p; > 0, and hence, pp > 0 o p i > 0.

(This is reminiscent of a theorem of Kummer stating that: h: o pl h; . That result can also be proved by using the Reflection Principle.) The fields Fn are abelian over Q. Iwasawa transforms the classical formula for the first factor of the class number of Fn, and also uses Stickelberger's theorem giving a nontrivial annihilator in Z[Gal (Fn/Q)] for S;, to obtain necessary and sufficient conditions for the nonvanishing of p i . We will just state here one necessary condition which turns out to be quite effective. (Iwasawa uses it to show that p; = 0 for p = 37,59,67-the three irregular primes < 100. This condition actually suffices to prove the vanishing of p; for all p < 16,000,000. See [BCEMS] for the latest information on such computations.)

If p i > 0, then there exists an even integer j, 2 5 j 5 p - 3, such that

(4) Bj = 0 (mod pZp) and Bj+p-l = - 3 (mod p 2 q ) . j + p - 1 j

Here Bj denotes the j-th Bernoulli number, which we recall is defined by the generating function

The Bj's are clearly rational numbers and are nonzero precisely when j is even or j = 1. If j $ 0 (modp - 1)) then Bj/j E Z, and so the congruences in (4) just involve elements of Z,. Kummer's famous criterion for regularity states that: plh, @ Bj = 0 (mod pZ,) for some even j, 2 5 j _< p - 3. The first congruence in (4) then follows because p; > 0 certainly implies that plh,. The second congruence is stronger than the well-known Kummer congruence:

Bj' - Bj (5) - - - - (mod pZp) if if1 r j $ 0 (mod p - 1),

j1 j

but it can sometimes hold. For example, we have (Bls/16) - (B4/4) = -7 - 1 3 ~ 1 2 ~ - 5 - 17, which implies the second congruence in (4) for the pair (p, j ) = (13,4).

344 R. Greenberg

A subsequent paper On the theory of cyclotomic fields (published in 1959) continues with the case F = Q(C,), F, = Q(Cp, Cp2, . . . ), introducing several new Galois groups into the topic. Let M, denote the maximal, abelian extension of F, which is pro-p (i.e., Y = Gal (M,/F,) is a projective limit of finite p-groups) and in which only the prime of F, lying over p is ramified. If L, is as before, then obviously L, c M,, and so X = Gal (L,/F,) is a quotient of Y = Gal (M,/F,), as A- modules. In contrast to X , Y is not A-torsion. Iwasawa shows that rankn(Y) = f (p - I), although he doesn't use the terminology of A- modules. Let N, be the field obtained by adjoining to F, all ppower roots of units of F,. Then N, c M, and Gal (M,/N,) is shown to be A-torsion. Let S, = LimS,, where the maps S, -+ Sm for m > n

defining this direct limit are as follows: if c E Sn is the class of the ideal a of the ring of integers On of Fn, then c is mapped to the class of aOm in Sm. Then S, can be regarded as a discrete A-module, which Iwasawa shows is isomorphic to Hom(Ga1 (M, IN,), ppw ). (Here ppw is the group of p-power roots of unity. The isomorphism preserves the action of Gal (F,/Q), and in particular the action of r, on both groups.)

We will state several important results from this paper. The Ga- lois action on p,- gives a canonical isomorphism Gal (F , /Q) r Z,X =

pP-1 x (l+pZp) for any odd prime p. Here p,-1 denotes the (p-1)st roots of unity in ZpX. We write Gal (F,/Q) = A x r, where r = Gal (&IF) as before. We regard A as Gal (F,/Q,), where Q, is the unique subfield of F, such that Gal (Q, /Q) = Z,. (In fact, Q, is the unique Z,-extension of Q.) There is also a canonical isomorphism w : A -+

p,-1 c Z,X defined by the action of A on ppw. If A is any Z,-module P - 2 k

on which A acts, then we have a canonical decomposition A = @ A" , k=O

where A " ~ = {ala E A,6(a) = wk(6)a i / 6 E A}. We consider this decomposition for A = X , Y, and S,. Iwasawa refers to this as the A-decomposition. Since Gal (F,/Q) acts on these groups, the corresponding actions of A and F commute and so we can regard xwk, ywk

(4 and skk as A-modules. For each k, 0 5 k 5 p - 2, let AF) and p,

denote the Iwasawa invariants for xwk, which are determined by the P - 2 P - 2

polynomial f,,k (T). Then A, = C AF), p, = C pF). The results we

want to mention are the following.

Proposition (2.3). Suppose that 0 5 i, j 5 p - 2 are integers such that i + j - 1 (mod p - 1) and i is odd (so that j is even). Then

A!) > - A P ) and p;) > &). Consequently, A; > A: and p; 2 $.


Proposition (2.4). Assume that i and j are as in the previous proposition. Then there exists a perfect pairing

which is compatible with the action of r .

(Note: This means that yWJ 2 H O ~ ( S ~ , yw ) as A-modules. The pairing is also compatible with the action of A because wiwj = w.)

Proposition (2.5). For odd i, xwi has no nonzero, finite A- submodules. For all k, ywk has no nonzero, finite A-submodules.

Proposition 2.4 is a refined version of the Reflection Principle. The pairing is defined roughly as follows. Let s E S; have order pm. Suppose that s is the class of an ideal a (coming from some level F,I of F,) such that ii = a-' and apm = (a) , a E F:,. One can choose a so that - a = a - l . Let y E Y. Define (s, y) E p,m by (s, y) = y(,O)/P, where p p r n = a. (One checks easily that ,O E MG.) This can be verified to induce a well-defined perfect pairing

and one obtains Proposition 2.4 by studying the action of A. Proposi- tion 2.3 is a consequence of Proposition 2.4. First note that

and so one has a surjective A-module homomorphism Ywk -+ xwk for all k. Hence, for even j, we see that fx,j (T) divides fy, j (T) in A.

Therefore A F ) and &) are bounded above by the A- and pinvariants of

Y"', which the above pairing shows are actually equal to A:) and p:), respectively. This equality follows from Iwasawa's theory of adjoints, which allows one to relate the structure of the discrete A-module S< to

that of the compact A-module x w i . In particular, the invariant A!) can be identified as the Z,-rank of xwi or as the 5-corank of sL. (That

is, the maximal divisible subgroup of S< is ( ~ 5 ) ~ ~ " " (Qp/Zp)*).) As we will mention below, Proposition 2.4 and the theory of adjoints gives an important relationship between xwi and Y"'.

Now we discuss briefly the proof of Proposition 2.5. The first part asserts that X - has no nonzero finite A-submodules. Iwasawa shows that this assertion is equivalent to the fact that the maps S; -+ S; for m > n are injective, which he verifies using properties of the units of

346 R. Greenberg Iwasawa Theory - Past and Present 347

F,, interpreted in terms of Galois cohomology. In a footnote, Iwasawa states that the injectivity of the map Si -+ ST was proved by F. Pol- laczek in [Po] where, Iwasawa writes, one may trace the germ of other results proved in the present paper. Iwasawa also refers to Pollaczek's paper in [ I w ~ ] . As for the second part, one crucial ingredient is the fact that S, has no proper A-submodules of finite index. (The pairing in Proposition 2.4 would then immediately give the result for even values of k.) Actually, it is true for every Zp-extension F,/F (with F arbitrary) that S, = Lim Sn has no proper A-submodules of finite index. This is

+

not hard to deduce from the fact that the norm map Nm,, : Sm + Sn is surjective for m > n > no, where no is large enough so that at least one prime (over p) is totally ramified in F, / Fn, . The surjectivity of the norm maps follows from class field theory together with the surjectivity of the restriction maps Gal (Lm/Fm) + Gal (Ln/Fn) for m > n > no.

One more result that was already alluded to above relates x"' and Y"' when i and j are as in Propositions 2.3, 2.4. Let K : r 2 1 + pZp denote the canonical isomorphism giving the action of r on ppm. There is an involution of A defined by sending y E r to ~ ( y ) E A '. If 70 is a fixed topological generator of r, then T = yo - 1 is sent to T = ~(yO)( l + T)-' - 1. Now if Z is a A-module, we define a new A- module 2 by letting 6 E A act on z E Z just as 02, where 0 is the image of 6 under the above involution. Then combining the theory of adjoints in [Iw4] with Proposition 2.4 (and the fact that S& is the adjoint of xwi ) , Iwasawa obtains the following theorem.

Proposition (2.6). Let i and j be as in Proposition 2.3. Then, yw3 is pseudo-isomorphic to xW2 . Hence f y w j (T) generates the same ideal as fxwi (n(yo) (1 + T)-' - 1) in A.

3. Returning to the question of determining A,, p,, and up, Iwasawa shows in [Iw2] that p, = 0 and A, = 1 for p = 37,59, and 67. We've already discussed the vanishing of p,. Now it is known that p) [ h, for these three primes. Hence pllh; and so So = Sc is cyclic of order p. Recall that X/TX g So. Therefore, X is a cyclic A-module (by Nakayama's Lemma). Since X = X - has no nonzero, finite A-submodules by Propo- sition 2.5, it follows easily that X is isomorphic to A/(g(T)), where g(T) = fx (T). By (2), we have g(0) = p, but it is also known that IS1 I = p2 for the above three primes, and so we have n g ( [ - 1) = p2,

where [ runs over all p t h roots of 1. By (2), it follows that pp = 0 and that A, = up = 1, for p = 37, 59,. and 67.

More generally, suppose that So is cyclic as a Z[A]-module. Equiv- alently this means that stk is a cyclic group for all k. Nakayama's

Lemma then implies that x"* is a cyclic A-module. We obtain the following result, again using Proposition 2.5.

Proposition (3.1). Suppose that So is cyclic as a Z[A] -module. Then, for every odd integer i , 1 < i 5 p - 2, we have

where I is the principal ideal (fxu. (T)) of A.

The ideal I is called the characteristic ideal of x"'. Under the hypotheses of Proposition 3.1, Iwasawa proves in [Iw7] that a certain generator gi(T) of I can be chosen in a completely explicit way, which can be used quite effectively for computation. Write

g, (T) = C b:)~m

where bm E Zp for m > 0. It is clear (since gi(T) and fx,"T) differ by

multiplication by an element of AX) that pf) = 0 if and only if p t bk)

for some rn 2 0. In this case, hf) is equal to the smallest such value of

m. The constant term of gi(T) is given by b t ) = -B1,,-. where, for a Dirichlet character x of conductor f , one defines

One thinks of x = w-i as a Dirichlet character of conductor p by the canonical isomorphism A =(ZlpZ) X . A congruence argument shows

that b t ) o pl Bj, where i + j = 1 (mod p - 1) as before. Therefore,

if p 'i Bj, then gi(T) E AX and A:) = p:) = v!) = 0. On the other

hand, if plBj, then either A f ) or pg) must be positive. If Vandiver's Conjecture (that p { h:) is valid for the prime p, then it follows from

the Reflection Principle that s:' is cyclic for each odd i, and so the hypothesis in Proposition 3.1 holds. In [I-S], Iwasawa and Sims find that for all primes p 5 4001 and for all even j (2 5 j 5 p - 3) such that

pl Bj, the coefficient by) turns out to be in Z,X . Vandiver's conjecture

was known to hold for these primes. Thus pg) = 0 and A;) = 1 (i.e., XU' " Z,) for those pairs (p, i). One also finds that s:' h. Z/pZ and

ug) = 1, both following from the fact that p2 1 b r ) for p < 4001.


Similar computations have been carried out by many others, extending to date up to p < 16,000,000. (We refer the reader to [BCEM] , [BCEMS], and to the references given there.) But so far nothing essentially different has been found. That is, for p < 16,000,000, one has: (i) p t h,C, (ii) pp = 0, (iii) X i ) = vg) = 1 when plBj, and (iv) So has exponent p. Concerning (ii), Ferrero and Washington succeeded in proving in 1978 that pp = 0 for a11 primes p. Their proof is based on

a careful study of the explicit expressions for the b:),s. As for (iii) and

(iv), this amounts to verifying that p l l 6;) and p { bv) for the pairs (p, i) where p(Bi. One then has the equality

However, it seems reasonable to conjecture (on probabilistic grounds)

that A!) > 2 holds for infinitely many pairs (p, i), i.e., p lb t ) and b?). But no such pair has yet been found. We have already mentioned that p lb t ) if and only if Bj I 0 (mod pZp), which is the first congruence in

(4). As we will explain later (by using padic L-functions), pjb(:) if and only if the second congruence in (4) holds. It also seems reasonable to

conjecture that p2 1 b t ) holds for infinitely many pairs (p, i ) . Assuming that p 1- h,C, that would mean that So is not of exponent p.

Suppose now that F is any finite extension of Q. Let p be any prime. We will consider the A- and pinvariants associated to the cyclotomic Zp-extension F,/F, which is defined by F, = FQ,. Concerning the pinvariant, Iwasawa made the following well-known conjecture.

Conjecture (3.2). Let F,/F be the cyclotomic Zp-extension. Then p(F,/F) is equal to 0.

We mentioned earlier that S, = Lim S, has no proper A-submod- --+

ules of finite index. If p(F,/F) = 0, then it would follow that

as a Zp-module, where X = X(F,/F). This is an illustration of an interesting analogy with the theory of algebraic function fields of one variable. Iwasawa discusses this analogy in several places and it seems to have been an important source of motivation from the start. (It is already mentioned in [ Iw~] . ) Suppose that K = k(x, y) is the function field of an absolutely irreducible- curve C over a finite field k. Let g be the genus of C. Let % denote an algebraic closure of k. Then it is known that the pprimary subgroup of the divisor class group (of degree

0) for z(x, y) is isomorphic to (QP/Zp)'g, assuming that p # char(k). (One can identify this divisor class group with J(E) , where J denotes the Jacobian of C. If p = char(k), then the pprimary subgroup is isomorphic to (Qp/Zp)a, where 0 5 a 5 g.) Now K does have a Zp-extension K, = k, (x, y), where k, denotes the unique subfield of z containing k such that Gal (k, / k) 2 Zp. (Recall that Gal (Elk) 2 2.) The divisor

class group (of degree 0) can be identified with J(k,) = ~ ( z ) ~ . ' ('Ikm). Its pprimary subgroup is easily seen to be divisible and hence isomorphic to (QP/Zp)', where 0 5 X 5 29 (or 0 5 X 5 a if p = char(k)).

The only general result to date concerning Conjecture 3.2 is the following theorem of Ferrero and Washington [F-W] which we already alluded to above in the special case F = Q(Cp).

Theorem (3.3). Assume that F /Q is abelian and that F,/F is the cyclotomic Zp-extensions. Then p(F,/F) = 0.

The proof for F = Q(Cp) is based on one of the criteria given in [ I w ~ ] , together with results about normality of the padic expansion of padic integers. A rather different proof was discovered by Sinnott [Si]. If q is a prime and q # p, then, under the same hypotheses as in Theorem 3.3, Washington proves in [Wall that the number of elements of order q in the ideal class group of F, is bounded as n + oo. (The vanishing of p(F,/F) is just this same statement when q = p.) It is not hard to deduce that the power of q dividing the class number of F, must then be bounded as n -t oo.

It is interesting to speculate about the behavior of X(F,/F), but quite hard to prove anything of a general nature. The analogy with function fields suggests the following possibility: if F is a fixed number field, but the prime p is allowed to vary, then X(F,/F) is bounded. For F = Q, this is certainly true since Proposition 2.1 implies that X(Q,/Q) = 0 for all p. But it has not been verified for any other number field F . The equality (6) suggests the question of how X(F,/F) and dimzlpz(So/pSo) might be related. These quantities are certainly not necessarily equal. For example, there are many real quadratic fields F such that X(F,/F) = 0, but So # 0, where p is either 2 or 3. We will mention some examples later. On the other hand, suppose that F is an imaginary quadratic field and that p is an odd prime. Then one has the inequality

This is because So = SF, S, = S,, and the map S; + S& is injective. Since S, - (Qp/~p)X(F, lF)7 the inequality is obvious. It is often a

350 R. Greenberg Iwasawa Theory - Past and Present 35 1

strict inequality. For example, suppose that So = 0 and that p splits in the field F. By Proposition 2.2, we have X(F,/F) > 1. There are infinitely many such primes p. T. Fukuda [Fu] has done extensive and systematic calculations of X(F,/F) when F is imaginary quadratic and p is 3, 5, or 7. It seems reasonable to believe that: if p is a fixed prime and F varies over all imaginary quadratic fields, then X(F,/F) is unbounded. For p = 2, this is not difficult to prove. For p > 3, it is an open question. The record to date is due to Fukuda, namely X = 14 for p = 3. He found three such fields, one of which is F = Q( J-956238) which has class number 3.

The following conjecture was proposed and studied in [Grl].

Conjecture (3.4). Assume that F is a totally real number field and that F,/F is the cyclotomic Zp-extension. Then X(F,/F) =

p(F,/F) = 0. That is, the power of p dividing the class number of Fn is bounded as n + oo.

According to Leopoldt's conjecture, L,/F, should be the only 25,- extension of F . The above conjecture states that X = Gal (L,/F,) should be finite. In [Grl], several sufficient conditions for this to be true are proved and a few examples are given where one can verify that X is finite, but nontrivial. (That is, X = p = 0, but v > 0.) An expanded version of [Grl] was published in 1976 ([Gr3]), including many more examples. Since then this conjecture has been studied by T. Fukuda, K. Komatsu, H. Taya and many others. If F has just one prime lying over p, totally ramified in F,/F, then a necessary and sufficient condition for Conjecture 3.4 to hold is that ker(So -+ S,) = So for some m > 0. This is proved in [Grl, 31, but we will now give a simpler proof using the fact that X/w,X E S, for all n. This isomorphism is equivariant for the action of Gal (F, IF) . By class field theory, the norm map NhIF : Sn -+ SO is surjective. Let qn = wn/wo E A. The image of qn in A/w,A = ZP[l?/rn] is just the norm element and so it follows that Im(So -+ S,) = qnSn. If ker(So -+ Sm) = So, then qmSm = 0 and therefore qmX C wmX. Since qrnlwm in A, we must have q,X = w,X. Letting Y = w,X, it follows that woY = Y and Nakayama's lemma then implies that Y = 0. Thus, we see that X % S, if ker(So -+ Sm) = So. For the necessity, we just remark that, for an arbitrary 2,-extension, S, has no proper A-submodules of finite index. If X is finite, then S, would also be finite, and therefore S, = 0. It would follow that, for any n > 0, there exists an m > n such that ker(S, -+ S,) = Sn. An interesting example illustrating the above criterion is F = Q ( a ) and p = 3. Then So 2/32. In this case, Kurihara, Ichimura-Sumida, and Kraft-Schoof independently found that ker(So -+ S,) = So holds for

m = 5, but not for m = 4. Thus, X E S5, which is cyclic of order 35. For more on this general topic, we refer the reader to [F-K] and also to [Ic-S], [0-TI and the numerous references which are given there.

Suppose now that F is an arbitrary Galois extension of Q. We suppose also that F n Q, = Q. Then F, = FQ, is Galoisian over Q and Gal (F,/Q) E A x I?, where A = Gal (F,/Q,) can be identified with Gal (F/Q). Let x be the character of an irreducible representation of A over op, with underlying representation space V,. Let d, =

dimap (V,). Let X = Gal (L,/F,) as before. Then VF = X 8 z p $ is

a finite-dimensional representation space for A over $. Its dimension is X(F,/F). We define Ax to be the multiplicity of V, in VF. That is, A, = dim- Homn (VF, V,)). Then we have

QP (

where x runs over all irreducible characters of A. (Note that in defining each A,, one can assume that x is faithful, changing F if necessary.) Now we can write dx = d l + d;, where d: denote the dimensions of the (f 1)-eigenspaces for the action of a complex conjugation So E A. (One fixes an embedding of F into @ to define So. The dimensions d: are independent of this choice.) If dx = d',, then one can assume that F is totally real. Conjecture 3.4 then implies that Ax = 0 for all such X . If d, = d;, then one can assume that F is a totally complex quadratic extension of a totally real number field F + . (That is, F is a so-called CM field.) In this case, Ax is often nonzero. As we will mention later, the value of A, is related to the number of zeros of a padic Artin L- function which can be associated to X . The simplest case is when F is an imaginary quadratic field (i.e. F+ = Q and x is an odd Dirichlet character of order 2). Then A, = XF.

The "mixed" case (where dz and d; are both positive) seems quite mysterious. Virtually nothing is known. One can use Proposition 2.2 to give examples where A, is nonzero. To explain this, note that A =

Gal (F/Q) acts on Gal (F/F) by inner automorphisms and therefore Gal (PIP) @zp$ becomes a representation space for A. One can verify that Vx occurs in this representation space with multiplicity at least d;. (That's the exact multiplicity if Leopoldt's conjecture is valid for F and p.) If p splits completely in F/Q, the proof of Proposition 2.2 then shows that A, 2 d;. More generally, let Ap denote the decomposition subgroup of A corresponding to a prime p of F lying over p. Then one


can show that

It would be interesting to find examples where this inequality is strict. Are there such examples if one requires that p 1( [F : Q]?

Consider the case where F is a totally complex field and A = Gal (FIQ) is dihedral of order 2m, where m is odd. If m > 1, any faithful irreducible representation of A is 2-dimensional and of mixed type. Also F contains a unique imaginary quadratic subfield K . If k denotes the compositum of all Zp-extensions of K, then Gal (K/ K) 2 Z;. Con- sidering the action of Gal (KIQ) on this group, one can find a unique Z,-extension K g / K such that K z / Q is Galoisian and the nontrivial element of Gal (KIQ) acts by -1 on Gal ( K Z I K ) . One often refers to K z as the "anti-cyclotomic" Zp-extension of K . Assume that p is a fixed odd prime and that K is also fixed. For a positive n, let F = KEC, the n-th level in the Zp-extension K g / K . Then F n Q, = Q and Gal (F/Q) is dihedral of order 2pn. It seems reasonable to believe that A, = 0 if x is a faithful character of Gal (FlQ) and n >> 0. Equivalently, this means that A(K,""Q,/K,"") is bounded as n + oo. There is another interesting interpretation. Let K& = KQ, be the cyclotomic Zp-

extension of K . Then K = KZKk and Gal ( k / ~ ) = r- x I?+, reflecting the action of Gal (KIQ) on Gal ( k / ~ ) . Here r- = Gal (Z/K&), I'C = Gal ( k / ~ = ) . Let i denote the maximal, abelian, unramified, pro-p extension of (or, more briefly, the pro-p Hilbert class field of - - K). Let 2 = Gal (LIK), which can be viewed as a module over the ring i? = Zp[[Gal (E/K)]] . This ring can be identified with a formal power series ring over Zp in two variables. It is known that 2 is a finitely gen- - erated, torsion A-module. It might in fact be pseudo-null, which means that it has two relatively prime annihilators in % (which is a UFD). (Note: The finitely generated pseudo-null modules over A = Zp[[T]]

are simply the finite A-modules. But over i? = Zp[[Tl, TZ]], pseudo-null modules can be infinite.) Now we have the following equivalence:

Ax = 0 for all n >> 0 t, 2 is a pseudo-null x-module.

We will just sketch the reason for this. As a module over A- = Z,[[r-I], one can show that 2 is still finitely generated. (The crucial ingredient is to show that z/(y; - 1 ) g is finitely generated as a Zp-module, where 7,- is a topological generator of r - . This follows from the fact that - p(K&/K) = 0.) Let w; = ( s ) p n - 1. NOW K;Q, = ~ ~ b n d one

can show that A(K;Q,/K;) = rankg(%/w;2) + O(1) as n --+ oo.

But it is easy to see that rankzp(z/w;2) is bounded if and only if - rankA- (X) = 0, and that this will be true precisely when 2 is pseudo- null as a %-module.

For several different reasons, including the remarks in the previous paragraph, we have been tempted to make the following conjecture.

Conjecture (3.5). Suppose that F is a number field and that p is a prime. Let denote the compositum of all Zp-extensions of F . Let Z - - denote the pro-p Hilbert class field of F and let 2 = Gal (LIP) , regarded - as a module over the ring % = Zp[[Gal ( P ~ F ) ] ] . Then X is a pseudo-null

i?-module.

We refer to [N] and to [L-N] for some equivalent versions of this conjecture and some additional references.

4. In his paper, On some modules in the theory of cyclotomic fields ( [ I w ~ ] ; published in 1964), Iwasawa proved two versions of what would later be known as Iwasawa's "Main Conjecture" under a certain hypothesis. This paper concentrates on the case F = Q(Cp), F, = Q(Cp, C p 2 , . . . ). The hypothesis that he makes is the following.

Cyclicity Hypothesis: So is a cyclic Z[A]-module.

Under this same hypothesis, one version of the Main Conjecture is already proven (in essence) in the earlier paper A class number formula for cyclotomic fields ( [ Iw~] ) . We will discuss this first. Under the cyclicity hypothesis, it follows that Sn is cyclic as a module for Zp[Gal (Fn/Q)] for any n 2 0, and then Iwasawa proves that, for any odd i, 3 5 i 5 p - 2, one has

(8) pn+l

1 S;' 2 Z , [ ~ a l (F,/F)]/(o;)), where 0;) = - pn+ - 1 x aw-"(a(ua)-l

a= 1

Here oa E Gal (Fn/Q) is determined by ua(CP~+l) = C;n+l, (ua) is the projection of a, to Gal (Fn/F) in the direct product decomposition Gal (Fn/Q) = A x Gal (Fn/F), and w-'(u,) is determined by the projection of a, to A, regarding w - b s a character of that group. It is not hard to verify that Of) E Zp[Gal (Fn/F)]. (For i = 1, this isn't true,

but it is shown that s,W1 = 0.) The fact that 0;) annihilates s,Wi is a consequence of Stickelberger's Theorem.

354 R. Greenberg

A crucial observation is that 0:) is mapped to 0;) under the Zp- algebra homomorphism Zp[Gal (F,/F)] -t Zp[Gal (Fn/F)] for m > n 2 0. We let = Lim 0:) E A = Zp[[r]]. Iwasawa proves the following

C

result in [ I w ~ ] .

Theorem (4.1). Suppose that the cyclicity hypothesis holds. Let i be odd, 3 5 i 5 p - 2. Then xwi Z A / ( Q ( ~ ) ) as a A-module.

Here is a sketch of the proof. If one identifies A with Zp[[T]] as before, then o ( ~ ) is identified with a power series gi(T). This is the power series

which we referred to in Section 3. Note that gi(0) = b t ) = -B1,,-i.

Now Proposition 3.1 asserts that xW2 % All , where I is the principal ideal generated by fi (T) = fx,i (T). Stickelberger's Theorem implies

that o ( ~ ) annihilates XW' % Limszl . This means that gi(T) E I . That C

is,

in the ring A. Using (1) and (2) for n = 0, it follows that fi(0) z Isg21. Using (8) for n = 0, one has s,"~ E ZP/B1,,-, Zp and so gi (0) .; IS,"' 1 too. It then follows from (9) that gi(T)/ fi(T) E AX, which implies Theorem 4.1.

Without the cyclicity hypothesis, there seems to be no simple way to prove the divisibility (9). However, Iwasawa later (in Chapter 7 of [IwlO]) proves the following proposition by using the formulas for the

first factor of the class number of the fields Fn for n > 0. Let Aga,,, . , and Pz),na, denote the A- and pinvariants for A/(gi(T)), which can be

- ,

easily described in terms of the coefficients b:) of gi(T). Let Ab",)alg and (4 y,,l, be the A- and pinvariants of xwi, which we previously denoted

( ' (4 more simply by Ad) and p, . Proposition (4.2). For any odd prime p, we have the following

equalities:

i=3 i=3 i odd i odd

i=3 i=3 i odd i odd

Thus, if we somehow know that. (9) holds for all odd i, 3 5 i 5 p - 2, then it would still follow that the ideals (fi(T)) and (gi(T)) are equal. This is of course weaker than Theorem 4.1 in that one does not obtain


the precise structure of x W z . On the other hand, if one could prove the divisibility gi (T) 1 fi (T) for all these i's, then one would again obtain ( fi(T)) = (gi(T)). In 1981, Mazur and Wiles succeeded in proving this divisibility, as we will discuss below.

Several years later Iwasawa discovered that the power series gi(T) is intimately connected to the p-adic L-function Lp(s, wj ) which was constructed by Kubota and Leopoldt in [K-L] (also published in 1964). Here i and j are related as before: i + j G 1 (mod p - 1) and p is an odd prime. This padic L-function can be characterized as the unique continuous function from Zp to Qp such that

for all m 2 1 with m - j (mod p - I). Here [ ( z ) denotes the Riemann zeta function. It is known that [(I - m) = -B,/m for all m > 1, where B, denotes the m-th Bernoulli number. Kubota and Leopoldt prove that Lp(s, wj) is actually analytic for all s E Z,, except for a simple pole at s = 1 when j = 0 (which corresponds to i = 1). They also give the values Lp(l - m, wJ) for all m > 1, and in particular one has ~ ~ ( 0 , wj) = -B1,"-z.

We will state Iwasawa's result in terms of ~ ( ~ 1 . We assume that i # 1, and hence j # 0. Let K : r + 1 + pZp be the isomorphism giving the action of l? on pPm. That is, K = x 1 r, where x is the usual cyclotomic character. For any s E Zp, we can define a continuous homomorphism K" : r -+ 1 + pZp by ~ ' ( 7 ) = ~ ( 7 ) " for 7 E I'. One can extend K" to a continuous Zp-algebra homomorphism cp,: A + Zp. (If one identifies A with Zp[[T]] by setting T = yo - 1, then cp, can be defined as follows: cp,(g(T)) = g(rc(yo)' - 1) for any g(T) E A.) Iwasawa proves the following result in [Iwg].

Theorem (4.3). Suppose that j is an even integer, 2 5 j 5 p - 3. Then Lp(s, wj) = c p , ( ~ ( ~ ) ) for all s E Zp. Equivalently, gi(T) satisfies the following interpolation property: gi(~(yo)l-m - 1) = -(1 -pm-l) Bm/m for all m 2 1 such that m EE j (mod p - 1).

A nonzero element of A has only finitely many zeros and so the above interpolation property determines gi (T) uniquely. The Kubota-Leopoldt padic L-function Lp(s, wj) is obtained from gi(T) by the substitution T = &(yo)" - 1.

This may be a good place to discuss the congruences in (4) again. 00

Writing gi(T) = C b f ) ~ " as before, it is clear that n=O


for all t E pZ,. It follows that B,/m = - b t ) (mod pZp) for all m E j (mod p - I ) , taking t = t ~ ( y ~ ) ' - ~ - 1, and so we have

Here, as before, i and j are related by i + j = 1 (mod p - I), 2 5 i, j 5 p - 2 with i odd, j even. (Thus, wiwj = w, where wi is an odd character, wj is an even character.) On the other hand, if t l , t2 E pZ,, then

It follows that if Bjt / j' - B j / j (mod p2Zp) for some j' = j (mod p - 1)

where j', j > 4 and j' $ j (mod p), then p(b(,"). Conversely, if plb(;), then gi(t) = gi(0) (mod p2Zp) for all t E pZ,. Thus, we have

provided that j > 4. In summary, the two congruences in (4) hold if (4 and only if A:),,,, > 2 (since we know that pp,a,al = 0). AS we have

mentioned, this does not happen for p < 16,000,000. The first version of Iwasawa's Main Conjecture can be stated as

follows. For each odd i, 3 5 i 5 p - 3, let fi(T) be the characteristic polynomial for the A-module x"'. Let gi(T) be the power series which is characterized by the interpolation property in Theorem 4.3. (It is related to Lp(s, wj ) by a simple change of variable.)

Conjecture (4.4). The ideals (fi(T)) and (gi(T)) of A are equal.

As Iwasawa discusses in another article [ I w ~ ] , one can view this conjectural relationship between fi (T) and gi (T) (which Iwasawa proved under the cyclicity hypothesis) as another aspect of the analogy between algebraic function fields and algebraic number fields mentioned earlier. There is an important theorem of Weil which states that the zeta function of a curve C over a finite field k is closely related to the action of the Frobenius automorphism in Gal (Elk) on the ppower torsion points of the Jacobian variety for C, where p is any prime such that p # char(k). The analogy arises from the fact that gi(T) is related to values of the Riemann zeta function [ ( z ) by an interpolation property. This analogy can in fact be made quite precise.

Theorem 2.6 shows that Conjecture 4.4 can be formulated in the following equivalent form.

Conjecture (4.5). The characteristic ideal (fywl (T)) for the A-

module Y"' can be generated by &(T) = gi(n(yo)(l + T)-' - 1).

Later we will point out that the power series gi(T) can also be characterized by a nice interpolation property. We want to discuss a third version of Conjecture 4.4, which Iwasawa also proves is equivalent. We first observe that there is a natural factorization of the polynomial fywj (T). Recall that F, c L, c M, and one therefore has an exact sequence

of finitely generated, torsion A-modules. (Torsion because j is even.) Here Z = Gal (M,/L,). It follows that

Iwasawa proves that gi (T) has a factorization parallel to (12). If n > 0, let Un denote the group of units in the completion (Fn)Pn, where p, is the unique prime of Fn above p. Let En and C, denote the group of units and the subgroup of cyclotomic units for the field F,. Let En and C, denote the closures of En and C, respectively in the topological group U,. Let !lJ = Lim un/Cn, where the maps defining the inverse

C

limit are induced by the norm maps N,,, : Urn -+ Un for m > n. Note that N,,,(C,) c C, (in fact, equal) and N,,,(E,) C En. Also, note that u,/C, is a Z,-module since all nonzero residue classes modulo p, have representatives in C,. (The residue field is just IF,.) Let X = ~irnE,/C, and 3 = Lim u,/E,. Then Iwasawa shows that one has an C t

exact sequence

of finitely generated, torsion A-modules and furthermore one has the following theorem.

Theorem (4.6). For even j , 2 5 j 5 p - 3, there is a A- isomorphism

where i + j = 1 (mod p - 1).

Consequently, one does have a natural factorization of gi(T), namely

R. Greenberg Iwasawa Theory - Past and Present 359

where u(T) E AX. Iwasawa also proves that 3"" 2"" A-modules, the isomorphism

coming from class field theory: one identifies Lim Un/En with the in- C

ertia subgroup for p in Gal (M,/F,), which of course coincides with Gal (M,/L,). Comparing (12) and (14), one is then led to a third equivalent formulation of Conjecture 4.4.

Conjecture (4.7). For even j , 2 5 j < p - 3, the characteristic ideals of xw3 and xW3 are equal.

We should mention that, under the assumption of Vandiver's conjecture, -

one has xW3 = 0. But Iwasawa shows that xW3 /TX"~ S E;;" /??t3, which is also trivial under the assumption of Vandiver's conjecture. It would follow that xw3 = 0 too. Thus, Conjecture 4.7 is then obvious and Conjecture 4.4 holds. One could also deduce Theorem 4.1 again.

These conjectures can be formulated in a more general setting. Let F be a finite, abelian extension of Q. For simplicity of exposition, we will assume that p is an odd prime, that A = Gal (FIQ) has exponent dividing p- 1 (so that irreducible characters of A will have values in Z: ), and that F contains a primitive p t h root of unity. These assumptions are not at all essential. Let x and $ be two irreducible characters of A such that X+ = w and x is odd (so that $ is even). Let F, = FQ, and let X = Gal (L,/F,), Y = Gal (M,/F,) where L,, M, are defined as before. We can define XX and Y $ which turn out to be related just as in Proposition 2.6 (which is the special case x = w" + = wJ ). Propositions 2.3-2.5 are also true. There is also a padic L function Lp(s, $) defined by a certain interpolation property and which corresponds to a power series gx(T) E A. (We use x as a subscript to be closer to the previous notation gi(T) corresponding to x = wi.) One can easily state the analogues of Conjectures 4.4, 4.5, and 4.7, which again turn out to be equivalent. We refer the reader to [Co2] or [Gr2] for more details. In this generality, the conjectures were proved by Mazur and Wiles in [M-Wl]. We should mention that Iwasawa7s arguments work quite well (and describe XX or Y* up to pseudo isomorphism) if one makes a certain hypothesis which is slightly weaker than the cyclicity hypothesis (which is false in general). We will state it in a form which makes sense whenever F / Q is Galois and F, = FQ,. Then F,/Q is Galois. We can define Zp [[Gal (F,/Q)]] as Lim Zp [Gal (Fn/Q)].

t

Pseudo-cyclicity Hypot he&. There is a cyclic Zp [[Gal (F, /Q)]] - submodule Z of X = Gal (L,/F,) such that X/Z is finite.

We do not know of any examples where this hypothesis fails to be true. If F / Q is abelian, then Conjecture 3.4 (applied to the maximal real subfield of F(Cp)) would imply the pseudo-cyclicity hypothesis. (See [Gr3].) It also would be true if we somehow knew that all the roots of gx(T) were simple for all odd characters x of A = Gal (F(Cp)/Q), an assertion which is quite likely to be valid.

Even more generally, one can formulate the analogues of Conjec- tures 4.4 or 4.5 for abelian characters x or $ of any totally real number field K under the assumption that x is totally odd or + is totally even. If X+ = W K , where w~ = wlG,, then the two conjectures are again equivalent, as one shows by using the Reflection Principle. The padic L-functions Lp(s, $), which satisfy an interpolation property involving the numbers L(1 - m, +wKm) for m 2 1, were constructed by Deligne and Ribet [D-R] using Hilbert modular forms for K and independently by D. Barsky and by P. Cassou-Noguks [Ca] using explicit formulas of Shintani. In this generality, Wiles succeeded in proving these "Main Conjectures" in 1988. The proof appeared in [Wi2]. The approach uses 2-dimensional padic representations associated to Hilbert modular forms for K and is inspired partly by ideas of Hida [HI]. As a consequence of this result of Wiles, an analogous main conjecture for padic Artin L-functions can be deduced. These functions are associated to representations of Gal (FIQ), where F is any finite, totally real, Galois extension of Q, and can be characterized by an interpolation property involving values of the corresponding complex Artin L-function. The invariant Ax discussed in Section 3 (for an irreducible x which is not of "mixed7' type) then has an "analytic" interpretation as the number of zeros of a certain padic Artin L-function.

Iwasawa gave a course at Princeton University during the academic year 1968-69 in which he explained many of the ideas that have been mentioned so far. That course was my first introduction to the subject. Iwasawa7s lectures were beautiful, and usually given without consulting any notes. I recall that the notes that I took of his lectures were quite in demand and circulated for many years afterwards. The results in the course were proved in complete generality and eventually became incorporated in Iwasawa's 1973 paper On Zl-extensions of algebraic number fields. The course and this paper included the study of a skew-symmetric pairing which was inspired by the analogy with algebraic function fields and the Weil pairing.

5. Barry Mazur gave a series of lectures in Paris during the Spring of 1970, where he developed a theory aimed at proving the following kind of result.

360 R. Greenberg

Conjecture (5.1). Suppose that A is an abelian variety defined over a number field F . Assume that p is a prime such that A has good, ordinary reduction at all primes of F lying above p. Let F,/F be the cyclotomic Zp-extension. Then A(F,) is finitely generated.

The details of his theory were published in [Mazl]. One case in which Mazur succeeded in proving this conjecture is under the following assumption:

(15) A(F) and IIIA(F)~ are both finite.

Here LUA(F), denotes the pprimary subgroup of the Tate-Shafarevich group for A over F . We will formulate several of Mazur's results and conjectures in terms of the classical Selmer group, although he uses a certain variation of this group. Recall that if K is an algebraic extension of F, then the pprimary subgroup SelA(K)p of the Selmer group fits into an exact sequence

Thus (15) means that SelA(F), is finite. One of the main results of [Mazl] is the following.

Theorem (5.2). Assume that A I F has good, ordinary reduction at all primes of F lying over p. Let F,/F be the cyclotomic Zp-extension. Then the kernel and cokernel of the natural maps

are finite and have bounded order as n t oo.

This is often referred to as Mazur's "Control Theorem" and is valid for every Zp-extension F,/F. Now assume that SelA(F), is finite. SelA(F,), is a discrete, pprimary group on which r = Gal (F,/F) acts. We can regard SelA(F,), as a discrete A-module and its Pontryagin dual XA(F,) as a compact A-module. If we are assuming that SelA(F)p is finite, Theorem 5.2 implies that XA(F,)/TXA(F,) is finite. Thus XA(F,) is a finitely generated, torsion A-module. The classification theorem then implies that XA(F,) has finite Zp-corank, which we denote by XA (F, IF). Therefore, the maximal divisible subgroup (SelA (Fm)p)div of SelA(F,), is isomorphic to ( Q , / Z ~ ) ~ A ( ~ ~ ~ ) from which it follows that A(F,) 8 (Q,/Z,) 2 (Qp/Zp)' where 0 5 r 5 XA(F,/F). On the other hand, if F,/F is the cyclotomic Zp-extension, then it is known that A(F,)t,,, is finite. Mazur proves this in [Mazl] under certain hypotheses. By a simple argument given in Mazur's paper, it then follows

Iwasawa Theory - Past and Present 36 1

that A(F,) is indeed a finitely generated group, provided that (15) holds.

More generally, the same conclusion (i.e., Conjecture 5.1) follows from the following conjecture.

Conjecture (5.3). Under the same assumptions as in Conjec- ture 5.1, the A-module XA (F,) = selA (F,L is finitely generated and torsion.

We would then say that SelA(F,), is cofinitely generated and cotorsion as a A-module. In fact, for any Zp-extension F,/F and for any abelian variety A (with no restriction on the reduction-type at p), SelA(F,), is always a cofinitely generated A-module, but can fail to be A-cotorsion. For example, let F be an imaginary quadratic field. Let A be an elliptic curve over Q. Suppose that F, is the anti-cyclotomic Zp-extension of F . Then it often happens that rankz(A(F,)) is unbounded as n -+ oo. This interesting phenomenon is discussed in [Maz2]. In such a case, it is clear that SelA(F,), cannot be A-cotorsion.

Mazur also states a Main Conjecture somewhat analogous to Con- jecture 4.4 or 4.5. It is for the case where A is an elliptic curve E l Q which is modular and where F, is the cyclotomic Zp-extension of a subfield F of Q(cp). The prime p is assumed to be odd and such that E has good, ordinary reduction at p. For simplicity, we will discuss F = Q. For such a prime p, Mazur and Swinnerton-Dyer constructed a padic L-function Lp(s, E) in [M-SwD]. If r = Gal (Q,/Q) and A = Zp[[r]], then Lp(s, E ) = 9s-1 (BE) for all s E Zp, where BE is an element of 5 A for some t 2 0. Here 9, : A -+ Zp is just as in Theorem 4.3. The element BE is characterized by a certain interpolation property involving the values at z = 1 of the twisted Hasse-Weil L-series L(z, E, p) for E lQ, where p varies over all Dirichlet characters of ppower order and conductor. (They can be regarded as characters of I?.) It is now known under very mild assumptions that BE E A. This should be true in general. One important idea in [M-SwD] is that BE can be identified with a Qp-valued measure on the Galois group r. The measure of any open subset of r is in $5, and presumably should be in % itself. If p~ is this measure, the i

where K ~ - ' is viewed as a function on r . Mazur's Main Conjecture is the following statement.

362 R. Greenberg

Conjecture (5.4). The characteristic ideal of XE(Qm) = SelE (Qm )a is generated by BE.

Even without assuming Conjecture 5.3, this conjecture makes sense. It could be interpreted as asserting that SelE(Qm), is A-cotorsion if and only if BE # 0. It is now known that indeed BE # 0, a consequence of a theorem of Rohrlich [Ro] which states that L(l , E, p) # 0 for all but finitely many characters p of I?. Just to give a simple illustration of how Conjecture 5.4 can be applied, we will mention one corollary, namely the following piece of the Birch and Swinnerton-Dyer conjecture:

(16) L( l , E) # 0 E(Q) and LUE (Q), are both finite.

This would follow because the interpolation property implies that

where T = yo - 1 E A as before. If Conjecture 5.4 is valid, then T 1( BE is equivalent to the assertion that XE(()m)/TXE(()oo) is finite. Mazur's Control Theorem (Theorem 5.2) shows that this last assertion is indeed equivalent to the finiteness of SelE (Q),. We should also add that, if L( l , E) # 0, then Conjecture 5.4 would imply the ppa r t of the Birch and Swinnerton-Dyer conjecture. (See Chapter 4 of [Gr5] for an exposition of this result.)

If E(Q) is infinite, then Conjecture 5.4 implies the following inequality:

This is because XE (0,) /TXE (0,) has Z,-rank equal to the Z,-corank of SelE (Q),. This is at least rankz(E(Q)) (with equality if UIE (Q), is finite). The Birch and Swinnerton-Dyer conjecture asserts that ~rd , ,~(L(z , E ) ) = rankz(E(Q)). In order to deduce this from Con- jecture 5.4 one would need to prove three results:

(i) LUE (0) is finite. (ii) TXE (Qm)/T2XE (0,) is finite.

(iii) ord,,l (L(z, E)) = ordSzl (L,(s, E ) )

The first result is of course a well-known conjecture, proved by Koly- vagin if ~rd , ,~(L(z , E)) < 1. In this case, Kolyvagin also proves the equality of rankz(E(Q)) and ~ r d , = ~ ( L ( z , E)) . The second result is an easy consequence of Theorem 5.2 if SelE(Q), is finite. (For then XE(Qoo) /TXE(~oo) is finite and this implies (ii).) More generally, it is equivalent to the nondegeneracy of a certain p-adic height pairing.


This equivalence is proved in [Pel] for elliptic curves with complex multiplication and in [Sch2] in a more general context. The nondegeneracy is trivial if SelE (Q), is finite. It has been proven by D. Bertrand if E has complex multiplication, E(Q) has rank 1, and UTE (Q), is finite. B. Perrin-Riou [Pe2] has also proven it if ~rd,,~(L,(s, E)) = 1. But nothing is known about the nondegeneracy if ~rd , ,~(L(z , E)) > 1.

As for the equality in (iii), it is obvious if L( l , E) # 0 and would follow from the Gross-Zagier theorem together with Perrin-Riou's p adic analogue [Pe2] if ~ r d , , ~ ( L ~ ( s , E)) = 1. It is also known that ordZzl (L(z, E)) and ord,,l (L,(s, E)) have the same parity since one can compare the signs in the functional equation for L(z, E ) and its analogue for L,(s, E) . Beyond this, we know nothing about the relationship between these orders of vanishing.

We should also mention the interesting case where E has split, multiplicative reduction at p. The corresponding padic L-function L,(s, E) has been constructed in [M-T-TI. But it has a "trivial zero7'. That is, the natural interpolation property given in [M-T-TI implies that Lp(l , E) = 0, and, concerning the order of vanishing, it is conjectured there that ~rd ,=~(L,(s , E ) ) = 1 + ~ r d , = ~ ( L ( z , E)). The functional equation proved in [M-T-TI show that these orders have opposite parities.

t 5 Conjectures 5.3 and 5.4 have been proven by Rubin [Ru2] when E / Q ' has complex multiplication and p is any odd prime where E has good,

ordinary reduction. For a modular elliptic curve El Kato has proven

1 Conjectures 5.3 and has also proven that BE is at least contained in the

I characteristic ideal of XE(Qm), up to multiplication by a power of p.

If E is a modular elliptic curve over Q having good, supersingular reduction at p, then a padic L-function Lp(s, E) still exists, but now

1 corresponds to an unbounded (),-valued measure /.LE on I'. (This means that the measures of open subsets have unbounded denominators.) Also,

6 1 the Selmer group SelE(Qm), will definitely not be A-cotorsion. This 1 topic has been studied by Perrin-Riou and by Schneider. We refer the

I reader to [Pe3], where one can even find a formulation of a Main Con-

/ jecture in the supersingular case. We want to mention just one specific question, which seems to still be open. Assume that UTE(Qn), is finite for all n. What can one then say about the growth of ILUE (On),[ as

! n --t oo? If E has good, ordinary reduction at p and if Conjecture 5.3

1 holds for A = E and the Z,-extension Q,/Q, then one can prove that

1 i [UIE(Qn),[ = p""+pn+" for n >> 0, where A, p, and v are suitable

I integers. But if E has supersingular reduction, we do not even have a good guess. More generally, one can consider the analogous question for

j an arbitrary abelian variety A I F and an arbitrary Z,-extension Fm/F. In 1976, Coates and Wiles proved the following theorem.

I i

364 R. Greenberg

Theorem (5.5). Assume that E is an elliptic curve defined over Q with complex multiplication and that L( l , E) # 0. Then E(Q) is finite.

The proof involves a beautiful argument based on adapting some of the results in Iwasawa's paper [Iw7] to a different, but quite analogous, situation. We will outline this argument and also take the opportunity to state another main conjecture which Coates and Wiles formulated.

Suppose that E is an elliptic curve defined over Q such that Endc(E) = 0, the ring of integers of an imaginary quadratic field K . We will assume that p is an odd prime and that E has good, ordinary reduction at p. Then p splits completely in K . Since K must have class number 1, we can write that p = IT*, where T, ?i E O (complex conjugates). Let E[IT"] denote the group of IT-power torsion points on ~ ( 0 ) : E [ I T ~ ] = Un E [IT"+'], where E[7rnf'] is the kernel of the endomorphism .rrn+l of ~ ( 0 ) . Adjoining coordinates to K, we obtain the fields F, = K(E[.nW]) = Un Fn, where Fn = K(E [7rn+']). Considering the action of Gal (F,/K) on E[IT"] (which is isomorphic to Qp/Zp as a group), one obtains an isomorphism

$JE : Gal (F,/K) -Z Z: .

Therefore, Gal (F,/K) 2 A x r, where r = Gal (F,/Fo) is isomorphic to 1 + pZp and A 2 (ZlpZ) ' . The situation is quite analogous to that for Q(pPm)/Q, where ppm denotes the group of p-power roots of unity. The prime IT is totally ramified in F,/K. (But ?i is unramified.) If F = Fo, then A can be identified with Gal(F/K) and there is a canonical isomorphism

which gives the action of A on E [TI. (We also regard WE as having values in ZpX .) The extension F,IF is a Zp-extension, and only the prime of F lying above 7r is ramified. (Note: In F / K , the primes of K where E has bad reduction are also ramified.)

Now suppose that E(Q) is infinite and that P is a Q-rational point on E of infinite order. We will assume that P @ ITE(K). For each n 2 0, let Pn E ~ ( 0 ) be such that .?rnflPn = P. Then Po @ E(K) . Let Tn = Fn(Pn), T, = Un Tn. It turns out that Tn/Fn is cyclic of order pn+l and is unramified except at the unique prime of Fn above IT. The extension T,/K is Galoisian, Gal (T,/F,) Z Z,, and the action of Gal (F, / K) on Gal (T, IF,) by in.ner automorphisms is given by $E.

This can be seen by considering the 1-cocycles a, : GK -+ ~ [ 7 r " + l ] defined by an(g) = g(Pn) - Pn for all g E GK. One can check that an[~,-


induces a compatible set of isomorphisms Gal (F,(Pn)/F,) E[rn+l] for n 2 0, equivariant for the actions of Gal (F,/K). This implies that Gal (T,/F,) 2 T,(E), the IT-adic Tate module for E, as Gal (F,/K)- modules. Since T,/F, is ramified only at IT, we have T, c M,, where M, denotes the maximal, abelian pro-p extension of F, which is unramified everywhere except at IT.

Let X = Gal (L,/F,), where L, is the pro-p Hilbert class field of F,. Let Y = Gal (M,/F,) and Z = Gal (M,/L,), noting that L, c M,. Then X, Y, and Z are A-modules, where A = Zp[[r]]. Since A acts on them too (because M, and L, are Galoisian over K), we can consider the A-components corresponding to W E and obtain an exact sequence

of A-modules. Coates and Wiles verify that T,/F, is ramified at IT, a crucial fact for their proof. It then follows that T, @ L, and so T, n L, is a finite extension of F, since all nontrivial subgroups of Gal (T,/F,) have finite index. Therefore Z has a quotient Gal (T,/T, n L,) which is isomorphic to Zp and on which Gal (F,/K) acts by $JE. Let KE =

$JEl r . Then it follows that ZWE has a quotient which is isomorphic to A/(yo - yo)), as a A-module where yo denotes any topological generator of r.

If F' is any algebraic extension of K , then the Selmer group for E over F' is an 0-module. One can consider its T-primary subgroup SelE (F'),, which is a subgroup of H1 (GF,, E[IT"]). Now, let F' = F,. Then, since GFoo acts trivially on E [ P ] , SelE(F,), is a subgroup of Hom(Ga1 (F~/F,), E[-rrm]). Coates proves that

Thus Sel (F, ), is closely related to the Pontryagin dual Hom(Y, Q,/Zp) of the Galois group Y. They are isomorphic as groups, but the action of

: Gal (F,/K) is twisted by $JE. Now let r = rank(E(Q)) = ranko(E(K)). Then SelE(K), has a subgroup isomorphic to (Qp/Zp)T and the restric-

tion map SelE(K), -+ el^(^,):"(^"'^) can be shown to have finite kernel and cokernel. This means that Homr (Y W E , E [IT"]) has Zp-corank at least r. If r > 0, then the fact that T,/F, is ramified at IT implies that the image of SelE (K), in Homr (ZWE , E [PI) has Zp-corank at least 1. It is possible to show that this image then has Zp-corank exactly 1 and hence Homr (XWE , E [ P I ) has Zp-corank at least r - 1. Therefore it follows that A(F,/F) 2 r - 1. That is, if r > 1, then Iwasawa's A-invariant for the non-cyclotomic Zp-extension F,/F will

366 R. Greenberg

be positive. Letting S = T - ( ~ ~ ( 7 ~ ) - I) , we see that Srl fYwE (T), Sl fzwE (T), and Sr-ll f ~ w E (T). These divisibilities should conjecturally be exact. Perrin-Riou proves in her thesis [Pel] that this is so precisely when UTE(K), is finite and a certain p-adic height pairing on E ( K ) 80 KT is nondegenerate. The finiteness of U E ( K ) , implies that YWE/SYWE has Zp-rank exactly r . The nondegeneracy is shown to imply that SYWE/S2YWE is finite, i.e., in the classification theorem applied to the A-module YWE, there is no factor of the form A/(Sa) with a 2 2.

In their paper [C-Wl], Coates and Wiles show that if E(Q) is infinite, then the rational number L( l , E/Q)/flE (where flE denotes the real period of E) is divisible by all primes in a certain infinite set, concluding that L( l , E/Q) = 0. In another paper [C-W2] they prove a perfect analogue of Theorem 4.6 (which Iwasawa proved in [ I w ~ ] ) . This result gives another proof of Theorem 5.5, which is the one we will briefly explain. Let Un denote the group of principal units in the p n - a d ' 1c completion of Fn, where pn is the unique prime of Fn lying over IT. Let En and Cn denote respectively the groups of global units and elliptic units in Fn which are congruent to 1 modulo p,. Let En, Cn denote the corresponding closures in Un. Let X = ~ i m En/Cn, ZJ = Lim un/Cn,

C C

and 3 = Lim un/En. Then X, ZJ, and 3 are torsion A-modules on C

which A = Gal (F /K) acts. To state the result of Coates and Wiles, we must mention the padic L-functions that they consider, which were first constructed by Manin-Vishik [M-V] and, in a much more precise form, by Katz [Ka]. Let $E denote the grossencharacter of K associated to the elliptic curve E by Deuring. It has the property that L(z, E/Q) = L(z, gE). Suppose that 1 5 j 5 p - 2. For each such j,

there is a power series G ~ ) ( T ) with the property that

for all positive integers k such that k - j (mod p - 1). Here uo = t-cE (yo) and Ak is a certain explicit, nonzero factor which involves the real period flE and a certain "p-adic period" fl(EP). The coefficients of Gg) (T) as well

as fl(EP) belong to the ring of integers Z in the completion of the maximal unramified extension QpUnr of Qp. Furthermore, there is a power series

gg)(T) with coefficients in Zp such that G k ) ( ~ ) / g g ) ( T ) is an invertible

element in the formal power series ring Z[[T]]. Only the ideal ( g g ) ( ~ ) ) of A is uniquely determined. The result of Coates and Wiles can be stated as follows.


Theorem (5.6). Let p be a prime such that p 2 5. Suppose that E has good, ordinary reduction at p, and y ! ~ ~ (p)/N(p) $ 1 (mod p). Then for j, 1 5 j 5 p - 2,

If qE(p)/N(p) - 1 (mod p), then p is called an anomalous prime for E . (Equivalently, E(F,) has an element of order p, where E is the reduction of E modulo p.) Then some modification of Theorem 5.6 holds. There are infinitely many primes satisfying the hypotheses of Theorem 5.6. One deduces Theorem 5.5 as follows. If E(Q) is infinite, then, as we have discussed, one obtains that S = T - (uo - 1) divides f Z w E (T). But class field theory shows .that Z 2 3 as Gal(F,/K)- modules. Thus S also divides f 3 w E (T) and therefore divides fg w E (T) .

(1) By Theorem 5.6, it is then clear that ~lg;)(T) and so gE (uo - 1) = 0 since S = T - (uo - 1). Therefore, by the interpolation property (17), L(1, qbE) = L(l , E/Q) is indeed zero.

I

In [C-W2], Coates and Wiles state the following conjecture, which is often referred to as the one-variable main conjecture for elliptic curves

' 1 with complex multiplication.

B Conjecture (5.7). With the above notation and assumptions, the

characteristic ideal of yWb is generated by g (c ) ) (~ ) . Equivalently, the 1 i A-modules xWL and XWb have the same characteristic ideal.

I Later on, Yager [Y] proved a two-variable analogue of Theorem 5.6 and formulated an analogous conjecture, referred to as the two-variable main

f conjecture. The corresponding two-variable padic L-function was con-

i structed by Katz and has an interpolation property involving the num- e , bers L(1, $;$',) where k and 1 are in fixed residue classes modulo p - 1

a n d k > 1 , 1 5 0 .

1 6. In their paper Class fields of abelian extensions of Q published in 1984, Mazur and Wiles gave a proof of Conjecture 4.4. They also prove the more general version for any finite abelian extension F/Q. If $ is an even Dirichlet character, their result gives an interpretation of the Kubota-Leopoldt padic L-function Lp(s, $) (or more precisely its zeros) in terms of the X-component of Gal (L,/F,), where x = w$-' (which is an odd Dirichlet character) and F is chosen so that x can be

: identified in the usual way with a character of A = Gal (F/Q). Their approach was inspired by Ribet's proof of the converse of a theorem

368 R. Greenberg

of Kummer-Herbrand in that they use the structure of certain finite groups of torsion points on abelian varieties arising as quotients of the Jacobian varieties of some modular curves. An important role is played by the cuspidal group whose structure is related to Stickelberger ideals, and hence to Bernoulli numbers, by results of Kubert and Lang. By using the fields generated by these groups of torsion points, Mazur and Wiles construct a sequence of finite extensions of F, contained in L,. A crucial part of their proof depends on the theory of Fitting ideals to prove the divisibility statement that they need. In the special case where F = Q(<,), it states that gi(T)l fi(T). (This corresponds to the case x = w2, where i is odd.) As we mentioned earlier, such a divisibility result would be sufficient because of Proposition 4.2. One can find a good outline of their proof in the introduction of their paper, and also a good expository account in the Seminaire Bourbaki lecture on this topic given by Coates [ C O ~ ] .

We would like to give some idea of why modular Jacobian varieties and modular forms provide a natural approach to such questions. For this we will just discuss a proof of the converse to the result of Kummer-Herbrand alluded to above. Let F = Q(<,). Suppose that 2 5 i , j 5 p - 2, that i + j = 1 (mod p - I) , and that i is odd (so that j is even). The Kummer-Herbrand result asserts that if s,"' # 0, then plBj. As the discussion in Section 4 shows, this follows easily from Stickelberger's theorem giving an annihilator in Z[A] of So. Ribet [Ri] proves the converse by showing that if plBj, then Gal ( L ~ / F ~ ) " ~ # 0, where Lo denotes the pHilbert class field of Fo = F . To do this, he constructs a nontrivial, unramified pextension L / F such that Gal (LIF) is abelian, LlQ is Galoisian, and A = Gal (FIQ) acts on Gal (L/F) by the character wZ.

An idea which had been proposed by various people in the 1970s was to construct such a field L by using the padic representations associated to modular forms. The existence of these representations had been conjectured by Serre and proved by Deligne. The motivation for approaching the question in this way was suggested by one of the famous congruences proved by Ramanujan, namely that ~ ( n ) = all (n) (mod 691) for all n 2 1, where ~ ( n ) denotes the n-th coefficient in the q- expansion (or Fourier expansion) of f = q n,"==, (1 - qm)24, the unique normalized cusp form of level 1 and weight 12, and a1 (n) = Ed", where d runs over the positive divisors of n. The above congruence is derived directly from the fact that 6911B12. One then obtains a congruence between the Eisenstein series of weight 12 which has all (n) as its n-th Fourier coefficient and a cusp form which must be f12. In general, if


pl Bj, then a similar congruence must exist involving some cusp form of level 1 and weight j.

Let p be any prime. Let C = {p, oo) and let Qc denote the maximal extension of Q unramified outside C. Deligne constructs a 2- dimensional representation space V, of Gal (Qc/Q) associated to f la such that Trvp(Frobl) = ~ ( 1 ) for all primes 1 # p. Here Frobl E Gal (Qc/Q) is the Frobenius automorphism for any prime of Qc lying above 1 and TrVp is the trace. Now let p = 691. Choose a Gal (Qc/Q)- invariant Zp-lattice T, in V,. Then one obtains a 2-dimensional representation space Tp/pTp for Gal (Qc/Q) over IF, = Z/pZ such that Frobl has trace equal to 1 + 1'' (mod p), i.e., equal to 1 + wl' (1) (mod pZp). The Chebotarev Density Theorem then implies that Tp/pTp is reducible and has composition factors Fp = Fp(wO) (on which Gal (Qc/Q) acts trivially) and IF, (wl') (on which Gal (Qc/Q) acts by w" ). If one knew that V, were irreducible, then it would be easy to show that Tp could be chosen so that one has a nonsplit exact sequence

of Gal (Qc/Q)-modules. (But in this specific case, it is possible to verify ; this directly.) In matrix form, the corresponding IF,-representation looks f

like [ Jll 1 , where r is nontrivial. i t follows that there is a cyclic

/ extension L of F of degree p such that this IF,-representation factors

1 through Gal (LIQ) and its restriction to Gal (L/F) gives a A-equivariant isomorphism

I b ! Gal (L/F) 1 ~ o m ( ~ , ( w " ) , F,) = F,(w-"). 5 f Thus, starting from the fact that pl B, for j = 12 and p = 691, one ' obtains a field L as above such that A = Gal (FIQ) acts on Gal (L/F) 1 by x = wl-J = wz where i, j are related as before. In this case, i = 679. I It turns out that the extension L / F is automatically unramified. 1

1 The easiest way to explain this is to use a later result of Wiles which 1 (in a much more general formulation) actually plays an important role ! in his proof of the Main Conjecture over totally real number fields. The i prime p = 691 is a s~ca l l ed "ordinary" prime for f12. This means that

p t ~ ( p ) , as Ramanujan's congruence shows. Wiles' result implies that 1 for any such prime p, if one regards Vp as a representation space for GQP, then it is reducible. More precisely, there is an exact sequence

t

370 R. Greenberg

where Wp and Up are 1-dimensional representation spaces for GQp such that

W Qp(l l ) and Up 2 Qp(0)

as representation spaces for the inertia subgroup Ip = GQ;~=. Here we use the notation Qp(k) for the 1-dimensional space on which any Galois group acts by the k-th power of the ppower cyclotomic character. Thus Up is unramified as a GQp-module. This implies that Tp/pTp has a GQp-submodule isomorphic to Fp(wl1). Since it also has IFp(wO) as a GQp-submodule, we have

as GQp-modules. Therefore GQp(Cp) acts trivially on Tp/pTp which means that the unique prime of F lying over p splits completely in L /F . Since L c Qc and C = {p, GO), L is indeed a subfield of the pHilbert class field of F.

Vandiver's conjecture is true for p = 691. Hence the cyclicity hypothesis of Section 4 is valid for this prime and so the Main Conjecture has been proven by Iwasawa in this case. The Ribet-Kummer-Herbrand theorem is an easy consequence and can be viewed as a first approximation to the Conjecture 4.4. This is because s,"' # 0 u fi(T) @ AX, whereas pJ Bj a gi(T) @ AX, following the notation of Section 4. Ri- bet proves the converse of the Kummer-Herbrand theorem for all p and j by pursuing the idea of finding unramified extensions L / F in the 2- dimensional representations associated to modular forms. He succeeds in making this work by using modular forms of weight 2 which have the advantage that the associated 1-adic representations arise from abelian varieties. He then still obtains a congruence between an Eisenstein series and a cusp form if plBj. He can prove the irreducibility of the associated 2-dimensional representation, and then the existence of a suitable GQ-invariant lattice. To prove that L / F is unramified, he reduces the necessary splitting for GQp-modules to a theorem of Raynaud concerning finite commutative group schemes. In the work of Wiles proving the Main Conjecture for padic L-functions attached to totally real number fields, unramified extensions are constructed in the 2-dimensional representations associated to Hilbert modular forms. Under the assumption of ordinariness, he proves the reducibility as a GQp-representation space, just as we mentioned for flz. The argument adapts ideas of Hida and again somehow reduces to the case of 2-dimensional representations obtained from abelian varieties (i.e., from weight 2).


There are now other proofs of Conjecture 4.4 which proceed by using Kolyvagin's Euler systems. This approach was first inspired by Thaine's discovery of a method to relate the order of ( E ~ / c ~ ) $ to the order of

s,"' for every j, where F = Fo = Q(Cp) and the notation is just as in Section 4. Thaine's technique involves studying the cyclotomic units in certain abelian extensions of Q containing F. In retrospect, Thaine uses the first step in an Euler system. Rubin carries this method through in [ R u ~ ] , proving the equivalent Conjecture 4.7. Rubin also gives an Euler system proof of Conjecture 4.4 in [ R u ~ ] . In his paper The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Rubin proves the conjectures formulated by Coates and Yager mentioned at the end of Section 5. The approach is to study Euler systems formed from elliptic units in abelian extensions of an imaginary quadratic field. As a consequence, Rubin obtains the best results to date concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication.

The method of Wiles using systematically 2-dimensional representations associated to ordinary modular forms (for a prime p) allows him to prove the main conjecture corresponding to abelian characters of totally real number fields. The method of Euler systems has given the same result only for abelian characters (i.e., Dirichlet characters) of Q. Ru- bin has succeeded in making the Euler system method work for abelian characters of imaginary quadratic fields, as we mentioned in the previous paragraph, obtaining the conjectures of Coates and of Yager. One wonders if these conjectures can also be obtained from a modular form approach.

7. There is now a large literature concerning padic L-functions. The padic analogues of various classical complex L-functions have been constructed. We refer to [ C O ~ ] , [C-PI, [C-s], [H2] and to their references as a guide to this topic. We have already described in some detail the conjectural interpretation for the Kubota-Leopoldt padic L-functions which was proposed by Iwasawa and proved by Mazur and Wiles. We have described more briefly the conjecture of Mazur which gives an interpretation of the padic L-function associated to a modular elliptic curve over Q with good, ordinary reduction at p. This padic L-function is the padic analogue of L(z, fE), where fE is the modular form corresponding to E, a newform of weight 2 and level equal to the conductor of E . But a padic analogue of the L-function L(z , f ) associated to a newform f of weight k 2 2 and any level not divisible by p had also been constructed in the 1970s by Manin-Vishik, and Amice-Vklu. Under an

372 R. Greenberg Iwasawa Theory - Past and Present

"ordinariness" hypothesis, this padic L-function corresponds to an element in the Iwasawa algebra A = 0[[T]] , where 0 denotes the integers in the finite extension of Qp generated by the coefficients of f . The hypothesis is that the p t h Fourier coefficient is a unit in 0 . At the time it seemed quite mysterious how to interpret this padic L-function when k > 2. That is, could one formulate an appropriate Main Conjecture?

In 1987 I gave two lectures on this topic at the conference Iwasawa Theory and Special Values of L-functions which took place at M.S.R.I.. I then described a rather simple, general, and natural way to formulate such a conjecture under a certain "ordinariness" hypothesis. This conjecture gave a possible interpretation for the padic analogue Lp(s, V) of the complex L-function L(z, V) attached to a compatible system V = (6) of 1-adic representations of GQ. The ordinariness hypothesis for V and p is that there should exist a filtration FiVp of 0,-subspaces of Vp (for i E Z) with the properties:

( a )F i+ 'VPcFiVp; F i V P = V p i f i < O , FiVp=Oifi>>O. (b) FiVp is invariant for the action of GQp and the inertia subgroup

I, of GQP acts on F 'V~/F*+'V~ by Xb.

Here xP : GQ, --+ Z,X is th ppower cyclotomic character. Let A = Vp/Tp, where Tp is a GQ-invariant Zp-lattice in Vp. Then A is a discrete GQ- module isomorphic to ( Q ~ / Z ~ ) ~ as a group, where d = dimqp(Vp). We define Ff Vp to be FIVp and F+A to be the image of F+Vp in A. Thus F+A is -a divisible subgroup of A invariant under the action of GQp.

We can now define a certain A-module SA(Q,), which we refer to as the Selmer group for A over Q,. It is defined by

where T denotes the unique prime of Q, lying over p, I, denotes the inertia subgroup of GQm for a fixed prime of 0 over T, v varies over all primes of Q, except T, and I, denotes the inertia subgroup of GQ- for any prime of 0 over v. Also, as usual, H1 (K, *) denotes H1 (GK, *) for any field K . SA(Q,) is a pprimary group on which r acts naturally, and hence it is a discrete A-module, where A = Zp[[r]]. It is always cofinitely generated as a A-module, but not always A-cotorsion. We let XA (Q,) denote the Pontryagin dual of SA (Q,).

The gamma-factors in the conjectural functional equation for L(z, V) have a pole at z = 1 with order r v , say. Then one would expect that L(z, V, p) will have a zero at z = 1 of order exactly r v for all but finitely many characters p of I?, where L(z, V, P) is the L-function for V twisted

by the character p. The natural conjecture is that:

X*(Q,) has A-rank equal to rv

Let V* = {&*I, where &* = Hom(6, Q1 (1)). Th.en V* is another compatible system of I-adic representations of GQ. Assume now that r v = rv* = 0. This means that L(1, V) is a "critical value" of L(z, V) in the sense defined by Deligne. (And so is L( l , V*).) Under this assumption, as well as the ordinariness assumption, Coates and Perrin-Riou [C-P] formulate a precise conjecture about the existence and the interpolation property of a padic analogue Lp(s, V). It should correspond to an element Ov in A (which is unfortunately only defined up to multiplication by an element of O X ) . The interpolation property involves the numbers L( l , V, p) with p € P and one would expect that Bv # 0. Here then is the Main Conjecture.

Conjecture (7.1). The characteristic ideal of XA (Q,) is generated by ov.

There is an ambiguity in this conjecture. In addition to the fact that Ov and hence the ideal (Bv) are not well-defined, the Selmer group SA(Q,) depends on the choice of the Zp-lattice Tp. Both ambiguities involve only the p-invariant. The p-invariant of SA(Q,) can indeed be positive, but it is possible to make a precise conjecture about its value.

An obvious question to ask (and which stumped us for quite a while) was whether the above conjecture is consistent with the functional equation for the corresponding Lfunctions, which relates the values L( l , V, p) to L( l , V*, p-l). For the padic L-function one obtains a functional equation which can be expressed as Ov* = BG, where L : A -+ A is the involution of A induced by ~ ( y ) = y-' for all y € r . For the Selmer groups, the question was then whether the characteristic ideals of XA(Q,) and Xa* (0,) are also related by the involution L. Here A* = V;/T,' where T; = Hom(Tp, Zp(l)), which is a GQ-invariant Zp-lattice in V; . Our first attempts to prove this were based on the Reflection Principle (which works in the case where V is a compatible system of 1-dimensional representations), but then we found that the Duality Theorems of Poitou and Tate were just the right tool. In my paper, Iwasawa theory for p- adic representations ([Gr4]), one can find a detailed description of the conjectures, results about the structure of Galois cohomology groups and Selmer groups as A-modules, various examples, and the proof of the compatibility with the functional equation.

Consider the compatible system Q(k) = {Ql(k)) for k E Z, where Q ( k ) is the 1-dimensional Ql-vector space on which GQ acts by X : ,

374 R. Greenberg

~1 being the 1-power cyclotomic character. Then Q,(k) satisfies the ordinariness condition and Ff Q,(k) = Q,(k) if k > 1, F+Qp(k) = 0 if k 5 0. Let C = {p, oo) and let A = Q,(k)/Z,(k). Then

where H:,, (Q,, A) = ker(H1 (Q,, A) - n H' (I,, A)), where v varies 'u

over all primes of Q,. This is the group of everywhere unramified cocycle classes. Assuming that p is odd, the restriction map H1 (Q,, A) -+

H1(F,, A)A is an isomorphism, where F, = Q ( P , ~ ) and A = Gal (F,/Q,). Now H' (F,, A ) ~ = H o m A ( ~ a l (F~/F,), A). The action of A on A is by the character wk. Using the notation of Section 1, we have

where the isomorphisms are A-module isomorphisms and come from the natural restriction maps.

If k 2 1 and is odd, then SA(Q,) has A-corank 1 since it is just the

Pontryagin dual of YY* with the A-module structure twisted in a simple way (by K - ~ ) ) . For all other k E Z, SA(Q,) is A-cotorsion. (Vandiver's conjecture implies that SA(Q,) = 0 if k 5 0 and is even. Conjecture 3.4 would imply that SA(Q,) is finite in this case.)

Now L(z, Q(k)) = <(z - k). The corresponding gamma-factor is F ( 9 ) and we see that r v = 1 if k 2 1 and is odd, but that r v = 0 otherwise. Also, Q(k)* = Q(1- k). Thus L(1, Q(k)) is a critical value if and only if k is either a positive even integer or a negative odd integer. In either case one can use the Kubota-Leopoldt padic L-function to define both L,(s, Q(k)) and the corresponding element OQ(k) (which is in A if wk + wO or w') in a precise way. One also finds that Conjecture 7.1 is then equivalent to Conjecture 4.4 when k 5 0 and to Conjecture 4.5 when k > 1. For more details about this equivalence we refer the reader to Section 1 of [Gr4].

Let E be a modular elliptic curve over Q with good, ordinary reduction at p. This means equivalently that p I a,, where a, = a,(E) denotes the p t h Fourier coefficient for the newform fE attached to E . Consider the compatible system V(E) = {K(E)) , where K(E) = Z ( E ) C3 Q1

and Tl(E) is the 1-adic Tate module for E. As a GQp-representation

space, Vp(E) does have a natural filtration. If J!? is the reduction of E


- modulo p, then T,(E) E Z, since E is ordinary. The natural reduc-

tion map Vp (E) - . V, (E) is surjective and the inertia group I, acts trivially on vp(E). If F'V,(E) denotes the kernel of this map, then I, acts by X, on FIVp (because of the Weil pairing). Thus one can take F'v,(E) = V,(E), F~v,(E) = 0, and so p is indeed an ordinary prime for V(E). We have A = V,(E)/T,(E) E [pm] , the ppower torsion on E, and SE[pOO] (Q,) is a certain A-module. It turns out that

This will be explained later. On the other hand, we have L(l , V(E), p) =

L(l , E, p) for all p E P, and hence we can just define L,(s, V(E)) to be L,(s, E) , the padic L-function constructed by Mazur and Swinnerton- Dyer. This also gives the right normalization: the period involved in the interpolation property defining Lp(s, V(E)) should be the real Neron period QE for E (which also occurs in the precise formulation of the Birch and Swinnerton-Dyer conjecture). Therefore, Mazur's conjecture is equivalent to Conjecture 7.1 when V = V(E) and the padic L-function is as defined in [M-SwD].

Now consider V = V(f12) = {&(flZ)), the compatible system of l-adic representations defined by Deligne for the unique newform f12 of weight 12 and level 1. The corresponding complex L-function is

where ~ ( n ) is Ramanujan's tau-function and where the Dirichlet series expression is valid for 'Re(z) > 7. The functional equation relates the values L(z, V) and L(12 - z, V) . The gamma-factor is simply I'(z). The critical values of L(z, V) are therefore L(j, V) for integral j, 1 5 j 5 11. For each such j and for any prime p, there is a padic L-function defined by an interpolation property involving the values L(j, V, P ) for p E P. But one can view these values as L(1, V ( l - j), p) , where V(t) = {q (t) ) denotes the t-th Tate twist. (That is, &(t) = & 63 x:). The functional

: equation then relates L( l , V(1- j ) , p) to L(1, V(1- j'), p-') for j + j' = I 12, reflecting the fact that V( l - j)* = V(l- j') because the determinant

of & is X;l. Manin and Vishik found that the corresponding padic L- functions L,(s, V(l - j)), 1 < j < 11, are associated to a bounded

i measure on F and hence to an element of A (choosing a suitable period) precisely when p t T (p) . Thus if p r (p) , one can define elements Ov( - J )

in A for each such j . h In early 1986 I asked Ken Ribet the following question: if p is a 1 prime such that p 1 ~ ( p ) , then does Vp = Vp(fiz) have a I-dimensional


unramified quotient when considered as a 0,-representation space for GQp, just as is the case for V,(E) = Vp(fE) when E is a modular elliptic curve and p + a,(E)? He told me that Mazur and Wiles had just recently proved such a result. (It can be found in [M-W2] and also in a more explicit and general form in [Will.) This result was crucial to my speculations at the time because it would then follow that Vp(l - j ) had a 1-dimensional quotient on which I, acts by xi-j. That is, if V = V(f12) and p + ~ ( p ) , then V is ordinary in the sense defined earlier. Furthermore, since the determinant for Vp is xi', it would follow that F+Vp(l - j) is l-dimensional precisely when 1 5 j 11 (because I, acts on the composition factors for Vp(l - j) (as a representation space for GQp) by Xi-j and X F - ~ . If Tp is a GQ-invariant Zp-lattice in V,, then we let A(1- j) = Vp(l - j)/Tp(l - j ) . We then have A(1- j)' E A(1- j') where j + j' = 12. If j 5 0, then F+A(1 - j) = A(1 - j) and it is not hard to show that SA(l-j) (Q,) cannot be A-cotorsion. On the other hand, if j 2 12, then j' 5 0 and SA(j-')* (Qm) cannot be A-cotorsion. Both of these Selmer groups could possibly be A-cotorsion if 1 5 j 5 11. This seemed quite encouraging.

It may be worthwhile to recount some of the considerations which led me to ask Ribet that question about VP(fl2). During the academic year 1985-86 I was visiting 17Universit6 de Paris-Sud. In the Fall of that year, John Coates described to me his recent work with Claus Schmidt in which they construct a padic analogue of L(z, s y m 2 ( ~ ) ) and formulate a corresponding main conjecture under the assumption that E is a modular elliptic curve with good, ordinary reduction at p. They could verify that if E is an elliptic curve over Q with complex multiplication, then the two-variable main conjecture (mentioned at the end of Section 5) would imply their conjecture. Their formulation involved an Iwasawa module defined in terms of the Selmer group for E over the field Q(E[pm]) and did not suggest a way to formulate a main conjecture for L,(s, f12), an example which especially interested me. But that Winter I recall looking at some numerical data given in Manin's paper [Man]; namely, he writes

(j-iyi3 where rj-1 = L(j, f12) and the expression indicates ratios of these numbers. What did those 691s mean? Were they related to the fact that Tp/pTp is a reducible IFp-representation of GQ if p = 691? Manin also gives similar data for the other newforms f16, fls, f20, f22 and f26 of level 1 with rational Fourier coefficients and the same pattern continued.

It seemed reasonable to guess that the corresponding padic L-functions might have a positive p-invariant, i.e., Ov(l-j) E pA for j = 3 and 5. This would assume that one chose a period S2 so that L(1, f12)/R =

1, but even then the interpolation property defining L,(s, f12) would imply that plLp(l, f12) for p = 691 because (1 - a;') would be a factor. Here a, E Z,Y is the padic unit root of x2 - T(P)Z + pl' and so a, E ~ ( p ) - 1 (mod pZp) for p = 691. Thus, perhaps Ov(l-j) E pA for j = 1 too. For an elliptic curve E / Q with good, ordinary reduction a t p, Mazur had given many examples where the A-module selE(Q,& has a positive p-invariant. In those examples, E[p] = T,(E)/pT,(E) is always GQ-reducible and, more precisely, possesses a GQ-invariant subgroup isomorphic to pp. It occurred to me that there would then be a natural map with finite kernel from H1(Q,, p,) to a subgroup of H1(Q,, E[pm]) and perhaps that might be the source of the positive p-invariant. The fact that Tp( f 12)/pTp( f '2) would have a GQ-invariant subgroup isomorphic to pF1 ' if the Z,-lattice Tp( f 12) was chosen suitably and that this might also account for a positive p-invariant turned out to be another helpful clue.

These hints led me to look closely at the definition of the Selmer group for an elliptic curve E over 0,. Its pprimary subgroup is defined

by

where K, : E ((Q,),) 8 (Qp/Zp) + H1 ((Q,), , E [pm]) is the local Kum- mer homomorphism for E over (Q,),. If vll where I # p, then it turns out that Im(n,) = 0. This is quite easy to prove. For if L is any finite extension of Ql, then E(L) contains a subgroup of finite

index isomorphic to ~ i ~ : ~ ~ ~ . This subgroup is divisible by p and so it follows that E(L) 8 (Q,/Z,) = 0. This immediately implies that E((Qm),) 8 (Qp/Zp) = 0 and hence Im(k,) = 0 for v + p. Let 7r be the unique prime of Q, lying over p. Let I, denote the inertia subgroup of G(q,), . Then it turns out that

The equivalence of these descriptions follows from the easily verified fact that the map H' ((Q,),, Ebml) -+ H1 (I,, E[pml) is injective. One inclusion can be proved by observing that if g E I, = GQ;nr


and if P E ~(n , ) , the g(P) - P must be in the kernel of the re-

duction map ~ ( 0 ~ ) - E(FP). It then follows that any element of

Im(6,) becomes trivial in H' (I,, k[pm]). Coates and I managed to prove the equality. For a complete proof see [Gr5], or [C-GI where the local Kummer maps are studied in a more general context. The fact that SelE(Q,), = SElpm1(Q,) follows from these considerations. In particular, if E [p] contains a GQ-invariant subgroup isomorphic to p,, then it becomes rather clear that the image of H' (Qc/Q,, p,) (where

c = {P , 00)) in H1 (Q,, E[pW]) is contained in SelE (Q,),. Elements of this image are unramified at all v p and are also contained in Im(6,) because of the above description. But it is quite easy to see that H' (Qz/Q,, pp) is isomorphic to the A-module H o m ( ~ " ' lpxw I , p,) and therefore has (A/pA)-corank equal to 1. This shows that selE (Q, & has positive p-invariant. Similarly, H' (QE/Qm, pF'l) is isomorphic to

Hom(xwl lpxwl I , ,$" ' ) and this also has (A/pA)-corank equal to 1. Since, as Ribet informed me, Vp( f12) does have a suitable filtration and we would clearly have p?'' c F+A (when A = Vp(f12)/Tp(f12) and Tp( f2') is chosen as before), it again follows that the p-invariant of s A ( ~ , r i s positive.

8. In the past decade Perrin-Riou has made considerable progress in developing Iwasawa theory in the "non-ordinary" case. The first example to consider is SelE(Q,)p when E is an elliptic curve over Q with good, supersingular reduction at p. It was realized in the early 1970s that SelE(Q,), is not A-cotorsion, and so ~ e l ~ ( Q , ) ; ~ has unbounded Zp-corank as n + m . In contrast, it is reasonable to conjecture that SelE(Q,), has bounded Zp-corank for n 2 0. In [Pe3], Perrin-Riou proves this under the hypothesis that SelE (Q), is finite and p 2 5. If E is modular, the boundedness of corankzp (SelE (Q,),) has been proven by Kato. In either case, it is clear that the analogue of Theorem 5.2 would be false. On the analytic side, a padic L-function Lp(s, E ) for E was also constructed in the early 1970s. (Vishik, Manin, and Amice-Vela studied the existence of padic analogues of the complex L-functions attached to new forms of arbitrary weight. See [Man] and [M-T-TI.) But Lp(s, E) is not an Iwasawa function if E has supersingular reduction at p. (That is, L,(s, E) does not correspond to an element BE E A as it does in the ordinary case.) In fact, Vishik proved that L,(s, E) must have infinitely many zeros for s E $, Isl, < 1, and Rohrlich's nonvanishing theorem [Ro] shows that Lp(s, E) is not identically zero. .

Let a and p denote the two inverse roots of the zeta function for J@, the reduction of E modulo p. Since is supersingular, a and P both have

padic valuation 1. There are actually two distinct padic L-functions,

LP) (s, E) and Lip) (s, E) . Perrin-Riou constructs algebraic analogues for these p-adic L-functions in IPe3] and formulates a main conjecture. There are numerous unsolved questions. It is not clear what the zeros of these padic L-functions or their algebraic analogues really mean, except for those zeros corresponding to a character of I? = Gal (Q,/Q) of finite order. What do the common zeros mean? (They should conjecturally be a finite set.) It would be important to verify some special cases of the main conjecture. (For example, if E has complex multiplication, then Rubin has proven some deep results in [Rul, 21 which would seem to be closely related. Also, recent work of Kato connecting these p- adic L-functions to certain Euler systems should be helpful.) In Perrin- Riou's subsequent papers, she refines and extends her theory, developing a rather elegant formulation in terms of the Bloch-Kato logarithms which map Galois cohomology groups to Dieudonn6 modules. The details are difficult and we refer the reader to [Pe4], [Pe5], and the references to be found in those articles.

We have neglected to discuss the link between algebraic K-theory for rings of integers of number fields and Iwasawa theory. This was discovered by Tate for K2 in the early 1970s. More generally, the relationship arises from the Chern maps

for i 2 2. Here OF denotes the ring of integers of a number field F, FE denotes the maximal extension of F unramified outside C = {p, m) , p is any odd prime, and Zp(i) denotes a free Zp-module of rank 1 on which Gal (Fc/F) acts by xi, where x is the ppower cyclotomic character. It has been proven that the Chern maps are surjective. (Soul6 for i 5 p, Dwyer-Friedlander for arbitrary i.) This means that theorems about the K-groups will give results about the above Galois cohomology groups which can then be interpreted in terms of Iwasawa theory. We will mention two specific results which I believe have never been proven in any other way. Assume for simplicity that p, c F . Then F, = F(ppm ) is the cyclotomic Zp-extension of F . Let X = Gal (L,/F,), just as at the beginning of this article. Let fx (T) be the characteristic polynomial of X (with T = yo - 1) and let x(yo) = uo. As a consequence of a theorem of Bore1 asserting that Km(OF) is finite for even m 2 2, it follows that H 2 ( ~ c / F , Zp(i)) is finite, and Soul6 proves in [So] that this implies that fx(u;-' - 1) # 0 for all i > 2. Secondly, Lee and Szczarba proved in 1978 that the order of K4(Z) was not divisible by any prime p > 3.

380 R. Greenberg Iwasawa Theory - Past and Present 38 1

Thus, H2 (Fc I F, 2,(3)) = 0 for p > 5. In [Kul] , Kurihara deduces from

this the useful result that srPp3 = 0. Here the notation is the same as in Section 2, where F = Q(p,).

There are many other topics which have been overlooked in this article. The literature in Iwasawa theory has become quite vast over the years. The following list of references includes just a sampling of this literature. In addition to papers cited in the text, we have included various others which provide an introduction to important topics and also include many valuable references themselves. Thus, indirectly, we hope that this list will be rather comprehensive.

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K. Iwasawa, On I?-extensions of algebraic number fields, Bull. Amer. Math. Soc., 65 (1959), 183-226.

K. Iwasawa, On some properties of r-finite modules, Ann. of Math., 70 (1959), 291-312.

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K. Iwasawa, A class number formula for cyclotomic fields, Ann. of Math., 76 (1962), 171-179.

K. Iwasawa, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, 16 (1964), 42-82.

K. Iwasawa, Analogies between number fields and function fields, in Some Recent Advances in the Basic Sciences, 2 (1969), 203- 208, Yeshiva University.

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K. Iwasawa, Lectures on padic L-functions, Ann. of Math. Stud., 74, Princeton University Press (1972).

K. Iwasawa, On Zl-extensions of algebraic number fields, Ann. of Math., 98 (l973), 246-326.

K. Iwasawa, On the pinvariants of Z1-extensions, Number Theory, Algebraic Geometry, and Commutative Algebra, in honor of Y. Akizuki, Kinokuniga, Tokyo (l973), 1-11.

K. Iwasawa, Riemann-Hurwitz formula and padic Galois representations for number fields, T6hoko Math. J., 33 (1981), 263-288.

K. Iwasawa, C. Sims, Computation of invariants in the theory of cyclotomic fields, J. Math. Soc. Japan, 18 (1966), 86-96.

[Kal

[Kil

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;, [Man]

[M-Vl

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[M-SwD]

[M-TI


K. Kato, Lectures on the approach to Iwasawa theory for Hasse- Weil L-functions via BdR, in Arithmetic Algebraic Geometry, Lecture Notes in Math., 1553 (1994), 50-163.

N. Katz, padic L-functions for CM fields, Inv. Math., 49 (1978), 199-297.

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B. Mazur, J. Tate, J. Teitelbaum, On padic analogues of the conjectures of Birch and Swinnerton-Dyer, Inv. Math., 84 (1986), 1-48.

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W. McCallum, Greenberg's conjecture and units in multiple Z,- extensions, preprint.

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T. Nguyen Quang Do, Galois module structure of pclass formations, this volume.

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Department of Mathematics University of Washington Seattle, WA 981 95-4350 U.S.A. E-mail address: greenberhath. washington. edu


Iwasawa Invariants of Zp-Extensions over an Imaginary Quadratic Field

Manabu Ozaki

1 Introduction

Let k be a number field and p > 2 a prime number. For a Z,- extension K / k we denote by A(K/k) and p(K/k) the Iwasawa A- and p-invariants, respectively. If k is not totally real, k has infinitely many different Z,-extensions. We therefore are interested in the behavior of X(K/k) and p(K/k) as K varies over all Z,-extension fields over the number field k. Greenberg initiated the study of this problem in [4], and obtained some results on the behavior of A(K/k) and p(K/k) . For example he proved the boundedness of p(K/k) for fixed k and p under some assumption on the base field k and the prime p. After Greenberg's work, Babaicev and Monsky independently established the boundedness of p(K/k) without any assumption ( [ I ] , [ 1 2 ] ) .

The behavior of A-invariants is more difficult to study than that of p-invariants. In the present paper, we shall investigate the case where the base field is an imaginary quadratic field, and give the following theorem:

Theorem 1. Let k be an imaginary quadratic field and p 2 2 a prime number. Assume that the prime p splits i n k and the class number of k is prime to p. Then A(K/k) = 1 and p(K/k) = 0 for all but finitely many Z,-extensions K over k.

We shall make some remarks on the theorem. (1) If p does not split in a number field F and the class number of F is prime to p, then A(K/F) = p(K/F) = u ( K / F ) = 0 for every Z,- extension K / F by Iwasawa's result (161). Hence only the case where p splits in the imaginary quadratic field k is interesting under the assumption that p does not divide the class number of k.

Received September 4, 1998. Revised January 11, 1999.

388 M. Ozaki Iwasawa Invariants of Z,-Extensions

(2) If K/k is a Zp-extension such that every prime of k lying above p is totally ramified in Klk , then A(K/k) 2 1. There exist exactly two Zp- extensions N and N* over k in which one of the primes of k lying above p does not ramify (see Section 3). For these Zp-extensions, we have A = p = 0 by Iwasawa7s result mentioned above. Therefore the above theorem says that the A- and p-invariants take the minimal values for almost all Zp-extensions over k. (3) For the p-invariant, Bloom and Gerth have obtained stricter result ([2]). For any fixed imaginary quadratic field k and prime p, they proved that

#{Zp-extension fields K over k such that p(K/k) # 0) I Ap(k) + 1,

where Xp(k) denotes the A-invariant of the cyclotomic Zp-extension over k. (4) In Theorem 1, we surely have exceptional Zp-extensions, namely, Z,-extensions with A > 1 or p > 0, for many imaginary quadratic fields k and primes p. For example, let k = Q( J-23834) and p = 3. Then the class number h(k) of k is 232, which is prime to p = 3, and the A-invariant Ap(k) of the cyclotomic Zp-extension over k is 10. We can give a few more examples as follows:

0 For k = ( ) ( d m ) and p = 5, p j h(k) = 311 and Ap(k) = 8. 0 For k = ( ) ( d m ) and p = 7, p j h(k) = 12 and Ap(k) = 7.

However, we have no examples of exceptional Zp-extensions different from the cyclotomic Zp-extensions. It is a very interesting problem to find out an exceptional Zp-extension different from the cyclotomic Zp- extensions.

As stated in Theorem 1, the Iwasawa A- and p-invariants of Zp- extensions over an imaginary quadratic field with the class number prime to p tend to be very small. This phenomenon is caused by the smallness of the Iwasawa module for the Zi-extension over such an imaginary quadratic field. In Section 2, we shall briefly look at the Iwasawa module for Zg-extensions and Greenberg's conjecture on it, which predicts that the Iwasawa module is not so "large". Also we shall introduce Minardi7s result on Greenberg's conjecture for imaginary quadratic fields, which is a key to the proof of Theorem 1. In Section 3, we shall prove Theorem 1, in fact, we shall give a more general result which implies Theorem 1. In the final section, we shall prove Minardi's theorem in Section 2 for the convenience of the reader, because his proof was published in his thesis.

52. Greenberg's conjecture for the Iwasawa module

We fix a prime p throughout this section. For a number field k, denote by k the composite of all Zp-extension fields of k. Then Gal(h/k) - Zg with d = rz(k) + 1 + 6(k,p), where r2(k) is the number of complex archimedean places of k, and S(k,p) 2 0 is the "defect" of Leopoldt7s conjecture for k and p. In other words, Leopoldt7s conjecture for k and p holds if and only if S(k,p) = 0. Let ~ ( k ) be the maximal unramified pro-p abelian extension field over k , and define the Iwasawa module Xi to be Ga l (~ (h ) /h ) . Put A = ZP[[Gal(h/k)]], which is (non-canonically) isomorphic to the ring of d-variable power series with coefficients in Zp. Then XI, is a finitely generated torsion A-module by Greenberg's result ([4, Theorem I]), where Gal(k/k) acts on Xi by the inner automorphism as usual. Greenberg proposed the following conjecture, which states that the A-module Xi is not so "large" :

Greenberg's conjecture For a number field k and a prime p, Xi is a pseudo null A-module, namely, the height of the annihilator AnnA(XL) is greater than one.

Assume that k is totally real. Then we have d = 1 under the validity of Leopoldt's conjecture. Hence h/k is the cyclotomic Zp-extension k,/k, and the pseudo nullity of Xi is equivalent to the finiteness of Xi, which in turn is equivalent to that both Iwasawa A- and p-invariants of k,/k vanish. Thus we see that if we assume the validity of Leopoldt's conjecture, the above conjecture implies Ap(k) = pp(k) = 0 for any totally real number field k and any prime p (see [5]).

Minardi studied Greenberg7s conjecture especially for imaginary quadratic fields, and obtained the following theorem:

Theorem A (Minardi). Let k be an imaginary quadratic field and p a prime. If the class number of k is prime to p, then Greenberg's conjecture is valid for k and p.

He gave the proof of the theorem in his thesis [lo] (see also [ l l ] ) . We shall prove Theorem A in Section 4 below for the convenience of the reader.

He also verifies the pseudo-nullity of the Iwasawa module for the Zg-extension over many imaginary quadratic fields. See [lo] for details.

388 M. Ozaki

(2) If Klk is a Zp-extension such that every prime of k lying above p is totally ramified in Klk, then A(K/k) 2 1. There exist exactly two Zp- extensions N and N* over k in which one of the primes of k lying above p does not ramify (see Section 3). For these Zp-extensions, we have A = p = 0 by Iwasawa's result mentioned above. Therefore the above theorem says that the A- and p-invariants take the minimal values for almost all Zp-extensions over k. (3) For the pinvariant, Bloom and Gerth have obtained stricter result ([2]). For any fixed imaginary quadratic field k and prime p, they proved that

#{Zp-extension fields K over k such that p(K/k) # 0) I Ap(k) + 1,

where Ap(k) denotes the A-invariant of the cyclotomic Zp-extension over k. (4) In Theorem 1, we surely have exceptional Zp-extensions, namely, Z,-extensions with A > 1 or p > 0, for many imaginary quadratic fields k and primes p. For example, let k = Q( J-23834) and p = 3. Then the class number h(k) of k is 232, which is prime to p = 3, and the A-invariant Ap(k) of the cyclotomic Zp-extension over k is 10. We can give a few more examples as follows:

For k = Q(d-52391) and p = 5, p 1/ h(k) = 311 and Ap(k) = 8. For k = Q(d-1) and p = 7, p 1/ h(k) = 12 and Ap(k) = 7.

However, we have no examples of exceptional Zp-extensions different from the cyclotomic Zp-extensions. It is a very interesting problem to find out an exceptional Zp-extension different from the cyclotomic Zp- extensions.

As stated in Theorem 1, the Iwasawa A- and p-invariants of Zp- extensions over an imaginary quadratic field with the class number prime to p tend to be very small. This phenomenon is caused by the smallness of the Iwasawa module for the Zi-extension over such an imaginary quadratic field. In Section 2, we shall briefly look at the Iwasawa module for Zg-extensions and Greenberg's conjecture on it, which predicts that the Iwasawa module is not so "large". Also we shall introduce Minardi's result on Greenberg's conjecture for imaginary quadratic fields, which is a key to the proof of Theorem 1. In Section 3, we shall prove Theorem 1, in fact, we shall give a more general result which implies Theorem 1. In the final section, we shall prove Minardi's theorem in Section 2 for the convenience of the reader, because his proof was published in his thesis.

Iwasawa Invariants of Z,-Extensions

$2. Greenberg's conjecture for the Iwasawa module

We fix a prime p throughout this section. For a number field k, denote by & the composite of all Z,-extension fields of k. Then ~ a l ( & / k ) 2

Zg with d = r2(k) + 1 + 6(k,p), where r2(k) is the number of complex archimedean places of k, and S(k,p) 2 0 is the "defect" of Leopoldt's conjecture for k and p. In other words, Leopoldt's conjecture for k and p holds if and only if 6(k, p) = 0. Let ~ ( h ) be the maximal unramified pro-p abelian extension field over i, and define the Iwasawa module Xi

to be ~ a l ( L ( h ) / % ) . Put A = i ~ , [ [ ~ a l ( i / k ) ] ] , which is (non-canonically) isomorphic to the ring of d-variable power series with coefficients in Z,. Then Xi is a finitely generated torsion A-module by Greenberg's result ([4, Theorem I]), where ~ a l ( i / k ) acts on Xi by the inner automorphism as usual. Greenberg proposed the following conjecture, which states that the A-module Xi is not so "large" :

Greenberg's conjecture For a number field k and a prime p, Xi is a pseudo null A-module, namely, the height of the annihilator AnnA(Xi) is greater than one.

Assume that k is totally real. Then we have d = 1 under the validity of Leopoldt's conjecture. Hence i l k is the cyclotomic %-extension k,/k, and the pseudo nullity of Xi is equivalent to the finiteness of Xi, which in turn is equivalent to that both Iwasawa A- and p-invariants of k,/k vanish. Thus we see that if we assume the validity of Leopoldt's conjecture, the above conjecture implies Xp(k) = pp(k) = 0 for any totally real number field k and any prime p (see [5]).

Minardi studied Greenberg's conjecture especially for imaginary quadratic fields, and obtained the following theorem:

Theorem A (Minardi). Let k be an imaginary quadratic field and p a prime. If the class number of k is prime to p, then Greenberg's conjecture is valid for k and p.

He gave the proof of the theorem in his thesis [lo] (see also [I 11). We shall prove Theorem A in Section 4 below for the convenience of the reader.

He also verifies the pseudc+nullity of the Iwasawa module for the i%:-extension over many imaginary quadratic fields. See [lo] for details.

390 M. Ozaki Iwasawa Invariants of Z,-Extensions 39 1

$3. Proof of Theorem 1

In this section, we shall prove Theorem 1. In fact, we shall give a more general result, from which one can derive Theorem 1 by using Theorem A.

The notation used here is the same as in the preceding. For a prime p and an imaginary quadratic field k, let 3 = F(k,p) be the set of all Zp-extension fields K over k such that at least one prime of k lying above p does not split in Klk. We shall prove the following theorem in this section.

Theorem 2. Let k be an imaginary quadratic field and p 2 2 a prime number. Assume that the Iwasawa module XI;: is a pseudo-null A-module. Then the following hold: (i) If p splits in k, then X(K/k) = 1 and p(K/k) = 0 for all but finitely many Zp-extensions K E 3. (ii) If p does not split zn k, then X(K/k) = p(K/k) = 0 for all but finitely many Z,-extensions K E 3 .

Before starting with the proof, we shall make some remarks on this theorem. (1) We note that if a Zp-extension K/k has the property K n L(k) = k, then K E 3, where L(k) is the Hilbert pclass field of k. Hence F coincides with the set of all Zp-extensions over k if L(k) n & = k, for instance. Therefore, combining Theorem 2 and Theorem A in the preceding section, we obtain Theorem 1. (2) In general L(k) n Elk is a cyclic extension (including the case L(k) n & = k) unless p = 2 and p ramifies in k. Hence 3 is an infinite set in this case. (3) Assume that p splits in k. Let K/k be a Zp-extension. Then the statement K # F is equivalent to that p splits completely in the first layer kl of Klk. Suppose that K # F, and let il be the composite of all Zp-extension fields over kl . Then Gal(& /kl) $+' and the inertia

subgroup of Gal(&/kl) for a prime of kl lying above p is isomorphic to Zp. Hence we have X(K/k) > p if both primes of k lying above p ramify in K/k, because kl c K C % Kl and i l / ~ is unramified.

Example. Let k = Q(d=), p = 3. Then L(k) 2 &, [L(k) : k] = 3 and p splits completely in L(k) (See [lo, Table 6.11). Hence we obtain X(K/k) 2 3 if both primes of k lying above p ramify in K/k and L(k) C_ K. In particular, we have X(kgti/k) 2 3 for the anti-cyclotomic Zs-extension kZti/k, because the anti-cyclotomic Zp- extension field always contains L(k) n & if p # 2.

The restriction "K E 3" in case (i) of the above theorem is therefore indispensable. (4) We shall call a Zp-extension K/k with K E 3 such that either X(K/k) > eK - 1 or p(K/k) > 0 exceptional Zp-extension, where eK denotes the number of primes of k which ramify in Klk . As in the case of Theorem 1, we have no examples of exceptional Zp-extensions different from the cyclotomic Zp-extensions.

We fix a prime p once for all in what follows. Let a and T be - - independent generators of Gal(%/k) E Z:, Gal(%/k) = (a) x (T ) , and we identify A with the ring of power series &,[[s, TI] by regarding a = 1 + S and T = 1 + T. In the case where p splits in k, say p = pp*, we write N and N* for the p-ramified (i.e., unramified outside p) and the p*- ramified Zp-extension fields over k, respectively. For a field F C 0, we write L(F) and XF for the maximal unramified pro-p abelian extension field over F and Gal(L(F)/F), respectively. When an element x of a ring operates on a module M, we write M, = MlxM. Also, when a group G operates on a module M , we denote by M~ (resp. MG) the G-invariant submodule (resp. the G-coinvariant quotient module) of M.

We study various quotient modules of Xi to obtain information about the Iwasawa invariants of a Zp-extension of k from them:

Lemma 1. Let k be an imaginary quadratic field and K/k a Zp- extension dijferent from N/k and N*/k. We assume that K E 3 if p does not split in k. Then we have the following exact sequence of Zp [[Gal(K/ k)]] -modules:

where C is gal(%/^) - Zp (if p splits in k) or a Zp-module of finite order (otherwise).

Proof. First we treat the case where p splits in k and K # N, N*. In this case, %/K is unramified since the inertia subgroups of Gal(k/k) for the primes lying above p are isomorphic to Z,. Hence L(K)/K is the maximal abelian subextension of L(&)/K and G ~ ~ ( L ( K ) / & ) - (XG)Gal(~lK). Thus we have the lemma in this case.

We next assume that p does not split in k. Denote by F/K the maximal abelian subextension of L(&)/K, then L(K)& c F and L(K)/K is the maximal unrarnified subextension of F /K . It follows from K E F that there is a unique prime of K lying above p. Let I, Gal(F/K) be the inertia group for the unique prime of K lying above p. Since F/& is unramified, we have Ip n gal(^/%) = 1. Hence we asserts that

392 M. Ozaki

L ( K ) ~ = F by Gal(F/L(K)) = I,. Therefore Gal(L(K)/L(K) n k) - ~ a l ( F / k ) E (Xb)Gal(LIK). Since the inertia subgroup of ~ a l ( b / k ) for

the prime of k lying above p is isomorphic to Zg by class field theory,

Gal(L(K) n h / ~ ) is finite. Thus we have the lemma. 0

For an a E Z,, put T, = (1 + S)-"(1 + T) - 1 = o-,r - 1 E A, and let K, C L be the fixed field of (o-,r). Then K,/k is a Z,-extension. Here, we note that A = Z,[[S, T,]] for every a E Z,. Let A C Z, be the set of all padic integers with K, E F. Each Z,-extension field over k is the fixed field of ( c a r ) or of (or-") for some a E Z,. Hence we shall show that #(Xi)T, < oo for all but finitely many a E A, which implies Theorem 2 by Lemma 1.

Let

be a shortest primary decomposition of the module 0 in the Noetherian A-module Xi, namely, Ass~(Xi lY,) = {Pa) and P, # P, if i # j ,

I AssA(M) denoting the set of associated primes of M for a A-module M (see [9] for example). Note that d ~ n n A (x~/Y,) = Pa and htP, 2 2 from the assumption of the theorem. Then we have the following:

Lemma 2. Under the assumption of Theorem 2, we have the exact sequence

where D is a Zp-module of finite order and 4 is the natural projection map.

Proof. We can see that

Hence it is enough to show that #(Xi/ njzi Y, + Y,) < oo. Put Ii = AnnA(Xk/ nj-+ Y, + Y,) for 1 < i 5 r. Since AnnA (Xi/Y,)

C Ii, we have

Iwasawa Invariants of Z,-Extensions 393

which assures that htIi 2 2. If we assume that htIi = 2, then we have = Pi and htPi = 2. Since

AnnA(Xi/Y,) C. Pi for some j # i. We therefore obtain Pj =

JAnnA(xk/Y,) C Pi, which asserts that Pi = Pj since htPi = 2 and htPj 2 2. This is a contradiction. Thus we have htIi = 3 for 1 5 i 5 r . Then #Alli < oo since A is a local ring of dimension three with the maximal ideal ( S , T, p) and #A/(Sn , Tn , pn) < oo for any n 2 1. Since Xi/ njzi Y, + Y , is a quotient of a direct sum of finitely many copies of Alli, we have the lemma. 0

From the exact sequence of Lemma 2, we get an exact sequence

where D[T,] = { x E DJT,x = 0). Hence we shall show that #(XK/Y,)Ta < oo for fixed i and all but finitely many a E A.

Lemma 3. For an a E Z,, #(X&/Y,)T~ is infinite i f and only if T, E Pi and htPi = 2.

Proof. We assume that #(Xil-/x)T, is infinite. There exists a surjection

for some n 2 1, where Ji = AnnA(Xi/Y,). This surjection yields a surjection

Hence #A/(Ji + T,A) = oo. As in the proof of Lemma 2, we can see that ht (Ji + T, A) < 2. From the inclusion of ideals

and that htPi 2 2 and h t ( d m ) 5 2, we have htPi = ht(P, + T,A) = 2, which implies T, E Pi.

Conversely, we assume that T, E Pi and htPi = 2. Then Tg E A n n ~ (Xi/K) for some n 2 1 since Pi = ,/AnnA (Xi/Y,). If we assume #(x,/<)T, < oo, then XL/Y, = ( X ~ J ~ ) ~ ; < oo, contradicting to htp' = 2. Therefore #(X&/x)T,I = oo. 0

394 M. Ozaki

Now we assume that # ( X ~ / Y , ) T ~ = oo for some a E A. We may assume that K, # N , N*. (If there is not a E A such that #(Xic/Y,)~, = oo and K, # N, N*, we have nothing to do.) Then T,A C Pi and htPi = 2 by Lemma 3. Hence Pi/T,A is a prime ideal of height one of A/T,A. We have A/T,A N Zp[[S]] by (f (S, T,) mod T,A) ++ f (S, 0) (note that Zp[[S, TI] = Zp[[S, T,]]). Since every prime ideal of height one of Zp[[S]] is a principal ideal generated by an irreducible element of Zp[[S]], we find some irreducible element g, (S) E Zp [[S]] such that

Since Pi = J A ~ ~ A ( X ~ / Y , ) , we have g,(S)" € A n n Z p [ [ s l l ( ( x ~ / ~ ) T a ) for some n > 1, where we identifies A/T,A with Zp[[S]] via the above isomorphism. Hence g, (S) divides a generator of the characteristic ideal of Zp [[S]]-module (Xi/ X)T, because # ( X L / ~ ) T ~ = o;) (in fact, a power of g,(S) is a generator); this implies g,(S) divides a generator of the characteristic ideal of Zp[[S]]-module (Xi)Ta by (2):

In the following, we shall show #(Xk/Y,)T, < oo for any ,B E Zp different from a. Sppose that #(Xi/Y,)Tp = oo for some /? E Zp different from a. Then T, and Tp are contained in Pi by Lemma 3. Hence T, - Tp = (1 + ~ ) - p ( l + T)(( l + s)P-, - 1) E Pi, from which we derive (1 + s ) p n - 1 E Pi for some n 2 0 since ,B - a # 0. Because Pi = (T,, g,(S)), we can see that g,(S) divides (1 + s ) p n - 1 in Zp[[S]]:

Lemma 4. Let k be an imaginary quadratic field and K E .F a Zp-extension field over k with the Galois group I? = (yj. Put v,, = ( 7 ' ~ ~ - I)/(? - 1) E Zp[[r]] for n > 0. Then a generator of charzp[[rllXK is prime to v, for any n > 0.

This lemma is well-known if K/k is a totally ramified Zp-extension. However, in our situation, K E .F is not necessarily totally ramified over k at ramified primes.

Proof of Lemma 4. We first note that if we have #M/gM < oo for a finitely generated torsion Zp [[I'll-module M, a generator of charZp[[rll M is prime to any of g E Zp[[r]]. Let n 2 0 be a fixed integer. If p does not split in k, then there exists a unique prime of the n-th layer kn of K/k lying above p. Hence the genus formula (see [8, p.307 Lemma


4.11) says that # A z n = # A , / ( ~ P ~ - l)Am is bounded for all m 2 n; here A, denotes the p-Sylow subgroup of the ideal class group of k,. Since xK/(?P" - l )XK N proj lim A,/(~P" - l)Am (the projective limit is taken with respect to the norm maps), xK - l)XK is finite. Therefore we have the lemma (in fact, a generator of charZp[[rllXK is

prime to ?pn - 1). Next we assume that p splits in k. We first note that if either

K = N or K = N*, the ramified prime of k in K/k does not split in K/k by the assumption K E 3; this is because p and p* decompose in the the same way in any cyclic unramified extension over k. (Note that p = pp* is principal.) Hence we have the lemma in the same way as above. Therefore we may assume that both p and p* ramify in Klk.

From the assumption K E 3, we may assume that p* does not split in K . Let k, be the n-th layer of K/k as above. Denote by M ( K ) and M(k,) the maximal pro-p abelian extension fields over K and k, which are unramified outside the primes lying above p, respectively. -

Then we have the isomorphism Gal(M(k,)/L(k,)) E JJYn ,, u$! /E?' by class field theory, where 13, is a prime of k, lying above - p, u$: is

the p r e p part of the local unit group Uvn of (Ic,),. and E?) is the

closure of E, n nQn I P u$: in n, (1) ,, U, , En being the group of units in k,. (We embed En diagonally in nqnl, U,,, as usual.) Since kn is an abelian extension field over an imaginary quadratic - field, p-adic

Leopoldt's conjecture is valid for k,, namely, rankzp E?) = rankzEn =

[k, : Q] - 1 (see [3]). Hence we see that Gal(M(k,)/k,) is finitely generated over Zp and

Let X = Gal(M(K)/K) and Fn the maximal intermediate field of M ( K ) / K which is abelian over k,. Then Gal(F,/K) N X/w,X, where w, = yP - 1. We denote the inertia group the unique prime of k, lying above p* by Ip* C Gal(F,/k,). Then Ip* N Zp and Gal(F,/M(kn)) = I,*. Therefore Gal(F,/k,) is finitely generated over Zp and rankzpGal(Fn/kn) = 2 by (6), which implies that X/w,X is finitely generated over Zp and

for all n 2 0. It follows from (7) that charzp~[rllX/woX = wOZp[[r]], which implies

396 M. Ozaki

Here a generator of charzp[~rllwoX is prime to vn for each n 2 1 because woX/vnwoX = woX/wnX is finite by (7). Thus we conclude from (8) that a generator of charzp[lrllX is prime to vn for each n 2 0. Since XK is a quotient of X, we obtain the lemma. 0

It follows from Lemma 1 that

By (4), (5), Lemma 4, and the above formula, we have

Lemma 5. Let k be an imaginary quadratic field and K/k a Zp- extension with Galois group r = m. Then (i) ~ h a r ~ ~ [ [ ~ ~ ~ X ~ g (y - l)Zp [[I?]] if p does not split in k, (ii) ~ h a r ~ , [ [ ~ X ~ g (y - I)~Z,[[I']] if p splits in k.

Proof. In the case where p does not split in k, we can obtain the lemma by using the genus formula in a similar way to the proof of Lemma 4.

We assume that p splits in k. Let M(k)/k be the maximal pro-p abelian extension which is unramified outside the primes lying above p. We first note that the primes of k lying above p finitely decompose in M(k) by class field theory. Then we find that the module XK is semisimple at y - 1, i.e., XK has no submodule isomorphic to Zp[[r]]/(y - 1)2Zp[[r]], by [7, Proposition 61. Furthermore, since XK /(y - l )XK is a quotient of Gal(M(k)/K) and rankzpGal(M(k)/K) = 1 by class field theory, we have rankzpXK /(y - l )XK 5 1. Thus we have the lemma (see [7] for details). 0

It follows from (4) and (10) that charzprrsll(Xi)Ta c SZp[[S]]. In formula (9), charzp [[s]] C = SZ, [[S]] or Zp [[S]] according to p splits in k or not. Thus we have a contradiction by Lemma 5. Consequently, we have shown that if there exists a E A such that #(Xi/x)Ta = m and K, # N, N*, then #(Xi/Y,)T, < m for any ,O E Zp different from a. Since this holds for S, = (1 + S ) ( l + T)-, - 1 instead of T,, we find that the number of exceptional Zp-extensions in .F different from N, N* is at most 2r, where r is the number of primary components of 0 C Xi in (1). Thus we have proved Theorem 2.

54. Proof of Minardi's theorem

In this section, we prove Minardi's theorem (Theorem A) in Section 2 following his thesis [lo].


In fact, we shall show Theorem B below. Recall that in the case where p splits in k, say p = pp*, N/k (resp. N*/k) denotes the unique Zp-extension over k in which only the prime p (resp. p*) ramifies.

Theorem B (Minardi). Let p be a prime and k an imaginary quadratic field.

(i) Assume that p does not split i n k. If there exists a Zp-extension K/k such that X(K/k) = p(K/k) = 0 and K E .F = F(k,p) , then Xi;, is a pseudo-null A-module. (ii) Assume that p splits i n k. If X(N/k) = p(N/k) = 0 and N E F , or there exists a Zp-extension K/k # N/k, N*/k such that X(K/k) = 1 and p(K/k) = 0, then XL is a pseudo-null A-module.

Suppose that the class number of k is prime to p. If p does not split in k, then the prime of k lying above p is totally ramified in every Zp-extension K/k and we have X(K/k) = p(K/k) = 0 in this case. In the case where p splits in k, the prime p is totally ramified in N/k and X(N/k) = p(N/k) = 0. Hence the above theorem certainly implies Theorem A in Section 2.

Proof. We shall show that < m for some subgroup r Gal(k/k) with Gal(%/k)/r - 5. Then we obtain the pseudo-nullity of Xi because

where a is the natural projection map from A to Ar and char,(*) denotes the characteristic ideal. (Note that the pseudo-nullity of Xi and charAxi = A are equivalent, which in turn is equivalent to a(charAXi) = Ar; see [13, 1.1.Lemma 41.)

We first treat the case where p does not split in k. Let K/k be a Zp-extension with the property stated in (i) of the theorem. It follows from Lemma 1 that (Xi)Gal(i/K) Gal(L(K)/K). Hence we have #(Xi)Gal,i,K, < m by X(K/k) = p(K/k) = 0. This concludes the proof of (i).

Next we assume that p splits in k. Suppose that there exists a Z,-extension K/k # Nlk, N*/k such that X(K/k) = 1, p(K/k) =

0. Then we obtain (XZ)Gal(L/K) = G a l ( L ( ~ ) / k ) by Lemma 1. Since Gal(L(K) /K) is a finitely generated Zp-module of rank 1 by X(K/k) =

1, p(K/k) = 0, we have #(XZ)Ga1(6/K) = # ~ a l ( L ( ~ ) / k ) < m. Finally we shall derive (XL)Gal(l/N) < m from the fact that X(N/k) = p(N/k) =

0 and N E 3. Let F be the maximal intermediate field of L(k)/k which is abelian over N. Then we see G a l ( ~ / i ) = (Xi)Gal(L,N) and

398 M. Ozaki

L(N) c F . Let k, (the n-th layer of N/k) be the decomposition field of N/k for the prime p* and yi * (1 5 i 5 pn) the primes of kn lying above p*. (Note that a prime of k lying above p is finitely decomposed in h by class field theory.) Denote by Iyl * c Gal(F/N) the inertia group for the prime of N lying above !&Ii*. We note that Gal(N/k,) acts on Gal(F/N) as usual, and that Iyc is stable under this action. The inertia group IyI * is mapped into Gal(k/N) - Z , injectively via the restriction Gal(F/N) - ~ a l ( h / ~ ) because F/k is an unramified extension. Hence it follows that Iyl * 21 Z p and Gal(N/k,) acts trivially on Iy , * . Since F I N is unramified outside the primes lying above p*, we have

It follows from the assumption [L(N) : N] < oo and (11) that Gal(F/N) is finitely generated over Z,, and that Gal(N/k,) acts trivially on Gal(F/N)/Gal(F/N)to,, Gal(F/N)t,, being the Z,-torsion submodule of Gal(F/N). Let F1 C F be the fixed field by Gal(F/N)to,. Then F1/kn is an abelian extension because Gal(N/k,) acts trivially on Gal(F1/N) - Gal(F/N)/Gal(F/N)t,,. Since N E 3, the prime p does not split in N/k. We write Ip for the inertia subgroup of Gal(F1/kn) of the unique prime of k, lying above p. Then Ip II Z p since p is unramified in F1/N and Gal(N/lc,) Z,. Let M*(k,) be the maximal pro-p abelian extension field over k, which is unramified outside the primes lying above p*. Because F'" M* (k,) and rankzpGal(M*(kn)/kn) = 1 as we have seen in the proof of Lemma 4, we conclude that rankzpGal(F1/k,) = 2,

which implies [F' : h] < oo. Therefore we have #(Xi)Gal(i,N) = [F :

h] < 03. 0

In conclusion, combining Theorem 2 and Theorem B, we obtain the following:

Theorem 3. Let p be a prime and k an imaginary quadratic field. For a Zp-extension Klk , we denote by e~ the number of primes of k which ramify in Klk . Assume that .F = 3 ( k 7 p ) is not empty. Then the following three statements are equivalent: (i) There exists a Zp-extension field K E 3 over k with X(K/k) = eK - 1 and p(K/k) = 0, (ii) X(K/k) = e~ - 1 and p(K/k) = 0 for all but finitely many K E 3 , (iii) Xi, is a pseudo-null A-module.


References

V.Babaicev: On the boundedness of Iwasawa's p-invariant (Russian), Izv. Acad. Nauk. SSSR, Ser. Mat., 44 (1980), 3-23; Translation: Math. USSR. Izvestia, 16 (1980), 1-19.

J.Bloom, F.Gerth: The Iwasawa invariant p in the composite of two Zl- extensions, J . of Number Theory, 13 (1981), 262-267.

A.Brumer: On the units of algebraic number fields, Mathematika, 14 (1967), 121-124.

R.Greenberg: The Iwasawa invariants of r-extensions of a fixed number field, Amer. J . of Math., 95 (1973), 204-214.

R.Greenberg: On the Iwasawa invariants of totally real number fields. Amer. J . of Math., 98 (1976), 263-284.

K.Iwasawa: A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258.

J.-F.Jaulent, J.Sands: Sur quelques modules d'Iwasawa semi-simples, Compositio Math., 99 (l995), 325-341.

S.Lang: Cyclotomic Fields I and 11 (2nd ed.), Graduate Texts in Mathe- matics 121, Springer-Verlag, New York, 1990.

H.Matsumura: Commutative ring theory, Cambridge studies in advanced mathematics 8, Cambridge University Press, Cambridge, 1986.

J.Minardi: Iwasawa modules for z$-extensions of algebraic number fields, Thesis (l986), University of Washington.

J.Minardi: Iwasawa modules for Zi-extensions of number fields, CMS Conf. Proc., 7 (1987), 237-242.

P.Monsky: Some invariants of Zi-extensions, Math. Ann., 255 (1981), 229-233.

B.Perrin-Riou: ArithmBtique des courbes elliptiques et thkorie d'Iwasawa, MBm. Soc. Math. France, 17 (1984), 1-130.

Department of Information and Computer Science, School of Science and Engineering, Waseda University, 3-4-1, Ohkubo Shinjuku-ku, Tokyo 169, JAPAN

Current address : Department of Mathematics, Faculty of Science and Engineering, Shimane University, Nishikawatsu-Cho 1060, Matsue 690-8504, JAPAN E-mail address: ozakihath. shimane-u. ac . jp


On p-Adic Zeta Functions and Class Groups of Z,-Extensions of certain Totally Real Fields

Hisao Tayal

Abstract.

Let k be a totally real field and p an odd prime number. We assume that p splits completely in k and also that Leopoldt's conjecture is valid for k and p. In this note, focusing on Greenberg's conjecture, we will report on our recent results concerning padic special functions and ideal class groups in the cyclotomic Z,-extension of k.

1 Introduction

For a number field k and a prime number p, we denote by k, the cyclotomic Zp-extension of k, by k, the n-th layer of k, over k , and by A, the pSylow subgroup of the ideal class group of k,. Also, for a finite set S , we denote by #S the number of elements in S. Then Iwasawa [Iw59] has proved that there exist three integers Ap(k), pp(k) and vp(k), depending only on k and p, such that

for sufficiently large n , where up denotes the pad ic valuation normalized by vp(p) = 1. These integers Xp(k), pp(k) and vp(k) are called the Iwa- sawa A-, p- and v-invariants, respectively, of the cyclotomic Zp-extension of k.

Concerning these invariants, Iwasawa mentions that it would be an important problem to find out if Xp(Q(Cp)+) = pp(Q(Cp)+) = 0 for any prime number p in [Iw70, page 3921, or to find out when the "plus-part" of Ap(k) is positive for CM-fields k in [Iw73a, page 3161, where Q(CP)+

1991 Mathematics Subject Classzfication. Primary llR23; Secondary llR42, llR29.

Key words and phrases. Z,-extensions, Iwasawa invariants, padic zeta- functions, ideal class groups.

Received August 31, 1998. This research was supported in part by the Grants-in-Aid for Encour-

agement of Young Scientists, The Ministry of Education, Science, Sports and Culture, Japan.

402 H. Taya

is the maximal real subfield of the cyclotomic field Q(Cp) of p t h roots of unity. Probably arising from this, the following was posed by Greenberg [Gr76] as a problem. This is now known as Greenberg's conjecture.

Conjecture 1.1 (Greenberg's conjecture). If k i s a totally real field, then Ap(k) = pp(k) = 0 for every prime number p.

As for the p-invariants, we know by the Ferrero- Washington theorem [FW79] (or [Wa97, Theorem 7.151) that pp(k) always vanishes for every prime number p when k is abelian (not necessarily totally real) over Q. More generally, the following is conjectured.

Conjecture 1.2. For any number field k and any prime number p, we should have pp(k) = 0.

The Ferrero-Washington theorem says that Conjecture 1.2 is true in the case of abelian number fields. Besides this, we know by the theorem of Iwasawa [Iw73b, Theorem 31 that pp(k) = 0 for any finite Galois p-extension k. Here we note that Iwasawa also constructed in [Iw73b] examples of non-cyclotomic Zp-extensions with arbitrarily large p-invariant.

On the other hand, as for the A-invariants, we know Kida's formula ([Kid80], [Kid82]) which describes the behavior of the "minus-parts" of Ap(K) and Ap(k) when K/k is a finite Galois p-extension of CM-fields. Also we have examples of k with arbitrarily large A-invariant if k is not totally real (e.g. [Gr76, page 2641). However, very little is known about the "plus-part" of Ap(k) except for k = Q. The field Q of rational numbers is so far the only number field that Greenberg's conjecture is known to be true for every prime number p (in fact A,(()) = pp(Q) = vp(Q) = 0 for any p). Recently, some efficient criteria for Greenberg's conjecture to be true are given from different points of view when k is abelian over Q and p is an odd prime number (see the paper [IS96-71 and the papers cited in it, especially Kraft-Schoof [22] and Kurihara [23] in part I1 of [IS96-71). However, at present we do not know any algorithm to determine whet her Greenberg's conjecture is true, or more precisely, to determine the order of A, for sufficiently large n, after a finite amount of times.

Let k be a totally real field and p an odd prime number. We denote by r the Galois group Gal(k,/k), and by A: the subgroup of A, consisting of ideal classes which are invariant under the action of r. As- sume that p splits completely in k. and also that Leopoldt's conjecture is valid for k and p. In this note, we will report on our recent results, most of which are appearing in [Taggal, [Ta99b] and [TaOO], concerning

On p-Adic Zeta Functions and Class Groups of Z,-Extensions 403

p-adic special functions and ideal class groups of k,. First, we give in 52 a simple formula for the order of A: for sufficiently large n in terms of the residue at 1 of the p-adic zeta function of k. This enables us to calculate the order of A: as a practical matter. When k is a real abelian field with degree prime to p, as mentioned in $3, a formula for the order of @-component of A; can be also given in terms of a special value of the padic L-function associated to @, where @ is an irreducible Qp-character of the Galois group Gal(k/Q). In the case where p splits completely in k, the order of A: is closely related to Greenberg's conjecture. Hence these formulas imply an alternative formulation (resp. the @-component version of it) of Greenberg's criterion [Gr76, Theorem 21 on the vanishing of the Iwasawa A- and p-invariants of k,/k. In 54, as an easy application of these formulas, we give a simple proof of Ozaki's theorem [0z97, Theorem I] which is regarded as a totally real version of classical Kummer's criteria for pdivisibility of the class number of Q(Cp), and also show the @-component version of Ozaki's theorem. As another application, we mention in 55 the result that there exist infinitely many real quadratic fields in which the prime 3 splits and whose Iwasawa &-invariant vanishes.

52. A simple genus formula (totally real case)

Here and in what follows, we use the same notation as in the previous section. Assume that k is a totally real field and p an odd prime number. Let Cp(s, k) be the p-adic zeta function of k, which is continuous on Zp - (1) and has simple pole at s = 1 if Leopoldt's conjecture is valid for k and p (cf. [C0188]). Let us put

to cancel the simple pole at s = 1. Then we have the following genus formula in terms of the residue of the p-adic zeta function.

Theorem 2.1 (cf. [Taggal). Let k be a totally real field and p a n odd prime number. Assume that p splits completely in k and also that Leopoldt's conjecture is valid for k and p. Then

for suficiently large n.

Proof. It follows from a lemma on the order of Gal(M/k,) proved by Coates [Coa77, Lemma 8 in Appendix], a limit formula for padic

404 H. Taya On p-Adic Zeta Functions and Class Groups of Z,-Extensions 405

zeta functions proved by Colmez [Co188, Main theorem], some property of a finite unramified extension over k,, and (local or global) class field theory, where M denotes the maximal abelian pro-p-extension of k unramified outside p. For the details, see the paper [Taggal. 0

Remark 2.2. A limit formula for padic zeta functions [Co188, Main theorem] implies that the right hand side in the theorem above is given by

where Rp(k) denotes the padic regulator of k and [k : Q] the degree of k over 0. Using this formula, we can practically calculate the order of A: for sufficiently large n, even if k is non-Galois (see [Ta99a, Example 4.21).

Remark 2.3. The formula in Theorem 2.1 is regarded as a generalization of a formula for real quadratic fields obtained by Fukuda and Komatsu [FK86a, Proposition 11 (or [FK86b]), and also as an explicit version of a formula for real abelian fields obtained by Inatomi [In89, Proposition 21. Comparing our formula in Remark 2.2 and Inatomi's one, we find that vp(Rp(k)) is equal to the integer m, which is defined in [In891 and used to describe his formula. Therefore, it follows that

#A: = ( ' 3 " ) for all n >_ vp(%(k)),

because Inatomi shows his formula holds for all n 2 m and his proof works equally as well for our situation.

Remark 2.4. In the proof of Theorem 2.1, we also use the prop erty that M/k, is an unramified extension, where M is the same as in the proof above (see [0z97, Proposition 11 or [Ta99a, Lemma 2.31). However, this property does not hold for p = 2.

Remark 2.5. The formula in Theorem 2.1 does not hold in general without the assumption on the decomposition of p in k/Q. For example, in the case where k = ~ ( m ) and p = 3 (so p = 3 remains prime in k), we find vp(Rp(k)) = 2. Hence we have

by a limit formula for padic zeta functions. On the other hand, since only one prime of k lies over p and this prime is totally ramified in k,/k, we have

for all n 2 0. Therefore we see #A: # p v p ( c ; ( l ~ k ) ) for p = 3.

In the case where p splits completely in k, A: is an important object in the study of Greenberg's conjecture. Let D , be the subgroup of A, consisting of ideal classes represented by products of prime ideals of k, lying above p. Then it is clear that D , c A:. Now we recall a theorem of Greenberg on the vanishing of the Iwasawa invariants (including p = 2).

Theorem 2.6 (Theorem 2 in Greenberg [Gr76]). Let k be a totally real field and p a prime number. Assume that p splits completely in k and also that Leopoldt7s conjecture i s valid for k and p. Then the following two conditions are equivalent :

(1) Xp(k) = Pp(W = 07 (2) #A: = # D , for suficiently large n.

Now, by Theorem 2.1 and Remark 2.2, we can obtain an alternative formulation of Theorem 2.6.

Theorem 2.7 (cf. [Taggal). Under the same assumptions as in Theorem 2.1, the following three conditions are equivalent:

(1) = Pp(k) = 0 , (2) # D , = p u ~ ( C ; ( l . for some n > 0 , (3) # D , = # A O ~ " P ( ~ P ( ~ ) ) - [ ~ ' Q ~ + ~ for some n 2 0.

Although Theorem 2.7 seems to be only a little different from the original theorem of Greenberg, it suggests that the validity of Green- berg's conjecture can be regarded as based on a certain arithmetic relation between an analytic object and an algebraic object.

$3. A simple genus formula (real abelian case)

In this section, we assume that k is a real abelian field and p an odd prime number. Then, Leopoldt's conjecture is valid for such a case by a theorem of Brumer [Br67], and also we know pp(k) = 0 by the Ferrero-Washington theorem [FW79]. Put A = Gal(k/Q). We further assume that the order of A is not divisible by p. Let * be an irreducible Qp-character of A, and A: the *-component of A,, namely,

Note that eq is an idempotent of Zp[A].

Remark 3.1. If !Po is the trivial character of A, then A:O = {I).


Let (A:)~ be the subgroup of A: consisting of ideal classes which are invariant under the action of I?. We define the padic L-function associated to B by

where + runs over all irreducible components of * over the algebraic closure 0, of Q,, and L,(s, $) denotes Kubota-Leopoldt's padic L- function associated to +. Here we regard + as a primitive padic Dirichlet character. Note that L,(s, *) has non-zero values at s = 1 if \I, is non- trivial. Then we have the following genus formula which is the *-part version of Theorem 2.1.

Theorem 3.2 (cf. [Taggb]). Let k be a real abelian field with Ga- lois group A and p an odd prime number. Assume that the order of A is prime to p and also that p splits completely in k. Then, for each non-trivial irreducible Q,-character B of A, we have

for suficiently large n.

Proof. It follows from Iwasawa main conjecture proved by Mazur and Wiles [MW84], some property of a finite unramified extension over k, and (local or global) class field theory. For the details, see the paper [Tag9 b] . 0

We denote by $'(k) the *-component of Xp(k), namely, the Iwa- sawa Xinvariant associated to A:. Note that Xp(k) = C, doXF(k), where * runs over all irreducible 0,-characters of A and de denotes the degree of B over Q,. In the case of real abelian fields, Greenberg's conjecture is equivalent to the statement called Greenberg's conjecture for B-components that X;(k) = 0 for any (non-trivial) B. Let D: be the *-component of D,,. Since Theorem 2.6 holds by replacing each object with its *-component, we obtain by Theorem 3.2 the following B-part version of an alternative formulation of Greenberg's theorem.

Theorem 3.3 (cf. [Taggb]). Under the same assumptions as i n Theorem 3.2, for a non-trivial irreducible Qp-character B of A, the following two conditions are equivalent :

(1) q k ) = 0, (2) #D: = pvp(Lp(l*e)) for some n 2 0.

Remark 3.4. Let ci ( s , k) be as in the previous section. Since

where the product runs over all non-trivial irreducible Q,-characters \Ir of A, we see by Remark 3.1 that Theorems 3.2 and 3.3 imply a special case of Theorems 2.1 and 2.7, respectively.

54. A simple proof of Ozaki's theorem

Ozaki showed the following theorem which can be regarded as a "totally real" analogue of classical Kummer's criterion for pdivisibility of the class number of Q(<,). In this section, we give its simple proof by using Theorem 2.1, and further, show the B-component version of his theorem.

Theorem 4.1 (Theorem 1 in Ozaki [0z97]). Let k be a totally real field and p an odd prime number. Let K, be a Z,-extension of k, with n- th layer Kn, in which the primes of k lying over p are totally ramified. Assume that p splits completely i n k. Then the following two statements are equivalent :

(1) The class number of Kn is divisible by p for all n 2 1, p& Wh (2) pCp(0, k) = 0 (mod P ) .

Remark 4.2. In the theorem above, since K,/k is totally ramified at p, the negative proposition of statement (1) is equivalent to the following one:

(a) The class number of Kn is not divisible by p for all n 2 0. Indeed, if statement (a) does not hold, namely, if the class number of Kn is divisible by p for some n 2 0 (so, for every n sufficiently large), then Gal(L,/K,) # {I), where L, is the maximal unramified abelian pro-pextension of K,. Note that Gal(L,/K,) is a finitely generated torsion Z,[[T]]-module (cf. [Iw73a, Theorem 51). Let vn = ((1 + T)," - 1) /T E Zp [[TI]. By Nakayama's lemma, we have Gal(L,/K,) /v,Gal(L,/K,) # (1) for all n 2 1. It is known that Gal(L,/K,) /vnGal(L,/K,Lo) is isomorphic to the pSylow subgroup of the ideal class group of Kn, where Lo is the maximal unramified abelian p extension of k (cf. [Iw73a, Theorem 61). Hence, this implies statement (1) in Theorem 4.1, because Gal(L,/K,Lo) c Gal(L,/K,). The converse is obvious. Therefore we conclude the desired equivalence.

Proof of Theorem 4.1. Assume that Leopoldt's conjecture for k and p is valid. Then K , is the cyclotomic Zp-extension of k. By Remark 4.2,

408 H. Taya

it suffices to prove that

A, = (1) for all n 2 0 ++ pCp(O, k) $ 0 (mod p).

Since A, = (1) if and only if A: = (1) and since K,/k is totally ramified at p, it follows from Theorem 2.1 that

A, = (1) for all n 2 0 +=+ A, = (1) for sufficiently large n

+=+ A: = (1) for sufficiently large n - C,*(l, k) + 0 (mod P).

On the other hand, we can see that

Cp(o' k, and pC, (0, Q) G 1 (mod p) . C p , k) = --- CP(0, Q)

Hence it follows from this that

Therefore we get the desired result. If Leopoldt's conjecture for k and p is not valid, then we can find that

statement (1) holds by the fact that the maximal abelian pro-p-extension of k unramified outside p is unramified over K, [0z97, Proposition 11 (or [Ta99a, Lemma 2.3]), and also that statement (2) holds by the Iwasawa main conjecture proved by Mazur and Wiles [MW84].

The following is one of immediate consequences of Theorem 4.1.

Corollary 4.3. Let p be a jixed odd prime number and q an odd prime number satisfying p = 1 (mod q). Put k = Q(Cq)+ and A = Gal(k/Q). For the cyclotomic Zp-extension of k, let k, be its n-th layer and A, the p-Sylow subgroup of the ideal class group of k,. Then the following two statements are equivalent :

( I ) A, # (1) for all n 2 1, (2) nX,, B~,,,-~ - 0 (mod P),

where the product runs over all non-trivial p-adic Dirichlet characters x of A, w the Teichmuller character for p, and B1,,,-I the generalized Bernoulli number.

Proof. Since p splits completely in k, this easily follows from T h e e rem 4.1 because

in this situation.

On p-Adic Zeta Functions and Class Groups of Z,-Extensions 409

Remark 4.4. Using the theory of cyclotomic units, Kim proved in [Kim95, Theorem 21 the assertion that (2) implies (1) in Corollary 4.3. Also, another result of Kim [Kim95, Theorem 31 is regarded as a consequence of the contraposition of the assertion that (1) implies (2) in the corollary.

Similarly as in the proof of Theorem 4.1, we can show the following theorem which is the @-part version of Theorem 4.1 in the case of real abelian fields.

Theorem 4.5. Let k be a real abelian field with Galois group A, p an odd prime number and k, the cyclotomic Zp-extension of k with n-th layer k, . Assume that the order of A is prime to p and also that p splits completely in k. Then, for a non-trivial irreducible 0,-character @ of A, the following two statements are equivalent :

(1) A: + (1) for all n 2 1, (2) L,(O,@) r 0 (mod p).

Again, the following is an immediate consequence of Theorem 4.5.

Corollary 4.6. Let p, q, k, A and A, be as in Corollary 4.3. Then, for a non-tr-ivial irreducible 0,-character @ of A, the following two statements are equivalent :

7, (1) A: # {I) for all n 2 1, ??; ,a L (2) n,,, B ~ , ~ ~ - ~ = 0 (mod P),

where the product runs over all irreducible components $ of Q over ,, and w and B1,+,-l be the same as in Corollary 4.3.

Proof. Since p splits completely in k and the order of A is not divisible by p, this easily follows from Theorem 4.5 because Lp(O, $) = -Bl,,,-l in this case. 0

55. A weaker problem o n t h e A-invariants

Althought there are some efficient criteria for Greenberg's conjecture to be true (see the paper [IS96-71 and the papers cited in it), we do not have general results on Iwasawa A-invariants of totally real fields, like as the FerrereWashington theorem, except for k = Q. So, in this section, we concentrate on the simplest case where k is a real quadratic field, and consider the following weaker problem than Greenberg's conjecture.

P rob lem 5.1. For a given prime number p, do there exist infinitely many real quadratic fields k satisfying Xp(k) = 0 and "some additional conditions"?


Needless to say, if there exist infinitely many real quadratic fields satisfying such "some additional conditions", then we clearly have an affirmative answer to Problem 5.1 under the validity of Greenberg's conjecture. Nevertheless, it seems that little is known about this problem, though we already have some results on it for p = 2, 3. Before recalling such known results, we describes the following theorem of Iwasawa which is often useful for this kind of problem.

Theorem 5.2 (Iwasawa [Iw56]). Let p be a prime number and L a finite extension of 0 . Let L, be the cyclotomic Zp-extension of L and L, its n - th layer. Assume that p dose not split i n L and that the class number of L is not divisible by p. Then the class number of L, is not divisible by p for all n 2 0. I n particular, we have Xp(L) = pp(L) = vp(L) = 0.

First, in the case where p = 2, genus theory implies that there exist infinitely many real quadratic fields k such that the class number of k is odd and 2 does not split in k. Hence, it follows from Theorem 5.2 that there exist infinitely many real quadratic fields k such that X2(k) = 0 and 2 does not split in k. Also, in [OT97, Theorem], we gave several infinite families of real quadratic fields k with X2(k) = 0. According to this, we immediately see the following:

(i) There exist infinitely many real quadratic fields k such that X2(k) = 0, the prime 2 splits in k and the class number of k is odd (or even),

(ii) There exist infinitely many real quadratic fields k such that X2(k) = 0, the prime 2 does not split in k and the class number of k is even,

(iii) Let N be a given positive integer. Then there exist infinitely many real quadratic fields k such that X2(k) = 0, the class number of k is even (or the prime 2 splits in k) and theuminus" X- invariant corresponding to k is greater than N , where theuminus" /\-invariant corresponding to k means X2(0(G)) if k = ~(4) .

Further, Ozaki recently constructed a new infinite family of real quadratic fields k with X2(k) = 0 (see [0z98, Theorem 61). The following is a conclusion from his result.

(iv) There exist infinitely many real quadratic fields k such that X2(k) = 0, the prime 2 splits in k (resp. does not split in k) and the 2-rank of the ideal class group of k is equal to 2 (resp. 3).

Here we note that we cannot apply Theorem 5.2 to cases ( i ) ~ ( i v ) . Next, we recall the case where p = 3. For a positive integer N, we

denote by K(N) the set of real quadratic fields with discriminant less

than N. Then, Nakagawa and Horie showed in [NH88] the following, using a theorem of Davenport and Heilbronn [DH71].

Theorem 5.3 (Theorem 3 in Nakagawa and Horie [NH88]). For a real quadratic field k, let d(k) be the discriminant of k and h(k) the class number of k. Then

From Theorems 5.2 and 5.3, it follows that

lim inf #{ k E K ( N ) 1 d(k) f 1 (mod 3), X3(k) = p3(k) = v3(k) = 0 ) N-+m #K(N)

Therefore we obtain the following corollary.

Corollary 5.4. There exist infinitely many real quadratic fields k such that X3(k) = 0 and the prime 3 does not split i n k.

On the other hand, we have not known whether there exist infinitely many real quadratic fields k with Xp(k) vanishing and p splitting there, for a given odd prime number p. But, we can now give an answer to

7

5 this problem in the case where p = 3 splits in k as an application of Theorem 4.1 (or Theorem 4.5). We introduce this in the rest of this I

note. First, in the case where k is a real quadratic fields and p = 3, we can

rewrite Theorem 4.1 (or Theorem 4.5) as follows, by using Remark 4.2 and a relationship between the class numbers of imaginary quadratic fields and generalized Bernoulli numbers (or special values of padic L- functions).

Proposition 5.5 (cf. [TaOO]). Let d be a square-free positive in-

teger with d = 1 (mod 3). Put k = ()(&) and k* = (')(a). For the cyclotomic &-extension k, of k, let k, be its n- th layer and A, the 3-Sylow subgroup of the ideal class group of k,. Then the following two statements are equivalent :

(1) A, = (1) for all n 2 0, (2) The class number h(k*) of k* is not divisible by 3.

Remark 5.6. Proposition 5.5 can be directly proved by a purely algebraic argument. For the details, see [TaOO, Remark 21.

412 H. Taya

Proposition 5.5 says that the problem of finding infinite families of real quadratic fields k with X3(k) vanishing and the prime 3 splitting is reduced to the problem of finding infinite families of imaginary quadratic fields k whose class number is not divisible by 3. And the latter problem can be solved by results of Nakagawa and Horie [NH88]. Consequently, we obtain the following theorem.

Theorem 5.7 (cf. [TaOO]). For a real quadratic field k, let d(k) be the discriminant of k, k, the n- th layer of the cyclotomic Z3-extension of k, and h, the class number of k,. Then

#{ k E K(N) I d(k) r 1 (mod 3), 3 Xh, for all n L 0 ) lim inf N--too #Ic(N>

Proof. This follows from Proposition 5.5 and [NH88, Theorem 1 and Proposition 21. For the detail, see the paper [TaOO]. [7

Now, we can give an affirmative answer in the case where p = 3 splits in k. Namely, we obtain the following.

Corollary 5.8. There exist infinitely many real quadratic fields k such that X3(k) = 0 and the prime 3 splits in k.

Combining Theorems 5.3 and 5.7, we get the following.

Theorem 5.9. W e have

Remark 5.10. Recently, K. Ono [On991 showed by Theorem 5.2 that for each prime number between 3 < p < 5000, there exist infinitely many real quadratic fields k such that Xp(k) = pp(k) = v,(k) = 0 and p does not split in k. In fact, he succeeded in estimating the number of real quadratic fields k with discriminant d(k) less than a given positive integer, such that its class number is not divisible by p, the prime p

ramifies in k, and up ( ~ , ( k ) / m ) = 0.

Remark 5.11. We do not know any infinite families of real quadratic fields k with X,(k) = 0 and vp(k) # 0, for a give prime number P 2 3.

On p-Adic Zeta Functions and Class Groups of Zp-Extensions 413

Remark 5.12. We do not know any infinite families of real quadratic fields k such that Xp(k) = 0 and the prank of the ideal class group of k is (arbitrarily) large, for a give prime number p 2.

References

[Br67] A. Brumer, On the units of algebraic number fields, Mathematika, 14 (1967), 121-124.

[Coa77] J. Coates, p-adzc L-functions and Iwasawa's theory, in Algebraic Number Fields, Durham Symposium, 1975, ed. by A. Frohlich, Academic Press, 1977, 269-353.

[Co188] P. Colmez, Rbzdu en s = 1 des functions z2ta p-adzques, Invent. Math., 91 (1988), 371-389.

[DH71] H. Davenport and H. Heilbronn, On the denszty of discriminants of cubic fields 11, Proc. Roy. Soc. London Ser. A, 322 (1971), 405-422.

[FW79] B. Ferrero and L. C. Washington, The Iwasawa invariant pp vanishes for abelian number fields, Ann. of Math., 109 (1979), 377-395.

[FK86a] T . Fukuda and K. Komatsu, On the X invamants ofZp-extensions of real quadratic fields, J. Number Theory, 23 (1986), 238-242.

[FK86b] T . Fukuda and K. Komatsu, On Zp-extenszons of real quadratzc fields, )

J . Math. Soc. Japan, 38 (1986), 95-102. + k t [Gr76] R. Greenberg, On the Iwasawa invariants of totally real number fields,

Amer. J. Math., 98 (1976), 263-284. [IS96-71 H. Ichimura and H. Sumida, On the Iwasawa A-invariants of certain

real abelian fields, Tbhoku Math. J., 49 (1997), 203-215; part 11: International J. Math., 7 (1996), 721-744.

[In891 A. Inatomi, On Zp-extensions of real abelian fields, Kodai. Math. J., 12 (1989), 420-422.

[Iw56] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258.

[Iw59] K. Iwasawa, On I'-extensions of algebrazc number fields, Bull. Amer. Math. Soc., 65 (1959), 183-226.

[Iw70] K. Iwasawa, On some infinite abelian extenszons of algebmic number fields, Actes, Congres. Intern. Math. (Nice, 1970), Tome 1, 391- 394.

[Iw73a] K. Iwasawa, On Zl-extensions of algebmic number fields, Ann. of Math., 98 (1973), 246-326.

[Iw73b] K. Iwasawa, On the p-znvanants of Zl-extensions, Number theory, Algebraic Geometry and Commutative algebra (in honor of Y. Ak- izuki), ed. by Y. Kusunoki, et al., Kinokuniya, Tokyo, 1973, 1-11.

[Kid801 Y. Kida, 1-extensions of CM-fields and cyclotomic znvariants, J. Number Theory, 12 (1980), 519-528.

[Kid821 Y. Kida, Cyclotomic Zz-extensions of J-fields, J. Number Theory, 14 (1982), 340-352.

414 H. Taya

[Kim951 J. M. Kim, Class numbers of certain real abelian fields, Acta Arith., 72 (1995), 335-345.

[MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179-330.

[NH88] J. Nakagawa and K. Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc., 104 (1988), 20-24.

[On991 K. Ono, Indivisibility of class numbers of real quadratic fields, Com- positio Math., 119 (1999), 1-11.

[0z97] M. Ozaki, The class group of Z,-extensions over totally real number fields, Tohoku Math. J., 49 (1997), 431-435.

[0z98] M. Ozaki, On Greenberg's conjecture (Japanese), Siirikaisekikenkyii- sho K6kyuroku, 1026 (1998), Res. Inst. Math. Sci. Kyoto Univ. 20-27.

[OT97] M. Ozaki and H. Taya, On the Iwasawa A:!-invariants of certain families of real quadratic fields, Manuscripta Math., 94 (1997), 437- 444.

[Taggal H. Taya, On p-adic zeta functions and 25,-extensions of certain totally real number fields, Tohoku Math. J., 51 (1999), 21-33.

[Ta99b] H. Taya, On p-adic L-functions and 25,-extensions of certain real abelian number fields, J. Number Theory, 75 (1999), 170-184.

[TaOO] H. Taya, Iwasawa invariants and class numbers of quadratic fields for the prime 3, Proc. Amer. Math. Soc., 128 (2000), 1285-1292.

[Wag71 L. C. Washington, Introduction to Cyclotomic Fields, Second edition, Graduate Texts in Math. Vol. 83. Springer-Verlag, New York, Hei- delberg, Berlin 1997.

Added in proof. Concerning Problem 5.1, D. Byeon recently showed by refining Ono's ideal that for a given prime number p 2 5, there exist infinitely many real quadratic fields k such that A,(k) = 0 and p splits in k. For the details, see his paper "Indivisibility of class numbers and Iwasawa A-invariants of real quadratic fields", which is to appear in Compositio Math.

Graduate School of Information Sciences Toholm University Katahira, Sendai, 980-8577, Japan E-mail address: tayahath. is .tohoku.ac. jp


Class Numbers of Imaginary Quadratic Fields

Winfried Kohnen

1 Introduction

Starting with Gauss, class numbers of quadratic fields always have been very interesting and quite mysterious objects. Here we would like to survey some more recent results concerning indivisibility of class numbers of imaginary quadratic fields by prime numbers. For more details we refer the reader to [I].

In the following, we denote by D < 0 the discriminant of an imaginary quadratic field. We let h (D) be the class number, i.e. the order of

the class group CL(D) of ~ (a ) . We want t o study the question "how often" h (D) is not divisible by

a given prime number 1.

§2. Classical results

Theorem (Gauss). Let t be the number of different prime divisors of D. Then h(D) is odd if and only if t = 1.

In fact, by genus theory one has CL(D) /cL(D)~ " (Z/2Z)t-1 . Now use the structure theorem for finite abelian groups.

Theorem (Hartung, 1974). Let 1 be an odd prime. Then there ezist infinitely many D < 0 such that h(D) f 0 (mod 1).

Let us sketch the proof. For a natural number N with N = 0 , 3 (mod 4) let H ( N ) be the Hurwitz-Kronecker class number, i. e. the class number of positive definite binary quadratic forms of discriminant -N where each class C is counted with multiplicity &. If - N = D f with f E N and D a fundamental discriminant, then


416 W. Kohnen

where w (D) is half the number of units of ~ (a ) and ol (n) = xdln d. By the Hurwitz-Kronecker class number relation one has

n x H(4n - x2) = x max{d, ;i} (n not a perfect square).

Now choose n to be a prime different from 1 such that n = 7 (mod 8) and n is not congruent to a perfect square modulo q for all odd primes q < P, where P is an arbitrary large number. We then conclude that there is x with x2 < 4n and H(4n - x2) $ 0 (mod 1). Writing 4n - x2 = - D f we see that also h(D) is not divisible by 1, and from the conditions posed on n one must have ID] > P . This concludes the proof.

$3. More recent results

Theorem (Horie, 1990). Let el, . . , en E {f 1,O) and ll , - - . , 1, be diflerent odd primes. Then for all primes 1 large enough there exist infinitely many D < 0 such that

D h(D) $ 0 (modl) , (-) = E , ( u = l , . . . , n ) .

1"

The proof uses the theory of modular forms and the trace formula for Hecke operators.

Horie's theorem in general is not effective, and various refinements concerning effectiveness using the theory of modular forms of half- integral weight modulo 1 have recently been given, including works by Jochnowitz (1997) and most recently Bruinier (1997198). Bruinier's results assert that the statement in Horie's theorem is true whenever I does not divide 11~=11,(1, + 1)(1, - 1).

However, these methods do not seem to give any reasonable lower bound for the number of D with -x < D < 0 and h(D) f 0 (mod I) for large x. On the other hand, there is the following

Conjecture (Cohen-Lenstra, 1983). Let 1 be an odd prime. Then the probability that 1 does not divide h(D) is IIE=l(l - 1-").

One knows that {D < 0 I h(D) $ 0 (mod 3)) has a positive probability (Davenport-Heilbronn, 1971 and Nakagawa-Horie, 1988). For primes 1 > 3, however, nothing is known.

Theorem (Kohnen-K. Ono, 1997). Let 1 > 3 be a prime. Then the following assertions hold.

Class Numbers of Imaginary Quadratic Fields 417

i) Let p be any prime with p f (2) (mod 1). Then there exists

dp E N with dp < a(p + 1) such that D := -pdp or D := -4pdp is a fundamental discriminant and 1 does not divide h(D).

ii) Let E > 0. Then

2(1 - 2) #{-x < D < 0 I h(D) +! 0 (mod 1)) 2 ( fi

- -(x >>c 0). ( 1 - 1 E, log x

The proof of i) uses properties of certain Hecke operators on spaces of modular forms of half-integral weight modulo 1 together with a theorem of Sturm which gives a bound on the dimension of spaces of modular forms reduced modulo 1. The assertion of ii) follows from i) by using the prime number theorem. For details we refer to [I].

References

[l] Kohnen, W. and Ono, K., Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math., 135 (1999), 387-398.

Mathernatisches Institut Universitat Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail address: winf r i edbath i . mi-heidelberg . de


On Parities of Relative Class Numbers of certain CM-Extensions

Ryotaro Okazaki

Abstract.

Let k / F and kl /F be CM-extensions, K = kk' and K+ the maximal totally real subfield of K. It holds h-(k) ( h-(K) if k / F is unramified at all finite primes or K + / F is unramified. Hence, h-(k) is an obstacle that prevents h-(K) from being 1. On the other hand, analytic class number formula implies the class number relation h-(K) = h-(k)h-(kl)/c(K/F), where c(K/F) is an integer determined by units. Consistency of the first assertion and the class number relation is guaranteed by h-(kl). As a reason of the consistency, the parity equality h-(k) = h-(kl) (mod 2) is formulated under the situation of the first assertion. (See Theorems 1 and 2.) Non-trivial Examples (ss3.2) and a proof of the parity equality are given. The tool behind the first assertion and indices related with

2 & the class number relation are discussed in detail.

1 Introduction

A finite extension of Q is called a number field. The number h(L) of ideal classes of a number field L is called the class number of L. When k is a totally imaginary quadratic extension of a totally real number field F, the extension k / F is called a CM-extension and k is called a CM- field. The subfield F is identified as the maximal totally real subfield of k, which is denoted by k+. Hence, the ratio h- (k ) = h(k ) /h (F) is determined by the CM-field k. We call h- ( k ) the relative class number of a CM-field k.

An important property of relative class numbers comes from class field theory. We denote by K ( L ) the Hilbert class field of a number field L, i.e., the maximal unramified abelian extension of L. Then, we have h(L) = [ X ( L ) : L] by class field theory. Let k be a CM-field. Since x ( k + ) / k + is a totally real normal extension and since the quadratic

I

i Received September 30, 1998.

420 R. Okazaki Parities of Relative Class Numbers of CM-Extensions 42 1

extension k/k+ is ramified at an infinite prime, we have [kX(k+) : k] = [X(k+) : k+]. Since X(k+)/k+ is an unramified abelian extension, so is kX(k+)/k. Hence, kX(k+) c X(k) holds. The mentioned equality of degrees now implies

(1) h-(k) = [X(k) : kX(k+)].

In particular, h-(k) is an integer. We shall prove the following two theorems on parity equality:

Theorem 1. Let k / F and k l /F be CM-extensions. Assume that k / F and kl /F are unramified at all finite primes. Then, the following parity equality holds:

h (k) h ( I ) (mod 2).

Theorem 2. Let k / F and kl /F be CM-extensions. Set K = kk'. Assume that K+/F is unmmified. Then, the following equivalence holds:

h (k) h ( I ) (mod 2).

As examples of '$53.2 will show, the interesting cases indeed exist. In some cases, exponents of 2 in h-(k) and h-(k') are equal. (See Ex- amples 35 and 37.) In some cases, they are different. (See Examples 28, 29 and 38.)

The background of these Theorems is a problem of divisibility of relative class numbers in relation with inclusion of CM-fields, i.e., a problem of obstacle for class number one. Our Theorems 1 and 2 came up from investigation in the tools for approaching that problem.

Let k and K be CM-fields satisfying k c K. Hasse proved that h-(k) divides h-(K) multiplied by 2 to a suitable exponent when K/Q is abelian [I]. (Dependence of the exponent on K and k was mysterious.) Hirabayashi and Yoshino calculated relative class numbers of imaginary abelian number fields whose conductors are less than or equal to 200 and placed them in Hasse's diagrams [2, 31. In the diagrams, Horie observed h- (k) 1 4 h- (K). He proved his observation true under the assumption that K/Q is abelian [5, Theorem 11. He also gave an example of a pair k = Q( J-4 -3. -7) and K = Q ( a , G, fl) with h-(k) = 4, h- (K) = 1 and h- (K)/h- (k) = 114, which illustrates that the assertion of the following Theorem 3 is best possible.

In [ll], we proved a generalization of his theorem:

Theorem 3. Let k c K be CM-fields. Then, h-(k) divides 4 h-(K).

Hence, hT(k)/4 is an obstacle that prevents h-(K) from being 1. The basic ideas of proofs of [5] and [ll] are totally different. Horie

used well-known factorization of relative class number of imaginary abelian number fields into product of generalized Bernoulli numbers, and regrouped factors for proving his theorem. However, we do not even assume K/k to be normal in Theorem 3. Therefore, a completely different method is required for proving it. We combined three algebraic tools, which are listed below:

1. Group Theoretic Tool; 2. Field Theoretic Tool; 3. Class Number Relation.

We shall briefly explain the three tools, and define the strategy of this paper.

Group Theoretic Tool We denote by e(L) the ideal class group of a number field L. When k is a CM-field, we denote by LI, : e(k+) -+ e(k) the natural lift of ideal classes. We have

~ ( k ) h-(k) = # coker LI,

where ~ ( k ) denotes the order of ker LI, and #A denotes the cardinality of a given set A. As we shall review in Lemma 17, ~ ( k ) is either 1 or 2. Hence, coker ~k has much information on hV(k).

Exploiting information from coker LI,, we proved several statements. One of such statements is on a delicate relation on h- (k) with a property of k+:

Lemma 4 (Lemma 26 of [Ill). Let k be a CM-field and r the 2- rank of the strict class group of k+. Set u(k) = 2 if k/k+ is unramified at all finite primes and u(k) = 1 otherwise. Then, 2'1 ~ ( k ) u ( k ) i s an integer, and divides h- (k) .

Let r' be the 2-rank of the class group of k+. Then, 2'( divides 2'/u(k). Hence, Lemma 4 implies the well-known divisibility: 2'(/ ~ ( k ) I h- (k). (See [14, Theorem 10.121 .) However, Lemma 4 turned out useful for dealing with combined structure of unit group and class group.

If K is a CM-field satisfying k c K, the norm map N : coker LK 4

coker ~k is well-defined. If the order of the cokernel of N is either 1 or 2, it follows easily that h-(k) divides 4 h-(K). Investigating the structure of the cokernel of N , we obtained a nicer statement:

Proposition 5 (Corollary 28 of [ll]). Let k c K be CM-fields. Assume K does not contain a bicyclic biquadmtic extension of k. Then, h- (k) divides 4 h- ( K ) .

422 R. Okazaki

Field Theoretic Tool When k is a CM-field, we denote by 3C0(k) the maximal CM-field that is contained in 3C(k). We have the following:

Lemma 6. Let k be a CM-field. Then, XO(k) i s well-defined and the extension X0 (k)/k+ is abelian.

An illustration of a relation of this concept with relative class numbers is the following:

Lemma 7. Let k c K be CM-fields. Then, the degree [X(k) : Xo(k)] divides h- (K) . Therefore, h- (k) divides [3C0(k) : kX(k+)] h-(K).

Extending this idea, we proved the following two Propositions: (See Propositions 22 and 23 of [Ill.)

Proposition 8. Let k C K be CM-fields. Assume k/k+ is unramified at all finite primes. Then, h- (k) divides h- (K) .

Proposition 9. Let k c K be CM-fields. Assume K+/k+ is unramified. Then, h - (k) divides h- (K) .

Relation of unramifiedness and divisibility of relative class numbers will be examined in more detail in 552.1.

Class Number Relation When L is a number field, W ( L ) denotes the group of roots of unity of L, w(L) the order of W (L) and E(L) the group of units of L. When k is a CM-field, Q(k) denotes Hasse's unit index [E(k) : W(k)E(k+)]. Property of Q(k) and the aforementioned index ~ ( k ) shall be reviewed in 552.2, in which the notion of Vibte ideal will be recognized.

Let k c K be CM-fields and assume that Klk is quadratic. Set F = k+. Then, K+/F is also quadratic. Hence, K / F is bicyclic biquadratic. Therefore, KIF contains a CM-extension k l /F other than k/F. Analytic class number formula implies the following class number relation:

where

In the current paper, we call c(K/F) the denominator constant of class number relation for K/F . It is known that the fist factor in the right hand side of (3) is either 1 or 2. (See Lemma 23.) It is also known that

Parities of Relative Class Numbers of CM-Extensions 423

a Hasse's unit index is also either 1 or 2. (See Lemma 17.) Hence, it looks as if c(K/F) can take all values from $ , I , 2,4,8.

Lemmermeyer gave a new formula for c(K/F) in [81:

where v = 1 if the both of k and k' are obtained by adjoining to F square roots of units in F or v = 0 otherwise. This formula and the original formula imply c(K/F) # 8. Indeed, we shall see

in Lemma 24. One consequence is the following:

Proposition 10. Let k c K be CM-fields. Assume that Klk i s quadratic. Then, h- (k) divides 4 h- (K) .

The class number relation was also used in the critical part of our proof for Theorem 3.

Cooperation of Tools We illustrate importance and cooperation of the three tools explained by giving a sketch of a proof of Theorem 3, in which the three tools cooperate covering different areas.

Let k c K be CM-fields. Let M be a maximal intermediate field of Klk such that hP(k) divides h- (M). It suffice to show h-(M) I 4 h-(K).

The first step is an application of the group theoretic tool. If K I M does not contain a bicyclic biquadratic extension of M , Proposition 5 implies h-(M) 1 4 h-(K). We turn to the other case: we assume that K I M contains a bicyclic biquadratic extension of M.

The second step is the most involved one. It is a search for a quadratic extension M' of M in K such that h-(k) divides h-(M'). Tools for the search are Proposition 9 and the class number relation (2) together with (5). Lemma 4 is also applied to avoid certain obstacle coming from delicate structure of the strict class group of M+. It was shown that the search will be successful unless the following three conditions hold:

(a) The strict class number of M+ is odd; (b) The CM-field K contains three CM-extensions MI /M+ , M2/M+,

M3/M+ of prime power conductor over M+; ( c ) And M c MI M2 M3.

On the other hand, the choice of M implies that the search cannot be successful. Hence conditions (a), (b) and (c) hold. Then, the genus

424 R. Okazaki

theory, possibly combined with Lemma 7, applied to M/M+ implies the desired result.

Competition of Tools We have seen that the field theoretic tool and the class number relation play important roles in an interesting problem of divisibility of relative class numbers. In certain situation, these tools can be used independently to study divisibility in the same pair of relative class numbers. A delicate competition of the two tools, which arises in such a situation, motivated the current work.

Let k / F and k'/F be distinct CM-extensions. Set K = kk'. The class number relation (2) is equivalent to

h-(K) - h-(k') -

h- (k) c(K/ F) '

1 Assertion (5) implies that the ratio in the right hand side lies in -Z.

4 This shows the ratio in the left hand side to be an integer when h-(kt) is a multiple of 4 or c(K/F) is 1. However, Examples 11, 12, 13 and 14 show that the ratio may not be an integer in some cases.

On the other hand, Propositions 8 and 9 assert

under certain situation. The class number relation and the field theoretic tool with their

own coverages are competing. Their own coverages intersect at a small area of Integrality (7). It looks as if the two tools contradict in the intersection: (6) suggest a non-trivial denominator while (7) exclude a non-trivial denominator. An effort, i.e., Theorems 1 and 2, to fill in the seeming gap is the current work.

Strategy As our target is comparison of the field theoretic tool and the class number relation, we need a review of the two tools. Our interest on the latter is in the denominator constant. We shall discuss properties and complicatedness of the denominator constant. For this purpose, we need a review of properties of indices related with CM-fields, in which we shall recognize the notion of Vi6te ideals. We shall put them in the bottom-up order in 52: we shall review the field theoretic tool in 552.1; indices in 552.2; and the denominator constant in $52.3. (Note that some part of 552.2 are added for interpreting Examples.) In 53, we shall give an answer to the problem of the previous paragraph: we shall identify the problem with a Suspicion in $53.1 and formulate the Suspicion as


the parity equality of Theorems 1 and 2; we give non-trivial example of the problem in $53.2; and prove Theorems 1 and 2 in 553.3.

52. Detail of the Competing Tools

We shall review detail of the two competing tools (the field theoretic tool and the class number relation) out of the aforementioned three tools. We shall firstly review the field theoretic tools in 552.1. We shall secondly review indices related with CM-extensions in 552.2, in which the notion of Vi6te ideal shall be proposed. We will lastly apply 552.2 to the denominator constant of class number relation in 552.3.

2.1. Field Theoretic Tool We shall prove Lemmas 6, 7, and Propositions 8 and 9 for illustration

of the concept of X O ( ~ ) . We shall also give examples that illustrate relation of unramifiedness and divisibility of relative class numbers. We shall lastly illustrate another competition of tools (the group theoretic tool and the field theoretic tool) by giving a proof of a well-known lemma through a use of XO(L).

Let L be a CM-field. Then, the non-trivial conjugation of L/L+ is called the complex conjugation of L.

Proof of Lemma 6. Let k be a CM-field and M the maximal totally real subfield of ?C(k). Then, it is obvious that kM is a CM-field. We show that kM is the maximal CM-field that is contained in X(k), i.e., XO(k) = kM.

Let L be a CM-field contained in X(k). Choose 6 E k such that k = k + ( G ) and 6' E L such that L = ~ + ( m ) . Then, 6 and 6' are totally positive. On the other hand, k+, L+ c M follows from the choice of M. Hence, we have 46 ' E M. Hence, ~ ( m ) is totally real. On the other hand, M(&%) C X(k) follows from k, L C X(k). The maximality of M now implies M (m) = M, i.e., 66' E ( M )2. Hence, L M = kM holds. We get L c kM as desired.

Since the first assertion of the lemma is established, we show the second assertion. By class field theory, normality of k/k+ implies normality of X(k)/k+. Hence, the choice of M implies normality of M/k+. On the other hand, M/k+ is disjoint with k/k+. Therefore, we get an isomorphism Gal(kM/k) e Gal(M/k+). The left hand side is abelian by the choice of M. Hence, the right hand side is also abelian. By corn-

! posing abelian extensions, we get an abelian extension kM/k+. Since

1 ?CO(k) is constructed as kM, we now conclude Ho(k)/k+ is abelian.

426 R. Okazaki Parities of Relative Class Numbers of CM-Extensions 427

Step 1. (Composition): By class field theory, normality of k/k+ implies normality of X(k)/k+. Noting that k+ c X(K+), we get normality of

Step 2. (Galois Action): Let a be the complex conjugation of KX(K+). Since k c K is totally imaginary, comparison of degrees implies K X(K+) = kX(K+), and hence KX(K+) C 3C(k)X(K+). Therefore, Step 1 implies that a extends to an element a' of Gal(X(k)X(K+) / X W + 1). Step 3. (Intersection): By the choice of a', we see that a' preserves

KX(K+). On the other hand, a' belongs to Aut(X(k)X(K+)/k+). Noting that X(k)/k+ is normal, we see that a' preserves X(k). There- fore, a' preserves the intersection X(k) n KK(K+). We denote by a" E Gal(X(k) n KX(K+)/k+) the restriction of a' to X(k) n KX(K+). Then, a" turns out to be a restriction of a . Hence, we get all2 = 1. On the other hand, X(k) n KX(K+) contains k, on which a acts non- trivially. Therefore, a" # 1 holds. Let M be the fixed field of a". Then, X(k) n KX(K+)/M is quadratic. We get X(k) n KX(K+) = kM. Since k is totally imaginary and M , which is fixed by a, is totally real, this identity implies that X(k) n KX(K+) is a CM-field. By definition of XO(k), we now get X(k) n KX(K+) c XO(k). Step 4. (Tower): By class field theory and Step 3, we get the following tower:

We see [X(k) : XO(k)] divides [X(k) : KX(K+) n X(k)]. The latter degree equals [KX(K+)X(k) : KX(K+)], which divides [X(K) : KX(K+)]. Recalling (I), we get [X(k) : Xo(k)] ( h- (K), which is the first assertion of the Lemma. Recalling (1) again, we get h-(k) = [X(k) : XO(k)] [XO(k) : kX(k+)] I [XO(k) : kX(k+)] hP(K).

Proof of Proposition 8. Since k/k+ is unramified at all finite primes, class field theory implies that X(k)/k+ is unramified at all finite primes. Hence, Xo(k)+/k+ is unramified. By Lemma 6, we see XO(k)+ c %(I;+). We now have XO(k) = kXo(k)+ c kX(k+). Since the reverse inclusion is already shown, we get kX(k+) = 3C0(k). Now, Lemma 7 implies the proposition. 0

Proof of Proposition 9. We use the following fact that is shown in Step 4 of the proof for Lemma 7: [X(k) : KX(K+) n X(k)] I h-(K). We also use the tower shown in the same step.

Let M = (KX(K+) n X(k))+. Then, M c X(K+) follows. Since K+/k+ is unramified, so is X(K+)/k+. Therefore, M/k+ is also unramified. On the other hand, we have M/k+ c XO(k). By Lemma 6, we see MIL+ is abelian. Therefore, we get M c X(k+) and hence K3C(K+) nX(k) c k3C(k+). Since the reverse inclusion is already shown in the tower, we get KX(K+) n X(k) = kX(k+). Now, the divisibility at the beginning of this proof implies that h-(k) = [X(k) : kX(k+)] divides h- (K) 0

Examples in $53.2 will illustrate the truth of Proposition 8 and Proposition 9, i.e., h-(k) I h-(K) holds when k/k+ is unramified at all finite primes or when K+/k+ is unramified. In some cases, however, h-(k) fail to divide h-(K) while one or two of K/k and K/K+ are unramified at all finite primes:

Example 11. Let k = Q ( J m ) and K = k(J--;?). Then, h-(k) = 2 and h-(K) = 1 hold. Hence, h-(K)/h-(k) = 112 4 Z. Note that K/k is unramified.

Example 12. Let k = Q(J-4. -3. -7) and K = k ( G ) . Then, h-(k) = 4 and h- (K) = 2 hold. Hence, h- (K)/h- (k) = 112 6 Z. Note that K/k is unramified.

Example 13. Let k = Q( J-4 -3 - -7) and K = k ( G ) . Then, h-(k) = 4 and h-(K) = 2 hold. Hence, h-(K)/h-(k) = 112 4 Z. Note that K/K+ is unramified at all finite primes. (K+ = Q

Example 14. Let F = 0 ) . Then, h(F) = 1. Let k = F ( G ) and k' = F ( a ) . Then, we have w(k) = 2 3, w(kl) = 4, Q(k) = 1, Q(kl) = 2, ~ ( k ) = ~ ( k ' ) = 1, h-(k) = 2 and hP(k') = 1. Here, k / F is ramified above (2) and k'/F is unramified at all finite primes. Set K = kk'. Then, we have K+ = Q ( d n , d n ) , h(K+) = 1, w(K) = 4 . 3 , Q(K) = 2, K(K) = 1, c(K/F) = 2 and h-(K) = 1. Hence, h-(K)/h-(k) = 112 4 Z. Note that K/k is unramzfied and K/K+ is unramified at all finite primes.

In Examples 11, 12 and 14, XO(k) n kX(K+)/k is quadratic. This fact and the field theoretic method just explained imply h-(k) I 2h- (K). (See Step 4 of the proof for Lemma 7.) However, XO(k) n kX(K+)/k is quartic in Example 13. Hence, the field theoretic method only explains h-(k) 14 h-(K).

We denote by E+(F) the group of totally positive units of a number field F. The following Lemma is well-known:


Lemma 15. Let k / F be a CM-extension. Let r" be the 2-rank of E + ( F ) / E ( F ) 2 . Assume that k / F is ramified at some finite prime. Then, 2'" divides h- ( k ) .

Remark. This Lemma can be deduced from Lemma 4. For illustration of another competition of tools, we review an alternative proof.

Proof of Lemma 15. Let H be the maximal abelian extension of F that is unramified at all finite primes. Then, kH c X ( k ) holds. Therefore, [kH : k X ( F ) ] divides h- (k ) = [%(It) : k X ( F ) ] . It suffices to show [kH : k X ( F ) ] = zr". The quadratic extension k / F is disjoint with H/F since k /F is ramified at some prime. Further, H / F is normal. Therefore, we get [kH : k] = [H : F ] . Similarly, we get [ k X ( F ) : k] = [ X ( F ) : F ] . Hence, we have [kH : k X ( F ) ] = [H : X ( F ) ] . By class field theory, we have [H : X ( F ) ] = 2"'. 0

2.2. Indices Related with CM-Extensions This subsection will be devoted to a review of the standard the-

ory on Hasse's unit indices of CM-fields and capitulation kernel of CM- extensions since those objects are basic obstacles closely related with the denominator constants. There are nice descriptions in [ l ] and [9] . We follow the latter reference. We also propose to recognize the notion of Vikte ideal. The result shall be used for controlling the denominator constant of class number relation. It is also used for interpreting Examples.

Hasse's unit indices of CM-fields and capitulation in CM-extensions are information on certain delicate structure of ambiguous ideals:

Definition 16. Let k / F be a CM-extension. A n ideal of k is called ambiguous i f it is invariant under Gal(k /F) . The group of generators of ambiguous principal ideals is denoted by A(k) . The group of elements of k that generate ideals of F is denoted by P ( k / F ) .

We are interested in the order ~ ( k ) of the kernel of the natural lift ~k : e ( F ) -+ e ( k ) of ideal classes. It is obvious that ~ ( k ) equals the index [ P ( k / F ) : F X E ( k ) ] . We now characterize indices Q ( k ) and tck:

Lemma 17. Let k / F be a CM-field and a the complex conjugation of k. Then, we have the following inclusions:

Moreover, Hasse's unit index Q ( k ) = [E(k ) : W ( k ) E ( F ) ] and ~ ( k ) = [ P ( k / F ) : F E(k)] are characterized by

Therefore, Q ( k ) ~ ( k ) divides 2. W e also have

Proof. Let NkIF be the norm map from k X to F X and o(k) the ring of integers of k. We firstly recall Kronecker's characterization of W ( k ) :

W ( k ) = o(k) n ker Nk/ F.

The inclusion W ( k ) c o(k) n ker NkIF is obvious. The reverse inclusion is the heart of Kronecker's Theorem: an algebraic integer a is necessarily a root of unity if all archimedean valuation of powers of a are uniformly bounded.

We secondly prove the first assertion. The mentioned characterization of W ( k ) implies the first identity. The three inclusions follow from W ( k ) C E(k) C P ( k / F ) c A(k) . The last identity is proven by verification of inclusions in the both directions: The group A(k)'-" obviously lies in ker NklF. It also lies in E(k) C o(k) . Hence, we get A(k) lFQ c W ( k ) by the characterization of W ( k ) . We turn to the reverse inclusion. By Hilbert 90 and the characterization of W ( k ) , any E E W ( k ) is of the form for some a E k. We get a" = [- 'a and hence a E A(k) . It follows < E A(k)'-a. Now, we see W ( k ) c A(k) '-".

We thirdly prove the second assertion. By the first assertion, the map 1 - a induces a homomorphism

of quotients. Let q E ~ ( k ) satisfy qW ( k ) E ( F ) E ker i.e., q'-u =

5l-0 for some < E W ( k ) . Then, q /< is invariant under a and hence lies in E ( F ) . It follows q E W (k) E ( F ) . Therefore, #E(k) is injective. Noting that I m h ( k ) = ~ ( k ) ' - ~ / W ( k ) ' - ~ , we get the first identity.

By the first assertion, the map 1 - o induces a homomorphism

of quotients. Let a E P ( k / F ) satisfy a F X E ( k ) E k e ~ + p ( ~ / F ) , i.e., c r l - ~ = ql-" for some q E E(k) . Then, a / q is invariant under a and

hence lies in F X . It follows a E F X E(k ) . Therefore, 4 p ( k I F ) is injective. Noting that Im4P(kIF) = P ( k / F ) l - a / E ( k ) l - u , we get the second identity.

We nextly deduce the third assertion from the first two assertions.


We lastly prove the last assertion. By the first assertion, the map 1 - a induces a surjective homomorphism

- of quotients. Let a E A(k) satisfy a P ( k / F ) E ker4A(*), i.e., a'-" - yl-" for some y E P(k/F) . Then, a / y is invariant under a and hence lies in F . It follows a E P(k /F) . Therefore, 4P(kIF) is an injection and hence is an isomorphism. Identity [A(k) : P(k/F)] = [A(k)'-" : P(k/F)'-"1 follows. The first two assertions and this identity imply the last assertion. 0

Lemma 18. Let k C K be CM-fields. Then we have w(k) I w(K),

w(k>Q(k> I w(K>Q(K> and w ( k ) Q W ( W I w ( K ) Q ( K ) 4 0

Proof. The first assertion is obvious. The second and the third assertions are proven in a similar way. Hence, we give a proof for the third assertion.

Since w(k) = 2 [W(k)'-" : 11, the second assertion of Lemma 17 implies ~(k)Q(k)w(k) = 2 [~(k/k+) ' -" : 11. Similarly, we can obtain K(K)Q(K) w(K) = 2 [P(K/K+)'-" : 11. On the other hand, we obviously have P(k/k+) c P(K/K+). Hence, [P(k/k+)'-" : 11 divides [P(K/K+) -" : 11. The desired assertion follows immediately. 0

Remark. If w(K)/w(lc) is odd, the latter two divisibilities of Lemma 17 imply Q(k) I Q(K) and Q ( k ) ~ ( k ) 1 Q(K)K(K). If further we have Q(k) = 2, we get Q(K) = 2. Hence, we can sometimes calculate a Hasse's unit index of a CM-field through calculation of a Hasse's unit index of a smaller CM-field. However, Q(K)/Q(k) = 112 sometimes happens when w(K)/w(k) is even. Lenstra's example in the preface to 1985-edition of [l] (see also [9]) is

Example 19. Let F = ~(J8.17). Then, h(F) = 2. Let k =

F(-) and k' = F(J-8). Then, we have w(k) = 4, w(k') = Q(k) = Q(kl) = 2, ~ ( k ) = ~ ( k ' ) = 1 and h-(k) = h-(kl) = 4. Set K = kk'. W e have K+ = Q(J8, m), h(K+) = 1. (K+/F is unramified.) W e have w(K) = 8,Q(K) = 1, K(K) = 1, r ( K / F ) = 2, c(K/F) = 4 and h-(K) = 4. Therefore, h-(k) and h-(k') divide h-(K). Note that Q(K)/Q(k) = Q(K)/Q(k1) = 112 holds.

See Hirabayashi and Yoshino [4] for further discussion and examples. Determination of indices Q(k) and ~ ( k ) is relatively easy if k does

not contain G . However, it becomes delicate if k contains p . Therefore, we prepare a tool for dealing with CM-fields which contain p .

Definition 20. W e define Vie'te numbers Vo, V2, V3, . . . by

The Vie'te index I of a number field L is the maximal index i such that V , E L. The ideal V = (VI) is called the Vie'te ideal of L.

Remark. ViCte7s historical formula for 7r is

- 7r - - gg/'Ig.. i with square roots taken in positive real numbers. (See e.g. [7, p. 2511 .)

' I , t 4 Vikte numbers are algebraic integers. We see that Q(V,, G ) is the %,, 2*2-th cyclotomic field. Or more precisely, V, = (1 + C)(1 + C-') holds $+ $$ for some 2if 2-th root C of unity. The ViCte ideal V is characterized by t y V = ((1 + C) (1 + C-')) for a generator C of the 2-part of w ( L ( G ) ) .

With notion of ViCte ideals, we determine Q(k) and ~ ( k ) of CM- fields k:

Lemma 21. Let k / F be a CM-extension. Choose 6 E F such that k = F(-). If k # F(-), indices Q(k) and ~ ( k ) are determined as follows:

Q(k) ~ ( k ) condition;

6) 1 1 if (6) i s not a square of any ideal of F; (ii) 1 2 if (6) i s a square of a non-principal ideal of F;

(iii) 2 1 i f (6) is a square of a principal ideal of F. If k = F(-), indices Q(k) and ~ ( k ) are determined as follows:

Q(k) ~ ( k ) condition; (iv) 1 1 if V is not a square of any ideal of F ;

(v) 1 2 i f V is a square of a non-principal ideal of F; (vi) 2 1 if V is a square of a principal ideal of F.

, where V is the Vie'te ideal of F .

Proof. We denote the complex conjugation of k by a. Case (i): We assume (6) is not a square of any ideal of F . Then, E

A(k) and a 4 P(k /F) hold. By the last statement of Lemma 17, we conclude Q(k) ~ ( k ) = 1. Case (ii): We assume (6) is a square of a non-principal ideal of F . Obviously, ~ ( k ) > 1 follows. By the third assertion of Lemma 17, we conclude ~ ( k ) = 2 and Q(k) = 1. Case (iii): We assume k # F ( G ) and (6) is a square of a principal ideal (P) in F with P E F X . Then, E = SIP2 is a unit in F and

= -113 E k holds. Let < be a generator of W(k) and set 1) =

432 R. Okazaki

C f i . Then, we get ql-" = -C2. Since a $ k, (q1-u)2 = C4 generates the subgroup ~ ( k ) ~ of index 2 in W(k). Since c2 E W(k)2 and -1 $ ~ ( k ) ~ , the unit ql-" = -C2 does not belong to ~ ( k ) ~ . We get [ ~ ( k ) ' - " : ~ ( k ) ~ ] > 1. By the second assertion of Lemma 17, we conclude Q(k) = 2 and ~ ( k ) = 1.

We now assume k = F ( G ) . Let C be the generator of the 2-part of W(k) and t the generator of the odd-part of W(k). Then tl-" = t2 is also a generator of the odd-part of W(k). On the other hand, (1 + <)I-" = C generates the 2-part of W(k). Therefore, a'-" generates W(k) for a = (I+[)[. On the other hand, (a2) = ((l+C)(l+c-l)) = V holds. (Recall comment after Definition 20.) Case (iv): In addition to k = F ( G ) , we assume that V is not a square of any ideal of F . The fact = V and the current assumption imply [A(k) : P(k/F)] > 1. By the last assertion of Lemma 17, we conclude Q(k) = ~ ( k ) = 1. Case (v): In addition to k = F(&i), we assume that V is a square of a non-principal ideal of F . Then, a generates an ideal of F . Hence, we get ~ ( k ) > 1. By the third assertion of Lemma 17, we conclude Q(k) = 1 and ~ ( k ) = 2. Case (vi): In addition to k = F ( G ) , we assume that V is a square of a principal ideal (P) in F with P E F . We see that a lp is a unit in k and (a/P)'-" = a l - O generates W(k). By the second assertion of Lemma 17, we conclude Q(k) = 2 and ~ ( k ) = 1.

We determined Q(k) and ~ ( k ) in all cases. 0

The following lemma is also well-known and is useful for calculation.

Lemma 22. Let k be a CM-field. If E+(k+) = E(k+)2, we have Q(k) = 2. If h(k+) is odd, we have ~ ( k ) = 2.

Proof. Let F = k+. We prove contrapositive of the assertions. Assume Q(k) = 2. Then, Case (iii) or (vi) of Lemma 21 holds:

In Case (iii), there is an element ,B of F and q E E+(F) such that 6 = qP2. Hence, we have F ( f i ) = k # F ( e ) . Hence, we get q E Ef (F) - E(F)2 and hence E+(k+) # E(k+)2 In Case (vi), there is an element p of F and q E E + ( F ) such that V , = qP2, where I denotes the Vibte index of F . By definition of Vibte index, V, is not a square in F . Hence, we get q E E+ ( F ) - E(F)2 and hence E+ (k+) # E (k+)2. A proof of the first assertion completes.

Assume ~ ( k ) = 2. Then, Case (ii) or (v) of Lemma 21 holds. In either case, there is a non-principal ideal of F whose square is principal. Therefore, h(F) is even. A proof of the second assertion completes. 0


Lemmata of this subsection is silently used for calculation of examples through out the current paper.

2.3. Denominator Constant of Class Number Relation We shall prove that the denominator constant of class number rela-

tion belongs to {1,2,4). We also investigate the real delicacy of combi- nation of indices by several examples.

Lemma 23. Let k / F and kl/F be distinct CM-extensions. De- note by i the Vie'te index of F . Set r ( K / F ) = 2 if K = F(&i, a) and r ( K / F ) = 1 otherwise. Then, we have

Moreover, we have

Proof. We firstly reduce the second assertion to the first assertion. We note that k n k' = F implies W(k) n W(kl) = W(F) = {f 1). Therefore, we have W(k) W(kl)/{f 1) -- W(k)/{f 1) x W(kl)/{f 1). In particular, we get 2 #(W (k) W(kl)) = w(k) w(kl). Therefore, the second assertion is reduced to the first assertion.

We now prove the first assertion. Let p be the non-trivial conjugation of K/k and a the complex conjugation of K . Then, pa becomes the non-trivial conjugation of K/kl. We consider the maps $ : Z E W(K) et (Z1+p, Z1+pu) E W(k) x W(kl) and cp : (c, 5') E W(k) x W(kl) - EE' E W(K). Identities E1+P = e2, E1+p" = NkIFE = 1, t1lf = NkrIF[ = 1, and <'l+P" = t12 imply $cp(e, [I) = (t2, El2). On the

other hand, we have cp$(Z) = z2+p(l+") = Z2. Therefore, $ and cp induces isomorphisms between the odd-parts of W (k) x W (k') and W (K) . (They do not necessarily give a pair of inverse isomorphisms.) Since $ factors through W (k) W (k'), we see that the odd-parts of W (k) W (k') and W(K) are identical. Comparison of the 2-part of W(k) W(kl) and W(K) is left.

If a $ K , then the 2-parts of W(k), W(kl) and W(K) are all identical to {f 1). Therefore the 2-part of W(k)W(kl) and W(K) agrees. On the other hand, r ( K / F ) = 1 holds in this situation. Therefore, we get [W(K) : W(k)W(kl)] = 1 = r (K/F) .

We now assume E K. We assume E k without loss of generality. The 2-part of W(k) is generated by a 2i+2-th root of unity. The 2-part of W(kf) is {f 1). Hence, the 2-part of W(k) W(kl) is generated by a 2i+2-th root of unity. Let I be the Vikte index of K+.


Then, the 2-part of W ( K ) is generated by a 21+2-th root of unity. Under our situation, r ( K / F ) = 2'-i. Comparison of the order of 2-parts of W ( k ) W ( k l ) and W ( K ) now implies [ W ( K ) : W(k)W(k l ) ] = r ( K / F ) . Our proof for the first assertion completes. 0

We are now ready to prove the following lemma.

Lemma 24. Let k /F and k l /F be distinct CM-extensions. Then, we have

Moreover, the denominator constant c (K/F) of class number relation satisfy

If not both of k and k' are obtained by adjoining square roots of units in F to F, the denominator constant c (K/F) satisfies

Proof. We firstly reduce the second and the third assertions to the first assertion. By (4) , we have

with v E (0, l) . Hence, the second assertion is reduced to the first assertion. The condition of the third assertion implies v = 0. Hence, the third assertion is reduced to the first assertion.

We now prove the first assertion. We have the following inclusions:

Identity [ E ( K ) : W(k)W(k l )E(K+)] = r (K /F)Q(K) follows. On the other hand, we have the following inclusions:

Therefore, the index [ E ( K ) : E(k)E(kl)E(K+)] divides r (K /F)Q(K) . The assertion of the theorem follows if r ( K / F ) or Q(K) is 1. (Recall that Lemma 17 and 23 imply r ( K / F ) , Q ( K ) E {1,2).)

We now assume r ( K / F ) = Q ( K ) = 2. Let i be the Vi6te index of F. Under the current assumption, K+ = ~ ( f l ) holds by Lemma 23. Therefore, i + 1 is the Vi6te index of K+. By Lemma 21, Q ( K ) = 2 implies that the Vi6te ideal (V,+l) of K+ is a square of a principal ideal

of K+. Choose a E K+ such that (V,+l) = (a)2. Taking norm to F , we get (V,) = (NKtIF&+1) = ( N K + j F a ) 2 . By Lemma 21, we get Q(k) = 2.

Let C be a generator of the %part G of W ( k ) . Set /3 = NKtIF a and y = (1 + C)//3. Then, y E E(k ) holds. On the other hand, P / f l E E(K+) follows from K+ = F ( f l ) . Set C = y . / 3 / f l . Then, E E E(k) E(K+) holds. On the other hand J = (1 + C ) / f l generates the 2-part of W ( K ) . Therefore, W ( K ) E (Kc ) c E(k) W ( k l ) E (K+) follows. Hence, we get W ( K ) E(K+) c E(k) E(k l ) E(K+). Now, we see [ E ( K ) : E(k) E(kl)E(K+)] divides [ E ( K ) : W ( K ) E(K+)] = Q ( K ) = 2. 0

In the proof, the following fact became apparent:

Lemma 25. Let kl/F be a CM-extension other than F(-) . Set k = F ( m ) and K = kk'. Then, the following implications hold.

Here, E+(F) denotes the group of totally positive units of F

When [E+(F) : E(F)2] = Q(k) = Q(kl) = 2, equality Q ( K ) = 2 is possible but not necessary. An example of Q(K) = 1 is Example 19. Two examples of Q(K) = 2 are below:

Example 26. Let F = Q ( J m ) . Then, h(F) = 1. The Vie'te index i of F is 0 and (V,) = (2 + 1/6)~ . Let k = F(-) and k' = ~ ( m ) . Then, we have w(k) = 4, w(kl) = 2 . 3 , Q(k) = Q(kl) = 2, ~ ( k ) = n(kl) = 1, h-(k) = 2 and h-(k') = 1. Here, k /F is ramified above (2) and kl/F is unramified at all finite primes. Set K = kk'. The Vie'te index I of K+ is 1 and (VI ) = ((1 + + W e have K+ = F(&), h(K+) = 1, w(K) = 8 - 3 , Q(K) = 2, n ( K ) = 1, r ( K / F ) = c(K/F) = 2 and h- (K) = 1. Therefore, h-(k') divides h - (K) .

Example 27. Let F = Q( J-8.-7). Then, h (F) = 1. The Vie'te index i of F is 0 and (V,) = (4 + m ) 2 . Let k = F(-) and k1 =

I F ( G ) . Then, we have w(k) = 4, w(kl) = Q(k) = 2, Q(kl) = 2, ~ ( k ) = ~ ( k ' ) = 1, h-(k) = 4 and h-(k') = 1. Here, k /F is ramified

I

above (2) and kl/F is unramzfied at all finite primes. Set K = kk'. W e : have h(K+) = 1. The Vie'te index I of K+ is 1 and (V1) = ( (1 + fi + 1 W e have K+ = F(&), h(K+) = 1, w (K) = 8 , Q ( K ) = 2,

436 R. Okazaki

K ( K ) = 1, r ( K / F ) = c ( K / F ) = 2 and h - ( K ) = 2. Therefore, h-(k ' ) divides h- ( K ) .

In Examples 26 and 27, the Vi&e ideal of K+ is a square of a principal ideal of K+ which is not a lift of any ideal of F . In some cases, the Vi6te ideal of K+ is a square of a lift of an ideal of F :

Example 28. Let F = Q(d-8.-3>, d a ) . Then, we have h ( F ) = 2. The Vie'te index i of F is 0 . W e have (V,) = ( 2 + &)2 =

( 3 + a)2 = (8 + m)2. The prime ideal ( 2 ) ramifies totally in F / Q and the prime ideal of F above ( 2 ) is generated by 2/ (1+ &+ a). Let k = F(-) and k' = F ( J - 8 ) . Then, we have w ( k ) = 4 , w ( k l ) = 2 - 3, Q ( k ) = Q ( k l ) = 2, ~ ( k ) = ~ ( k ' ) = 1, h - ( k ) = 4 and h-(k ' ) = 2. Here, k / F and k l / F are unramified at all finite primes. Set K = kk' . Then, we have K+ = ~ ( f i ) . The Vikte index I of K+ is 1. W e have (VI) =

( 2 + + a)). W e have h ( K + ) = 1, w ( K ) = 8 3, Q ( K ) = 2, K ( K ) = 1, r ( K / F ) = c ( K / F ) = 2 and h - ( K ) = 4. Therefore, h - ( k ) and h - (k ' ) divide h - ( K ) .

Example 29. Let F = Q(d-8.-3, d-). Then, we have h ( F ) = 2. The Vikte index i of F is 0. W e have (V,) = ( 2 + &)2 =

( 3 + Ji)2. However, the prime ideal of Q ( d 8 - -3 . -7) above ( 2 ) is non-principal as it i s verified by use of Legendre symbol. The prime ideal ( 2 ) ramifies totally in F / Q but the prime ideal of F above ( 2 ) is non- principal. (Note that its norm to Q ( J 8 - -3 e -7) is non-principal.) Let k = F(-) and k' = F ( J - 8 ) . Then, we have w ( k ) = 4 , w ( k l ) = 2 3, Q ( k ) = Q ( k l ) = 2, ~ ( k ) = ~ ( k ' ) = 1, h - ( k ) = 2 and h - ( k t ) = 4. Here, k / F and k l / F are unramified at all finite primes. Set K = kk' . Then, we have K+ = ~ ( f i ) . The Vie'te index I of K+ is 1. W e have (V,) = ( ( 1 + f i + = ( ( 1 + fi + f i ) / f i )2 . ( T h e prime ideal of F above ( 2 ) capitulates in K+/F.) W e have h ( K + ) = 1, w ( K ) =

8 3, Q ( K ) = 2, K ( K ) = 1, r ( K / F ) = c ( K / F ) = 2 and h - ( K ) = 4. Therefore, h- ( k ) and h- ( k t ) divide h- ( K ) .

(Proof of h ( K + ) = 1 for these two examples is in [13]. Since K + / F is unramified, this implies h ( F ) = 2 by class field theory.)

We cannot infer r ( K / F ) = 2 from Q ( k ) = Q ( k l ) = 2 and k =

k+ (G) alone, although r ( K / F ) = 2 is possible as Examples 19, 26 and 27 show.

Example 30. Let F = ~ ( f i , J-8.-7). Then, we have h ( F ) = 1. W e also have [ E + ( F ) : E ( F ) 2 ] = 4. The Vie'te index i of F is 1. W e have (V,) = ( ( 1 + a + f i ) / f i )2 . Let k = F(-) and

k' = F (J- ( 3 + Ji) ( 2 + & ) / 2 ) . Then, we have w ( I ) = 8 , w ( k l ) =


Q ( k ) = Q ( k l ) = 2, ~ ( k ) = ~ ( k ' ) = 1, h - ( k ) = 2 and h-(k ' ) = 4. Here, k / F is unmmified at all finite primes and k l / F is ramified above ( 2 ) . Set K = kk' . Then, K + / F is ramified above (2). The Vie'te index I of K+ equals i . W e have h ( K + ) = 1, w ( K ) = 8 , Q ( K ) = 2, K ( K ) = 1, r ( K / F ) = 1, c ( K / F ) = 4 and h - ( K ) = 2. Therefore, h - ( k ) divides h- ( K ) .

Below is slightly difficult part of calculation of Example 30: Data of F : The group E ( F ) is generated by - 1, 1 + fi, (4 + a)/a, ( 3 + Ji ) / f i . Therefore, E + ( F ) is generated by ( 1 + ( 1 + f i ) ( 4 + a)/fi, ( 1 + f i ) ( 3 + f i ) / f i . Hence have [ E + ( F ) : E ( F ) ~ ] = 4.

A quartic subfield of k': Let q = -(3 + f i ) ( 2 + a ) / 2 and 77' # f 77

a conjugate of 77 over ~(a) . Then, we have (rlq')2 = 1. Since f 77' are conjugate integers of 77 over ()(a), we get that 77'' = 1/77 is a conjugate of 77 over ~(a) . It is easy to verify (7 + $"'2 = -4 - a and (17-r111)2 = - 8 - a = (-4+J14)(3+J14)2. Therefore, k' contains

the normal closure of L = Q ( ~ r a ) . Comparing degrees, we see

that k' is the normal closure i f L. Since the maximal abelian subfield of L is ~(a) , we have w ( L ) = 2. Since d- belongs to A ( L ) - P ( L / F ) , the last assertion of Lemma 17 implies Q ( L ) = K ( L ) = 1. We have h - ( L ) = 2. (See 110, p 11431.) Data of k': Since k' is obtained by composing conjugate fields of L , class number relation ( 2 ) and ( 3 ) imply h- (k ' ) = w ( k l ) ~ ( k ' ) h- ( ~ ) ~ / 4 =

w(k1)Q(k ' ) . Since the maximal abelian subfield of k' is Q(&, d m ) , we have w ( k l ) = 2. On the other hand, k l / Q is non-abelian while k / Q is abelian. Hence, k' # k follows. Moreover, 77 is a unit. These two points imply Q ( k l ) = 2 by Lemma 21. We now see h- ( k t ) = 4. Pari (ver. 2.06) confirms h ( k l ) = h ( Q [ x ] / ( x 8 + 1 2 x 6 + 24X4 + 12X2 + I ) ) = 1. Data of K+: Since the maximal abelian subfield of K+ is F , the Vibte index I of K+ equals i. Pari (ver. 2.06) computes h ( K + ) = h ( Q [ x ] / ( x 8 -12X6 + 24X4 - 1 2 x 2 + 1 ) ) = 1. Data of K : The previous assertion implies Q ( K ) = Q ( k ) = 2 and r ( K / F ) = 1. Since the maximal abelian subfield of K is k , we have w ( K ) = w ( k ) = 8. Now, we have enough data to calculate c ( K / F ) = 4 and h - ( K ) = 2.

When k = F ( G ) , Q ( k ) = 2, Q ( k l ) = 1 and r ( K / F ) = 1 hold, Lemma 18 implies Q ( K ) = 2 and hence c ( K / F ) = 2. However, the situation is again complicated when k = F(-), Q ( k ) = 1, Q ( k l ) = 2 and r ( K / F ) = 1. There is an example of Q ( K ) = 1 ( c ( K / F ) = 4 ) and examples of Q ( K ) = 2 ( c ( K / F ) = 2.)

438 R. Okazaki

Example 31. Let F = Q ( 4 - 3 . 5 . -7). Then, we have h ( F ) = 2. The Vie'te index i of F is 0 . The Vie'te ideal (V,) of F is not a square of any ideal of F . Let E = 41+4@. Then, we have = 5.5. Let k = F ( G ) and k' = ~ ( d z ) = F ( 6 ) . Then, we have w ( k ) = 4 , w ( k l ) = 2, Q ( k ) = 1, Q ( k l ) = 2, ~ ( k ) = ~ ( k ' ) = 1, h - ( k ) = 4 and h - (k ' ) = 8 . Here, k / F and k ' / F are unramified above ( 2 ) . Set K = kk'. Then, K + / F = F(&) /F is unramified. The Vie'te index I of K+ is 0. The Vie'te ideal (Vz) of F is not a square of any ideal of F . W e have h ( K + ) = 1, w ( K ) = 4 , Q ( K ) = K ( K ) = r ( K / F ) = 1, c ( K / F ) = 4 and h- ( K ) = 8. Therefore, h- ( k ) and h-(k ' ) divide h - ( K ) .

Example 32. Let F = Q ( d - 4 . - 3 . 5 ) . Then, we have h ( F ) = 2. The Vie'te index i of F is 0 . The Vie'te ideal (V,) of F is a square of a non-principal ideal of F . Let E = 4 + a. Then, we have ( 3 + fi)2 = 6.5. Let k = F(Q) and k' = ~ ( 4 n ) = F ( 6 ) . Then, we have w ( k ) = 4, w ( k l ) = 2, Q ( k ) = 1, Q ( k f ) = 2, ~ ( k ) = 2, ~ ( k ' ) = 1, h - ( k ) = 1 and h - ( k t ) = 4. Here, k / F is unramified at all finite primes and k ' /F is unramified above ( 2 ) . Set K = kk'. Then, we have K + / F = F(d-8.-3-)/F i s ramified above ( 2 ) . The Vie'te index I of K+ is 0 . W e have (Vz) = (4+ &)2. W e have h ( K + ) = 2 , w ( K ) = 4 , Q ( K ) = 2, K ( K ) = r ( K / F ) = 1, c ( K / F ) = 2 and h - ( K ) = 2. Therefore, h - ( k ) divides h- ( K ) .

Example 33. Let F = Q( 4-8.-3.5). Then, we have h ( F ) = 2. The Vie'te index i of F is 0 . The Vie'te ideal (V,) of F is a square of a non-principal ideal of F . Let E = 11+2&6. Then, we have (5+&%)2 =

5.5. Let k = F ( G ) and k' = F( 4-4.5) = F ( 6 ) . Then, we have w ( k ) = 4 , w ( k l ) = 2, Q ( k ) = 1, Q ( k l ) = 2, ~ ( k ) = 2, ~ ( k ' ) = 1, h - ( k ) =

2 and h-(k ' ) = 4. Here, k / F and k ' / F are unramified above (2 ) . Set K = kk l . Then, we have K+ = F(&). Then, K + / F is unramified. The Vie'te index I of K+ is 0. W e have (Vz) = ( 2 + &)2. W e have h ( K + ) = 1, w ( K ) = 4, Q ( K ) = 2, K ( K ) = r ( K / F ) = 1, c ( K / F ) = 2 and h- ( K ) = 4. Therefore, h- ( k ) and h- ( k ' ) divide h- ( K ) .

53. Consistency of the Two Competing Tools

We shall firstly formulate the essential part of the proposed problem of consistency in $53.1. The formulation will be the parity equality of Theorems 1 and 2. We shall secondly give non-trivial examples of parity equality in $53.2. We shall lastly prove parity equality in 553.3.


3.1. Parity Equality as Consistency

We shall formulate the essential part of the proposed problem of consistency.

Let k / F and k ' / F be distinct CM-extensions. Set K = kk'. We display Identity ( 6 ) and ( 5 ) :

h- ( k ' ) - h- ( K ) + - - c ( K / F ) h - ( k ) '

where v is 1 if k and k' are obtained by adjoining square roots of units to F and 0 otherwise.

Identity ( 9 ) together with (10) suggests h - ( K ) / h - ( k ) can have non- trivial denominator. Indeed, it does have non-trivial denominator in some cases as Examples 11, 12, 13 and 14 show.

On the other hand, Propositions 8 and 9 implies ( 7 ) , i.e.,

under the situation

( A ) k / F is unramified at all finite primes or ( B ) K+ / F is unramified.

This looks contradicting the mentioned suggestion. We analyze delicate relation of ( 9 ) and (11). 1. If h - ( k ) is odd, comparison of denominators in the both sides of ( 9 ) implies ( 1 1) . 2. If h- ( k ) is even under situation ( A ) or ( B ) , we need either 2 1 h- (k ' ) or c ( K / F ) = 1 for consistency of ( 9 ) and (11). Indeed, c ( K / F ) is often 2. Therefore, we are lead to

Suspicion: Some principle forces h- (k ' ) to be even when h - ( k ) is even under situation (A) or ( B ) .

Of course, possibility of c ( K / F ) = 4 poses a further difficult problem. (See Example 19.) However, Suspicion explains some part of consistency of (11) with (9) . 3. Under situation ( A ) , we have E+(k+) # E ( I C + ) ~ . If k ' / F is ramified at some finite prime under situation ( A ) , h - (k ' ) is even (by Lemma 15) and Suspicion is explained. 4. Therefore, interesting cases are (A') and ( B ) , where (A') is the following situation:


(A ' ) k / F and k ' /F are unramified at all finite primes.

Since situations (A') and ( B ) have symmetry with respect to exchange of k and k', Suspicion is formulated as the following equivalence

h- ( k ) is even h-(k t ) is even

under situation (A ' ) or (B) . The equivalence is stated as parity equality in Theorems 1 and 2.

In conclusion, Theorems 1 and 2 are interpretation of some part of a delicate competition and consistency of the field theoretic tool and class number relation.

3.2. Examples of Parity Equality We shall give delicate examples for Theorems 1, 2, Propositions 8

and 9. The examples shall illustrate that the formulated problem indeed makes sense.

We begin with Theorem 1 and Proposition 8.

Example 34. Let F = Q ( d - 4 . -3.5). Then, h ( F ) = 2. The Vie'te index i of F i s 0. The Vie'te ideal (V,) is a square of a non- principal ideal of F . Let k = F(Q) and k' = F(-). Then, we have w ( k ) = 4, w(k l ) = 2 .3 , Q ( k ) = Q(kl ) = 1, ~ ( k ) = &(kt ) = 2 and h - ( k ) = h-(k t ) = 1. Moreover, k / F and k ' /F are unramified at all finite primes. Set K = kk'. The Vie'te index I of K+ equals i. W e have (V I ) = ( 1 + W e have K+ = F(&), h(K+) = 1, w ( K ) = 4 . 3 , Q ( K ) = 2, K ( K ) = 1, T ( K / F ) = 1, c ( K / F ) = 1 and h - ( K ) = 1. Therefore h - ( k ) and h- (k ' ) divide h- ( K ) .

Example 35. Let F = Q ( d - 4 . -3.17). Then, h ( F ) = 2. The Vie'te index i of F i s 0. W e have (V,) = (7 + f l)2. Let k = F(-) and k' = F(-). Then, we have w ( k ) = 4, w(k l ) = 2 . 3 , Q ( k ) = 2, Q(k l ) = 1, ~ ( k ) = 1, ~ ( k ' ) = 2, and h - ( k ) = h-(k t ) = 2. Moreover, k / F and k ' /F are unramified at all finite primes. Set K = kk'. The Vie'te index I of K+ equals i. W e have K+ = ~ ( m ) , h(K+) = 1, w ( K ) = 4 . 3 , Q ( K ) = 2, K ( K ) = 1, T ( K / F ) = 1, c ( K / F ) = 2 and h - ( K ) = 2. Therefore h - ( k ) and h-(k ' ) divide h - ( K ) .

The CM-field k' in Example 35 shows that the order of 2 in h-(k t ) can be greater than the lower bound imposed by Lemma 4.

We turn to Theorem 2 and Proposition 9.

Example 36. Let F = ~(a). Then, h ( F ) = 2. The Vie'te index of F is 0. The Vie'te ideal (V,) is a square of a non-principal ideal of F . Let k = F ( a ) and k' = F ( J - 8 ) . Then, we have w ( k ) = 4,

w(k l ) = 2, Q ( k ) = Q ( k l ) = 1, ~ ( k ) = ~ ( k ' ) = 2 and h - ( k ) = h-(k ') = 1. Extensions k / F and k ' /F are ramified above (2) . Set K = kk'. Then, K+/F = F(&)/F is unramified. The Vie'te index I of K+ is 1. The Vie'te ideal ( V I ) is not a square of any ideal of K+. W e have h (K+) = 1, w ( K ) = 8 , Q ( K ) = 1, K ( K ) = 1, T ( K / F ) = 2, c ( K / F ) = 1, and h - ( K ) = 1. Therefore h7 (k ) and h - ( k t ) divide h - ( K ) .

Example 37. Let F = ~(m). Then, h ( F ) = 2. Let k =

F(-) and k' = F ( 4 m ) . Then, we have w ( k ) = 2 . 3 , w(k l ) = 2, Q ( k ) = Q ( k l ) = ~ ( k ) = ~ ( k ' ) = 1 and h - ( k ) = h-(k') = 2. Extensions k / F and k ' /F are ramified above (3 ) . Set K = kk'. Then, K + / F =

F(&) /F is unramified. W e have h(K+) = 1, w ( K ) = 2.3 , Q ( K ) = 1, K ( K ) = 1, r ( K / F ) = 1, c ( K / F ) = 2, and h - ( K ) = 2. Therefore h - ( k ) and h-(k ' ) divide h- ( K ) .

Example 38. Let F = ~(m). Then, h ( F ) = 2. Let k =

F(-) and k' = ~(48.-7). Then, we have w ( k ) = w(k l ) = 2, Q ( k ) = Q(k l ) = ~ ( k ) = ~ ( k ' ) = 1 , h- (k) = 2 and h-(k') = 4. Ex- tensions k / F and k ' /F are ramified above (7) . Set K = kk'. Then, K+/F = F(&)/F is unramified. W e have h(K+) = 1, w ( K ) = 2, Q ( K ) = 1, K ( K ) = 1, r ( K / F ) = 1, c ( K / F ) = 2, and h - ( K ) = 4. Therefore h- ( k ) and h- (k ' ) divide h- ( K ) .

Example 19 is also an example of Theorem 2 and Proposition 9. We have seen non-trivial examples of Theorems 1, 2, Propositions 8

and 9.

3.3. Proof of Parity Equality We shall firstly reduce Theorem 1 to Theorem 2 and then prove

Theorem 2.

Proof of Theorem 1. CM-extensions k / F and k ' /F are unramified at all finite primes. Then, K / F is unramified at all finite primes. Hence, K+/F is unramified at all finite primes. On the other hand K + / F is unramified at the infinite primes since K+ is totally real. Therefore, K+/F is unramified. Theorem 2 now implies the desired equivalence.

Proof of Theorem 2. We introduce some notation and reformulate the assertion. We denote by ? d 2 ) ( L ) the maximal Zextension of L in X ( L ) for a number field L. Since X ( L ) / L is abelian, the order of [ X ( L ) : x ( ~ ) ( L ) ] is always odd. When L is a CM-field (i.e., L+ makes sense), the ratio [ X ( L ) : L X ( L + ) ] / [ X ( ~ ) ( L ) : L X ( ~ ) (L+)] is odd. Therefore, the parity of [ X ( L ) : LX(L+)] and that of [ x ( ~ ) ( L ) : L X ( ~ ) ( L + ) ] are identical. By the identification (I), the former index is he(L) . On

Parities of Relative Class Numbers of CM-Extensions

the other hand, 2 1 [x (~) (L) : L?d2)(L+)] is equivalent to [ ~ c ( ~ ) ( L ) :

L X ( ~ ) (L+)] > 1. Therefore, we get:

The assertion of the Theorem is now equivalent to

By symmetry, it suffice to prove the implication from left to right. We assume the left hand side and prove the right hand side in several steps: Step 1. (Isolation of essential case): If k = k', our conclusion is trivial. Therefore, we assume k # k'. By class field theory, unramifiedness of the quadratic extensionK+/F implies that the 2-rank of e(F) is positive. If the 2-rank of e(F) is greater than 1, Lemma 4 implies that h-(k') is even, which is equivalent to our conclusion through (12). We now assume that the 2-rank e(F) is 1. By class field theory, (F)/F is a non-trivial cyclic extension. Step 2. (Construction of extension) : Since the quadratic extension K+

/F is unramified, we have k' c K = kK+ C ~ x ( ~ ) ( F ) . Inclusion ~ ' K ( ~ ) ( F ) c ~ K ( ~ ) ( F ) follows. By symmetry, we get the reverse inclusion and hence ~ ' K ( ~ ) ( F ) = ~ K ( ~ ) ( F ) . Hence, our assumption implies [ K ( ~ ) (k) : k' W2) (F)] > 1. Step 3. (Unramifiedness): On the other hand, 3 ~ ( ~ ) ( k ) / K is unramified

since K is an intermediate field of an unramified extension ~ ( ~ ) ( k ) / k . The extension K/k' = k'K+/kl is also unramified since K+/ F is unramified. Therefore, Jd2) (k) /k' is unramified. Step 4. (Galois property): Since k / F is normal, class field theory

implies normality of ~ ( ~ l ( k ) / F . It follows that ~ ( ~ ) ( k ) / k ' = k'9d2)(k) /k lF is also normal. On the other hand, k ' ~ ( ~ ) ( F ) / k ' is cyclic since

( F ) / F is cyclic by Step 1. Step 5. (Abelian extension): Let G = ~ a l ( X ( ~ ) ( k ) / k ' ) . Let H be the

maximal abelian extension of k' in (k). It turns out [H : k'?d2) (F)] > 1. Suppose contrary H = ~ ' x ( ~ ) ( F ) . Then, the maximal abelian quotient of G is cyclic. Hence, Burnside Basis Theorem implies that G is cyclic. (See e.g. [12, Theorem 1.16 (p. 92)] for Burnside Basis The- orem.) Hence ~ ( ~ ) ( k ) / k ' is abelian, i.e., H = !Jd2)(k) holds. Now, the conclusion of Step 2 contradicts the supposition on H . By contradiction, we see [H : k ' ~ ( ~ ) (F)] > 1. Step 6. (Class Field): On the other hand, Step 3 and the definition of

H implies H c ?d2)(k'). Therefore, [9d2)(k') : k'?d2)(F)] > 1 follows.

54. Conclusion

We reviewed Horie's theorem on divisibility of relative class numbers, i.e., a theorem on an obstacle for class number one. It was explained that a generalization of Horie's theorem has been proven by cooperation of three tools: the group theoretic tool, the field theoretic tool and class number relation. A certain competition of the latter two tools was explained. The competition arose when a pair of distinct CM- extensions k / F and k l /F with K = kk' are in one of the following situations: k / F is unramified at all finite primes or; K+/F is unramified. The second tool gave apparently stronger obstacle for class number one. In $1, the reason for consistency of application of the two tools, i.e., for integrality of the ratio h-(kl)/c(K/F), is asked.

The two tools were discussed in detail in 52 before analysis of the problem.

Suspicion was responsibility of h-(k') for consistency and hence for the obstacle. (See 553.1.) Suspicion was formulated as parity equality of Theorems 1 and 2. The parity equality and the real problem of consistency were illustrated by an example in $53.2. The two theorems were proven by the field theoretic tool in $53.3. Unfortunately, the proof was one-sided. Hence, it was delicate if the consistency was really explained. However, responsibility of h-(k') for the obstacle to class number one was established. It was also confirmed by examples.

A further problem is caused by the possibility of c(K/F) = 4. In- deed, Examples 19 and 31 show that c(K/F) = 4 sometimes happen in the situation of Theorem 2. As Example 19 of $52.2 and Examples 26 through 33 of 552.3 show, the value of c(K/F) is hard to understand. Therefore, Theorems 1 and 2 constitute a meaningful answer to the

: problem of consistency although they might not constitute the perfect answer.

, Acknowledgment The author would like to thank the referee who carefully read the paper and gave suggestions for improvement.

References

[ 1 ] H. Hasse, ~ b e r die Klassenzahl abelscher Zahllcorper, Akademie Verlag (1952); Springer Verlag (1985).

[ 2 ] M. Hirabayashi and K. Yoshino, "On the relative class number of the imaginary abelian number field I," Mem. College Liberal Arts,

L

Kanazawa Medical Univ., 9 (1981), 5-53. I

444 R. Okazaki

[ 3 ] -, "On the relative class number of the imaginary abelian number field 11," Mem. College Liberal Arts, Kanazawa Medical Univ., 1 0 (1982), 33-8 1.

[ 4 ] -, "Remarks on unit indices of imaginary abelian number fields," Manus. Math., 60 (1988), 423-436.

[ 5 ] K. Horie, "On a ratio between relative class numbers," Math. Z., 211 (1992), 505-521.

[ 6 ] -, "On CM-fields with the same maximal real subfield," Acta Arith., 67 (1994), 219-227.

[ 7 ] M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press (1972).

[ 8 ] F. Lemmermeyer, "Kuroda's class number formula," Acta Arith., 66 (1994), 245-260.

[ 9 ] -, "Ideal class groups of cyclotomic number fields I," Acta Arith., 72 (1995), 347-359.

[lo] R. Okazaki, "On evaluation of L-functions over real quadratic fields," J. Math. Kyoto Univ., 31 (1991), 1125-1153.

[ll] -, "Inclusion of CM-fields divisibility of relative class numbers," Acta Arith., 92 (2000), 319-338.

[12] M. Suzuki, Group Theory vol. I, Springer Verlag (1982). (131 H. Wada, "On the class number and the unit group of certain algebraic

number fields," J. Fac. Sci. Univ. Tokyo, 13, Part 2 (1966), 201-209. [14] L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., GTM 83,

Springer Verlag (1997).

Department of Mathematics, Doshisha University, Kyotanabe-shi, Kyoto-fu, 61 0-0321 JAPAN E-mail address: rokazakiQdd. ii j4u. or . jp


Some Congruences for Binomial Coefficients

Sang Geun Hahn and Dong Hoon Lee

Abstract.

Suppose that p = tn + r is a prime and that h is the class number of the imaginary quadratic field, Q(-). If t E 3 (mod 4) is a prime, just r is a quadratic residue modulo t and the order of r modulo t is k$, then 4ph can be written in the form a2 + tb2 for some integers a and b. And if t = 4k where k = 1 (mod 4), r G 3 (mod 4), r is a quadratic non-residue modulo t and the order of r modulo t is k - 1, then ph = a2 + kb2 for some integers a and b. Our result is that a or 2a is congruent modulo p to a product of certain binomial coefficients modulo sign. As an example, we give explicit formulas for t = 11,19,20 and 23.

1 Introduction

Let p be a prime number throughout the paper. Gauss [I, 3, 41 proved that if p = 4n + 1 then p = a2 + b2 where a = 1 (mod 4) and

Jacobi [4, 61 proved that if p = 3n + 1 then 4p = a2 + 27b2 where a r 1 (mod 3) and

a - - ( 3 (modp).

Eisenstein [I, 21 proved several results. If p = 8n + 3 then p = a2 + 2b2 where a =- (-1)" (mod 4) and


446 S.G. Hahn and D.H. Lee

He also proved that if p is a prime of the form p = 7n + 2 or 7n + 4 then p = a2 + 7b2 where a - p2 (mod 7) and

In this paper we study similar problems for primes of the form p = tn+r.

52. t = 3 (mod 4) is a prime

l+J=i] Since t r 3 (mod 4), the ring of integers of Q ( 0 ) is Z[--2--

and Q(C) is the extension field of Q ( n ) with degree = '-'. 2 Set

s = 9. Let r be a quadratic residue modulo t such that 1 < r < t and the order of r modulo t is s. If s is a prime, then the order of a quadratic residue r modulo t such that 1 < r < t is s. Let p = tn + r be a prime. By Dirichlet's theorem, there are infinitely many primes of this type since (r, t) = 1. Then

where ( i) is Legendre symbol. So p splits in Q ( n ) as p = plp2. Let jii be the prime ideal of Q(&) over pi. Since the order of r modulo t is s, so is the order of p. Hence the residue class degree of pi/p is s and pi is inert in Q(Ct) lo(-). Let q = pS. If Pi is a prime in Q(&- lying above pi, then the residue class degree of v i / p is also s, hence we can identify Z[<,-l]/Y1 with Fq where F, is a finite field with q elements. Note that Yi/p is unramified.

Some Congruences for Binomial Coeficients 447

2.1. Gauss Sums

The unit group of the finite field F,X can be identified with the (q - 1)-st roots of unity via Teichmiiller character w:

satisfying

w(a) -. a (mod Y1) for all a E IF;

where is the primitive (q - 1)-st root of unity. Let x be a multiplicative character such that

where Ct is the primitive t-th root of unity. Note that tl(q - 1). Define the Gauss sum as follows:

where t r : IF, ---+ Fp is the trace map and C, is the primitive p-th root of unity. Note that g ( x ) E Q(&) since ~ ( a ) E Q(Ct).

Definition 2.1.1 (Adler [I]).

Lemma 2.1.2 ( Adler) .

¶5 g(Xu) = p r , for x = w

S.G. Hahn and D.H. Lee

Hence

Some Congruences for Binomial Coeficients

since xV is a non-trivial character, C xY(a) = 0

a€F;

For a E F,X, a% = (up-')& = 1 because t , (p - l)l(q - 1) and ( t ,p - 1) = 1. Hence

Therefore

Definition 2.1.3 ([I]).

t-1

Then$(Cr)= C c j ( ( j ) " = C x V ( a ) = r v . Weknowthat

Since cj is determined by j (mod t) and p r r (mod t), cj = cpj = c,j.

Proof. The Galois group Gal(Q(ct) /Q(fl)) is cyclic of order s generated by

Hence

So I?, E Q(J-t). By Lemma 2.1.2, the above lemma is proved. 0

Definition 2.1.5 (Washington [lo]).

where ob : Q(&) --+ Q(&) such that ob(&) = c;.

%$ 6 is called the Stickelberger element for Q(ct)/Q. Since x = w

-EL=; bob1 (dx-lit) = iie = P1


by Stickelberger 's theorem. G a l ( Q ( ~ ) / Q ( f l ) ) = {oblb is a quadratic residue modulo t } fixes i j l , and all the other abs send pl to p2. So

Lemma 2.1.6.

as ideals in iZ[w].

Proof. g (x ) t Q(0) by Lemma 2.1.4 and pi n Q ( G ) = p i . So we are done. 0

Consider the analytic class number formula

for abelian extensions. We know that if $(-1) = -1 then

f where f is the conductor of + and r($) = C +(a)ezni/f is a Gauss

a= 1 sum. Apply these formulas to Q ( 0 ) . Then rl = 0, r2 = 1, R = 1, w = 2, \dl = t , and + = (i). Hence f = t and I T ( + ) \ = &. So


By taking the absolute value

2.2. p-adic G a m m a Function

Definition 2.2.1 (Lang [7]).

r P ( z ) := lim j m+t

O < j < m

where m approaches z p-adically through positive integers.

Defini t ion2.2.2([10]) . I f O < d < q - l a n d d = d o + d l p + . . . + - ds-lps-l such that 0 < d j < p, define

I?, is called padic Gamma function. Note that if $J1 is a prime in Q(Cq- 1 , C p ) lying above then

where is $Il-adic valuation. Let ll be a (p - 1)-st root of -p. Then Gross-Koblitz formula is


Lemma 2.2.3. If d = q ( t - 1 ) = do +dlp+. . .+ds- lps- l and dl = Q-1 = dl + d i p + . + d:-lps-l, then

Proof. It is sufficient to show that s ( d ) = ( p - l ) P and s ( d l ) =

( P - 1 ) a . s ( d ) = v g l ( g V d ) ) = v g l ( d x ) ) .

Since g ( x ) E Q(-), 7 3 1 = and yl/pl is unramified, v g l =

( p - l ) v y l = ( p - l ) v p l . So by Lemma 2.1.5

Similary s (d l ) = ( p - 1 ) a . So we are done.

2.3. Main Result

r > Let

Theorem 2.3.1. Suppose that t - 3 (mod 4 ) is a prime and that 1 i s a quadratic residue modulo t and its order is s = 9 = 9. h be the class number of Q ( 0 ) and p = tn + r be a prime. Let

s-1 , s-1

d = (q) ( t - 1) = C d 3 9 and d' = (9) = C d:$ as i n the previous j=O j=O

section. Then

I n particular, if p l i s principal ideal, p l = -L , then 4p = A2 + (*+: -7 tB2 and A h = f a (mod p) .

Remark. Note that h = la - PI, and a and b are unique up to sign. Since s ( d ) and s (d l ) are multiples of ( p - I ) , n ( d i ) ! and n ( d ! , ) !

Some Congruences for Binomial Coeficients 453

can be expressed as some products of binomial coefficients by Wilson's Theorem.

Proof. The first statement is trivial. For the second, we will prove only a < p case because the other

case is done in a similar manner. By Lemma 2.1.6 and Gross-Koblitz formula,

Hence

Since

rP ( 1 - ( p b 3 - ( - l ) l d ( d ) ! (mod p) ,

a f n ( d j ) ! (modp).

If pi is principal, then

as ideals. So

B G - A b a Since - = - (mod p 2 ) and - - - (mod p2), we have Ah -

2 2 2 2 f a (mod p).


Example 2.3.2. Let t = 7, then s = 3, a = 1, 0 = 2 so h ( Q ( G ) ) = 1. Since h = 1 if 4p = a2 + 7b2 then a and b are unique up to sign. Let p be a prime of the form 7n+2 or 7n+4 and d = 6(p3 - 1)/7. Then

By Theorem 2.3.1, if 4p = a2 + 7b2, then

f(3n)!(5n + 1)!(6n + I)! (mod p) if p = 7 n + 2 ,

f (5n+ 2)!(3n+ 1)!(6n+3)! (mod P) if p = 7n+4 .

By Wilson's theorem, if p is a prime and p - 1 = x + y, then x! y! r (-l)y+l (mod p). So we get the Eisenstein's result.

1 f ) (mod p) (&(3n)!. - - - -

(2n)! (n)!

Since a 5 2,/5 < p/2, the sign can be uniquely determined.

Example 2.3.3. Let t = 11, then s = 5, a = 2, = 3, and so h ( Q ( a ) ) = 1. Jacobi[4,5] showed that if p = l l n + 1 is a prime and 4p = a2 + 11b2 where a = 2 (mod 11) then

l a =

(n) ! (3n) ! (4n) ! (5n) ! (9n) ! (mod P)

Suppose that p = l l n + 5 is a prime.

By Theorem 2.3.1, if 4p = a2 + l l b 2 then

a = &-(2n)!(7n+3)!(8n+3)!(6n+2)!(10n+4)! (modp) - 1 ) ( ) 2 1 ) 1 (modp).

Some Congmences for Binomial Coeficients 455

Since a 5 2,/5 < p/2 , the sign can be uniquely determined. In a similar manner, we can get the following corollary.

Corollary 2.3.4. Let p = l l n + r be a prime and 4p = a2 + l l b 2 where r is a quadratic residue modulo 11. Then

I f n ~ 1 ) ( 6 1 ) 2 1 ) 1 (modp) i f r = 3 ,

a. f(Snn+1)(;:::)(?n2:1)-1 (modp) i f r = 4 o r 5 ,

Example 2.3.5. Let t = 19, then s = 9, a = 4, = 5, and so h ( Q ( m ) ) = 1. Since s is not a prime, the order of a quadratic residue is not always s(= 9). Suppose p = 19n + 4 is a prime. Then

If 4p = a2 + 19b2, then by Theorem 2.3.1 and Wilson's theorem,

Similarly we get the following corollary.

Corollary 2.3.6. Let p = 19n + r be a prime where r is a quadratic residue and its order is 9. If 4p = a2 + 19b2, then a is congruent modulo p to a product of binomial coeficients modulo sign.


Example 2.3.7. Suppose p = 23n+4 is a prime. Then h ( Q ( m ) ) = 3 and

If 4p3 = a2 + 23b2 for p ;(a, then by Theorem 2.3.1 and Wilson's theorem,

a ( ) ( : ) ( l ) ( : ) ( 1 2 ~ 1 ) 1 (modp).

If pi is principal, then 4p can be written in the form A2 + 2 3 ~ ~ . For example, if p = 211, then 4p = 42 + 23 - 62. So we can verify Theorem 2.3.1 that 4p3 = 24682+23.11702 and 42 - -(2468) (mod p). Note that if t $ 3 (mod 8) then a and b are even. Similarly we get the following corollary.

Corollary 2.3.8. Let p = 23n + r be a prime and 4p3 = a2 + 23b2 for p JIC;1 where r is a quadratic residue modulo 23. Then a is congruent modulo p to a product of binomial coeficients modulo sign.

Some Congruences for Binomial Coefficients 457

10n + 5 l l n + 5 12n + 5 12n+ 5 a - ' ( 4 n n + 1 ) ( 4 n + 2 ) ( ~ n + ~ ) ( 4 n + 1 ) ( 5 n + 2 ) if r = 12;

1 0 n + 5 l l n + 6 12n+ 5 12n+ 5 8. a ~ ' ( 4 n n + 1 ) ( 4 n + 2 ) ( 2 n + l ) ( 4 n + i ) ( 5 n + 2 )

if r = 13; 10n+6 l l n + 7 12n + 7 12n+ 7

a ~ * ( 4 n n + 2 ) ( 4 n + 2 ) ( 2 n + l ) ( 4 n + 2 ) ( 5 n + 3 ) if r = 16;

53. t =4k for a prime k r 1 (mod 4)

Suppose p = 4kn + r is a prime where k E 1 (mod 4) is a prime and r - 3 (mod 4) is a quadratic non-residue modulo k, that is ( f ) = -1. Then the ring of integers of Q ( G ) is Z[G] and

So p splits in Q(-) as p = hp2. Suppose the order of r modulo t is k - 1. Then pi is inert in Q ( [ t ) / Q ( G ) . Let ki be the prime ideal of Q(ct) over pi and q = pk-l. If pi is a prime in Q(Cq-l) lying above 6i, then the residue class degree of pi is k - 1; hence we can identify Z[[q-l]/!$ll with IFq. Note that p i / p is unramified.


3.1. Gauss Sums and p-adic Gamma Functions

Let x be a multiplicative character such that

where w is the Teichmiiller character. g(x), r,,, 4(x ) and rp are defined as in the previous section.

Lemma 3.1.1.

9 5 Proof. It is sufficient to show that for a E F,X, a t = 1 as in the Lemma 2.1.2.

We will show that 4k(p- 1) 1 (q - 1). Since r = 3 (mod 4), 9 is odd. So 8, k and 9 are relatively prime. Clearly kl(q - 1) and K$l(q - 1).

If i > 2, then 81(4kn)'. If i = 1, then 21(";') = k - 1 and 414kn, and

hence 8 1 (k l) (4kn). Since r 1 3 (mod 4), r2 I 1 (mod 8), and hence

( r 2 ) y = 1 (mod 8). So if i = 0, then 81rk-I - 1. Thus we showed 81(q - I) , and hence 4k(p - l)l(q - I). So we are done.

The Galois group ~ a l ( Q ( c ~ ) / Q ( f l ) ) is cyclic of order k - 1 generated by

because the order of r modulo t is k - 1. Hence ~ ( r , ) = r, since cj = c,j as in the previous section. So g(xV) E Q(-). Let 9 denote the Stickelberger element for Q(ct)/Q. By Stickelberger's theorem

Gal(Q(ct)/Q(fl)) = {ablb = ri (mod t), i = 1,2, . . . , k - 1) fixes $1.

Let CFL; ( r v m o d t)) = a t and (C(b,t)=l b) - a t = (k - 1)t - a t = Pt for some integers a, p > 1. Then


Hence

as ideals in ~ [ a ] . Let $ be a multiplicative character $ : (ZItZ) --+ CX such that

1 if a = ri (mod t) for some i, $(a) =

- 1 otherwise.

Then ~ ( f l ) is the field belonging to $.

Lemma3.1.2. $(-I)=-1.

Proof. If - 1 = ri (mod 4k) for some i. Then i = because the order of r is k - 1. But

k - 1 r 1 3 (mod 4) + r2 = 1 (mod 4) 3 r 2 ( 7 ) = 1 (mod 4).

It is a contradiction. 0

Apply the analytic class number formula to ~(fl) and take the absolute value. Then

f S o h = la-PI. Let II be a (p - 1)-st root of -p. Then Gross-Koblitz formula is

Lemma 3.1.3. If d = 9 (t - 1) = do + dlp + . . . + dk-2pk-2 and d' = t = db + dip + . . + d;-2pk-2, then

460 S.G. Hahn and D.H. Lee Some Congruences for Binomial Coeficients 461

Proof. See t h e proof o f Lemma 2.2.3. 0

3.2. Main Result Theorem 3.2.1. Suppose that t = 4k for a prime k r 1 (mod 4 )

and that r - 3 (mod 4) is a quadratic non-residue modulo k and its order is = k - 1. Let h be the class number of ~ (a ) and p = tn + r

k-1 . be a prime. Let p = p l p 2 in ~[a], p: = ( a + b a ) , x i = l ( r z

k-2 ,

(mod t ) ) = at, = ( k - 1 ) - a, d = (G)(t - 1) = C dip' and j=O

k-2 ,

d' = (q) = C dip' as in the previous section. Then j=O

1. ph = a2 + kb2; 2.

k - 2 f n ( d j ) ! ( m o d p ) i f a < P ,

j =O 2a - k-2 f n ( d : ) ! ( m o d p ) i f f B < a .

j =O

Proof. See the proof o f Theorem 2.3.1. 0

Example 3.2.2. Let k = 5, then a = l , P = 3 so h(Q(-)) = 2. Let p be a prime o f the form 20n + 3 or 20n + 7 and d = 19(p4 - 1)/20. T h e n

(13n + 1) + ( l ln + l ) p + ( l 7 n + 2)p2 + (19n + 2)p3 i f p = 2 0 n + 3 ,

(17n + 5 ) + (11n + 3)p + (13n + 4)p2 + (19n + 6)p3 i f p = 2 0 n + 7 .

B y Theorem 3.2.1, i f p2 = a2 + 5b2, then

f ( 1 3 n + l ) ! ( l ln + 1)!(17n + 2)!(19n + 2)! (mod p) i f p = 20n+3,

2a - f (17n + 5)!(11n + 3 ) ! ( l 3 n + 4)!(19n + 6)! (mod p)

i f p = 2 0 n + 7 .

B y Wilson's theorem,

At t h e t ime o f the publication o f this paper, we could remove the sign ambiguity o f our results in [B ] . Moreover we generalized our results t o the primes o f the form o f p = tn + r such that p splits in Q(-) [9].

References

11 Allan Adler, Eisenstein and the Jacobian Varieties of Fermat Curves, Rocky Mountain Journal of Mathematics, 27 (no. 1) Winter (1997), 1-60.

[2] Gotthold Eisenstein, Zur Theorie der Quadmtische Zerfallung der Prim- zahlen 8n+ 3,7n+2 und 7n+4, Crelle, 37 (1948), 97-126 [Math. Werke 11, pp. 506-535, art. 331.

[3] Carl Friedrich Gauss, Theoria Residuorum Biquadmticorum, Comment. I, Comment. soc. reg. sci. Gottingensis rec., 6 (1828), 27 [Werke, vol. 11, pp. 89-90].

41 Richard H. Hudson and Kenneth S. Williams, Binomial Coefficients and Jacobi Sums, Trans. of the Amer. Math. Soc., 281 (no. 2) February (1984), 431-505.

51 C.G.J. Jacobi, De Residuis Cubics Commentatio Numerosa, J . Reine Angew. Math., 2 (1827), 66-69.

61 C.G.J. Jacobi, Uber die Kreistheilung und ihre Anwendung auf die Zahlen- theorie, J. Reine Angew. Math., 30 (1846)) 166-182.

[7] Serge Lang, Cyclotomic Fields I and 11, Springer-Verlag, GTM 121, 1990. [8] D.H. Lee and S.G. Hahn, Some congruences for binomial coeficients. 11,

Proc. Japan Acad., 76, Ser. A (2000), 104-107. [9] D.H. Lee and S.G. Hahn, Gauss sums and binomial coeficients, J . Number

Theory (2000); submitted. [10] Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer-

Verlag, GTM 83, 1997.

Sang Geun Hahn Department of Mathematics KAIST Yusong- Gu Kusong-Dong 373-1 Taejon 305- 701 South Korea E-mail address: sghahnhath. kaist . ac . kr

Dong Hoon Lee Department of Mathematics KAIST Yusong- Gu Kusong-Dong 373-1 Taejon 305- 701 South Korea E-mail address: dhleehath . kaist . ac . kr


Stably Free and Not Free Rings of Integers

Jean Cougnard

Let N/Q be a tame Galois extension of number fields with finite degree [N : Q] and Galois group G. One knows by the normal basis theorem that IV is a free rank one Q[G]-module. It is natural to look at the structure of the ring of integers ON as a Z[G]-module. The first known result is Hilbert7s theorem which asserts that If G is abelian and the discriminant of N/Q is prime to [N : Q] then ON has a normal integral basis ([Hi] satz 132); that is to say, there exists an algebraic integer a E N such that ON has a basis made of the set {g(a) I g E G). The following result is E. Nether's theorem which asserts that ON is Z[G]-projective if and only if N/Q is a tame extension; in fact Nether's result shows that ON is locally-free: for all prime p the extended module Z, @z ON is Z,[G]-free with rank one. We can associate to ON its image [ON] in the projective class group Cl(Z[G]) of Z[G]-modules; from now on, all the extensions will be tame. In 1968, J . Martinet proved that if G is a dihedral group of order 2p, p an odd prime then ON is Z[G]-free. A few years after, in the case where G = H8 (the quaternionic group of order 8), he was able to describe Cl(Z[G] ) - { f 1) and to give a criterion for ON free or not. Moreover, he produced rings of integers Z[G]-free and not free ([Ma2]). Almost in the same time, Armitage gave examples of L-functions of quaternionic fields with a zero at s = f ([A]).

Knowing the two results J-P. Serre did computations in [S] on examples and was surprised to see that the constant of the functional equation of the Artin L-series for the irreducible degree two character of Gal(N/Q) was 1 whenever ON was free and -1 in the other cases. This was proved to be a theorem by A. F'rohlich [Fl].

The following years A. Frohlich proposed a nice conjecture finally established by M.J. Taylor [TI. We give a few notations before we state this theorem.

The first step was a description of the projective classgroup as a quotient of a group of equivariant maps from RG (the group of characters

Received September 14, 1998

464 J. Cougnard Stably Free and Not Free Rings of Integers 465

of the Galois group) to the idkles group of a sufficiently large field E. One can represent ON in this group using a Q[G] basis of N and the Zp[G]-basis of Zp € 3 ~ O N . On the other hand one can construct a map from Rc; to E by sending symplectic irreducible characters to W(X) (the constant in the functional equation L ( s , X, N/Q)) and the others on +l . The image UN/Q of this map in Cl(Z[G]) is the Cassou-Noguks F'rohlich invariant (see [F2] for more details).

Theorem. If N/Q is a tame Galois extension with Galois group G, we have i n Cl(Z[G])

[ON] = UNIQ.

This result has a lot of consequences. As symplectic characters are real we have W(X) = f 1 and so UNIQ is of order 2.

Corollary. If N/Q is a tame Galois extension with Galois group G, we have an isomorphism of Z[G]-modules: ON @ O N 2 Z[G] @ Z[G].

When the class of a projective Z[G]-module M is 0 in Cl(Z[G]) we can only say that M @ Z[G] E Z[G] $ Z[G]; one says, in this case, that M is stably free. Unfortunately it is not always possible to cancel and to have M r Z[G] as was shown by R.G. Swan [Swl]. Jacobinski has given sufficient conditions to allow cancellation: If G has no binary polyhedral quotient and [MI = 0 then M E Z[G] . From this, we can deduce

Corollary. If N/Q is a tame Galois extension whose Galois group G has no binary polyhedral quotient and i f W(X) = +1 for all the irreducible symplectic characters, we have an isomorphism of Z[G]-modules: ON Z[G], that is to say, ON possesses a normal integml basis.

Let Isoml(ZIG]) be the of isomorphism classes of rank one projective Z[G]-modules; we have the map cp which send each (M) of Isoml(2[G]) to cp((M)) = [MI in Cl(Z[G]). We can ask the following question: can we represent each element in cp-l (0) by a ring of integers? In [SW2], R.G. Swan has extensively studied cancellation for binary polyhedral groups. Among the numerous computations (made thanks to Pari [PI) we quote that for the product G = H8 x C2 of a quaternionic group of order 8 by a cyclic group of order 2 the set Isoml(ZIG]) has 40 elements and Cl(Z[G]) 2 2 /42 x 2 /22 x 2/22; in this case cp-'(0) has four elements (note sixteen is the smallest order for which exists a group where stably free doesn't imply free). We can prove

Theorem [C]. If G = H8 x C2, each of the four classes of mnk one stably free Z[G]-modules can be represented infinitely many times by a ring of integers.

The ingredients to prove this results are the followings: The Witt's criterion to embed a biquadratic bicyclic extension in a quaternionic one, and Martinet's criterion to decide if the quaternionic extensions have or not a normal integral basis; Witt's construction of quaternionic extensions linked with Martinet's criterion allows us to construct normal integral basis of quaternionic extensions when they exist. It is then possible to use Swan's computations to decide what the class of ON is in Isom1 @[GI). We refer to [C] for the details.

References

J. V. ARMITAGE, Zeta functions with zero at s = i, Invent. Mat., 15 (1972), 199-205.

J . COUGNARD, Anneaux d'entiers stablement libres sur Z [ H s x Cz], Journal de Thkorie des nombres de Bordeaux, 10 (l998), 163-201.

A. FROHLICH, Artin root numbers and normal integral bases for quaternionic fields, Invent. Math., 17 (1972), 143-166.

A. FROHLICH, Galois module structure of algebraic integers, Ergeb. der Math. 3 Folge Band 1. Springer Verlag (1983).

D. HILBERT, Die Theorie der algebraischen Zahlkorper, Jahr. ber. der deutschen Math., 4 (1897)' 175-546.

H. JACOBINSKI, Genera and decomposition of latticesover orders, Acta math., 121 (1968), 1-29.

J. MARTINET, Sur l'arithmktique d'une extension galoisienne B groupe de Galois diedral d'ordre 2p, Ann. Inst. Fourier, 19 (1969)' 1-80.

J. MARTINET, Modules sur l'algbbre du groupe quaternionien, Ann. Sci. Ecole Norm. sup., 4 (1971), 399-408.

E. NOETHER, Normalbasis bei Korpern ohne hohere Verzweigung, J . reine und angew. Math., 167 (1932), 147-152.

C. BATUT, D. BERNARDI, H. COHEN, M. OLIVIER, User's Guide to Pari-GP version 1.39-12 (1995).

J-P. SERRE, lettre 8- Jacques Martinet. R.G. SWAN, Projective modules over group rings and maximal orders,

Ann. of Math., 76 (1962), 55-61. R.G. SWAN, Projective modules over binary p yhedral groups, J . reine

und angew. Math., 342 (1982), 66-172. M.J. TAYLOR, On Frohlich conjecture for rings of integers of tame

extensions, Invent. Math. (1983), 41-79.

466 J. Cougnard

ESA 6081 du CNRS Structures DiscrBtes et Analyse Diophantienne Esplanade de la Paix F14032 CAEN cedex E-mail address: cougnardQmath . unicaen. f r


The Capitulation Problem for certain Number Fields

Mohammed Ayadi, Abdelmalek Azizi and Moulay Chrif Ismaili

1 Abstract

We study the capitulation problem for certain number fields of degree 3, 4, and 6.

(I) Capitulation of the 2-ideal classes of Q(& i) (by A. AZIZI) Let d E N, i = f i, k = ~ ( 4 , i), k(,2) be the Hilbert 2-class field

of k, kr) be the Hilbert 2-class field of k?), Ckj2 be the 2-component

of the ideal class group of k and Gz the Galois group of kp)/k. We

/ suppose that Ck,2 is of type (2,2); then ky) contains three extensions f Fi/k, i = 1,2,3. The aim of this section is to study the capitulation of

/ the Pideal classes in Fi, i = 1,2,3, and to determine the structure of GO.

1 (11) On the capitulation of the 3-ideal classes of a cubic cyclic "eld (by M. AYADI)

Let k be a cubic cyclic field over Q, and ky) the Hilbert 3-class field of k. If the class number of k is exactly divisible by 9, then its 3-ideal class group is of type (3,3), and k(,3) contains four cubic extensions Ki/k in which we study the capitulation problem for the 3-ideal classes of k.

(111) On the capitulation of the 3-ideal classes of the normal closure of a pure cubic field (by M. C. ISMAILI)

Let I' = Q(*) be a pure cubic field, k = Q(*, j ) its normal closure ( j = e ), k(;l) the Hilbert 3-class field of k, and let Sk be the 3-ideal class group of k. When Sk is of type (3,3), we study the

Received July 30, 1998. Revised January 13, 1999.

468 M. Ayadi, A. Azizi and M. C. Ismaili The Capitulation Problem for certain Number Fields 469

capitulation of the 3-ideal classes of Sk in the four intermediate extensions of k(:)/k, and we show that if the class number of I? is divisible by 9, then we have some necessary conditions on n. We have also some informations about the unit group of k in some cases.

52. Intoduction

Let k be a number field of finite degree over Q and Ck be the class group of k. Let F be an unramified extension of k of finite degree and let OF be its ring of integers. We say that an ideal A (or the ideal class of A) of k capitulates in F if it becomes principal in F, i.e., if AOF is principal in F. The Hilbert class field kl of k is the maximal abelian unramified extension of k. Let p be a prime number; the Hilbert pclass field k p ) of k is the maximal abelian unramified extension of k such

that [ky) : k] = pn for some integer n. The first important result on capitulation was conjectured by D. Hilbert and proved by E. Artin and P. Furtwangler. It deals with the case F = kl.

Theorem 2.1 (Principal ideal theorem). Let kl be the Hilbert class field of k. Then every ideal of k capitulates i n kl.

The principal ideal theorem was generalized by Tannaka and Terada to the next one. Let b be a subfield of k such that k/ko is abelian and let ( k / b ) * be the relative genus field of k/ko.

Theorem 2.2 (Tannaka-Terada) . If k/ko is cyclic, then every ambiguous ideal class of k/ko i s principal i n ( k / b ) * .

The case where F /k is a cyclic extension of prime degree was studied by D. Hilbert in his Theorem 94:

Theorem 2.3 (Theorem 94). Let F / k be a cyclic extension of prime degree. Then there exists at least one class (no t trivial) i n k which capitulates in F .

We find in the proof of Theorem 94 this result:

Let (T be a generator of the Galois group of F / k and NFIk be the n o n n of F/k. Let EL be the unit group of the field L. Let EG be the group of units of norm 1 i n F/k. Then the group of classes of k which capitulates in F i s isomorphic to the quotient group EG/E;-O = H'(EF), the cohomology group of G = (a) acting o n the group EF.

With this result and other results on cohomology, we have:

Theorem 2.4. Let F / k be a cyclic extension of prime degree. Then the number of classes which capitulate i n F/k i s equal to [F : k] [Ek : NF/k (EF)] -

The case where F / k is an abelian extension was treated by H. Suzuki who has proved Miyake7s conjecture: I n a n abelian extension F / k the number of classes of k which capitulate i n F is a multiple of [F : k].

Let p be a prime number and let k?) (resp. k5 ) ) be the Hilbert

pclass field of k (resp. of k(P)). If L is a subfield of kl and A is an ideal class of k whose order is equal to pm for some integer m. Then A capitulates in L if and only if A capitulates in ~n k?). So we study only the capitulation of classes whose order is equal to pm in the subfields of kp) , and since the capitulation problem is solved when k p ) / k is cyclic,

we study only the cases where kp) /k is not cyclic. For more details see [Mi - 891, [Su - 911, [CF - 911, [Ism - 921, [Az -

931, [Ay - 951 and [Az - 971.

53. Capitulation of the 2-ideal classes of some biquadratic fields

Let k be a number field such that the 2-component Ck,2 of Ck is

isomorphic to 2 /22 x 2/22. Let G2 be the Galois group of kp)/k.

By class field theory, ~ a l ( k y ) / k ) E 2/22 x 2/22. Then k r ) contains three quadratic extensions of k denoted by F1, F2 and F3. Under these conditions, Kisilevsky [Ki-761 proved the following.

Theorem 3.1. Let k be such that Ck,2 E 2 /22 x 2/2Z. Then we have three types of capitulation:

Type 1: The four classes of Ck12 capitulate i n each extension

Fi , i = 1, 2, 3. This i s possible if and only if k y ) =

k?' . Type 2: The four classes of Ck,2 capitulate only in one ex-

tension among the three extensions Fi, i = 1, 2, 3. I n this case the group G2 is dihedral.

Type 3: Only two classes capitulate i n each extension Fi, i = 1, 2, 3. I n this case the group G2 is semi- dihedral or quaternionic.

In this section, we suppose that k = Q(&, i ) where d E N is such that Ck,2 N 2 /22 x 2/22, and we study the capitulation problem in the extensions Fi/k, i = 1, 2, 3.

M. Ayadi, A. Azizi and M. C. Ismaili

Diagram 1

The first step is to study the structure of Ck,2. Using genus theory, the class number formula for biquadratic fields, Kaplan's results on the 2-part of the class number for quadratic number fields and other results, we can prove

Theorem 3.2. Let Q be the Hasse unit index of k and let Ck,2 be the 2-component of the class group of k . Let k(*) be the genus field of k. Then the group Ck,2 is of type (2,2) if and only if one of the next cases occurs: (1) d = 2pq, p = -q = 1 (mod 4), at least two of the three sym-

bols (9) , ($) , (a) are -1 and Q is 1, in which case, k(*) = k y ) =

Q ( f i 7 &, a , i);

(2) d = 2qlq2, ql = q2 = -1 (mod 4), (g) = - 8 - 1 ( q l ) - ($1 =

- (2 ) = 1 and Q = 1, in which case, k(*) = k y ) = Q ( a , 6, a , i);

(3) d = plp2, pl = 1 (mod 8), p2 e 5 (mod 8) and

which case, k(*) = k(&) # k y ) ;

(4) d =pq , p~ 1 (mod 8), q = -1 (mod 4), (9) = -1 and Q = 2,

in which case, k(*) = k( f i ) # k y )

Remarks 3.1. If k(*) # k y ) , we set Fl = k(*) = k(f i) where

p = 1 ( mod 8), F2 = k ( J m ) and F3 = k(d-) where a and b are two integers such that p = a2 + b2, a -- 1 (mod 4) and b 0 (mod 4).

The Capitulation Problem for certain Number Fields 471

In order to determine the number of ideal classes which capitulate in F i /k , i = 1, 2, 3, we have to determine the unit group of each F i , i = 1, 2, 3, where Fi is a composite of three quadratic fields. So using the previous results and others, we obtain the next solution of the capitulation problem.

Theorem 3.3. Let CFi,2 be the 2-component of the class group of Fi and let ji : Ck,2 -+ CFi,2 be the canonical homomorphism.

(2) (1) If k(*) = k y ) , then k y ) # k2 , G2 E Q, or Sm (m > 3) and lker jil = 2 for i = 1,2 ,3 (capitulation type 3), where Q, and S, are respectively the group of quaternions and the semi-dihedral group of order 2m. (2) Let k(*) # k y ) . Then 1 ker jl 1 = 4. Moreover, (a) If d is divisible by a prime q = -1 (mod 4) and p # x2 + 32y2,

then k g ) = k?), G2 e 2 / 2 2 x 2 / 2 2 and lkerjil = 4 for i = l , 2 , 3 (capitulation type 1); (b) If d is not divisible by a prime q r -1 (mod 4) or if p = x2 + 32y2,

then k?) # k y ) , G2 E D,(m 2 3), the dihedral group of order 2,, and 1 ker ji 1 = 2 for i = 2 ,3 (capitulation type 2).

For more details see [Az - 931 and [Az - 971.

Numerical Examples.

1 Values of d I Capitulation tvpes I

Table 1

$4. On the capitulation of the 3-ideal classes of a cubic cyclic field

Let k be a cubic cyclic field over Q whose class number is exactly divisible by 9. Let k y ) be its Hilbert 3-class field and let k(*) be its absolute genus field. Then the 3-ideal class group of k is of type (3,3),

and kY)/k contains four subfields K l , K2, K 3 and K4. We want to study the capitulation problem of the 3-ideal classe of k.

For the details of all the proofs and results given in this section see [Ay-951.

M. Ayadi, A. Azizi and M. C. Ismaili

Diagram 2

We have to distinguish two cases.

First case: [k(*) : k] = 3. It turns out that this is equivalent to each of the following conditions:

- G a l ( k y ) / ~ ) is not abelian; - k(*) = Ki for some i E {1,2,3,4}; - Exactly two distinct prime numbers p and q are ramified in k.

Second case: [k(*) : k] = 9. This is equivalent to each of the following conditions:

- Gal(ky)/Q) is abelian; - k(*) = k(3).

1 ,

- Exactly three distinct prime numbers p, q and r are ramified in k.

(A) Case where [k(*) : k] = 3

In this case, exactly two prime numbers p and q are ramified in k, and there exists another unique cubic cyclic field denoted by k having the same conductor as k. Denote by hk (resp. by hi;) the class number of k (resp. of k).

Theorem 4.1. Let k be a cubic cyclic field of conductor divisible only byp and q. Then

911hk 911hi;.

If 9) 1 hk, then k and k have the same Hilbert 3-class field.

The Capitulation Problem for certain Number Fielab 473

Let a be a generator of Gal(k/Q) and let S = a - 1. From class

field theory we know that k r ) corresponds to sb2 and that s6' is trivial,

where S = ~ a l ( k y ) / k ) . Moreover the group of ambiguous classes is of order 3, and generated by the classes [PI and [&I, where P and & are the prime ideals of k lying above p and q. We have of course [Pln[Q]" = 1 for some n, m E {0,1,2) and (n, m) # (0,O); the nontrivial relation [PIn [&Im = 1 is obtained by calculating a constant of Parry denoted bk. Here bk = pnqm is caculated from a fundamental unit of k (generating over Z[a] the unit group of k) and its irreducible polynomial (see [Pa-

901 ).

Theorem 4.2. Let P, & (resp. P, a) be the prime ideals of k (resp. of k) lying above p and q. Then the following assertions are true: (1) Vn,m E N; [PIn[Qlm = 1 a [6In[ilm = 1. (2) [p] = 1 or [&I = 1 @ 911hk(,). Saying this, is equivalent to: [p] # 1 and [Q] # 1 @ 3(lhk(*) -

The fact that the prime P (resp. P) is inert in k(*)/k (resp. in

k(*)/k ) and that k(13) = k y ) , we get that the Artin maps (kf) /k, P), ( k ( ; ? ) / k , ~ ) , and (k(;?)/k*, P*) are equal, where P* is a prime in k(*) lying above p; so we obtain (1). The fact that the 3-class number of k(*)

I is equal to 3 or 9 is obtained by using a formula giving hk(.) where k(*) i is considered as the composite of cubic cyclic fields, so the assertion (2)

is proved by calculating some unit index involving Parry's constant (see [Pa-901).

Theorem 4.3. (1) All the 3-ideal classes capitulate in each of the

four intermediate fields of k(:)/k if and only if 311hk(*). In this case,

k y ) = ki3) for each n 2 2.

(2) Let L be a subeztension of ky ) which is cubic over k. Then only the ambiguous ideal classes capitulate in L if and only if 911hk(*). In this

case, k r ) = k r ) for each n 2 3.

The first assertion is obvious. For the second, the unit index in the extension k(*)/k is 1, so only the three ambiguous classes capitulate in

k(*) (see [Fr-931 and [Ja-881); we use the fact that the group Gal (kf ' lk)

has two generators and we prove that Gal(kr)/k) is metacyclic of order 27 (see [Bl-581); so the conclusion is obtained via the transfer for groups of order 27. See [Mi-891 for more information on transfer and [Ne-671 for all the different groups of order 27.

474 M. Ayadi, A. Azizi and M. C. Ismaili

Numerical Examples.

(B) Case where [k(*) : k] = 9

In this case, k y ) = k(*) and exactly three prime numbers p, q and r are ramified in k; there are exactly three other cubic cyclic fields having the same conductor as k. Using the cubic symbol, G. Gras distinguished 13 different situations (see [Gr-731). We solved the capitulation problem for four of them, namely under the following equivalent conditions: Let p, q, r be distinct prime numbers = 1 (mod 3), and allow p to be equal to 3; the Hilbert 3-class field of each cubic cyclic field of conductor dividing (pqr)2 is equal to its absolute genus field.

Under these conditions we have:

Theorem 4.4. If k is a cubic cyclic field of conductor pqr (or 9qr if p = 3) and if 911hk, then all the 3-ideal classes capitulate in each of

the four intermediate fields of ky ) /k .

By using Parry's constant and some unit index (see [Pa-SO]), we

prove that the 3-class number of each bicubic bicyclic field in ky ) /k is equal to 3.

Numerical examples. Suppose that k is a cubic cyclic field with conductor fk 5 16000. Then k satisfies the last theorem if and only if fk E (819, 1197, 1729, 1953, 2223, 2331, 2709, 2821, 2843, 3627, 3913, 4221, 4329, 5031, 5301, 5551, 5719).

$5. On the capitulation of the 3-ideal classes of the normal closure of a pure cubic field

Let = Q ( f i ) be a pure cubic field with class number hr, k =

Q ( f i , j) its normal closure ( j = e t ), k(:) the Hilbert 3-class field of k, and Sk the 3-ideal class group of k. Suppose that Ek is the group of units of k, Eo the subgroup of Ek generated by the units of all proper subfields of k, and u = [Ek : EO]. Let Gal(k/Q) = (a, 4, Gal(k/ko) = (0) , Gal(k/F) = (T), where u3 = r2 = 1, OT = ro2 and a2r = TO.

The relation between the class number hk of k and the class number hr of r is given by hk = h; $ (see [B-C-711).


Proposition 5.1. (1) Sk Z 2/3Z x Z/32 e 3 divides exactly hr and u = 3. (2) If Sk is of rank 2 and if 3 exactly divides hr, then u = 3, whereupon Sk E 2 /32 x 2/32.

The study of the structure of Sk and its rank is based on Gerth's results in [Ge-751, [Ge-761 and [Ger-761.

The action of the Galois group of k/Q on Sk and genus theory allow us to distinguish three different cases (see [Ism-921). We let ko = Q(j) and we define (k/ko)* to be the relative genus field of k over ko. Then (1) k is of type I if (k/ko)* = k r l , where rl is the Hilbert 3-class field of r; (2) k is of type I1 if (k/ko)* # k r l and (k/ko)* is a proper subfield of

k y ) ;

(3) k is of type I11 if (k/ko)* = ky) . When the 3-group Sk is of type (3,3), it has 4 subgroups of order

3, denoted by Hj, 1 5 j 5 4. Let K j be the intermediate extension of kl /k, corresponding by class field theory to Hj. As each K j is cyclic of order 3 over k, there is at least one subgroup of order 3 of Sk, i.e., a t least one HL for some 1 E {1,2,3,4), which capitulates in K j (by Hilbert's theorem 94).

Definition 5.1. Let Sj be a generator of Hj (1 5 j 5 4) corresponding to Kj . For 1 5 j 5 4, let i j E {0,1,2,3,4). We say that the capitulation is of type (il, 22, i3, i4) to mean the following: (1) when i j E {1,2,3,4), then only the class Sij and its powers capitulate in Kj ; (2) when i j = 0, then all the 3-classes capitulate in Kj.

Suppose that k is of type I; we show (see [Ism-921) that Sk =

{AT+SO 10 5 r, s 5 2) where A is such that AT = A. The four subgroups of Sk are given by: H1 = (A), H2 = (Aa), H3 = and H4 = (Aa-l) which corresponds to K4 = (k/kO)*.

Theorem 5.1. Let p and q be prime numbers and let u = [Ek : Eo] . (1) I f k is of type I, then the possible forms of n (where I' = Q(f i ) ) are

(i) n =pel , p = 1 (mod 3) with el E {1,2); (ii) n = 3"pe1, p = 4 or 7 (mod 9) with e, el E {1,2); (iii) n = peqel = z t1 (mod 9), p or -q = 4 or 7 (mod 9) and Gel E {1,21.

476 M. Ayadi, A. Azizi and M. C . Ismaili

( 2 ) Let n E N be as in (i i) (resp. ( i i i ) ) , let (g) # 1 (resp. ( g ) 3 # 1 ) ) 3

and assume 31 1 hr . Then u = 1, Sk is cyclic of order 3 and Ek = (E, cU, - j ) , where E is the fundamental unit of r .

Theorem 5.2. ( 1 ) All the 3-classes capitulate in K 4 = krl =

( k / W * . ( 2 ) The numbers of 3-classes capitulating in K 1 , K 2 and K g are the same. More precisely, the possible capitulation types are (0 ,0 ,0 , O ) ,

3,O) or (474, 470).

Diagram 3

Suppose that k is o f type 11; we show (see [Ism-921) that the four cubic fields Ki are given as follows: K 1 = (k/ko)* which corresponds

by class field theory t o H1 = SF) = < A >, K 2 = kr?, K g = kr; and K 4 = kr l , where rl (resp. I?', , l7;) is the Hilbert 3-class field o f r (resp. o f the two other cubic fields r', r" contained in k) .


Diagram 4

Theorem 5.3. ( 1 ) The class A capitulates in the four cubic extensions K i , 1 <_ i < 4. ( 2 ) The numbers of 3-classes capitulating in K S , K 3 and K 4 are the same. More precisely, the possible capitulation types are (0 ,0 ,0 , O ) , ( O , l , 1, I ) , ( L O , 070) or (171, 1 , l ) .

Theorem 5.4. Let qi be prime numbers r -1 (mod 3 ) . ( 1 ) If the field k is of type 11, then the possible forms of n (where r =

Q(*N are ( i ) n = 3eq;1 with ql r -1 (mod 9 ) and e , el E { 1 , 2 ) ; ( i i) n = q;'qg2 with ql = 92 = -1 (mod 9) and e l , e2 E { 1 , 2 ) ; (iii) n = 3"q;' qEZ with ql or 92 r 2 or 5 (mod 9) , el , e2 E { 1 , 2 ) , e E {O,1,2) and n $ kl (mod 9 ) ;

el ez es ( i v ) n = ql q2 q3 with q1 or 92 or 93 = 2 or 5 (mod 9), n = f 1 (mod 9 ) and el ,ez,e3 E { 1 , 2 ) .

( 2 ) If the integer n has one of the four forms of ( 1 ) and i f 311 h r , then the index u = 3, whereupon k is of type II. (3) The normal closure k of r = Q ( 6 ) is of type 11 if and only if n has one of the four forms of ( 1 ) and 31 1 hr .

Suppose finally that k is o f type 111. T h e n we have the following.

Theorem 5.5. Let p, q, ql and 92 be prime numbers such that p -q - -ql -92 -- 1 (mod 3) . The normal closure k = Q ( j , *) of r = Q(*) is of type 111 i f and only i f 31) hr , and n has one of the following forms:


(i) n = 3"pe1 with p - 1 (mod 9) and e,el E {1,2); (ii) n = qepel with -q - p - 1 (mod 9) and e, el E {1,2); (iii) n = peqY1q2e2 with p or -ql or -q2 - 4 or 7 (mod 9), n - f 1 (mod 9) and e, el, e2 E {1,2); (iv) n = 3"pe1qe2 with p or -q - 4 or 7 (mod 9), e E {O,1, 21, e l , e2 E {1,2) and n $ f 1 (mod 9).

Let us remark that k is of type I11 means that (k/ko)* = k?); in this case for VA E Sk we have Aa = A, i.e., all the 3-classes are ambiguous classes.

When n has one of the four forms of the last theorem, and if p - -q = -ql - -q2 - 1 (mod 3), we have p = 7rl7r2, -q = 7r7 -q1 = 7r3 and -q2 = 7r4, where 7r, 7ri (1 5 i 5 4) are prime integers of ko; we also have 30k, = ( x ) ~ with X = 1 - j. We denote respectively by P17 P2, Q, Q1, Q2 and I the prime ideal of k lying above 7r1, 7r2, 7r, 7r3, 7r4 and A. We summarize in the next theorem most of the results concerning the capitulation problem when k is of type 111.

Theorem 5.6. Suppose that the normal closure k = Q(*, j) of I? = Q( fi) is of type III. (A) If n has one of the four forms of last theorem with the property that the prime number p = 7rl7r2 dividing n satisfies p = 1 (mod 9), or if n has the fourth form with p $ 1 (mod 9) and (n, 3) = 1, then we have the following: (1) k?) = k ( m , @), PIP2 is not a principal ideal in k and Sk =

([PlP21, [Pll). (2) K1 = k ( q m ) , K 2 = k(@), K3 = k ( m ) and K4 = k r l =

k( V X ) - (3) [PI P2] capitulates in K1, [P2] capitulates in K2, [PI] capitules in Kg and all the 3-classes capitulate in K 4 . (4) The possible capitulation types are (0,0,0, O), (1,3,2, O), (0,3,2,0) or (1,0,0,0). (B) If n has the form of (iii) with p $ 1 (mod 9) or n has the form of (iv) with 3 1 n, then all the 3-classes capitulate in K4 and we have the following capitulation types depending on some conditions on the ideals Ql '7 Q1 and Q2 : (a) (0,4747 o), (1,474, o), (4,4,4, o), (1,0,0,0) Or (4,070, 0); (b) (0,0,0,0);

(0,372, 0) or (0,273, 0); ( 1 , O l 070); (1,372, 0) or (1,2,3,0).


Theorem 5.7. Let hr be the class number of the pure cubic field I' = Q(*). If n = cepel, where c = 3 or q, and p, q are prime numbers such that p - -q r 1 (mod 9) and e, el E {1,2), then

When n has the form (iii) or the form (iv), we prove seven other similar results. Each time, we construct, under certain conditions, a natural integer c such that:

The proof of all the results given in this section can be found in [Ism-921. In this work we used also the arithmetic properties of a pure cubic field (see [De-00]), Kummer theory and the cubic symbol (see [I-R-821). For the following numerical examples we used the tables given in [B-871 and [B-W-Z-711.

Numerical Examples. (1) For p E (61, 67, 103, 151) we have k = Q(j, fi) is of type I.

(2)

n

3 . 1 7 = 51 32 17 = 153 3 . 5 3 = 159

I I "

32 . 107 = 963 1 3 I type I1

32 - 7 1 = 639 3 .89 = 267 32 -89 = 801

Table 2

hr 3 9 = 32 3

k = Q(j, fi) type I1

Sk C9 x Cg tvpe I1

18 = 2 . 32 15 = 5 - 3 6 = 3 . 2

" .

Sk E Cg x Cg type I1 tvpe I1


(3) For each integer n in the next table, k = Q(j , fi) is of type 111.

Table 3

References

M. Ayadi, Sur la capitulation des 3-classes d'ide'aux d'un corps cubique cyclique. ThBse de doctorat. Universitd Laval - Qudbec - Canada. (1995).

M. Ayadi, Table d7Ennola de corps cubiques cycliques de conduc- teurs 5 15993. PrBpublication. DBpartement de Mathdmatiques et de Statistique U. Laval, 95-21.

A. Azizi, Capitulation des 2-classes d'ide'aux de Q(&, i). ThBse de doctorat. Universitd Laval - Qudbec - Canada. (1993).

A. Azizi, Capitulation des 2-classes d'ide'aux de Q(&, i). C. R. Acad. Sci. Paris, t. 325, Sdrie I, 127-130, (1997).

W. H. Beyer, Standard Mathematical Tables, 28th edition, (1987), by CRC. Press Inc.

P. Barrucand and H. Cohn, A Rational Genus, Class Number Divisibility, and Unit Theory for Pure Cubic Fields. J. Number Theory, 2 (1970), 7-21.

P. Barrucand and H. Cohn, Remarks on Principal Factors in a Relative Cubic Field. J . Number Theory, 3 (1971), 226-239.

N. Blackburn, On Prime Power Groups with two Generators. Proc. Cambridge Phil. Soc., 5 4 (1958), 327-337.

[B-W-Z-711 B. D. Beach, H. C. Williams, C. R. Zarnke, Some Computer Re- sults on Units in Quadratic and Cubic Fields. Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathe- matical Congress (Lakehead Univ., Thunder Bay, Ont. 1971), 609-648.

[De-001 R. Dedekind, ~ b e r die Anzahl der Idealklassen in reinen kubischen Zahlkorpern. J . reine angewandte Mathematik, Bd., 121 (1900), 40-123.

[Fr-931 G. Frei, The'orie des corps de classes. Notes de cours, U. Laval 1992-1993.

[Ge-751 F. Gerth 111, On 3-Class Groups of Pure Cubic Fields. J. reine angew. Math., 278/279 (1975), 52-62.

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[Ger-761

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The Capitulation Problem for certain Number Fields 48 1

F. Gerth 111, On 3-Class Groups of Cyclic Cubic Extensions of Certain Number Fields. J . Number Theory, 8 (1976), 84-98.

F. Gerth 111, Ranks of %Class Groups of non-Galois Cubic Fields. Acta Arithmetica, 3 0 (1976), 307-322.

G. Gras, Sur les 1-classes d'ide'aux dans les extensions cycliques relatives de degre' premier 1. I,II, Ann. Inst. Fourier (Grenoble), v. 23, no. 3 (1973), pp. 1-48; ibid. v. 23, no 4, (1973), pp. 1-44.

F. P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. J. reine angew. Math., 336 (1982), 1-25.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, 84, Springer- Verlag (1982).

M. Ishida, The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics Vol. 555, Springer-Verlag (1976).

M. C. Ismaili, Sur la capitulation des 3-classes d'ide'aux de la cl6ture normale d'un corps cubique pur. ThBse de doctorat. Uni- versitd Laval - Qudbec - Canada. (1992).

J. F. Jaulent, L'e'tat actuel du problkme de la capitulation. Sdminaire de thdorie des nombres de Bordeaux, 1987-1988, ex- posd no. 17.

H. Kisilevsky, Number Fields with Class Number Congruent to 4 mod 8 and Hilbert's Theorem 94. J. Number Theory, 8, (1976), 271-279.

P. Kaplan, Diwisibilite' par 8 du nombre de classes des corps quadratiques dont le 2-groupe des classes est cyclique et re'ciprocite' biquadratique. J. Math. Soc. Japan. vol. 25, No 4, (1973).

P. Kaplan, Sur le 2-groupe des classes d'ide'aux des corps quadratiques. J. reine angew. Math., 283/284, (1976), 313-363.

T. Kubota, ~ b e r die Beziehung der Klassenzahlen der Un- terkorper des bizyklischen Zahlkorpers. Nagoya Math. J., 6 , (1953), 119-127.

T. Kubota, ~ b e r den bizyklischen biquadmtischen Zahlkorper. Nagoya Math. J., 10, (1956), 65-85.

S. Kuroda, ~ b e r den Dirichletschen Zahlkijrper. J. Fac. Sci. Imp. Univ. Tokyo, Sec. I, vol. IV, part 5, (1943), 383-406.

K. Miyake, Algebraic Investigations of Hilbert's Theorem 94, the Principal Ideal Theorem and Capitulation Problem. Expos. Math., 7, (1989), 289-346.

J. Neubiiser, Die Untergruppenverbande der Gruppen der Ord- nungen 5 100 mit Ausnahme der Ordnugen 64 und 96. Publi- cations de 1'U. Kiel, (1967).

C. J. Parry, Bicyclic Bicubic Fields. Can. J. Math., vol. XLII , no. 3, (1990), 491-507.


[S-T-341 A. Scholz und 0. Taussky, Die Hauptideale der kubischen Klassen- korper imaginar-quadratischer Zahlkorper : ihre rechnerische Bestimmung und ihr Einflup auf den Klassenkorperturm. J. reine angew. Math., 171 (1934), 19-41.

[Su- 911 H. Suzuki, A Genemliration of Hilbert's Theorem 94. Nagoya Math. J. , vol. 121, (1991).

[Te-711 F. Terada, A Principal Ideal Theorem i n the Genus Fields, TBhoku Math. J . , Second Series, vol. 23, (4), (1971), 697-718.

[Wa-661 H. Wada, O n the Class Number and the Unit Gmup of Certain Algebraic Number Fields. Tokyo U. , Fac. of Sc. J., Series I, 13, (l966), 201-209.

Department of Mathematics, Faculty of Sciences, University Mohamed I , Oujda, MOROCCO.


Hiroshi Suzuki

In our previous paper [7], we proved a generalization of Hilbert's The- orem 94 which also contains the Principal Ideal Theorem. However, Tannaka-Terada's Principal Ideal Theorem was not contained in it. The purpose of this paper is to extend the main theorem of [7] in a natural way so that it contains Tannaka-Terada's Principal Ideal Theorem as a special case. Our main theorem (Theorem 1) now contains all of the three capitulation theorems: Hilbert's Theorem 94, the Principal Ideal Theorem and Tannaka-Terada's Principal Ideal Theorem.

Introduction.

For an algebraic number field k of finite degree, we denote the ideal class group of k by Clk and the Hilbert class field (namely the maximal unramified abelian extension) of k by H k . For a Galois extension K of k, we denote its Galois group by G(K/k). For a group G, we denote the commutator subgroup of G by Gc and we write G~~ = G/GC. We denote the integral group ring of G by Z[G] , and its augmentation ideal by IG = (g - 1 : g E GJzlGl For a finite group G we denote the trace of G by T ~ G = g E iZ[G]. For a Z[G]-module M, we denote the

sEG submodule consisting of G-invariant elements by

In Suzuki [7] we proved the following theorem.

Theorem (old version). Let K be a n unramified abelian extension of an algebraic number field k of finite degree. Then the number of ideal classes of k which become principal in K is divisible by the degree [K : k]

Received October 8, 1998. Revised December 16, 1998.

484 H. Suzuki

of the extension K l k . Namely we have

where i : Clk + CIK is the homomorphism induced by the inclusion map of corresponding ideal groups.

In the case where K l k is cyclic this theorem is nothing else than Hilbert's Theorem 94 (Hilbert [3] ).

Hilbert's Theorem 94. Let K be a n unramified cyclic extension of an algebraic number field k of finite degree. Then the number of ideal classes of k which become principal in K is divisible by the degree [ K : k ] .

Our old version contains the Principal Ideal Theorem (Furtwangler[2]), that is the case K = H k , because the degree [Hk : k] is equal to the order IClkl.

Principal Ideal Theorem. Every ideal of k becomes principal i n

Hk

The old version, however, does not contain Tannaka-Terada's Principal Ideal Theorem (Terada[8]).

Tannaka-Terada's Principal Ideal Theorem. Let k be a finite cyclic extension of an algebraic number field ko of finite degree and K be the genus field of k / k o (the maximal unramified extension of k which is abelian over k o ) Then every G(k/ko)-invariant ideal class of k becomes principal i n K .

The purpose of this paper is to prove the following main theorem.

Theorem 1. Let k be a finite cyclic extension of an algebraic number field ko of finite degree, and let K be a n unramified extension of k which is abelian over ko. Then the number of those G(k/ko)-invariant ideal classes of k which become principal i n K is divisible by the degree [ K : k] of the extension K l k . Namely

O n the Capitulation Problem 485

Our new theorem obviously contains the old version, that is the case k = ko. Now, suppose that K is the genus field of k / k o . Then we have [K : k ] = IClk/IG(k/ko)ClkI = I C Z , G ( ~ ' ~ O ) ~ . Therefore our theorem clearly implies

C I ~ ( ~ ' I * O ) c Ker ( C l k i C I K ) .

This is Tannaka-Terada's Principal Ideal Theorem. Hence our theorem contains Hilbert's Theorem 94, the Principal Ideal Theorem and Tannaka-Terada7s Principal Ideal Theorem.

Tannaka-Terada's Principal Ideal Theorem was generalized for endomorphisms in Miyake[5]. This method gives us an endomorphism version of Theorem 1. (About the history and the fundamental theorems of the capitulation problem see Miyake[G] .)

Theorem 2 (endomorphism version). Let K l k be an unmmified abelian extension, and let a be a n endomorphism of G ( H K / k ) such that a ( G ( H K / K ) ) c G ( H K / K ) and suppose that a induces the identity map on G ( K / k ) . Then a induces an endomorphism of C l k through the isomorphism Clk S G ( H K / ~ ) " ~ given by Artin's Reciprocity Law, for which we have

[ K : k ] I / { a E Ker (Clk + C I K ) ; a ( a ) = a ) [ .

To prove Theorems 1 and 2, we consider the group transfer of Ga- lois groups which corresponds to the homomorphism of lifting ideals (Artin[l]):

Thus Theorem 2 is equivalent to the following group theoretical version:

, Theorem 3 (group theoretical endomorphism version). Let a be an endomorphism of a finite group H , and N be a normal subgroup of H containing H c . Assume that a ( N ) c N and that a induces the identity map o n G = H I N . Then the order of the subgroup

i I { h H c E Ker V H - ~ : a(h)h- ' E H C }

j of the transfer kernel i s divisible by [GI = [ H : N ] . Here VH-N : H " ~ +

I N " ~ denotes the group transfer from H to N .

k. .

486 H. Suzuki

Now we summarize the method of Miyake[5] for the convenience of the reader. Consider the descending series

and take r large enough so that this series becomes stable. Put Ho = ar ( H ) , No = N n Ho and N' = Ker a r . Then we can write H and N as a -stable semidirect products H = Ho K N' and N = No K N'. In this case, we have

Ker (VHo+.No : ~g~ N:~) C Ker (VH-N : H~~ 3 N ~ ~ ) .

Moreover, the restriction alHo of a to Ho is an automorphism of Ho. By taking Ho instead of H , we may assume that cr is an automorphism.

Therefore we have only to prove the following group theoretical version of Theorem 1 which is the case of Theorem 3 in which a is an automorphism.

Theorem 4 (group theoretical version) . Let N be a normal subgroup of a finite group H containing the commutator subgroup H C of H . Suppose that a finite cyclic group A of automorphisms of H is given, and assume that N is stable under A and that A acts trivially on G = HIN. Then the order of the A-invariant part of the transfer kernel is divisible by the order /GI of G.

This theorem contains the group theoretical versions of Hilbert7s The- orem 94, the Principal Ideal Theorem and Tannaka-Terada's Principal Ideal Theorem.

Remark 1. If A is a non-cyclic abelian group, then the group theoretical version does not hold in general. For example, take a Z[A]- module M of finite order such that I M~ ( < I M / I A M 1 , and put H = M, N = IA M . (More interesting examples of transfer kernels with an action of non-cyclic abelian groups are seen in Miyake[G].)

In Section 1, we reduce Theorem 4 to the property for the divisibility of the order of a cohomology module (Proposition 1). In Sections 2 and 3, we give an annihilation mechanism on Z[G x A]-modules (Proposition

% On the Capitulation Problem r 487

+X ?2

2) by a careful calculation of determinants in the one-variable polyne mial ring Z[G][T] over Z[G]. In Section 4, we dualize this proposition to obtain Proposition 5. In the final section we translate this annihilation mechanism into a property for the divisibility of the order of a cohomology module by the technique used in Suzuki[7] which may be explained in the following way: "If a natural number annihilates a cyclic group,

.,I , then the order of the cyclic group divides the natural number". This completes our proof of Proposition 1.

1 Reduction to module theoretical version.

We do not bother to introduce Artin's splitting module, because we only need its kernel.

Lemma 1. Let H be a finite group and N be a normal subgroup of H . Put G = H / N and take a free presentation .rr : F + H of H . Then we have a commutative exact diagram

Put R = Ker (~I , - l (~)nb : ( N ) . ~ -+ NO^). Then

H'(G, R) " Ker (VH+.N : I fab + Nab) .

(Throughout this paper, cohomology is Tate cohomology of a finite

group. ) Proof. Since F~~ is Ztorsion free, the multiplication by the order \GI,

is injective. Hence the transfer map

is injective. Note that T - ' ( N ) " ~ is isomorphic to the kernel

488 H. Suzuki

rankF of the homomorphism which maps the canonical basis ej of @ Z[G]

to .rr(xj) - 1 for j = 1, . . . , rank F, where xj are the canonical free generators of F (see Lyndon[4]). Therefore we have

The group transfer VF,, - 1 ( N) coincides with the homomorphism

induced by the trace map

TrG : s - ' ( ~ ) " ~ -+ T-'(N)"~.

Then we easily see

Gab 2 V~,,-I(~)(F~~)/V~,~-~(~)(A-~(N)F~/F~)

" v~, , - I (~) ( F " ~ ) / T ~ ~ ( T - ' (N)"~)

HO(G, T-'(N)"~). -

Hence the image of VF,n-~(N) must coincide with (a-I ( N ) ~ ~ ) ~ . From the commutative diagram

we see

Ker ( V H , ~ : Hab -) Nab) N = v~,,-I(~)(F"~) f l K e r % l , - ~ ( ~ ) ~ b / ~ ~ , , - l ( ~ ) ( ~ e r r ' ~ )

= R n (T-' ( N ) " ~ ) ~ / TrGR

= R ~ / T ~ ~ R = HO(G,R).

0

Now assume that a finite group A acts on H as automorphisms and that N is an A-subgroup. We take a free presentation in the following manner. Let U = A K H be the semidirect product of A and H. Then we have a short exact sequence

On the Capitulation Problem 489

Take a free presentation po : Fo + U of U ; then we have a commutative exact diagram

pol - l ( N ) a b Po 1

1 + Nab -) U/NC I I

+ A K G + 1.

Then the subgroup F = p i ' (H) of Fo is a free group, and

P O I F : F ++ H

is a free presentation of H. Put

R = Ker (pol - Po l ( ~ ) a b

: %'(N)"~ --+ Nab).

Then, by Lemma 1, we have an isomorphism

Ker ( V H - ~ : Hab + Nab) HO(G, R).

By the choice of the free presentation, the commutative diagram

in the proof of Lemma 1 is a commutative diagram of Z[A]-homomorphisms. Therefore the above isomorphism is a Z[A]-isomorphism. Hence the A-invariant parts are also isomorphic:

Ker ( V H - ~ : Hab + Z HO(G, R ) ~ .

Since I ~ & ' ( N ) " ~ / R I = lNabl is finite, we have

R B ~ Q = P , - ' ( N ) ~ ~ B ~ Q

where Q is the rational number field. Since the sequence

is exact and Q[A K G] is a semisimple Q-algebra, we see that

492 H. Suzuki

Thus we finally have

Lemma 3. Let G and A be as in Proposition 2. Let M be a Z[G x A]- m

module such that M @z Q Z $Q[G x A] for some m > 0 and take a set

of generators wl, . . . , wm+,. Let (wi) be the column vector and put w =

(wi). Assume that a square matrix Q = (qij) E M(m + n,Z[G x A]), that is, Q is of size (m + n) x (m + n) with entries in Z[G x A], satisfies Q . w = 0. Then all the minors of Q of size greater than n are zero.

Proof. Let q be a prime ideal of Q[G x A], then the localization

Q[G x A], at q is a field. Since M @z Q Z ZQ[G x A], (M @z Q),

is a linear space of dimension m over Q[G x A], . Because Qw = 0 and wl, . . . wm+, spans ( M @z Q),, the rank of Q at q is at most n. Thus all the minors of Q of size greater than n are zero at all prime q of Q[G x A]. Therefore all the minors of Q of size greater than n are zero in Z[G x A]. The lemma is proved. 0

Let Z[G][T] be the polynomial ring in T over the group ring Z[G], and let p : Z[G][T] ++ Z[G x A] be a surjective homomorphism of Z[G]- algebras given by p(T) = a - 1. Note that Kerp = ((T + l)IAI - l)z[Gl[T]. Write (T + l)IAl - 1 = T . f (T). For a matrix, by abuse of notation, we denote the homomorphism obtained by applying p to every entry also by P.

Lemma 4. Let S be a noetherian ring, x be a n element of S such that x is not a zero divisor and (x) is equal to i ts radical. Let Q be an m + n square matrix such that Q modulo p has at most rank n at every minimal prime p of (x). Then xm divides det(Q).

Proof. Let pl, . . . , p, be the minimal primes of (2). Since the radical of (x) is equal to (x), (x) = pl n . - . n p,, and xSp = p j Sp j. Because the rank of Q mod pj is at most n, det Q is contained in the m-th power


xmSpj of the maximal ideal xSpj for all pj. Then det Q is contained in xmU-IS, where U = S \ (pl U . - . U p,) and U-'S is the localization of S by U. Therefore f det(Q) E (xm) for some f E U. Now f is in no minimal prime over (x), so the multiplication by f is injective on S/(x). Since x is not a zero divisor, the multiplication by x1 induces an isomorphism S/(x) Z (x1)/(x'+l) for every 1. The multiplication by f is injective on (x")lxl+ l) and also on S/(xl) for all 1. Thus det(Q) E (xm) as claimed. 0

Remark 2. Let G, A, M, w and Q be as in Lemma 3. Take a matrix Q = (@ij) E M(m + n, Z[G][T]) such that p ( ~ ) = Q. Then putting S = Z[G] [TI and x = ( T + l)IAl - 1 in Lemma 4, we have .

( (T + l)lAl - I det Q.

Furthermore all the cofactors of Q are divisible by ((T + l)IAl - I ) ~ - ' .

For x E Z[G][T], define x(<') and ~ ( 2 ' ) E Z[G] [TI by

and denote the coefficient of T' by x(') E Z[G]. For a matrix, we extend this definition to the whole matrix if it applies to all the entries. Denote the natural projection by

then x(O) is the image of x by pr o p.

Remark 3. Under the hypotheses of Remark 2, we have

494 H. Suzuki

Lemma 5. Under the hypotheses of Remark 2, we have

where E is the unit matrix and adj Q is the cofactor matrix of Q.

Proof. Since the cofactor matrix adj Q is divisible by Tm- l , we have

Since det Q is divisible by T m , (adj Q ) ( " - ~ ) Q ( ' ) must be equal to zero. This proves the lemma. [7

53. Proof of Proposition 2.

We may assume m > 0. Take al , . . . , am t ( a - 1) -' IG M as in Lemma 2, and put

Take bl, . . . , bn E M so that bl, . . . , bn, a l , . . . ,am generate M . Put

Then by Lemma 2, we find a square matrix B E M(n, Z ) such that

Bb E m & n ( ~ G x A ~ + Mo) and

det B = IM / IGxAM + MOI = I A ~ ~ ~ H - ~ ( G , M ) ~ I .

There exist matrices J1 E M(n, IGx A ) and L E M(n, m , Z[G x A ] ) such that Bb = Jlb + La. Since ( a - l )Mo c IGM, there exist J2 E


) t M(m + n , Z[G x A]) . -J3 ( a - l ) E - J 2 x = ( B - J 1

Then X . v = 0. Write X = ( x i j ) . Now take a lift jl E M(n, ( IG , T ) Z [ G ] [ T ~ ) , E M(n, m, Z[G][T] ) ,

j 2 E M(m, ( IG)z[G][T]) and j 3 E M(m,n , ( IG)z[G][T]) J l , L, J2 and J3

under the map p, respectively. Put

and write x = (&j) . Then x is a lift of X under p. By Remark 2, det x is divisible by ( ( T + l)lAl - l ) m . Put D = (det x ) ( b m ) . Then by Lemma 5,

Note that D is divisible by f ( T ) m and that

D = det B I ( A ~ ~ ~ H - ' ( G , M ) ~ I mod ( IGl T ) Z I G I M .

m+n Take an element c = ( c l , . . . ,cm+,) of @ Z[G x A] such that c . v E

m+n M ~ ~ ~ , and take a lift t = ( E l , . . . , Em+,) E @ Z[G] [TI of c. Put

Then we see that

H. Suzuki On the Capitulation Problem

is divisible by f (T)m. By Remark 3, we have

In fact,

We have

Therefore the determinant

is divisible by ( (T + l)IAl - I)", and hence

Here C is taken over all t l , . . . ,t, > 0 with tl + ... + t, = m - 1, all 1 < k2 < . . < k, < m+n except i and all distinct 1 < l l , . . . , 1 , < m+n. The indices 1 < il < . . . < is j m + n are taken as {il, . . . ,is} =

{ l , . . . ,m+n) \{ i ,k 2, . . . , k , ) , a n d l < jl < ... < j, <m+nare taken a s { j i , . - . , j s } = { l , . . . , m + n ) \ { l l , . . . ,l,). S i n c e r s t l + . . . + t , = m - 1, we see that s = m + n - r > n. Therefore by Remark 3, all of

498 H. Suzuki

the terms in C vanish. Hence we have

Thus Di is divisible by f (0)" = I A J m . Moreover since

On the Capitulation Problem

a similar argument to the above also implies

(C is again taken over all t l , . . . , t , > 0 with t l + . + t , = m - 1, all 1 < kz < - . . < k, < m+n except i and all distinct 1 < l l , . . . ,1, < m+n. The indices 1 < il < . . . < is < m + n are taken as { i l , . . . , i s ) =

{ I , . . . , m + n ) \ { i , k z , . . . , k,), and 1 < jl < . . . < j, 5 m+n are taken as { j l , . . . , j s ) = 11,. . . ,m + n ) \ 111 , . . . ,Z,).) By Remark 2, we have that

500 H. Suzuki

is divisible by Tm. Therefore the coefficient of Tm-l in the determinant is equal to zero. Thus we conclude that (g - l)Di are zero for all g E G. Hence

Since we have

each Di is congruent to zero modulo IG and hence equal to zero for 1 5 i 5 n. Take hk E Z such that D,+k = hklAlmTrG for k = 1, . . . ,m. Now put D = p ( d ) = p(det ~ ( 2 ~ ) ) . Then f (T)m divides D. Since (a - 1) f (a - 1) = 0, ( a - l ) D is equal to zero. Moreover X v = 0. Therefore

Since we have

we see that

This is true for every cv E MGxA. Since D E IA(" /H-~(G,M)~I mod IG A and since c . v runs through all the elements of MG A , we see that

By assumption M is Z-torsion free. Therefore we conclude

Proposition 2 is now proved.

On the Capitulation Problem 50 1

54. Dualization of Proposition 2.

In this section, we dualize Proposition 2 through the intermediary of finite modules.

Proposition 3 (finite dual annihilation version). Let G be a finite abelian group and A be a finite cyclic group generated by a . For a Z[G x A] -module M of finite order, we have

Proof. Let s be the exponent of M and denote the Pontrjagin dual of M by MA. Take a sufficiently large natural number m and take a surjective homomorphism

Then, since Z/sZ[G x A] A Z Z/sZ[G x A], we have an injective homomorphism

Let us consider M as a submodule of EZ/SZ[G x A]. Let R be the in-

verse image of M by the natural projection p : ~ Z [ G x A] ++

ZZ/SZ[G x A]. The kernel of p is isomorphic to ~ Z [ G x A]. There-

fore we have an exact sequence

Then H-I (G, R) Z H-l (G, M) as Z[A]-modules and

H - l ( ~ , R ) ~ z H - l ( ~ , M ) ~ .

Moreover, we have

The exact sequence given above induces an exact sequence

502 H. Suzuki

It is clear that

I G . g Z I G x A ] c g Z I G x A ] n I G R

c I G . g Z I G x A].

Therefore the term ~ Z [ G x A ] / ~ z [ G x A] n IG R in the previous ex- m

act sequence is isomorphic to the free Z[A]-module $Z[A]. Thus the

homomorphism R/IGR - M/IGM induced by R -t M induces the isomorphism

It is easy to see that

R ~ ~ ~ / T ~ G ( ( ~ - I)-' IGR) h. C O ~ ( n G : HO(A, RIIGR) - HO(G x A, R))

and

M G X A / n G ( ( a - l)-'IGM) g Cok (TrG : H'(A, M/IGM) - HO(G x A, M)).

Hence we have

R~~~ / TrG((a - I)-'IGR) 2 M~~~ / n G ( ( a - ~)-'IGM).

Proposition 2 for R now gives Proposition 3 for M. 0

Next we take the Pontrjagin dual of the preceding proposition.

Proposition 4 (finite annihilation version). Let G be a finite abelian group and A be a finite cyclic group. Let M be a Z[G x A]- module of finite order. Then

Proof. Take the Pontrjagin dual M A of M , then HO(G, M)" " HP'(G, MA). Since (HO(G, M ) ~ ) ' = IAH-'(G, MA) , we have

On the Capitulation Problem

and

I(HO(G, M ) ~ ) ~ I = IH-'(G, M " ) / I ~ H - ~ ( G , M")I

= I H - ~ ( G , M ~ ) ~ I .

Since ( M G ) I = IG(MA), we see

(IA . M G ) I = (a - 1)-'IG(MA), G I (TrG-'(IA . M )) = TrG((a - 1)-I IG(MA)) .

(Here a is a generator of the cyclic group A as before.) Combining this with (IGXAM)' = ( M " ) ~ ~ ~ , we have

( D ~ - ' ( I ~ . M ~ ) / I G ~ ~ M ) ^ 2 ( M ~ ) ~ ~ ~ / ~ I ~ ( ( o I - I ) - ' I ~ ( M ~ ) ) .

Thus Proposition 3 for M A implies Proposition 4 for M . 0

Proposition 5 (annihilation version). Let G be a finite abelian group and A be a finite cyclic group. Let M be a finitely generated

Z[G x A]-module such that M &, Q E ~ Q [ G x A], and suppose that M

is Z-torsion free. Then we have

IH'(G, M ) ~ I . T~G-'(IA - MG) C IGxAM.

Proof. By assumption M contains a Z[G x A]-submodule of finite index m

which is isomorphic to $Z[G x A]. Put N = M / ~ Z [ G x A]. Then

HO(G, M) 2 HO (G, N)

as Z[A]-modules and

HO(G, M ) ~ N HO(G, N ) ~ .

Moreover, we have

H - ~ ( G x A, M) h. H - ~ ( G x A, N).

The exact sequence

induces the exact sequence

504 H. Suzuki

Therefore we have

HO(A, M ~ ) = HO(A, N ~ ) .

It is easy to see that

~ G - ' ( I A M G ) l ~ c x a M

2 Ker (nG : H-I (G X A,

and

~ G - ' ( I A . N ~ ) / J G ~ A N

S Ker (TrG : H-'(G x A, N) - H-' (A, NG)).

Hence we have

nG-l(~A . M ~ ) / I ~ ~ ~ M " T ~ / ' ( I ~ . NG)/IGXAN.

Therefore Proposition 4 for N proves Proposition 5 for M. 0

55. Proof of Proposition 1.

In this section, we convert the annihilation property into a comparison property of orders, and this will complete the proof of Theorem 4. Now we begin the proof of Proposition 1.

Proof of Proposition 1. Denote the homomorphisms given by projections by

and

Put Ro = Kerp;! and R1 = pl(R). Then the short exact sequence

0 - Kerpl -, R - R1 --+ 0

gives us the long exact sequence

. - - H-'(G, R) - H-I (G, R1) - HO(G, Ker - HO(G,R) + HO(G,R1) - H 1 ( G , ~ e r p l ) - .


Since Kerpl S Z, we have HO(G, Kerpl) Z/IGIZ and H1(G,Kerpl) = 0. Thus we obtain an exact sequence

In the short exact sequence

p2(R) is isomorphic to Z. Therefore we have

Moreover, since pi 1 Ro : Ro - R1 is injective, we see that

Since IG Rl = PI (IG R) C pi (RO) , we have

Since IA R c Ro and R/& 2 Z, we have

Hence we obtain

where a is a generator of the cyclic group A. Note that, for r E R and g E G, (g - l ) r = 0 is equivalent to (g - l )pl(r) = 0. Since p l l R o is injective, (a - l ) r E TrGRO is equivalent to (a - l)pl(r) E TrGpl(&) for r E R. Hence we have

and

506 H . Suzuki

Since

we have

Since R l / p l ( R o ) r Z/ rZ for some r E Z , the subquotient

is a cyclic quotient of R~ n T ~ G - ~ ( I A . R P ) / I ~ ~ A R ~ . Since RI ~z Q EQ[G x A] , Proposition 5 shows that

Since P I ( & ) + R1 n T r G - l ( I ~ . R f ) I p l ( & ) is a cyclic group, this annihilation means that the order Ipl (&) + R1 n TrG-l ( I A . R P ) / ~ ~ (RO)I divides IHO(G, R ~ ) ~ I . Thus we have. shown that ]GI divides IHO(G, R ) ~ I.

Therefore the proof of Proposition 1 is completely done, and hence The- orem 4 is proved now. 0

References

[ I ] E. Artin, Idealklassen i n Oberkorpern und allgemeines Rezzprozitats-gesetz, Abh. Math. Sem. Univ. Hamburg, 7 (1930), 46-51; Collected Papers, 159-164.

[2] Ph. Furtwangler, Beweis des Hauptidealsatzes fur Klassenkorper algebmischer Zahlkorper, Abh. Math. Sem. Univ. Hamburg, 7 (1930), 14-36.


[3] D. Hilbert, Bericht: Die Theorie der algebraischen Zahlkorper, Jber. d t . Math.-Ver., 4 (1897), 175-546; Gesam. Abh. I., 63-363.

[4] R. C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. o f Math. (2)52 (1950), 650-665.

[5] K . Miyake, On the structure of the idele groups of algebraic number fields 11, Tbhoku Math. J . , 34 (1982), 101-112.

[6] K . Miyake, Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and capitulation problem, Expo. Math., 7 (1989), 289-346.

[7] H. Suzuki, A generalization of Hilbert's Theorem 94, Nagoya Math. J . , 121 (1991), 161-169.

[8] F . Terada, On a generalization of the principal ideal theorem, Thhoku Math. J . , 1 (1949), 229-269.

Department of Mathematics, School of Science, Nagoya University Chikusa-ku, Nagoya 4 64-01 Japan


Adele Geometry of Numbers

Masanori Morishita and Takao Watanabe

Dedicated to Professors Ichiro Satake and Takashi Ono

This is a historical and expository account on adele geometry. The word adele geometry appeared in the lectures ([W4], 1959-1960) by A. Weil which were stimulated by works of Siegel-Tamagawa on quadratic forms. Our main concern here is to exhibit the strings of thoughts in the development of this topic originated from Minkowski's geometry of numbers. The subject was originally related to integral geometry and some diophantine problems, and we discuss such aspects in adele geometry on homogeneous spaces.

Contents

1. From geometry of numbers to adele geometry 1. I . Minkowski's geometry of numbers 1.2. Siegel's main theorem and mean value theorem 1.3. Ono's G-idele and Tamagawa's interpretation of Siegel's

formula 1.4. Weil's integration theory and adele geometry

2. Tamagawa numbers and the mean value theorem in adele geometry

2.1. Tamagawa numbers 2.2. Mean value theorem in adele geometry

3. Geometry of numbers over adele spaces 3.1. Adelic fundamental theorems of Minkowski 3.2. Adelic Minkowski-Hlawka theorem 3.3. Generalized Hermite constants

4. Distribution of rational points in d e l e transformation spaces 4.1. Hardy-Littlewood variety and Weyl-Kuga's criterion

on uniform distribution

Received August 28, 1998 Revised October 5, 1998

510 M. Morishita and T. Watanabe

4.2. Hardy-Littlewood homogeneous spaces 4.3. Rational points on flag varieties

1 From geometry of numbers to adele geometry

1.1. Minkowski's geometry of numbers.

Following the footsteps of Gauss, Dirichlet, Eisenstein and Hermite, Minkowski developed the theory of quadratic forms and created the geometry of numbers. Suppose that f is a positive definite quadratic form on Rn whose determinant is assumed to be 1 for simplicity. Hermite showed, as a consequence of his reduction theory, the inequality

The function m( f ) of f is bounded and pn := mfm m( f ) is called the

Hermite constant. Minkowski improved this upper bound by a simple geometric idea which is the basis of his geometry of numbers. Consider the ellipsoid B = {x E Rn : f (x) < c) for some c > 0. It is a symmetric convex body. Then, Minkowski's first convex body theorem ([M, VIII], 1891) says that

if the Euclidean volume vol(B) > 2", then B contains an integral point other than the origin.

Since vol(B) = cnl2vn, where Vn stands for the volume of the n- dimensional unit ball, we have

Minkowski then introduced the notion of successive minima for any convex body to deepen the first theorem. Using the above notations, the i-th successive minima of f is defined by

Xi := inf{X > 0: XB contains i linearly independent integral points)

for 1 5 i 5 n (A: = m( f )). Minkowski's second fundamental theorem ([M, GZ], 1896) is then stated as

It should be noted that Minkowski's convex body theorem can be seen as an analogue of Riemann-Roch theorem for an algebraic curve and has

Adele Geometry of Numbers 511

basic applications in algebraic number theory such as the finiteness of an ideal class group and so on.

Related to the lower bound for the Hermite constant, Minkowski asserted, in his letter to Hermite ([M, XI], 1893), the following:

Suppose B is an n-dimensional star body whose volume is less than C(n), where C(s) is the Riemann zeta function. Then, there is a lattice L of determinant 1 such that B n L = (0).

This yields immediately the following inequality

Minkowski's assertion was proved by Hlawka about fifty years later. In his reduction theory of quadratic forms ([M, XXI], 1905), Minkow-

ski computed the volume of the set of all positive definite quadratic forms with determinant < 1. Namely, he gave a formula for the volume of SL, (R)/SLn(Z) in terms of the special values of the Riemann zeta function.

Finally, we should mention Minkowski's two contributions to the theory of quadratic forms, preceding the geometry of numbers. One is the local-global principle on the equivalence of quadratic forms ([M, VII] , 1890), which was completed by Hasse ([Ha], 1924) over any number fields based on the notion of a p-adic number field introduced by Hensel (1909). The other is the mass (weight) formula for a genus of a positive definite quadratic form ([M, IV] , 1885), which was vastly developed by Siegel. This is our next focus.

1.2. Siegel's main theorem and mean value theorem.

Siegel's main theorem on quadratic forms is a precise quantitative version of the Hasse principle. Let S be a positive definite symmetric integral n x n-matrix, n 2 3. Consider the diophantine equation

which defines a f n(n - 1)-dimensional affine variety over Z. For a prime P , set

[{X E Mn(Z/peZ) : t X S X = S mod pe)] cr,(S) := lim

7 e-oo 2p$n(n-l)e

where [*] stands for the cardinality of a finite set *. For an infinite prime oo, cr,(S) is defined as follows. Let U be a neighhborhood of S


in the space of positive definite symmetric matrices of size n and vol(U) denotes its Euclidean volume. Then, set

vol({X E Mn(R): tXSX E U}) a,(S) := lim

U-S vol( U)

Here, the product np ap(S) and a, (S) are essentially the singular series and integral respectively in the work of Hardy-Littlewood on the Waring problem ([H-L], 1920). Compared with Hardy-Littlewood's asymptotic formula, Siegel's formula gives an accurate balance between the size of integral solutions and these local densities. For a complete set S1, . . . , Sh of representatives of proper classes in the genus of S, let E(Si) denote the order of the unit group SO(Si)(Z). Then, the mass formula of Siegel ([Sl], 1935) is stated as

In fact, Siegel showed more general formulas including the case that S is indefinite where l/E(Si) is replaced by the volume of SO(Si)(R) /SO(Si)(Z) ([S2], 1936, 37, [S3], 1944).

After Hlawka proved Minkowski's assertion ([Hl], 1944), Siegel refined Hlawka's inequality to the equality ([S4], 1945). Let dg be an invariant volume element on SL, (R) normalized by void, (SL, (R) /SL,(Z)) = 1 and dx be the Euclidean measure on Rn. Denote the set of all primitive integral vectors by P. Then, Siegel's mean value theorem asserts: for a compactly-supported continuous function cp on Rn7

This theorem gives the most satisfactory explanation to both Minkowski- Hlawka's theorem and Minkowski's computation of the volume of the space of positive definite quadratic forms with determinant 5 1.

Here, we wish to explain how the mean value theorem can be seen as a formula in integral geometry. Set L = { g ( P ) : g E SL,(R)) that is identified with the space of all lattices of determinant 1, namely SL,(R)/SL,(Z). Let Z be the space {(L,x) E L x Rn: x E L} describing the incidence relation and look at the diagram:


Then, compute the integral

in two ways, noting that the volume vol({g: x E g(P)} ) is independent of x. Then we get the mean value theorem. In particular, we get Minkowski-Hlawka's theorem if we compute the volume p-l (B) in Z in two ways. This is the same idea as in Crofton's formula and others in integral geometry (cf. [San]) .

1.3. Ono 's G-idele and Tamagawa7s interpretation of Siegel's formula.

Inspired by Chevalley's work on class field theory and linear algebraic groups, T. Ono introduced the adele group (or adelization) of a linear algebraic group G ([Ol], 1957). It is called G-idele. Let G be a linear algebraic group in GL, defined over a number field k. The adele group G(A) is a locally compact group obtained as a restricted product of the local groups G(k,) and the group of global rational points G(k) is embedded in G(A) as a discrete subgroup. As applications of the adelizations, Ono generalized the finiteness of an ideal class group and Dirichlet's unit theorem to those for any solvable algebraic group ([02], 1959).

On the other hand, at the Tokyo-Nikko conference in 1955, M. Kuga already had formulated Siegel's mean value theorem in the adelic manner and posed a problem to interpret the mass formula in that context

([Kull). It was T. Tamagawa who discovered new interpretation and proof of

Siegel's mass formula by introducing the Tamagawa measure on the adele group in late fifties (cf. [TI). Notation being as in 1.2, let G = SO(S) be the algebraic group defined by

t~~~ = S and det(X) = 1 .

An invariant gauge form w on G defined over Q defines a local measure w, on G(Q,) for each p and w, on G(R). We then have the volume- theoretic interpretation of ap(S) and a,(S) as follows:

~ P W = dw, , (p = prime); a, (S) =

For the open subgroup G(Am) = G(R) x np G(%) of the adele group G(A), there is a bijection between the set of classes in the genus of S and


the double coset space G(Am)\G(A)/G(Q), under which Si is supposed to correspond to G(Am)giG(Q). Then, we have the decomposition

The product w~ = w, defines an invariant measure, called the Tam-

agawa measure, on G(A) and V O ~ , ( ~ ~ G ( A ~ ) ~ ~ ' ) = n, a, (S). Since

giG(Arn)g;' n G(Q) is isomorphic to the unit group SO(Si)(Z), we have

Hence, Siegel's formula is equivalent to

In fact, the adelic formulation works over any number field and (T) also yields Siegel's formulas in the indefinite case. Weil ([W5], 1962) showed that r(G) = 2 essentially contains Siegel's main theorems in general.

1.4. Weil's integration theory and adele geometry.

Weil's contribution to this subject is based on his theory of integra- tions on topological transformation spaces ([Wl], 1940). As an application of this general theory, Weil viewed Siegel's mean value theorem as a special case of the following Fubini-type theorem ([W2], 1946). Let X be a topological space on which a locally compact unimodular group G acts transitively. Let L be a discrete subspace on which a discrete subgroup I? of G acts. Let H be the stabilzer of x E L and set y = H n I?. Suppose we have a compactly supported continuous function cp on X. Here is a typical situation:

Then, Weil showed the equality


where dx and dg are suitable matching measures. Stimulated by Tamagawa's discovery (T), Weil introduced the Tam-

agawa number of a linear algebraic group over a global field and posed the Weil conjecture in [W3], 1959 (See 2.1 below for the precise forms). In 1959-60, Weil gave lectures on Tamagawa numbers ([W4]) which have played an important role in the development of the arithmetic of algebraic groups. In these lectures, first of all, Weil introduced the basic notions in adele geometry such as an intrinsic definition of the adelization of any algebraic variety defined over a global field k, convergence factors for a global measure on the adele group and so on. As for the Tamagawa numbers, he computed systematically those for classical groups applying the Poisson summation formula to the above equality in the adelic setting. Pushing this way, Weil later generalized the relation obtained by Siege1 ([S5], 1951, 52) between theta functions and Eisenstein series to the case of classical groups in the adelic and representation-theoretic language ([W6], [W7], 1964, 65).

In the following, we shall see that the topics dicussed above have been developed in the context of adele geometry in the latter half of this century.

$2. Tamagawa numbers and the mean value theorem in adele geometry

2.1. Tamagawa numbers.

From now on, k denotes an algebraic number field and A the adele ring of k. Let V, and Vf be the set of infinite and finite places of k, respectively. For v E V = V, U Vf, k, stands for the completion of k at v. As usual, the multiplicative valuation ( . 1, of k, is normalized so that la[, = p, (aC)/p, (C) for a E k, , where p, is a Haar measure of k, and C is an arbitrary compact subset of k, with nonzero measure. Then 1 . I A = nvEv I . 1, is the idele norm of A X . The precise definition of Tamagawa number is given as follows. Let G be a connected linear algebraic group defined over k. We denote by X* (G) and by Xi (G) the free Z-modules consisting of all rational characters and all k-rational characters of G, respectively. The absolute Galois group I?k of k acts on X*(G). The representation of rk in the space X*(G) @z Q is denoted by CG and the corresponding Artin L-function is denoted by L(s, oG) = nVEV, L,(s, o ~ ) . We set ok(G) = lima+l ( S - l)"L(s, uG), where n =

rankX;(G). For a left invariant gauge form wG on G defined over k, we associate a left invariant Haar measure wf on G(k,). Then, the

Tamagawa measure on G(A) is well defined by wz = Idk 1 - dim W d J f 7

516 M. Morishita and T. Watanabe Adele Geometry of Numbers 517

where wg = nvsv, w:, wfc = ok(G)-' nvEVf ~ ~ ( 1 , o~)w: and Idk[ is

the absolute value of the discriminant of k. For x E Xi(G), let 1x1~ be the continuous homomorphism G(A) + RT defined by I x l A ( g ) = 1 x(g) 1 A .

We write G(A)l for the intersection of kernels of all such 1 ~ 1 ~ ' s . If XI , . . . , Xn is a Z-basis of X i (G), then the mapping

yields an isomorphism from the quotient group G(A)/G(A)l to We put the Lebesgue measure dx on R and the invariant measure dxlx on R: . Then there exists uniquely a Haar measure WG(A) 1 of G(A) such

that the Haar measure on G(A)/G(A)' matching with w 2 and WG(A)I

is equal to the pull-back of the measure ny=l dxi/xi on (R:)" by the above isomorphism. The measure W G ( A ) ~ is independent of the choice of a Z-basis of Xl(G). Since G(k) is a discrete subgroup of G(A)l, we put the counting measure W G ( ~ ) on G(k). Then the Tamagawa number r (G) is defined to be the volume of the quotient space G(A)l/G(k) with respect to W ~ A ) 1 /wG(k).

The basic problem on the measure-finiteness and compactness of G(A)'/G(k), posed by Weil ([W3], 1959), was settled by Mostow- Tamagawa ([M-TI) and Borel-Harish-Chandra ([B-HC], [Bl]) in 1962, and by Harder ([HI], 1969) for the positive characteristic case. The following conjecture was also stated by Weil in [W3].

If G is a connected, simplyconnected semisimple group over a global field, the Tamagawa number r(G) = 1.

Inspired by the Weil conjecture on Tamagawa number, it had been a fundamental problem to get the 'arithmetic index theorem' for the Tam- agawa number after the model of the Gauss-Bonnet theorem for Rieman- nian manifolds and the Cauchy integral formula for complex manifolds. For the works of Ono, Demazure, Mars, Langlands and Lai on this problem, we refer to Ono's appendix to [W4]. Finally, the Weil conjecture was settled by Kottwitz ([KO], 1988) assuming the Hasse principle of H1 of the group of type E8, which was proved by Chernousov ([Ch], 1989). Combined with Ono's relative theory ([04]), the final formula for the Tamagawa number of a unimodular connected linear algebraic group G over Ic is given as follows. Let 7rl (G) be Borovoi's algebraic fundamental group ([Bo]) and (7rl (G)rk)t,r, denote the torsion part of the coinvariant quotient of 7rl(G) under the absolute Galois group rr, of k. As an abstract group, (G) is canonically isomorphic to the topological fundamental group of the complex Lie group G(C). The correspondence

G H 7rl (G) yields an exact functor from the category of connected affine k-groups to the category of rk-mOduk generated finitely over Z. Set

~ e r ' ( k , G) := ~ e r ( H ' ( k , G) --+ IT H1(kV7 G)) . v

Theorem 2.1.1. The Tamagawa number of G is given by

Hence, the Tamagawa number r(G) is determined by the Galois module a1 (G) , for ~ e r ' (k, G) is described by 7r1 (G) ( [Bo] ) .

Remarks 2.1.2. 1) The Weil conjecture in positive characteristic case is still open, except the case of Chevalley groups proved by Harder ([H2], 1974). We are informed that E. Kushnirsky recently proved the Weil conjecture for quasi-split groups in positive characteristic case in a part of his thesis ([Kus]).

2) The proof of the Weil conjecture for number field case involves several steps of quite different features: tori ([03]), quasi-split groups ([Lan], [Lai]), comparison between quasi-split groups and inner forms ([K]) and Hasse principle for H1 of simply-connected groups. It would be desirable to get a unified proof appealing to the simplyconnectedness of the group.

2.2. Mean value theorem in adele geometry.

Ono ([05]) introduced the notion of Tamagawa number for a homogeneous space in connection with the mean value theorem in the following adelic setting. Let G be a unimodular connected linear algebraic group and H be a unimodular connected subgroup of G and set X = GIH. Let wf be the canonical measure on the adele space X(A) so that the matching w z = wfwf holds. Assume further that G and H have no non-trivial k-rational characters. Then X is quasi-affine. Using Kottwitz's theorem, we have the following uniformity.

Theorem 2.2.1 ([Mo-Wall , [Mo] ) . Notations and assumptions being as above, there is a constant r(G, X ) SO that the following equality holds for any compactly supported continuous function cp o n G(A) X (k) :


Here, the constant T(G,X) is called the Tamagawa number of a homogeneous space (G, X) and given by the following formula.

Theorem 2.2.2 ([ibid]).

T(G, X ) = ( G ) r k 1 [ ~ ~ ( ~ ) ~ ~ ] [ ~ o k ( ~ e r l ( k , H) i ~ e r ' ( k , G))] '

In particular, if 7rl(X(@)) = r2(X(@)) = 1, then the mean value property r(G, X ) = 1 holds.

Remarks 2.2.3. 1) F. Sato ([Sat]) developed a theory in another direction which gives a natural interpretation of Siegel's mass in the case of indefinite forms and extends Siegel's main theorem to that for a homogeneous space.

2) As we explained in 1.2, the mean value theorem can be seen as a formula in the integral geometry. The familiar formulas such as Poincark's one in integral geometry deal with the integrals of intersection numbers of subvarieties in an ambient space (cf. [San]). It would be nice if there exist such kind of formulas in adele geometry.

53. Geometry of numbers over adele spaces

3.1. Adelic fundamental theorems of Minkowski.

A generalization of Minkowski's fundamental theorems in the geometry of numbers to those over any number field was first considered by Weyl ([We]) and later by Rogers and Swinnerton-Dyer ([R-S]). Mahler considered the geometry of numbers over rational function fields ([Mah]). The difficulty to work over a number field Ic arises from a fact that a lattice may not be free in general unless the class number of k is 1. This difficulty is automatically resolved by working over the adeles.

Now, let us explain adelic fundamental theorems in the geometry of numbers following Macfeat ([Mac]), Bombieri-Vaaler ([Born-V]) and Thunder ([ThZ]). In the following, 0 (resp. 0,) denotes the ring of integers of k (resp. k, for v E Vf) and r l (resp. 7-2) the number of real (resp. complex) places of k. Let V be an n-dimensional affine space over k. Fix a basis el, . . , en of V(k) and identify GL(V(k)) with GLn(k). Let L = Oel + . . . + Oen be a free 0-module and set L, = L 630 0, for v E Vf. f i r finite adele gf = (gv)vEVf E GLn(Af), define an 0-lattice

gf L by

9, L = (V(k) n 9vLv). vEVf


Then, the set of 0-lattices in V(k) is identified with

Choose a nonempty open bounded convex symmetric subset C, of V(k,) for v E V , and set C, = nu,,, C,. Take gf = (gY)vEVf E GLn(Af) and set

Then, S(C,, gf) is an open relatively-compact subset of V(A) and its volume is

An adelic first fundamental theorem is stated as follows.

Theorem 3.1.1 ([Mac, Theorem 21, [ThZ, Theorem 31). Notation

being as above, suppose 2n[k:Q] < w:(S(c,, Then, S(C,, gf) n V(k) # {o).

Next, for 1 5 i 5 n, define i-th successive minimum for S(C,, gf)

by

Xi(S(Cw 7 9f))

= inf{X > 0: S(XC,,gf) n V(k) contains

k-linearly independent i vectors.)

= inf {A > 0 : XC, n g L contains k-linearly independent i vectors.)

where XC, = nvEVm XC,. Setting Xi = Xi(S(C,, gf )), 1 5 i 5 n for simplicity, an adelic second fundamental theorem is stated as follows.

Theorem 3.1.2 ([Mac, Theorem 5,6], [Bom-V, Theorem 3,6]). One has the inequality

Further, assume that for each complex place v,

zC, = C, for z E k, with lzl, = 1.

Then, one has


Theorems 3.1.1 and 3.1.2 are proved by reworking with the proofs of the classical case in the adelic setting. They can be applied to deal with problems in diophantine approximation. Let us present typical ones.

Let Vm be the m-th exterior product of V for 1 5 m 5 n - 1. If we put el = ei, A - . . A eim, (il < . - < i,), for every subset I = {il,. . . , im) C {1,2, - . . , n), then {ez)lzl=m is a base of V,(k). For each v E V, the local height Hv on Vm(kv) is defined as follows:

If X I , - , x, E V(kv), we write Hv(xl, - . - , x,) for Hv(xl A - . A x,). The global height H on Vm(k) is defined by

Let W be an m-dimentional k-subspace of V(k) and wl, . . . , w, a basis of W. Since we have

for any y E GL(W), the height H(wl, - . , w,) is independent of the choice of a basis of W, and hence H ( W) = H(wl , . . , w,) is well defined. In other words, H is regarded as a height on the Grassmanian variety of m-dimentional subspaces of V(k). We set

The volume of Bg is given by


Theorem 3.1.3 ([Bom-V, Theorem 81, [Th2, Corollary 2 to Theo- rem 31). Let W be an m-dimentional subspace of V(k). Then there is a basis wl, - . . , w, of W such that

This kind of result is called Siegel's lemma, which usually asserts the existence of integral solutions with small height of a system of linear equations. Bombieri and Vaaler used the adelic second fundamental theorem and the cube slicing theorem to prove this. Incidentally, since the cube slicing theorem is also interesting from a viewpoint of adele geomety, we explain it here. We set for each infinite place v E V,

(if v is real) E V(kv): suplailV <

i= 1 i ( 2 (if v is complex)

and

The volume of C l is W: (c;) = Idk l - n / 2 . The cube slicing theorem is stated as follows:

Theorem 3.1.4 ([Bom-V, Theorem 71). If Vl be an afine subspace of V defined over k, then

Another application of the adelic fundamental theorems is the adelization of Jarnik's theorem. O'Leary and Vaaler [O'L-V] introduced an inhomogeneous minimum of S(C, , g f ) by

and they proved

Theorem 3.1.5 ([O'L-V, Theorem 51). Let Ai be the successive minima of S(C,, gf ). Then one has

522 M. Morishita and T. Watanabe Adele Geometry of Numbers

In general, the exact value of u(k) := p(B:) is not known. We have u(Q) = 112, u(Q(J-1)) = 1/fi and u ( Q ( G ) ) = 1 / a for example. O'Leary and Vaaler showed the following estimate ([O'L-V, Theorem

61); If k # Q(&f),Q(G), then

Theorem 3.1.5 was used to generalize Schinzel's theorem, which gives a refeinment of Theorem 3.1.3. Let W be an m-dimentional subspace of V(k). A basis wl, . . . , w, of W is called primitive if the equality

holds for all a l , . . . , a , E k, and all finite v E Vf .

Theorem 3.1.6 ([Bu-V, Theorem 21). There is a primitive basis wl, . . . , w, of W such that

( ( 2u(k) ) 2 ) 2, H(wl) H(w,) 5 (m- l)! 1 +

k ( ) l / k ~ ~ ( m ) l / [ " ~ ] H(W)

Besides these, the adelic second fundamental theorem is useful for seeking a nontrivial integral solution with small height of a homogeneous quadratic equation. To explain this, let cp be a quadratic form on V(k) and <P the symmetric matrix corresponding to cp with respect to el, . . - , en. The height H (a) of <P is defined in a similar fashion as the global height of V(k). The next theorem due to Vaaler is the adelization of Schlickewei's result.

Theorem 3.1.7 ([V, I, Theorem 11). Let W be a subspace of V(k). Assume that cp is not identically zero and not anisotropic on W. Then there exists a maximally totally isotorpic subspace X of W such that

As a corollary, it follows that there is a nonzero integral solution a1 el + . . + anen E L n W of a homogeneous quadratic equation cp(x) = 0

such that

Some generalization and refinement of Theorem 3.1.7 were given at [V, 111 and [Wa3].

3.2. Adelic Minkowski-Hlawka theorem.

As in the classical case, the mean value theorem 2.2.1 yields an adelic Minkowski-Hlawka type result. Suppose the situation is as in 2.2 and let C be a compact subset of X(A). Taking the characteristic function of C as cp in 2.2.1, we get the following

Theorem 3.2.1. Suppose wf (C) < r(G, X)/r(G). Then there is g E G(A) such that C n gX(k) = 0.

As a special case, we let X = V\{O), V being as in 3.1, G =

S L n , n 2 2 and H = {g E SL,: gel = e l ) . Then, r(G) = r ( G , X ) = 1 by 2.1.1 and 2.2.2. Following Thunder [Th2], let C be a relatively compact star domain in V(A). Namely, C = n, C, is a relatively compact open subset of V(A) containing 0 and satisfying the property

aC c C, for a E {(a,) E AX : la,lv 5 1) .

Then, F C = {ax : a E F, x E C) is also a relatively compact star domain and we can show (cf. [Th2, Lemma21)

where hk, Rk, wk and Ck denote the class number, the regulator, the number of root of unity and the zeta function of k, respectively. From this and 3.2.1, we have

Theorem 3.2.2. Suppose that C is a relatively compact star domain in an a f ine adele space V(A) such that


Then, there is g E SL,(A) such that C n gV(k) = (0).

3.3. Generalized Hermite constants.

Watanabe [Wa2] generalized the notion of Hermite constant to that for any strongly rational representation of a reductive algebraic group using the Borel reduction theory.

Let G be a connected reductive group over k and p: G + GL(V) be a strongly k-rational, absolutely irreducible representation of G on an affine k-space V. Let D be the highest weight space whose stabilizer is the parabolic subgroup P. Then, X = GI P is a smooth projective variety isomorphic to the closed image of the cone p(G)D in the projective space P(V) attached to V. We fix a suitable maximal compact subgroup K of the adele group G(A). We also fix a norm I I . I 1, on V(k,) compatible with I . 1, for each place v so that 1 I . I 1, is the maximum norm for almost all v when a base of V(k) is once fixed, and set 1 1 ~ 1 1 ~ = n, llxvllv for x E GL(V(A))V(k). We suppose that 1 1 . [ I A is K-invariant and normalized by llxol l A = 1 for xo E D(k)\{O). Let G(A)l be the subgroup of G(A) defined in 2.1. The situation is as follows:

By the Borel reduction theory ([B2], [GI), for each g E G(A)', the

twisted height function Hg (x) = I lp(gy)xol lil[k:Q1, x = p(y)xo, attains the mimimum at a point, in the intersection of G(k) and a Siege1 set in G(A), and g I+ min Hg(x) is a bounded continuous function on

x ~ X ( k )

Definition 3.3.1 ([Wa2, Proposition 21). The maximum

exists for a connected reductive group over k. We call it the generalized Hermite constant associated to (p, 1 ) . I IA).

When G = GL, and p is an exterior power representation, this notion is due to Rankin ([Ra], k = Q) and Thunder ([Th3], for any k). It was also studied by Baeza and Icaza [Ba-Ic], [Ic] when G = GL, and p is the natural representation.


Now, by Weil's integration theory in 1.4, for a suitable function cp on Rn, we have

The basic idea to give a lower bound for p(p, ( 1 . ( I A ) is same as in the classical case: Take a characteristic function of the interval [0, TI for T > 0 as cp and set B(T) = {g E G(A)'/P(A)': I lp (g )x~ l )~ 5 T). If T < M := sup{T: vol(B(T)) < r (G)/ r (P)) , then there is g E G(A)l so that Hg(x) > T'/[~:QI for all x E X(k). It means p(p, 1 1 . [ I A ) 2 M ~ / [ ~ : Q ] . The computation of the volume of B(T) was worked out in [W2] for the case that p is maximal, namely the restriction of the heighest weight of p to a maximal k-split torus is a positive integer multiple of a fundamental k-weight. In that case, P is a standard maximal parabolic subgroup and the integral over G(A)l /P(A)' boils down to easier one to handle. The main theorem in [Wa2] is stated in the following vague form.

Theorem 3.3.2. Assume that p is maximal. Then, the volume vol(B(T)) is of the form clTc2, c2 E Q, and hence

The constants cl can be computed using the argument in Langlands' computation [Lan] and given more explicitly for k-split G. But, it would take long here to give all definitions of quantities involved. So, we will just give simple examples.

Example3.3.3 ([Th3], [Wa2]). G = GL,(n> 2), p=m-thexterior power representation, 0 < m < n. Then X is a Grassmannian of m- planes in an affine n-space. The norm is Euclidean norm at infinite places and the maximum norm of coordinates at finite places. In this cases, we set y,,,(k) = p(p, I I - I [A). Then, we have


where Zk = (rr-s/21'(s/2))r1 ((27r) 1-sI'(s))r2& (s). Thunder gave the following upper bound:

Example 3.3.4 ([Wa2], [Wa3]). Let cp be a nondegenerate quadratic form on an n-dimentional vector space V(k) and @ the symmetric matrix corresponding to cp with respect to a basis e l , . . , en. We consider the special orthogonal group G = SO, of cp and a natural representation p: G --+ GL(V). Let q be the Witt index of cp. It is assumed to be n 2 3 and q 2 1. Then p(p, I I . I I A ) is interpreted as

min I J ~ ~ ~ J : / [ ~ : ~ ~ . '(P' I ' . 'la) = gFG"i:) zEV(k)\fO)

It was proved in [Wa3] that

where H(@) is a height of a. If n = 2q or 2q + 1, then G is split over Ic and we have

Similar estimates are proved for the fundamental k-representations of G.

'$4. Distribution of rational points in adele transformation spaces

4.1. Hardy-Littlewood variety and Weyl-Kuga's criterion on uniform distribution.

As for the asymptotic distribution of integral solutions under local- global principle, Hardy, Littlewood and Ramanujan invented the circle method in the course of their study of the Waring problem ([H-L]) which stimulated Siegel's work. The notion of a Hardy-Littlewood variety was


introduce by Borovoi and Rudnick ([Bo-R]) to describe the asymptotic distribution of integral points of an affine variety in terms of local densities, namely the product of singular series and singular integral the circle method expects. Let X be an affine variety defined over k embedded into an affine space V over k. Assume that there is a gauge form w X on X. The associated Tamagawa measure is denoted by wf (we attach a suitable convergence factor if necessary). Take a finite nonempty subset S of V such that X(kv) is non-compact for v E S, and fix a norm 1 1 . 1 1 , on V(kv) for v E S. We set ks = nvEs k, and

AS = the ring of S-adeles. Let Bs = nvEsnv, Bv x nSwf X(k,),

where Bv is a topological connected component of X(kv) for v E V,, and set, for T > 0, Bz = {(x,) E Bs: IIxvllv 5 T). Choose an open relatively-compact subset B~ of x (AS) and define the counting function

by Ns(T, X, B) = [X(k) n (B,T x BS)] .

Then, following Borovoi and Rudnick, we call X a relatively S-Hardy- Littlewood variety if there is a non-negative function 6 on X(A) satisfying the conditions

1) S is locally constant, not identically zero, 2) for any Bs and BS as above,

In addition to the above conditions, if we can take the constant 1 as 6, X is called strongly S-Hardy-Littlewood.

Borovoi and Rudnick investigated the Hardy-Littlewood property for the case k = Q and S = Vm, and in particular when X is an affine homogeneous space of a simplyconnected, semisimple group, based on the work of Duke, Rudnick and Sarnak ([D-R-Sar]). Among other things, they showed a certain affine symmetric space is indeed relatively Hardy- Littlewood. A typical example is the quadric {x: f (x) = a ) of an indefinite integral quadratic form f in n variables, n 2 4, a E Z\{O). In the above definition, the density function 6 may not be unique. In the next paragraph, we shall see that 6 is determined in a simple form for a wide class of affine homogeneous spaces and general S when we give a 'right' definition of an S- Hardy-Littlewood homogeneous space.

In the circle method, a key role is played by Weyl's inequality having its origin in the work of Weyl [Well on the uniform distribution of sequences. In [ K u ~ ] , Kuga extended Weyl's criterion for a sequence of numbers to that for a family of subsets of rational points on a linear


algebraic group, and later Murase ([Mu]) generalized Kuga's results in the adelic setting.

Remark 4.1.1. There have been some works to reconstruct adel- ically theHardy-Lit tlewood circle met hod for special cases ([Mar], [I], [Lac],..). As was discussed in [Pa], it would be interesting to develope a general framework for the adelic Hardy-Littlewood method.

4.2. Hardy-Littlewood homogeneous spaces.

An S-Hardy-Littlewood homogeneous space was introduced in [Mo- Wa2] as a framework to describe the asymptotic distribution of S- integral points on an affine homogeneous space.

Let G be an affine connected unimodular group. Suppose G acts on an affine space V and let X be a closed G-orbit in V. Suppose that G, V and the action are defined over a number field k and that X has a k-rational point xo. We assume the stabilizer H of xo in G to be connected and unimodular. We also fix invariant gauge forms wG, wH and a G-invariant gauge form wX on G, H and X, respectively, which match together, and the associated Tamagawa measures are denoted by wg, wf and wz. Let S be a finite nonempty subset of V so that G is isotropic over k, for v E S. We fix a norm ) I . 1 1 , on V(k,) for v E S. Choose a G(ks)-orbit Bs and set Bs(T) = {(x,) : I Ixvllv 5 T) for T > 0. Choose an open, relatively compact, 'convex' subset BS of x(AS) . Here, the 'convex' condition is added only when there is an infinite place v 4 S and, roughly, it means V,\S-component of BS is the intersection of X(kvoo\s) and a convex subset of V(kvm\s). Set

B(T) = Bs(T) x BS.

Definition 4.2.1. An affine homogeneous space X = G/H is called an S-Hardy-Littlewood homogeneous space if there exists a function S : X(A) -+ R20, called density function, such that

HI) S is locally constant, G(ks)-invariant, and not identically zero, H2) for any Bs and BS as above and g E G(A), one has

Further, if we can take S = 1, X is called strongly S-Hanly-Littlewood.

We easily see that the density function is unique by HI) and H2). Moreover, using the mean value theorem in 2.2, we can determine the density function in the following simple form.


Theorem 4.2.2. Let G, H and X be as above. Assume further that G has the strong approximation property with respect to S and that H has no non-trivial k-rational characters. Then, if X is S-Hardy- Littlewood, the density function 6 is given by

Our method to supply examples of S-Hardy-Littlewood homogeneous spaces is an extension of those of [D-R-Sar] and [Mu], namely based on Weyl-Kuga's criterion and Howe-Moore type vanishing theorem of spherical functions at infinity. Let the notations and assumptions be as in 4.2.2 and G is supposed to be semisimple and simplyconnected. We may assume Bs x BS is contained in a G(A)-orbit of x E X(k) and count # (gG(k)x n B(T)) , since the difference from # (gX (k) n B(T)) can be computed by Galois cohomology. For this, let XT be the characteristic function of B(T) and set

where V(T) = WAX (B(T)). Since FT (g) = V(T)-l#(gG(k)x n B(T)), the following condition (A) yields the S-Hardy-Littlewood property for each orbit.

(A) lim F ~ ( ~ ) = T(H). T-00

Next, consider the following condition. (B) For any compactly supported function 1C, on G(A)/G(k), one has

An extension of Weyl-Kuga-Murase's criterion is, roughly speaking, that the condition (A) holds if (B) holds for any c, where F: is a certain modified function of FT (For the precise, refer to [Mo-Wa21). To certify (B), we consider the vanishing condition of certain spherical functions at infinity. Let K, be a maximal compact subgroup of G(k,) for v E S so that the norm 1 1 . 1 1 , is Kv-invariant. For the stabilizer Hx of x in G, let 'FI(G(ks)/Hx(ks)) be the space of smooth functions on G(ks)/Hx(ks) which is nvES,V, Kv-finite and let a be

the representation of G(ks) via left translation on this space. For f E 'H(G(ks)/H, (ks)), let ' ~r f be the space generated by a(g) f , g E G(ks). Then, (B) follows from the following condition


(C) For f E 'H(G(ks)/H,(ks)), if .rrf is an infinite dimensional, unitarizable, irreducible representation, then f (g) + 0 as g 4 oo.

Now, we assume that G is anisotropic over k and that G has no anisotropic simple factor over k, for some v E S. Then we have

Theorem 4.2.3. Assumptions being as above, further assume that the condition (C) holds for all Hz, x E X(k). Then X = G / H is S - Hardy-Littlewood with density function given in Theorem 4.2.2.

For instance, using Rudnick-Schlichtkrull's vanishing theorem, we can show

Theorem 4.2.4. Assume that G is semisimple, simplyconnected, anisotropic over k, and X is afine symmetric. Further assume that S contains an infinite place v such that G is kv-almost simple, k,-isotropic and 1 1 - 11, is invariant under a maximal compact subgroup of G(k,). Then, X is an S-Hardy-Littlewood homogeneous space.

Remark 4.2.5. We expect that S-Hardy-Littlewood property still holds when G is k-isotropic. To show this, we need harder analysis involved with Eisenstein series.

Example 4.2.6. Let D be a skew field of index d and center kl, with an involution r such that k; = k. Let V be an n-dimensional space over D, which is also regarded as an affine k-space, and let cp be a non-degenerate e-hermitian form relative to r on V. Here E = f 1. It is assumed to be n 2 4. Let G be the simplyconnected covering group of the special unitary group SU,, which is regarded as an algebraic group defined over k. For a E D X , consider the affine variety X = {x E V: ip(x) = a). Assume that X(k) # 0. By Witt's theorem, X is a homogeneous space of G. Let S be a finite set of places of k. If cp is k-anisotropic and S contains an infinite place v such that cp is isotropic over k,, then X is strongly S-Hardy-Littlewood.

Example 4.2.7. Let cp be an integral k-anisotropic tenary quadratic form with discriminant d,. Consider the affine quadric X = {x =

(xl, 2 2 , x3) : p(x) = a) for a E k '. Assume X(k) # 0 and fix xo E X (k) . Let G = Spin, be the spinor group of ip and H the stabilizer of xo in G. Then H is k-anisotropic torus and X E G/H. If S contains an infinite place v, or that S contains a finite prime v which does not divide 2d,, where cp is isotropic over k, in both cases, then X is relatively S-Hardy- Littlewood with density function


To prove this, we use more general result ([Mo-Wa2, Theorem 7.21) than Theorem 4.2.4.

4.3. Rational points on flag varieties.

Let G be a connected reductive group defined over a number field k and p: G -+ GL(V) a finite dimensional k-rational representation. Suppose a parabolic k-subgroup P stabilizes a line D through a k- rational point xo E V(k). Thus p yields an embedding of the flag variety X = G / P into the projective space P(V). We fix a norm )I.I), on V(k,) for each place v of k and set 1)x1IA = n, llxvllv for x E GL(V(A))V(k). Suppose I I - ( I A is normalized by (Ixol = 1 and is invariant under a maximal compact subgroup K of G(A). Let x be the k-rational character obtained from the action of P on D via p. We then define a twisted height Hg on X(k) for g E G(A) by

We have the associated counting function

When g is the identity, we have the usual height and counting function and we shall drop g. Note H,(X)[~:QI = I x ( P ) ~ A if we write gy = kp with k E K and p E P(A).

A basic problem here is to describe the asymptotic behavior of N(X, T, g) as T + oo and some special cases were investigated. A theorem of Schanuel ( [Sca]) describes N (Pn-l, T ) for a projective space, where p is the natural representation of GL,. A generalization to a Grassmann variety Gr(n, m), where p is the rn-th exterior power representation of GL,, was carried out by Schmidt ([Scm], k = Q) and by Thunder ([Thl] , for any k) . Using the norm defined in 3.1, their results show

N(Gr(n,m),T,g) - cTn as T + oo

where c is.the constant depending only on k and m. As for general X = GIP , G being semisimple, Franke-Manin-Tschinkel ([F-M-TI) and Peyre ([Pel) obtained the asymptotic formula for N(X, T) by using the special height via the anticanonical bundle corresponding to x = -2pp, where pp is one half sum of k-roots occuring in the unipotent radical of P. For instance, this height is the n-th power of the usual one for Pn-'. Their method is to study the analytic property, the residue at s = 1, of the height zeta function

532 M. Morishita and T. Watanabe Adele Geometry of Numbers 533

which can be regarded as Eisenstein series owing to the ralation ~ , ( z ) [ ~ : Q l = lx(p)lA mentioned as above. Actually, their result holds for the twisted height H, and takes the following form

N(X,T,g) - CT(~O~T) ' - ' as T co,

where r is the k-rank of the center of a Levi subgroup of P and c is the constant which is independent of g.

Finally, we remark that for the set

B(T) = tgp(A)l E G(A)~/P(A)': Hg(xo) 5 T),

we have the equality as in 3.3,

So, it would be interesting to know when there is a function f (T) of T such that N(X, T, g) - f (T) uniformly with respect to g E G(A)' as T -+ co, since in that case N(X, T,g) is given asymptotically as T (P)r(G)-lvol(B(T)) by the above integral formula. For example, Sch- midt-Thunder's asymptotic formula is consistent with Theorem 3.3.2.

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R. A. Rankin, On positive definite quadratic forms, J . London Math. Soc., 28 (1953), 309-314.

K. Rogers and H.P.F. Swinnerton-Dyer, The geometry of numbers over algebraic number fields, Trans. Amer. Math. SOC., 88


(1958), 227-242. D. Roy and J . L. Thunder, On absolute Siegel's lemma, J . reine

angew Math., 476 (1996), 1-26. L.A. Santal6, "Introduction t o Integral Geometry", Hermann,

1953. F. Sato, Siegel's main theorem of homogeneous spaces, Comment.

Math. Univ. St. Paul., 41 (1992), 141-167. S. Schanuel, Heights in number fields, Bull. Soc. Math. France, 107 (l979), 433-449.

W. Schmidt, Asymptotic formulae for points of bounded determinant and subspaces of bounded height, Duke Math. J., 35 (1968), 327-339.

C.L. Siegel, ~ b e r die analytische Theorie der quadratischen For- men, Ann. of Math., 36 (1935), 527-606.

~ b e r die analytische Theorie der quadratischen Formen 11, Ann. of Math., 37 (1936), 230-263; 111, ibid 38, (1937), 212-291.

On the theory of indefinite quadratic forms, Ann. of Math., 45 (1944), 577-622.

A mean value theorem in geometry of numbers, Ann. of Math., 46 (1945)) 340-347.

Indefinite quadratischen Formen und Functionentheorie, I, Math. Ann., 124 (1951), 17-54; 11, ibid 124 (1952), 364-387.

"Lectures on the Geometry of Numbers", Springer-Verlag, 1989. T . Tamagawa, Adkles, Proc. Symp. Pure Math., 9, Amer. Math.

Soc., Providence (1966), 113-121. J.L. Thunder, Asymptotic estimates for rational points of bounded

height on flag varieties, Compostio Math., 88 (1993), 155-186. An adelic Minkowski-Hlawka theorem and an application to Sie-

gel's lemma, J . reine angew. Math., 475 (1996), 167-185. Higher-dimensional analogs of Hermite's constant, Michigan Math.

J., 45 (1998)) 301-314. J.D. Vaaler, Small zeros of quadratic forms over number fields,

Trans. Amer. Math. Soc., 302 (1987), 281-296; 11, ibid 313 (1989), 671-686.

T. Watanabe, Some elementary properties of Hardy-Littlewood homogeneous spaces, Comment. Math. Univ. St. Paul, 46 (1997), 3-14.

On an analog of Hermite's constant, J . of Lie Theory, 10 (2000), 33-52.

Upper bounds of Hermite constants for orthogonal groups, Comm. Math. Univ. Sancti Pauli, 48 (1999), 25-33.

A. Weil, "L'intkgration dans les groupes topologiques et ses applications" , Hermann, 1940.

Sur quelques resultats de Siegel, Summa Brasil. Math., 1 (1946), 21-39.


Adkles et groupes alghbriques, Skm. Bourbaki, 1959, no. 186, Paris. OEuv. Sci., 11, 398-404.

"Adeles and algebraic groups, Notes by M. Demazure and T. Ono", Birkhauser, 1982.

Sur la theorie des forms quadratiques, in "Coll. sur la Theorie des Groupes Algebriques", C.B.R.M., Bruxelles, (1962), pp. 9-22.

Sur certains groupes d'opkrateurs unitaires, Acta Math., 111 (1964), 143-211.

Sur la formule de Siege1 dans la thhorie des groupes classiques, Acta Math., 113 (1965), 1-87.

"Basic Number Theory (3rd ed.)" , Springer, 1974. H. Weyl, ~ b e r die Gleichverteilung von Zahlen mod Eins, Math.

Ann., 77 (1916), 313-352. Theory of reduction for arithmetical equivalence 11, Trans. Amer.

Math. Soc., 51 (1942), 203-231.

Masanori Morishit a Department of Mathematics, Kanazawa University, Kakuma, Kanazawa, 920-1 192 Japan

T a b Watanabe Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan


On Shafarevich-Tate Sets

Takashi Ono

Let K/k be a finite Galois extension of number fields with the Ga- lois group g = Gal(K/k). Let gp be the decomposition group at a prime P in K . Let G be a g-group. For each P in K, we have the restriction map rp : H(g, G) + H(gp, G) of 1-cohomology sets for which Ker r p makes sense. The Shafarevich-Tate Set for (Klk, G) is defined by UI(K/k, G) = np Ker rp .

Let X be a smooth curve of genus 2 2 over Q. Then G = Aut X is finite by Schwarz theorem and there is a finite Galois extension K/Q so that G is a finite g-group, g = Gal(K/Q). The set LU(K/k, G) becomes finite. As is well-known, the determination of the finite set amounts to an arithmetical refinement of geometrical classification of curves. In this paper, we shall show, among others, that for a hyperelliptic curve X : y2 = x5 - 12x, l = an odd prime, we have III(K/Q, G) = 1 (Hasse principle) if l - 3,5 mod 8, but #LII(K/Q, G) = 2 if l = 1,7 mod 8.

There is a way to associate an S - T set LLIH(g, G) for any group g and a g-group G once we specify a family of subgroups of g (such as the family of decomposition groups gp when g = Gal(K/k)). E.g., for any finite group G, let g = G, acting on itself as inner automorphisms, and let H be the family of all cyclic subgroups of G. One checks UIH(G, G) = 1 ("Hasse principle") for some easy groups. Here is an interesting question: Does the Monster enjoy the Hasse principle?

$1- U H ( ~ , G ) .

Let g be a group and G be a (left) g-group. A cocycle is a map f : g + G such that

f (st) = f (s)f (t)s, s, t E g.

We denote by Z(g, G) the set of all cocycles. Two cocycles f , f ' are equivalent, written f f ' if there exists an a E G such that

f l (s) = a-'f (s)as, s E g.

Received July 29, 1998.

538 T. Ono

We denote by [f] the class of a cocycle f . The quotient

is the cohomology set. Z(g, G) contains a distinguished map 1 defined by l(s) = 1 for all s E g. Then a map f N 1 is said to be a coboundary. Consequently, we have

f is a coboundary e f (s) = a-las for some a E G

Now, suppose we are given a family H of subgroups of g. For each subgroup h E H, we have the restriction map

induced by f f 1 h , f E Z(g, G). This map sends the distinguished class in H(g, G) to the one in H(h, G). Hence Kerrh makes sense. In this situation, we put

and call this the Shafarevich-Tate set for (g, G) with respect to H . For example, for any finite group G, let g = G, acting on itself as inner automorphisms, and let H = Hcyc the family of all cyclic subgroups of G. The determination of LLIcyc(G) = UIH(G, G) seems to be an interesting exercise in finite group theory. One verifies that all finite abelian groups, dihedral groups D2, and the quaternion group Q8 enjoy the Hasse principle ILIcyc(G) = 1.

Example 1. We shall find a pair (g, G) which fails to have Hasse principle with respect to H = Hcyc. So let

G = (a; as = 1) % 2 /82 ,

with the action a0 =ad1 , a7 =a5 .

Note that we have

Let [ f ] be an element of UIH (g, G) c H (g, G) . Since each cyclic subgroup (s), s E g, belongs to H , we have f (s) = a ( ~ ) - ' a ( s ) ~ , a(s) E G; so,

On Shafarevich-Tate Sets 539

on replacing f by a cocycle equivalent to it using a(a) , we may assume that

-1 7 f ( a ) = l , f ( r ) = x x , x = a Z , 0 2 ~ 2 7 .

Then, we find

If i is even, then obviously f = 1; if i is odd, then we have f + 1. In fact, if not, there should be y E G such that 1 = f (a) = y-' y" and f ( ~ ) = y-ly7, with y = a j for some j . The first equality implies that a2 j = 1; hence j must be even. Then we have a4 = f ( ~ ) = y-lyT = a-j 5 j - a - a4j = 1, a contradiction.

Conversely, one can easily construct a cocycle f which takes values shown in (1.1) at the generators a, T of g. So we found that

with a single nontrivial class [ f ] given by f (a) = 1, f (T) = a4

$2. LU(K/k, G).

Let K/k be a finite Galois extension of number fields and g be the Galois group: g = Gal(K/k). For a (finite or infinite) prime P in K, denote by gp the decomposition group of P for Klk:

gp = {S E g; PS = P).

Let H = Hdec = {gp; P primes in K).

For a g-group G, we can speak of the Shafarevich-Tate set UIH(g, G) in $1. Since the Galois group and the family H = Hdec are determined by the given Galois extension Klk , we can set

There are two extreme cases where we get the Hasse principle LII(K/k, G) = 1 without effort. First of all, let us call K/k trivial if g = gp, i.e., if g E Hdec. In this case, we have UI(K/k, G) = 1, trivially, for any g-group G. For example, every cyclic extension K/k is trivial by Chebotarev theorem. A counterexample for a noncyclic abelian extension will be given in the next example. Secondly, if g acts

540 T. Ono

trivially, then Z(g, G) = Hom(g, G) and Z(gp, G) = Hom(gp, G) for all P; hence, again by Chebotarev, we have UI(K/k, G) = 1.

The following important relation follows also from Chebotarev theorem.

Let us call K/k locally cyclic if gp is cyclic for all primes P. Since #gp 5 2 for primes at infinity, we have only to check the finite primes. For example, if K/k is unramified then K/k is locally cyclic. In view of (2.2), we have

(2.3) Hcyc = Hdec u K/k is locally cyclic.

Therefore in such a case an arithmetical problem of determining LLI(K/k, G) is reduced to an algebraic problem of UIH(g, G) with g = Gal(K/k) and H = Hcyc. The following example discusses these matters.

Example 2. Let C be an odd prime and C be a primitive 8th root of unity. Let k = Q(&), K = k(C) = Q(i, a, &). The extension K/k is Galois with

g = Gal(K/k) = (a, T ; a2 = T~ = 1, 70 = or) = 2 /22 x 2/22.

Let G = (c) C K X . The group g acts on G by the Galois action: Ca = C-' = C, CT = c5. Hence the g-group (g, G) is exactly the one in Example 1. Since g = 2 / 2 2 x 2/22, the only subgroup of g which is not cyclic is g itself. Hence

K/k is locally cyclic H Hcyc = Hdec H g 4 Hdec (2.4) u g # gp for all P H K/k is nontrivial.

From (1.2), (2.3), (2.4), we find

(2.5) K/k is nontrivial H #UI(K/k, G) = 2.

Now a criterion for the nontriviality of K/k can be obtained by the Kummer theory (cf. [2], Satz 119 and [5] ,(2.7) Theorem):

(2.6) C - 3 mod 4 and x2 - C mod 4(1- C)

K/k is nontrivial u has a solution in 2 [C] .

In particular, if C r 7 mod 8, the congruence has a solution x = i; so, from (2.5) and (2.6), we have

(2.7) #ILI(K/lc,G) = 2 for .t - 7 mod 8.


In terms of ordinary Galois cohomology, we have an isomorphism

(2.8) kX /(kX )8 Z H' (k, G), (similarly for kp for each p).

The Shafarevich-Tate group of G (in Galois cohomology) is

LU(k, G) d$ ~ e r ( H ' ( k , G) + I I p ~ ' ( k p , G)).

It can be shown that there is a natural bijection

where the set on the right hand side is the one in (2.1). In view of (2.7)-(2.10), we find that, when t - 7 mod 8, the Hasse principle for the equation

x8 = a, a E k = Q(&)

does not hold for some a. (In fact, one can take a = 16, as pointed out by Prof. Wada.) Instead of Klk , consider the absolute cyclotomic field F = Q(c) whose Galois group is the same as that for Klk. Since 2 is totally ramified in F, the extension F /Q is trivial; hence, unlike (2.7), we have LU(F/Q, G) = 1 for all t.

93. UI(K/k, Aut X).

Let X be a quasi-projective variety over a number field k. Assume that there is a finite Galois extension K/k so that every kautomorphism of X is a K-automorphism. When it is so, we shall call K a (finite) splitting field for G = Aut X over k. As in 52, we can talk about the Shafarevich-Tate set UI(K/k, G) which can be identified with the ordinary UI(k, G) as mentioned in (2.10). By the assumption on X , we have a well-known bijection:

1 (3.1) H' (k, G) 2 Twist (Xlk), (similarly for kp).

Consequently, determination of U ( K / k, G) amounts to an arithmetical refinement of a geometrical classification of varieties:

(3.2) UI(K/k, G) = {Y/k; Y 2 X over k and kp for all p) .

In particular, the Hasse principle for twists means that

(3.3) Y 2 X over 2 and kp for all p + Y X over k.

If X is a smooth curve of genus g 2 2, then G = Aut X is a finite group of order at most 84(g - 1) = -42E(X) by Hurwitz theorem. Therefore G is split by a finite Galois extension Klk.

542 T. Ono

Example 3. Consider the celebrated quartic

X : x3y+ y3z + Z ~ X = O over k = Q .

We have g = 3 and G = Aut X 2 PSL2(F7), a simple group of order 168 = 23 . 3 7. Klein [3] shows that

where

with

Note that = [+C4+C2 -C6 -C3 -C5 (Gauss sum). Consequently, K = Q(C) splits G. Since K/Q is cyclic, it is trivial and we have the Hasse principle LLI(K/Q, Aut X ) = 1 without effort.

However, Hasse principle cannot always be obtained without effort as the following example indicates. As for the details of this example see [6] .

Example 4. Consider the curve over k = Q:

X : y4 = x 4 - e 2 , t = anodd prime.

This curve is smooth and of genus 3. We have G = Aut X = A B, A n B = 1, A normal in G, where A = 2/42 x 2/42, B = S3, the symmetric group on 3 letters. So #G = 96 = 25 . 3. It can be shown that K = Q(i, a, a) splits G and g = Gal(K/Q) = 2 /22 x 2 /22 x 2/22. The determination of Hdec amounts to the exhibition of the Artin reciprocity for the abelian extension K/Q. Thus we find: K/Q

e - 1 is trivial * g E Hdec H e* = ( - 1 ) ~ l = 5 mod 8. So, if e* = 5 mod 8, we get the Hasse principle UI(K/Q, Aut X ) = 1 without effort. On the other hand, in the remaining case l* r 1 mod 8, we still have UI(K/Q, Aut X ) = 1, but with some effort.

Example 5. Let t be an odd prime and X be a hyperelliptic curve of genus 2 over k = Q:


Since X is not smooth, we mean by X the normalization (Riemann surface) over Q associated to the equation (3.4). When we compute the group G = Aut X , we can do it in the function field o ( x , y)/Q for the equation (3.4). It is natural to seek t E G : t(x, y) = (XI, yl) such that

a x + b x = - yl = eY m + d ' (cx + d)3 '

We find, for example, elements u, v E G as follows:

where I* = (- 1) 9 I. The group G acts on the space f l l (X) of holomorphic 1-forms. With

respect to the standard basis dxly, xdxly for fll (X), we obtain a faithful representation

(3.5) G = Aut X t G L ~ ( Q ) .

The matrices U, V corresponding to u, v by (3.5) are:

Let us put

S = VU; hence s2 = 1.

Call GI the subgroup of G generated by U, V:

G 3 GI = (U, V) = (V, S).

Put

K = IJ .

544 T. Ono

Then one verifies that Q8 = (I, J) is the quaternion group. Since V I = J V , SI = JS, we see that Q8 is normal in G'. Moreover, one verifies that

G1/Q8 2 S3

by the correspondence

[V] = v mod Qs ++ (123), [S] = S mod Qs * (12).

Hence #GI = 8 . 6 = 48. On the other hand, we have #G 5 48(g - 1) = 48(2 - 1) = 48. Therefore we have G = GI and so

As a splitting field for G = Aut X over Q, we can take

Then we have g = Gal(K/Q) = 2 /22 x 2/22 x 2 /22 = (a, 7, p) with

The group G = (U, V) is naturally a g-group and the action of g is given as follows:

Next we have to determine the family Hdec for our g = Gal(K/Q). This amounts to exposing Hilbert's Galois theory and Artin's law of reciprocity for the abelian extension K/Q. Note that we have an important inclusion Hcyc c Hdec in (2.2). Hence we have only to determine decomposition groups g p which are not cyclic. Since this part is the same


as the corresponding part of Example 4 (for the curve X : y4 = x4 - 12) we can copy the table (4.4) in [6]:

If l* r 5 mod 8, i.e., if l r 3,5 mod 8, then, in (3.8), g belongs to Hdec and so K/Q is trivial; hence UT(K/Q,G) = 1 without worrying about the action of g on G. On the other hand, if l* - 1 mod 8, i.e., if l = 1,7 mod 8, then g is not in HdeC; hence K/Q is not trivial and the action of g on G given in (3.7) plays a crucial role. It will turn out that

(3.9) If l* = 1 mod 8, then #UT(K/Q, G) = 2 and the single nontrivial cocycle [f] in LU(K/Q,G) is given by f (a) = f ( r ) = 1, f (p) = K = IJ E Qs.

The rest of the paper is devoted to prove (3.9). First of all, notice that, in the table (3.8), the subgroup (a, 7) appears simultaneously in Hdec when l* r 1 mod 8. Hence for any [f] in UT(K/Q, G), after a normalization, we may assume that

(3.10) f (a) = f(7) = 1 and f (p) = A-lAp for some A E G.

Since f A give the same f(p), we may assume that A is one of 24 elements: A = CB, B E (1, V, V2, S, SV, SV2), C E (1, I, J, K). Hence f (p) = B-I C-I CpBp = f B-I Bp. AS one verifies by (3.7) that B-lBp = 1, J, K , we find

(i) If f (p) = 1, then f = 1, i.e., [f] is trivial.

(ii) If f (p) = -1, then from (3.7) we find that K-'K" = 1 = f (a), K-'KT = 1 = f ( r ) and K-lKP = -1 = f(p); hence f .- 1, i.e., [f] is trivial, again.

(iii) If f (p) = EJ, E = f 1, then we have

546 T. Ono

which is absurd as J" = - J by (3.7). So there is no such cocycle f . To prove our assertion (3.9), it remains to verify the following statements (iv), (v) and (vi).

(iv) f (p) = EK, E = f 1, together with f (a) = f (7) = 1, really provides us with a cocycle which restricts a coboundary on each subgroup in Hdec.

(v) Call f , g the cocycles in (iv) corresponding to E = +1, -1, respectively. Then f - g, i.e., [f] = [g].

(vi) The cocycle f in (v) is nontrivial: f + 1.

Proof of (iv). We need to show that the function f defined as above on the generators a, r , p extends on the whole group g consistently. The cocycle condition, f (st) = f (s) f (t)', s , t E g, forces us to put f (or) = 1, f (ap) = f (rp) = f ( a ~ p ) = EK. The consistency such as f (ap) = f (pa), f (a rp) = f (par), f (p2) = f (1) = 1 follows from relations K" = K T = K and KP = - K in (3.7). E.g., f(p2) = f(p)f(p)P = EK(EK)P = KKP = -K2 = 1. All other cases are checked likewise. Therefore the value f (s) is E K or 1 according as s contains p or not, and we verify at once the cocyle conditions using (3.7). As for the coboundary condition, note that S-ISt = f K whenever t E g involves p. Replacing, S by x(t)S, if necessary, where ~ ( t ) = C-lCt = f 1, with C E {I , I, J, K),

-1 t we obtain E K = At At, At E G whenever t involves p; hence the cocycle f stated in (iv) induces a coboundary on each cyclic subgroup of g . There is one more group to be considered, i.e., (a, p) E Hdec in case l = 7 mod 8. So let X = S or S K according as E = +1 or -1. Then one finds that X-lX" = 1 and x- 'x~ = EK, which means f restricts a coboundary on (a, p) too.

Proof of (v). By (3.7), we see that X = K is a solution to the following simultaneous equations:

This means that g - f .

Proof of (vi) . Suppose, on the contrary, that f - 1. Then there is an X E G such that

f (a) = x- lXU = 1 , f (7) = X-lXT = 1 , f (P) = x - ~ X P = K .


Write X = QA with A E {I , V, V2, S, SV, SV2), Q E (1, I , J, K) . Since Q-lQT = 1 for all Q, we have AT = A by (3.7), a n d s o A = 1, V o r V2. Now,

a) A = 1 +- K = Q-lQp = f 1, absurd,

b) A = V + 1 = V - l Q - l ~ " ~ " = f V-'V" = f I , absurd,

c) A = v2 +- 1 = v - ~ Q - ~ Q " V ~ " = f VVPa, absurd because V" = V I by (3.7) . So the system (3.12) has no solution as required.

References

[I] V.I. Danilov and V.V. Shokurov, "Algebraic Curves", Algebraic Mani- folds and Schemes, Springer, Berlin-Heidelberg-New York, 1998.

[2] E. Hecke, "Vorlesungen iiber die Theorie der algebraischen Zahlen", Chelsea, New York, 1970.

[3] F. Klein, ober die Tranformation siebenter Ordnung elliptischen Funktio- nen, Math. Ann., 14 (1878/79), 428-471.

[4] B. Mazur, On the passage from local to global in number theory, Bull. Amer. Math. Soc., 29 (1993), 14-50.

[5] T. Ono, A note on Shafarevich-Tate sets for finite groups, Proc. Japan Acad., 74A (1998), 77-79.

[6] T. Ono, Shafarevich-Tate set for y4 = x4 - e2, Turkish J. Math., 23 (1999), no. 4, 557-573.

Added in Proof. After this paper had been written, I learned from Prof. K. Harada that the Monster enjoys the Hasse principle in the sense described in the last paragraph of the introduction of this paper. Later, Prof. W. Feit communicated to me that any finite simple group enjoys the Hasse principle. This is a consequence of Theorem C in the paper: W. Feit and G. M. Seitz, On finite rational groups and related topics, Illinois J. Math., 33 (1988), 103-131.

The Johns Hopkins University, Baltimore, Maryland 2121 8, U.S.A. E-mail address: ono(0chou .mat. j hu . edu


Class Field Theory in Characteristic p, its Origin and Development

Peter Roquette

Abstract.

Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E.Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of the 20th century, with emphasis on the emergence of class field theory for function fields. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.

Contents.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

52. Class field theory for number fields 1920-40 . . . . . . . . 552

2.1. Zeittafel: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 2.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553

2.2.1. E. Artin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 2.2.2. H. Hasse . . . . . . . . . . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . 554 2.2.3. The class field report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 2.2.4. Further development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

53. Arithmetic foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .561

3.1. The conference program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 3.2. F.K. Schmidt's thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

3.2.1. Arithmetic in subrings of function fields . . . . . . . . . . . 563 3.2.2. The n-th power reciprocity law . . . . . . . . . . . . . . . . . . . . 565

3.3. Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567

Received October 12, 1998. Revised March 30, 1999.

550 P. Roquette Class Field Theory i n Characteristic p, its Origin and Development 551

54. Analytic foundat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 4.1. F.K. Schmidt's letters to Hasse 1926 . . . . . . . . . . . . . . . . . . . 567 4.2. The preliminary announcement . . . . . . . . . . . . . . . . . . . . . . . . 570 4.3. Riemann-Roch theorem and zeta function . . . . . . . . . . . . . .573

4.3.1. The final version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 4.3.2. Theory of divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 4.3.3. Theory of the zeta function . . . . . . . . . . . . . . . . . . . . . . . . 577 4.3.4. General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .579

4.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .580

55. Class field theory: the first step . . . . . . . . . . . . . . . . . . . . . .581 5.1. General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1 5.2. The main theorems of class field theory 1927 . . . . . . . . . . .582 5.3. The L-series of F.K. Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . 584 5.4. Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 5.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .588

.................................... 56. The reciprocity law 588 . . . . . . . . . . . . . . . . . 6.1. Hasse7s paper on cyclic function fields .588

. . . . . . . . 6.1 . l . The reciprocity law: Theorems A, B and C 590 6.1.2. Hasse7s proof of Theorem A, in the cyclic case, by

. . . . . . . . . . . . . . . . . . . . . . . . . . means of algebras .591 . . . . . . . . . . . . . . . . . . . 6.1.3. The proof of Theorems B and C 594

6.1.4. Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 6.2. Witt: Riemann-Roch theorem and zeta function for

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .599 6.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .601

57. The final steps.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 7.1. H.L. Schmid: Explicit reciprocity formulas . . . . . . . . . . . . . 602 7.2. The existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606 7.3. Cyclic field extensions of degree pn . . . . . . . . . . . . . . . . . . . . 609 7.4. The functional equation for the L-series . . . . . . . . . . . . . . . .612 7.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .615

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58. Algebraization .616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. F.K. Schmidt's theorem 617

. . . . . . . . . . . . . . . . . . . . . . 8.2. The new face of class field theory 621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Summary .623

5 1 Introduction

What today is called "class field theory" has deep roots in the history of mathematics, going back to Gauss, Kummer and Kronecker. The

term "class field" was coined by Heinrich Weber in his book on ellip tic functions and algebraic numbers [I181 which appeared in 1891. It was Hilbert [67] who in 1898 proposed to establish class field theory as the theory of arbitrary abelian extensions of algebraic number fields. Although Hilbert himself discussed unramified abelian extensions only, i.e., what today is called the "Hilbert class field", it is evident from his introductory remarks that he clearly envisioned the possible generalization to the ramified case. And Takagi, giving class field theory a new turn, succeeded in completing Hilbert7s program to full extent [108], [110]. His work was crowned by Artin's general reciprocity law [6] together with Furtwangler7s proof of the principal ideal theorem [32].

Soon after Takagi7s fundamental papers, there arose the question whether algebraic function fields with finite base field could be treated similarly, i.e., whether class field theory could be transferred to function fields. Today we know that this is the case.

In this article I shall outline the origin and the development of those ideas, and I shall follow up the main steps until finally class field theory for function fields was well established. The initial steps were done by F.K. Schmidt, Hasse and Witt; other mathematicians will be mentioned in due course. The time period covered will be from 1925 to about 1940. Thereafter class field theory for function fields ceased to be a separate topic; it became possible to deal with number fields and function fields simultaneously; the common name for both became Global Field.

Class field theory for function fields was developed largely in analogy and parallel to class field theory for number fields. Hence, in order to understand what has happened in the function field case, it seems useful to give some comments to the development of class field theory in the number field case during the said time period. We shall do this briefly in the first preliminary section.

his has been pointed out by Hasse [42]. In some contrast to this is the statement of K. Takagi in his memoirs that Hilbert seemed to be interested in the unramified case only and, hence, Takagi was "misled" by Hilbert into the wrong direction of study. See Kaplan's article [73] where several passages of Takagi's memoirs are translated from Japanese into French.

2 ~ n more recent times, however, the theory of function fields was revived under new aspects, among them also a new class field theory. See e.g., the book by D. Goss [34]. But this is outside the scope of this article.

3~nformation on the history of class field theory can also be obtained from [301, WI, [561, [711, [721, ~ 3 1 , 1781.

552 P. Roquette

52. Class field theory for number fields 1920-40

2.1. Zeittafel:

1920 Takagi's first great paper [log], establishing class field theory in its full generality according to Hilbert's program

1922 Takagi's second paper [110], on reciprocity laws in number fields

1925 Hasse's report on Takagi's results at the Danzig meeting of the DMV (German Mathematical Society)

1926 Part I of Hasse's report "Klassenkorperbericht" [36]; the other two parts Ia and I1 appeared in 1927 and 1930 respectively, see [37] and [38]

1927 Publication of Artin's proof of the general reciprocity law [6], based on Chebotarev's ideas which were connected with his density theorem [ll], [12]

1928 Furtwangler proves the principal ideal theorem (the proof a p peared in print 1930 [32]; later simplifications by Magnus [77], Iyanaga [70] and Witt [l29])

1929 Kathe Hey's thesis: Class field theory on the basis of analytic number theory in non-commutative algebras [65]

1930 Hasse-F.K.Schmidt: Concept of local class field theory [39], [97] (later reorganized and simplified by Chevalley [15])

1931 Hasse determines the structure of the Brauer group over a local field [40], following ideas of E. Noether on crossed products;

1931 Herbrand: Essential simplification of computations pertaining to class field theory [63] [64]

1932 Hasse's Marburg lectures on class field theory [44]

1932 Brauer, Hasse, Noether: local-global principle for algebras [lo]; connection with the product formula for the Hilbert symbol

1933 Hasse: Structure of Brauer group of number fields [43]

1933 Publication of Chevalley's fundamental thesis [15] which contributed greatly to the simplification and adequate organization of class field theory

1934 Deuring's book on algebras [22], based on E. Noether's lectures, containing a treatment of class field theory by means of algebras

1935 Chevalley and Nehrkorn present algebraic-arithmetic proofs for many of the main theorems of class field theory

1940 Chevalley's purely algebraic-arithmetic proof of Artin's reciprocity law in the framework of iddes [17], without using analytic functions

Class Field Theory i n Characteristic p, i ts Origin and Development 553

2.2. Comments

In 1920, the same year when Takagi's first main paper [108] had appeared, he attended the International Congress of Mathematicians in Strasbourg where he reported about his results [log]. However it seems that he did not receive any visible reaction to his report. But two years later, 1922, after his second paper [I101 had appeared and became available in Western libraries, it turned out that there was a number of young mathematicians who were keenly interested in Takagi's results and methods. Among them were Emil Artin and Helmut Hasse.

2.2.1. E. Artin: Artin in two papers [3], [4], obviously inspired by Takagi's, investigated [-functions and his new L-functions, and on this occasion he conjectured what is now called Artin's reciprocity law. Artin's proof appeared in 1927 [6] but already in 1925 he knew how to prove it, as we can infer from a letter dated February 10, 1925 and addressed to Hasse 1281:

. . . Haben Sie die Arbeit von Tschebotareff i n den An- nalen Bd.95 gelesen? Ich konnte sie nicht verstehen und mich auch aus Zeitmangel noch nicht richtig dahinterklem- men. W e n n die richtig ist, hat man sicher die allgemeinen A belschen Reziprozitatsgesetze i n der Tasche . . .

. . . Did you read Chebotarev's paper in the Annalen, vol. 95? I could not understand it, and because of lack of time I was not able to dive deeper into it. If it turns out to be correct then, certainly, one has pocketed the general abelian reciprocity law . . .

Artin's reciprocity law can be considered as the coronation of Tak- agi's class field theory. It was soon completed by Furtwangler who proved the principal ideal theorem [32] which had been conjectured by Hilbert [67]. The actual proof of this theorem had been obtained some time before its publication date (1930). In our Zeittafel we have dated

4 ~ h i s may have been due to the fact that until then, the development of class field theory took place mainly in Germany, and that German mathematicians were not admitted at the Strasbourg congress (probably on political grounds shortly after World War I). Thus Takagi did not meet the experts on class field theory at that congress.

5 ~ 1 1 letters which we cite in this article are contained among the Hasse papers which are deposited in the Staats- und Universitatsbibliothek Gottingen - except when it is explicitly stated otherwise.

554 P. Roquette

it for 1928, because we have found a reference to its proof in a letter of Hasse to Mordell which is dated November 26, 1928:

. . . Ich lege eine Arbeit von Artin (Hamburg) bei, die einen ganz grundlegenden Fortschritt i n der Theorie der relativ-Abelschen Zahlkorper enthalt. Vielleicht ist es nicht ohne Interesse fur Sie, zu erfahren, dafl ganz kurzlich Furt- wangler, auf dem Boden dieser Artinschen Arbeit, den Hauptidealsatz der Klassenkorpertheorie (vgl. meinen Be- richt, S.45) vollstandig bewiesen hat, durch Reduktion auf eine Frage der Theorie der endlichen Gruppen . . . I enclose a paper by Artin (Hamburg) which contains a very important advance in the theory of relatively abelian fields. Perhaps it is not without interest for you to know that recently Furtwangler, based on this paper of Artin's, has completely proved the principal ideal theorem of class field theory (see my report, p.45), via reduction to a question of finite group theory.

The paper by Artin which Hasse was referring to, was Artin's proof of his reciprocity law [6]. And when Hasse mentioned his "report" then he referred to Part I of his "Klassenkorperben'cht" which had appeared in 1926.

2.2.2. H. Hasse: A detailed historical analysis how Hasse became interested in class field theory is given by G. Frei in his article which appears in this same volume [31]. Already in 1923, in a letter dated April 21, 1923 and addressed to Hensel, Hasse explained the relevance of Takagi's new results and methods with respect to their project of studying the local norms for abelian extensions. At that time he was 24 years of age and held the position of Privatdozent at the University of Kiel. He had just completed the manuscript of a joint paper with Kurt Hensel, his former academic teacher at Marburg. That paper was to appear in the Mathematische Annalen; it gives the description of the local norm group for cyclic extensions of prime degree el under the assumption that the !-th roots of unity are contained in the ground field [35]. Hasse now realized that Takagi's theory could be used to deal with the general case, without this assumption about roots of unity.

Moreover, Hasse leaves no doubt that he regards Takagi's papers as being of highest importance also for class field theory in general. He writes:

'This letter is contained among the Mordell papers at the archive of St. John's College, Cambridge.

Class Field Theory in Characteristic p , i ts Origin and Development 555

. . . Ich habe gemde die Ausarbeitung eines Kollegs uber die Klassenkorpertheorie von Takagi vor, die ich mit unseren Methoden sehr schon einfach darstellen kann . . . . . . Just now I am writing the notes for a course about

Takagi's class field theory, which I am able to present quite simply with our methods . . .

Clearly, when Hasse refers to "our methods" in this letter then he means the e-adic methods as employed in their joint paper.

We all know that a good way of learning a mathematical subject is to give a course about that topic; the necessity of a clear and coherent presentation to the participants of the course will prompt the speaker to look for a better understanding of the subject. As evidenced by the Vorlesungsverzeichnis (list of lectures) of the University of Kiel, Hasse's course about Takagi's class field theory was given in the summer term 1923, and was supplemented in the winter term 1923124 with a course on "Higher Reciprocity Laws". Hasse's manuscript still exists and is available among Hasse's papers. It became the basis of Hasse's great class field theory report (Klassenkorperbericht) which appeared in three parts I, Ia, 11. [36], [37] [38].

2.2.3. The class field report: As G. Frei states [29], it had been Hilbert who suggested to Hasse to write such a report, which then was conceived by Hasse as a follow-up of Hilbert's famous Zahlbericht [66]. Like Hilbert's report, Hasse's was commissioned by the DMV (Ger-

, man Mathematical Society), and it appeared in the Jahresbericht of the DMV; the last part as a supplement (Erganzungsband). The three parts were bound together as a single book which became known as the Klassenkorperbericht.

i Hasse delivered an excerpt from this report in a lecture at the annual DMV meeting 1925 at the town of Danzig.

The impact of Hasse's report, both the Danzig lecture and the

/ printed report, can hardly be overestimated. Hasse was not content t i 1 '1 am indebted to W. Gaschiitz for his help in obtaining the Vor-

I lesungsverreiehnis of Kiel University for those years. f '1t seems that Hasse's report was the last one which was commissioned by / the DMV. Whereas in its earlier years, the DMV had tried to initiate a number

of comprehensive reports in various mathematical disciplines, this usage came

[ to an end in the 20's. Later, the role of the DMV reports was taken up by the

1 publications in the series Ergebnisse der Mathematik und ihrer Grenzgebiete of : Springer-Verlag, edited by the editorial board of Zentralblatt fur Mathematik.

here are some corrections [45] which, however, have not been included ! into this book.

556 P. Roquette

with merely presenting Takagi's results. He set out to give a comprehensive and systematic overview of all of class field theory known at that time; his treatment included quite a number of simplifications and additions - including proofs.

Actually, Part I of the Klassenkorperbericht does not yet contain proofs. It seems that for these, Hasse had originally planned an additional, separate publication in the Mathematische Annalen. For there is a letter dated Nov 1, 1926, from Hilbert (who was editor of the Annalen at that time) to Hasse, in which Hilbert said:

Sehr geehrter Herr Kollege, Ihr Anerbieten, mir fur die Annalen ein Manuskript mi t dem Titel: "Takagi's Theo- rie der relativ-A belschen Zahlkorper, bearbeitet von Hasse" zur Verfiigung zu stellen, nehme ich mi t vielem Dank a n - zugleich auch namens der Annalenleser u. der zahlenth. Wiss., der Sie damit einen wichtigen Dienst enueisen. Ich habe soeben einen Brief a n Takagi aufgesetzt, darf aber von vorneherein seines Einverstandnisses sicher sein . . .

Dear Colleague, with many thanks I shall accept your offer to let me have a manuscript for the Annalen with the title "Takagi's theory of relatively abelian number fields, presented by Hasse" - also in the name of the readers of the Annalen and of the number theoretical science, to whom you will render an important service. Just now I have formulated a letter to Takagi but I am confident that he will agree . . .

Note that the date of this letter is late in 1926, hence after the appearance of Part I in the Jahresbericht der D M V . Hasse answered immediately, proposing to Hilbert several versions of his article. Thereupon Hilbert sent a second letter, dated Nov 5, 1926:

Ich bin gar nicht i m Zweifel, dafi wir I h w n ersten Vor- schlag annehmen sollten und eine unbedingt vollstandige Wiedergabe der Takagischen Theorie i n firer Ausfuhrung und Korrektur i n den Annalen bringen mussen; ich mochte Sie sogar bitten, nicht etwa auf Kosten der leichten Les- barkeit und Verstandlichkeit Textkiirzungen vorzunehmen; es kann in diesem Fall auf einige Druckbogen mehr nicht ankommen. Ich mochte eine solche Darstellung wunschen, dass der Leser nicht noch andere Abhandlungen von Ih- nen, Takagi oder anderen hinzuzuziehen braucht, sondern

Class Field Theory i n Characteristic p, its Origin and Development 557

- wenn er etwa mit den Kenntnissen meines Berichts aus- gestattet ist - Ihre Abhandlung verhaltnismafiig leicht verstehen und auch die Grundgedanken sich ohne g d e Miihe aneignen kann. Ich bin . . . iiberzeugt, dass das so entste- hende Heft (bez. Doppelheft) den Annalen zur Zierde gere- ichen wi rd . . .

I have no doubts that we should accept your first proposal and have to publish in the Annalen a fully complete presentation of Takagi's theory, in your treatment and cor- rection; in fact I would like to ask you not to shorten the paper on the expense of easy reading and understanding; in this case some more print sheets do not matter at all. I would prefer a presentation such that the reader does not have to consult other papers by yourself, by Takagi or by others but - if he is familiar with what is in my report - would be able to understand your article easily, and also become acquainted with the basic ideas without much trouble. I am confident that this fascicle (or double fascicle) will become a beautiful gem for the Annalen.

From these words we not only infer the high esteem in which Hilbert held the work of Hasse and his ability for presenting a good exposition. lo

We also see that Hasse was contemplating, at that time, to publish the full proofs for Takagi's theory in the Mathematische Annalen. Later this , idea was dropped.

We have said above already that Hasse's report had a great impact I

on the further development of class field theory. As a consequence of this report, class field theory had become freely and easily accessible, as Hilbert had wished it to become, in a way which did not assume

I

I any further knowledge beyond what was generally known from Hilbert7s I Zahlbericht. Indeed, Hasse in his preface to [36] explicitly states that

no essential prerequisites except chapters I-VII of Hilbert's Zahlbericht will be assumed. Alternatively, he said, the first six chapters of Hecke's

I 1

''One of the biographers of Hasse says that " . . . his books confirm Hasse's reputation as a writer who could be counted on to present the most dificult subjects i n great clarity . . . " [27]. We learn from Hilbert's letter that Hasse had that reputation already when he was young (and had not yet written any book at all).

558 P. Roquette

book on Algebraic Numbers [59] (which had just appeared) would be sufficient. l1

This triggered an enormous rise of interest in the subject, in particular in view of Artin's and Furtwangler's progress beyond Takagi, as mentioned earlier already. Both Artin's reciprocity law and Furtwangler's principal ideal theorem were included in Part I1 of Hasse's report.

It is remarkable, however, that local class field theory does not yet properly appear in Hasse's report. There is only a brief note in Part I1 57 (which is concerned with the norm residue symbol) to the effect that the result derived there can be regarded as establishing the main theorems of local class field theory ("Klassenkorpertheorie i m Kleinen" ). In this connection Hasse cites his own paper [39] and the related one of F.K. Schmidt [97] which had just appeared in Crelle's Journal (1930). In those papers, local class field theory is derived from the global, contrary to what we are used today. Hasse remarks, in the same context, that it would be highly desirable to have it the other way round, i.e., first to establish local class field theory and then, by some Local-Global- Principle, to switch to the global case. He informs the reader that, as a first step, F.K. Schmidt in a colloquium lecture at Halle l2 had developed local class field theory ab ovo, i.e., without the help of global class field theory. And he continues:

Von hier aus, durch Zusammenfassung der auf die einzelnen Primstellen bezuglichen Satze der Klassenkorper- theorie i m Kleinen zu den auf alle Primstellen gleichzeitig bezuglichen Satze der Klassenkorpertheorie i m Grogen, verspreche ich mir eine erhebliche gedankliche und vielleicht auch sachliche Vereinfachung der Beweise der Klas- senkorpertheorie i m Grojen, die ja i n ihrem bisherigen Zustande wenig geeignet sind, das Studium dieser in ihren Resultaten so glatten Theorie verlockend erscheinen zu lassen.

Starting from here, combining the theorems of local class field theory referring to the individual primes, in order to obtain the theorems of global class field theory which

11 However, in Part 11, sections I1 and IV there are some arguments which belong to Hensel's theory of local fields - and these were not mentioned neither in Hilbert's Zahlbericht, nor in Hecke's book, nor in Hasse's preface.

12 In the spring of 1925, Hasse had moved from Kiel to the University of Halle where he had been offered a full professorship. Rom the correspondence between F.K. Schmidt and Hasse we can infer that the colloquium lecture in question had been held in the first week of February, 1930.

Class Field Theory in Characteristic p, its Origin and Development 559

refer to all primes simultaneously, I hope to get an essential simplification, conceptual and perhaps also factual, of the proofs of global class field theory; in their present state these proofs are not particularly inviting to study this theory which is so elegant in its results.

From this we see clearly why local class field theory was not included in Hasse's report: because it did not yet exist. It seems that during the process of writing those parts of his report Hasse became conscious of the fact that, indeed, what he was doing could be regarded as local class field theory. And immediately he developed the idea that class field theory could be better understood if it would first be developed locally, and then globally by somehow combining all the local theories.

2.2.4. Further development: It did not take long until these ideas could be realized. As we see from the Zeittafel, already in 1931 there appeared Hasse's paper where he determines the structure of the Brauer group over local number fields. Although in that paper class field theory is not explicitly mentioned, it is clear from the context (and it was certainly clear to Hasse) that the results obtained on local algebras can be translated to yield local class field theory. Explicitly this is carried out in the papers by Hasse [43] and Chevalley [14], [15]. l3

The global theory then follows through the local-global prinple for algebras, proved jointly by Brauer, Hasse and Noether in 1932 [lo]; see also Hasse's systematic treatment [43] one year later.

As Hasse says in the introduction to [43], it was a suggestion of Emmy Noether which had led him to introduce the theory of non- commutative algebras into commutative class field theory. Due to Emmy Noether, algebras can be represented as crossed products which are given by so-called factor sets; today we would call them 2-cocycles which represent cohomology classes of dimension 2. Hence [43] can be regarded

1 3 F . ~ . Schmidt's foundation of local class field theory "ab ovo" as announced in Hasse's report has never been published. In a letter to Hasse dated Dec 27, 1929 F.K. Schmidt asserts that he is able to handle tame abelian extensions - and he realizes that wild extensions will present more difficulties. In a second letter of January 21, 1930 he confirms that he intends to talk about this subject in the colloquium at Halle, and that meanwhile he has some more results ( "zch habe mir einiges weitere uberlegtV). This does not sound as if he had obtained the full solution. In all of the following correspondence - and there are many letters - he never returns to this problem. Perhaps F.K. Schmidt, in his colloquium lecture at Halle, was quite optimistic that he could solve the problems with wild extensions but later he found that the difficulties were larger than he had expected.

560 P. Roquette

as the first instance where cohomology was introduced and used in class field theory. In the course of time it was discovered that the formalism of general cohomology theory was well suited to serve the needs of class field theory, and that the reference to algebras was of secondary importance and could be dropped at all. But this came later, after the period (1925-1940) which we are discussing here.

In the academic year 1932133, when Hasse was already in Marburg, he had the opportunity to deliver again a course on class field theory, as he had done nine years ago in Kiel. But now the methods employed were quite different from those in earlier times, reflecting the state of the art at the time (but without explicit use of algebras). There were notes taken from these lectures, which were widely circulated and for a long time constituted a valuable source for many mathematicians who wanted to become acquainted with class field theory without cohomology. l4

In these lectures Hasse still had to use complex analysis, namely in order to compute the norm residue index for cyclic extensions. More precisely, analytic properties of certain L-series were used in the proof of one of the two fundamental inequalities for the norm residue index. According to the trend of that time, the use of analytical tools in order to prove theorems of class field theory was not considered to be quite adequate. Since the main theorems of class field theory had become statements about algebraic structures, e.g., the reciprocity law as an isomorphism statement, it was desired to have a proof which would open more insight into the structures involved. The analytic methods of that time did not do this. Based on Hasse's methods, Chevalley and Nehrkorn [16] were able to go a long way towards this goal. Finally, the seminal paper by Chevalley [17] in which the proofs were given in the setting of idkles, marked a cornerstone in the development of class field

141n 1933, an English translation of the Marburg lectures was planned. It seems that Mordell was interested in such a translation, probably because class field theory had been used in Hasse's first proof (1933) of the &emann hypothesis for elliptic curves, and therefore Mordell wanted class field theory to become better known in England. In a letter to Mordell dated Nov 1, 1933, Hasse suggested that on the occasion of such a translation certain improvements should be carried out, the most important one being the inclusion of the theory of the norm residue symbol and the power residue symbol, which Hasse had covered in the lecture but which were not included in the notes. (This letter is found in the archive of St. John's College, Cambridge.) However, the translation plan had to be given up in 1934.


theory. Today it is generally accepted that the framework of idkles is most appropriate for questions concerned with class field theory. l5

Later in the sixties, the interest in Hasse's Marburg lectures rose again, and therefore the old lecture notes were printed and published in book form.

The foregoing comments refer to class field theory for number fields. They are meant to provide a background for the following discussion of the development of class field theory for function fields. That story begins in 1925 at the Danzig meeting of the DMV.

53. Arithmetic foundation

3.1. The conference program

As mentioned in the foregoing section already, in the year 1925 the DMV (German Mathematical Society) held its annual meeting at the town of Danzig. The meeting lasted from 11th to 17th of September. In the program we find the following entry for the session on Tuesday, September 15 afternoon: [25]

Dienstag, den 15. September, nachmittags 4,00 Uhr Vorsitz: Hensel.

1. H.Hasse, Halle a. S.: Neuere Fortschritte in der Theorie der Klassenkorper. (Referat, 60 Minuten)

2. F'riedrich Karl Schmidt, F'reiburg i.B.: Zur Korpertheorie. (20 Minuten.)

3. E. Noether, Gottingen: Gruppencharalctere und Idealthe- orie. (20 Minuten.)

4. Karl Dorge, Koln: Z u m Hilbertschen Irreduzibilitatssatz. (20 Minuten.)

The first entry represents Hasse's talk which we have discussed above already. At the time of the Danzig meeting Hasse was affiliated with the University of Halle, where he had just accepted a full professorship. Hasse's talk is labelled Refemt (report) which means that it was an invited lecture. The time allocated for it was 60 minutes, more than

1 5 ~ description and assessment of Chevalley's work on class field theory is given by S. Iyanaga in [72]. By this way, Iyanaga reports that the terminology of "idMeV is due to a suggestion of Hasse. Chevalley originally used two words: "616ments ideal".

562

for th

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le following talks. l6 Immediately after Hasse's report we see the announcement of a talk by F.K. Schmidt. l7

F.K. Schmidt was 24 at the time of the Danzig meeting Is, hence three years younger than Hasse. He had just completed his Doktorexa- m e n (Ph.D.) at the University of Freiburg. His formal advisor had been Alfred Loewy but in fact he had been guided in his work by Wolfgang Krull who at the time was assistent to Loewy in Freiburg. l9 It appears that the Danzig meeting was the first mathematical congress which the young F.K. Schmidt attended.

The title of his talk "On field theory" is not very informative. In the Jahresbericht der D M V [25] we find an abstract which says that arbitrary algebraic function fields F of one variable will be considered, over a base field K which is absolutely algebraic of prime characteristic p. Given a transcendental element x E F, it is announced that the speaker will present the ideal theory, the theory of units and the theory of the discriminant for the ring Rx of x-integral elements in F . The abstract ends with the words:

Eine erweiterte Fassung des Vortrags erscheint in diesem Jahresbericht.

An extended version of the talk will appear in this journal.

However, this "extended version" never appeared, neither in the Jahres- bericht der D M V nor elsewhere. Hence, in order to find out more about the content of F.K. Schmidt's talk we should consult his thesis, for it seems likely that he talked about the results which he had recently obtained there.

161n the final report about the meeting [25] it is said that the session started already at 3:25 p.m. instead of 4 p.m. as originally planned. It is conceivable that Hasse had asked for more time for his report which, after all, was a formidable task since it was to cover the whole of Takagi's class field theory.

171n Germany the name "Schmidt" is quite common. There are several known mathematicians with this name. In order to identify them it is common to use their first names, or first name initials. We shall follow this habit here too; this is the reason why we always use the initials when mentioning F.K. Schmidt, whereas with other mathematicians the initials are not used in general.

18 More precisely: Five days after the meeting he had his 24th birthday. 19 Biographical information about Loewy may be found in the article by

Volker Remmert [86]; about Krull in the obituary by H. Schoneborn [loll.


3.2. F.K. Schmidt's thesis 3.2.1. Arithmetic in subrings of function fields: Again, this thesis

has never been published. But the University of Freiburg still keeps the original and I could obtain a copy of it. 20 The thesis is written in clear, legible handwriting and contains essentially the following results. (The notation as well as the terminology is ours, not F.K. Schmidt's.) As already introduced above, R, denotes the ring of elements in F which are integral over K[x]. The base field K is assumed to be finite. 21

- Rx is a Dedekind ring. 22

- The discriminant of R, over K[x] contains precisely those primes of K[x] which are ramified in R,.

- The ideal class group of Rx is finite. - The unit group R,X is finitely generated, and the number of gen-

erators modulo torsion is one less than the number of infinite primes of F with respect to x.

- In K[x] there holds an n-th power reciprocity law under the assumption that the n-th roots of unity are contained in K (in analogy to Kummer's reciprocity law in the n-th cyclotomic number field if n is prime).

For the moment, let us disregard the last item which we shall discuss later. The other items belong today to the basic prerequisites for every student who wishes to study algebraic function fields. In the mid- twenties, however, it seems that these things were not general knowledge, at least there was no standard reference. Hence it was a good problem for a young Ph.D. student to develop this theory ab ovo, i.e. from scratch. F.K. Schmidt solved the problem by standard methods which were well known and used by that time, referring to the analogy with Dedekind's foundation of the theory of algebraic numbers. 23 The title of F.K. Schmidt's thesis reads:

Allgemeine Korper i m Gebiet der hoheren Kongruenzen (Arbitrary fields in the domain of higher congruences)

"1 am indebted to Volker Remmert for his help in this matter. ''some of the following results remain true and accordingly are proved

under the more general assumption that K is absolutely algebraic of prime characteristic p.

"F.K. Schmidt does not use this term which is common today. He speaks of "Multiplikationsring" (multiplication ring). This name should indicate that the non-zero fractional ideals form a group with respect to ordinary ideal multiplication.

2 3 ~ e could not refer to E. Noether's axiomatic characterization of Dedekind rings because her paper [84] appeared in 1927 only.

564 P. Roquette

This is a rather queer title, and the notion of "Gebiet der hoheren Kon- gruenzen" does not appear in the proper text of the thesis. But after reading the introduction it is clear why this title had been chosen. Namely, the author wished to refer to Artin's thesis [5] (which had been completed 1921 but appeared 1924 only), and which carried the title:

Quadratische Korper im Gebiet der hoheren Kongruenzen (Quadratic fields in the domain of higher congruences)

Artin had considered quadratic extensions F of the rational function field K(x) (with K = IFp). 24 Through the choice of the title F.K. Schmidt wished to signalize that he is generalizing Artin's work by considering not only quadratic but arbitrary field extensions F of K(x) of finite degree (with K algebraic over IFp). Artin in turn had chosen his title in order to refer to Dedekind's classical paper of 1857 whose title read:

Abriss einer Theorie der hoheren Congruenzen in Bezug auf einen reel Zen Primzahl- Modulus (Outline of a theory of higher congruences with respect to a real prime number module)

There Dedekind discusses the number theory of the polynomial ring FP[x] in analogy to the ordinary ring of integers Z.

So we see that F.K. Schmidt's thesis had been written with the aim of establishing the fundamental facts of the arithmetic in function fields over finite base fields - in analogy to the arithmetic of algebraic number fields, and in generalization of Artin7s thesis, in reference to ideas going back to Dedekind.

This gives us an explanation why F.K. Schmidt's thesis has never been published. For, the same results had appeared about the same time in another paper [I021 by the author Sengenhorst. As F.K. Schmidt explains in [96], at the time of completing his thesis he did not know about Sengenhorst's paper which already contained his results. Likewise, he did not know about the work of Rauter, a Ph.D. student of Hasse, who also at the same time (and also without knowledge of Sengenhorst) came to the same conclusions [85]. It seems that in those days the need for a solid foundation of the arithmetic of function fields was felt widespread, so that there were three dissertations, almost at the same time, dealing with the same subject. 25

-

24Hence Artin's thesis covered hyperelliptic function fields. 25~auter , in addition, dealt also with the Hilbert ramification theory for

Galois extensions of function fields.

Class Field Theory in Characteristic p, i ts Origin and Development 565

Both Sengenhorst and Rauter became gymnasium teachers, the first one in Berlin and the other in the town of Tilsit in East Prussia. They did not remain active in mathematical research. But F.K. Schmidt did; he realized that the results in his thesis could only be the beginning, and that the next aim should be to establish Takagi's class field theory in the function field case. And he started to work in that direction.

3.2.2. The n-th power reciprocity law: Actually, in F.K. Schmidt's thesis there is one chapter which already has some bearing on class field theory, namely the chapter on the n-th power reciprocity law in the rational function field K(x). (We had mentioned this above already.)

Let n be an integer not divisible by the characteristic p of K. S u p pose that K contains the n-th roots of unity, i.e., that n divides q - 1 where q is the order of K . Then, for any two elements a , b E K[x] which are relatively prime, the n-th power residue symbol ( x ) ~ can be defined in complete analogy to the number field case (i.e., when a, b are integers in a number field containing the n-th roots of unity). Suppose that a , b, when considered as polynomials in x, are monic of degree r and s respectively. Then the power reciprocity law according to F.K. Schmidt reads as follows:

In particular, if n is odd then the inversion factor is trivial and we obtain

The case n = 2 had been treated by Artin in his thesis [5]. But already Dedekind in 1857 [20] had written down the quadratic reciprocity law in K[x] (for K = Z,) with the comment: "Der Beweis kann ganz analog dem funften GauJschen Beweis fur den Satz von Legendre gefuhrt werden . . . " (The proof can be done in complete analogy to the fifth proof of Gauss for Legendre's theorem . . . ) - thereby Dedekind assumed that the reader is familiar with the various proofs of Gauss and their numbering.

F.K. Schmidt pointed out in his thesis that the proof of his n-th power reciprocity law in K[x] is elementary, in contrast to Kummer7s proof in the number field case over the field of n-th roots of unity. And one year later in [96] he presented a formula which made this law a triviality. Namely, if a = nl,,,,(x - a,) is the decomposition into linear factors of the polynomiaT a, and similarly b = n l< j<s (x - &)

- -


then he observed that

which is putting ( I ) into evidence. Later in 1934, Hasse [51] said that (I) is a "well known reciprocity

formula" (eine bekannte Reziprozitatsformel). Although he referred to F.K. Schmidt [96], Hasse did not specify whether he considered this formula to be known because of F.K. Schmidt's paper, or it had been "well known" before already. In any case, Hasse in his paper showed that this reciprocity formula finds it interpretation within the theory of cyclic class fields over a rational function field F = K ( x ) as ground field. One year later in 1935, Hasse's student H.L. Schmid then generalized this to an arbitrary function field F as ground field [91]. See section 7.1.

3.3. Further remarks As we have pointed out, it seems that during the Danzig meeting

F.K. Schmidt became aware that most of his results in his thesis had been obtained elsewhere already. On the other hand, he did realize that there was interesting and important work ahead in the form of a project to transfer class field theory from number fields to function fields. His result about the n-th power reciprocity law in rational function fields could be regarded as a beginning in this direction, however small.

He seems to have been stimulated by Hasse's Danzig lecture which, as we have seen, had been delivered just before his own talk, and he surely had attended Hasse's. Perhaps F.K. Schmidt had not known Takagi's class field theory before, and he became interested in it through Hasse's lecture. An indication for this is the fact that the notion of "class field" does not appear in his 1925 thesis - but in his 1926 paper [96] already he refers to Takagi'stheory of class fields as his main aim in the case of function fields. Another indication of Hasse's influence is the fact that starting in the spring of 1926, F.K. Schmidt regularly wrote to Hasse and informed him about his progress. Hasse seemed to have not only stimulated F.K. Schmidt's further work but he was continuously interested in its progress. 26

The proofs of Takagi's main statements on class field theory de- pended, at that time, heavily on analytic methods; more precisely: on

26~nfortunately, only one side of their correspondence is preserved, namely the letters from F.K. Schmidt to Hasse; they are to be found among the Hasse papers in the Gottingen library. The letters from Hasse to F.K. Schmidt seem to be lost.

the properties of the Dedekind C-function and the L-functions of the base field. Therefore, in order to transfer Takagi's theory to the function field case, as a first step one would have to transfer the relevant theory of [-functions and L-functions. Accordingly, F.K. Schmidt started to develop just such a theory, which became his first major and widely known paper.

3.4. Summary

F.K. Schmidt in his thesis (Freiburg 1925) proved the basic facts about the arithmetic i n function fields with finite base fields. Thereby he generalized the arithmetic part of Artin's thesis (Ham- burg 1921) where hyperelliptic function fields only were considered. But F.K. Schmidt's thesis was never published because the same results had been obtained independently by other authors, about the same time. F.K. Schmidt's thesis contained one section which had some bearing on class field theory; it contained the n- th power reciprocity law for polynomials in the ring K[x] i f K contains the n - th roots of unity. This generalized Dedekind's reprocity law (1857) for the case n = 2. Artin i n his thesis (1921) had also given a proof for n=2. It turned out that F.K. Schmidt's proof was very simple and almost trivial; nevertheless ten years later it was recognized, after suitable generalization, as an important ingredient of general class field theory. A t the D M V meeting in Danzig (1925) F.K. Schmidt met Hasse and attended his great lecture which reported about Takagi's class field theory. Stimulated by this experience he decided to direct his further work towards establishing class field theory for function fields.

54. Analytic foundation

4.1. F.K. Schmidt's letters to Hasse 1926

The first letter from F.K. Schmidt to Hasse which is preserved in the collection of the Hasse papers at Gottingen, is dated May 6, 1926. F.K. Schmidt wrote:

. . . Was die Grenzfomnel fur die <-Funktion i n Korpern von der Charakteristilc p angeht, so ist mir i n meiner Dissertation die ubertmgung auf den ersten Anhieb nicht gelungen . . .

P. Roquette

. . . Concerning the limit formula for the C-function in fields of characteristic p, on the first try I did not succeed in my dissertation to transfer it . . .

By "limit formula" he means a formula for the residue of [(s) at the point s = 1. And "transfer" means the transfer from the number field case to the function field case. It seems that Hasse had asked him whether he had obtained the result already, which as we see was not the case. But soon, on August 8, 1926 F.K. Schmidt could announce success:

Es ist m i r bei erneuter Betrachtung ziemlich bald moglich gewesen, die bekannten Dedekindschen Resultate i n vollem Umfang auf Korper der Charakteristik p auszudehnen . . .

After taking up the subject anew, I fairly soon succeeded in transferring completely Dedekind's well known results to fields of characteristic p . . .

And then he continues to report to Hasse about his definition of the zeta function and the limit formula. Given a function field F ( K with finite base field K , the zeta function [ ( s ) in his definition depends on the choice of a transcendental x E F, and it refers in the well known manner to the prime ideals of the ring R, of x-integers in F . The limit formula for this function, according to F.K. Schmidt, reads as follows:

where q = IKI is the order of the base field, n = [F : K (x)] the field degree, V is the discriminant of R, over K[x] with absolute norm (V/ = qdegv, and R is the "regulator" which F.K. Schmidt had some difficulty to define but finally succeeded, replacing the logarithms (which appear in the number field case) by the valuation degrees of the units at the infinite primes (the poles of x). h is the number of ideal classes of R,. The above formula holds only in the case when all infinite primes are of degree 1, which is the analogue to totally real fields in the number field case. In his letter, F.K. Schmidt had restricted himself to this "totally real" case for reasons of brevity only. At the end of the letter F.K. Schmidt writes:

Auf Veranlassung von H e m Prof. Haupt sol1 demnachst i n den Erlanger Berichten eine vorlaufige Mitteilung meiner Resultate und Methoden erscheinen; da der Druck dort sehr schnell geht . . .


On the suggestion of Prof. Haupt, a preliminary announcement about my results and methods is to appear shortly in the Erlangen reports; since there the printing will be very fast . . .

The reference to Erlangen shows that F.K. Schmidt had changed his place of activity from Freiburg to Erlangen, where he had accepted a position of assistent to Professor Otto Haupt. The latter, although his primary interests were in real analysis and geometry, was also keenly interested in the modern developments of algebra and number theory. Haupt kept contact with Emmy Noether who whenever she visited her home town Erlangen, was heartily welcomed in the Haupt residence. 27

From the remarks in F.K. Schmidt's letter we infer that Haupt was impressed by F.K. Schmidt's work and therefore wished to secure prior- ity for him in publication, in particular in view of F.K. Schmidt's earlier experiences with his thesis. The Erlanger Berichte could quickly publish but otherwise this journal was not so well known, devoted not only to mathematics but also to science at large, and not available in many university's mathematics libraries.

In his above cited letter F.K. Schmidt did not mention class field theory but in his next letter to Hasse, dated December 6, 1926 he does. Obviously replying to a question of Hasse, he writes that he did not plan a general axiomatic foundation of class field theory but he believes this could be done - similar to E. Noether's axiomatic characterization of rings which admit classical ideal theory. 28 However, he continues, there may arise difficulties concerning the existence theorem of class field theory in the case when the class number is divisible by the characteristic p. Then he offers to send a brief summary of his results on class field theory in characteristic p - but he does not mention any details in the letter; for those we are dependent on F.K. Schmidt's publications.

27~nspired by the discussions with Emmy Noether, Otto Haupt wrote a textbook on the then "modern" algebra [57], which appeared in 1929 and was the first such textbook, before van der Waerden's appeared. Haupt's book covered more material than van der Waerden's; the fact that the latter became more widely known than the former seems to be due to the style of writing.

2 8 ~ e is referring to the paper [84] of Emmy Noether of which he seems to know the content already, and he also assumes that Hasse knows it although the paper had not yet appeared in print (it appeared in 1927). Note that both Hasse and F.K. Schmidt had met E. Noether one year earlier in Danzig, as is evident from the program excerpt which we gave in section 3.1. - An axiomatic treatment of class field theory was given much later, in the early fifties, in the seminal lecture notes by Artin-Tate [9].

5 70 P. Roquette

4.2. The preliminary announcement

F.K. Schmidt's Vorlaufige Mitteilung [96] is signed by the author with the date "August 1926", soon after his letter where he announced this paper to Hasse. It appeared in November that year with the title:

Zur Zahlentheorie in Korpern der Chamkteristik p. (Vorlaufige Mitteilung.)

On number theory in fields of characteristic p. (Preliminary announcement .)

In the introduction he refers to his thesis and acknowledges that both Sengenhorst and Rauter had obtained identical results. But now, he says, he is going to start with the transfer of the analytic theory. In the quadratic case (i.e., quadratic extensions of rational function fields) the analytic theory had been covered in the second part of Artin's thesis. Now he (F.K. Schmidt) would generalize also the second part of Artin's thesis to the case of arbitrary function fields with finite base field (which had not been done neither by Sengenhorst nor by Rauter). And he cont inues:

Die hier angefuhrten Ergebnisse eroffnen u.a. die Mog- lichlceit, die Takagische Theorie der Klassenkorper und der hoheren Reziprozitatsgesetze [auf Funktionenkorper] zu ubertragen, womuf ich demnachst einzugehen gedenke.

The results given here open up, e.g., the possibility of transferring Takagi's class field theory and higher reciprocity laws [to the case of function fields]; I intend to discuss this soon.

Thus F.K. Schmidt announced publicly that he was aiming at class field theory in characteristic p. From his correspondence with Hasse as discussed above we may infer that in December 1926 he was already in the possession of the main class field theorems, at least in a first and maybe incomplete version. We shall return to this in section 5. Here we wish to discuss what seems to be the most important part of the preliminary announcement:

Namely, at the end of the paper we find a Zusatz bei der Korrektur (Added in proof), dated "October 1926". There F.K. Schmidt says that further considerations have led him to change his viewpoint, as follows.

Up to now, when transferring arithmetic or analytic notions to the case of a function field FIK, the theory had been developed with respect to a given transcendental element x E F. The ring R, of x-integral functions had been regarded as the analogue of the ring of algebraic integers


in a number field, and all the notions and theorems for function fields had referred to the structure of R, and its prime ideals. The same point of view had been taken also by the other authors, i.e., Artin, Sengen- horst and Rauter. But from the new viewpoint, F.K. Schmidt says, no transcendental element in F is distinguished. Today we would say that his new viewpoint was "birationally invariant" but F.K. Schmidt did not use this expression. Instead, he refers to the classical theory of complex algebraic functions, where the "birationally invariant" point of view means that one works with an abstract Riemann surface, independent of any of its representation as a covering of the complex plane. Let us cite F.K. Schmidt himself [96]:

. . . W i r nehmen also jetzt den Standpunkt ein, der i n der Theorie der algebraischen Funktionen zuerst bei Dedekind und Weber zu finden ist. Diese beiden Autoren haben bekanntlich fur die von ihnen behandelten Korper algebraischer Funktionen eine arithmetische Definition des Punk- tbegriffes gegeben, der von jeder Bezugnahme auf eine Va- riable frei ist . . . . . . Thus we now take the same viewpoint which in the

theory of algebraic functions had been taken the first time by Dedekind and Weber. As is well known, those two authors had given, for the fields of algebraic functions as considered by them, an arithmetic definition of the notion of point, which is free from any reference to a variable . . .

He is referring to the classical paper by Dedekind and Weber [21] on the algebraic theory of function fields over the base field C (the complex number field). The "arithmetic definition" he alludes to, is today's usual definition: 29 a point is given by a "place" of the function field or, equivalently, by a valuation which is trivial on the base field. Here again, we can verify the enormous conceptual influence which the paper by Dedekind and Weber has exerted in the course of time.

The remarkable fact is not so much that F.K. Schmidt had adopted the viewpoint of Dedekind-Weber which to us looks quite natural, but that it was not adopted earlier, neither by himself in his thesis nor by any of the other authors: Artin, Sengenhorst and Rauter. An explanation for this may be that the theory of algebraic function fields with finite base

2 g ~ e e , e.g., Stichtenoth's introduction to the theory of function fields [107]. The first systematic treatment in a textbook on the basis of F.K. Schmidt's viewpoint was given by Hasse [54] in his "Zahlentheorie" which had been completed in 1938 but appeared in 1949 only.

5 72 P. Roquette

field had first been developed in close analogy to the theory of algebraic number fields. In the latter case the ring of all algebraic integers in the field is a natural and distinguished object of study. In the first attempts to transfer number theory to function fields, one was looking for an analogue of this ring and found it in the ring R, of all x-integral elements in the field, with respect to a given transcendental x.

However in number theory it became more and more evident that the various "infinite primes" (as we call them today), which belong to the archimedean valuations of the number field, play an important role and should be treated, as far as possible, on the same footing as the "finite primes", which belong to the non-archimedean valuations and hence to the prime ideals of the ring of integers. These ideas were adopted via the analogy between number theory and the theory of complex algebraic functions on a compact Riemann surface - an analogy which had been pointed out on many occasions. We only mention Hilbert in his famous Paris lecture [69] in the year 1900; see also the report (831 by E. Noether on this subject, published 1919.

In particular during the development of class field theory for number fields the need to consider those "infinite primes" was strongly felt. For in the definition of a "class group" in the sense of Weber one has to consider modules which consist of finite as well as of those infinite primes. See, e.g., Part Ia of Hasse's Klassenkorperbericht [37].

We may imagine that F.K. Schmidt, during his attempts to transfer class field theory, observed that for a function field F ( K one has to consider a similar situation: given a transcendental element x E F its poles should be treated on the same footing as the finite places for x. And then he recalled that this viewpoint had been adopted much earlier by Dedekind-Weber in the case of complex algebraic functions. In this way it now became possible to appeal directly to the analogy with the fields of complex analytic functions on a compact Riemann surface - without the detour over the number field case. This then led F.K. Schmidt to the birationally invariant viewpoint, as announced in his "Note added in proof".

In that note he briefly outlined the basic definitions and results (but without proofs for which he referred to the forthcoming final version). Given a function field F over a finite field K with q elements, his new definition of the zeta function is as follows:

where the product is taken over all places (primes) p of F ("points" in F.K. Schmidt's terminology), regardless of whether p is a pole of any


given transcendental or not. a ranges over the positive divisors of the function field. la1 = qdega denotes the absolute norm. Since every a is composed uniquely by the places p it follows that the Euler product equals the Dirichlet series.

Indeed, this definition of [(s) is birationally invariant with respect to the function field F ( K . It is the analogue not to the classical zeta function of a number field, but to the modified zeta function which, besides of the Euler factors belonging to the finite primes, contains factors corresponding to the archimedean primes.

4.3. Riemann-Roch theorem and zeta function

4.3.1. The final version: The final version with the title Analytische Zahlentheorie i n Korpern der Charakteristik p (Analytic number theory in fields of characteristic p) appeared in 1931 only, in the Mathematische Zeitschrift [98]. The manuscript was received by the editors on April 30, 1929. But it was essentially finished already in the summer of 1927 because F.K. Schmidt had used it as the first part in his Habilitations- schrift (thesis for his second academic degree). He did his Habilitation at the University of Erlangen during the summer semester of 1927. The Habilitationsschrift carried the title: 30

A belsche Korper i m Gebiet der hoheren Kongruenzen. (Abelian fields in the domain of higher congruences)

and it consisted of two parts:

I. Analytische Zahlentheorie in Korpern der Charakteristik p (Analytic number theory in fields of characteristic p)

11. Die Theorie der Klassenkorper iiber einem Korper algebmischer Funktionen i n einer Unbestimmten und mi t endlichem Koefizien- tenbereich. (Class field theory over a field of algebraic functions in one variable and with finite coefficient domain.)

Part I1 appeared in 1931 in the Erlanger Nachrichten [99]; see section 5. And Part I is identical with the paper in the Mathematische Zeitschrift which we are discussing now.

The main object of the paper is to develop the analytic properties of the zeta funtion [ ( s ) of a function field FIK with finite base field - in a birationally invariant manner as sketched in the "Note added in proof" of the preliminary announcement. For this purpose, the results of his thesis seemed to F.K. Schmidt not well suited as a framework because

3 0 ~ am indebted to W. Schmidt (Erlangen) for providing me with a copy of F.K. Schmidt's Habilitationsschrift.

574 P. Roquette

they depend on the choice of a transcendental element x E F and hence are not birationally invariant. Accordingly F.K. Schmidt developed the entire arithmetic theory of function fields anew, in a birationally invariant setting. He had discovered, firstly, that the classical theorem of Riemann-Roch 31 can be transferred to function fields with finite base field, and secondly that this Riemann-Roch theorem is intimately connected with the main analytical properties of his new zeta function. Accordingly he divided the paper into two parts: In the first part he developed the theory of divisors, and in the second part the theory of the zeta function.

4.3.2. Theory of divisors: In the first part F.K. Schmidt relies heavily on the analogy with the theory of complex algebraic functions; for the latter he refers to the paper by Dedekind and Weber already mentioned above, and also to the book by Hensel and Landsberg [61] from the year 1902.

If we compare those sources with F.K. Schmidt's paper then we discover that in the latter almost the same methods and arguments are used as in the former; we are tempted to say that F.K. Schmidt just copies his classical sources. But we should not underestimate the conceptual difficulties which F.K. Schmidt had to overcome. Today we could just say that the arguments used by Dedekind-Weber are applicable mutatis mutandis in the cases discussed by F.K. Schmidt, i.e., for finite base fields and, more generally, for arbitrary perfect base fields. But such general statement is accepted today only because now i t is well known how to modify the arguments of Dedekind-Weber for the cases at hand - thanks to F.K. Schmidt. Before something is accepted to be "well known" it has to be done first.

What seems to be trivial or easy to us was by no means trivial to F.K. Schmidt at the time. Let us discuss the various steps which were to be taken in the transfer process from Dedekind-Weber:

1. Dedekind-Weber [21] already had mentioned that their whole theory remains valid if the base field C is replaced by, e.g., the field of all algebraic numbers. Today we read this remark as to say that their theory

3 1 F . ~ . Schmidt always writes "Roche" in his paper, instead of "Roch". This could possibly lead to confusion because the (German) mathematician Roch is not identical with the (French) mathematician(s) Roche. In a postcard to Hasse dated January 4, 1934 he apologizes for his mistake of constantly appending an "e" to the name of Roch. And he adds somewhat jokingly: Leider w i d diese Konstante "e" neuerdings, wohl i m AnschluP an mich, auch von anderer Seite geschrieben. Also wieder einmal der bekannte "Fluch der bosen Tat".


is valid over an arbitrary algebraically closed field of characteristic zero. This is evident (to us) by just looking at the Dedekind-Weber paper which is of purely algebraic nature.

2. A closer look convinces us (today) that the paper remains valid in characteristic p > 0 provided the choice of transcendental elements x will be restricted to separating elements whenever necessary, e.g., when computing formal derivatives. And we know today that separating elements do exist if the base field K is perfect. F.K. Schmidt was the first to prove this. 32 Thereafter he is able to define the genus g of the function field FIK in the same way as Dedekind-Weber:

where x E F is a separating variable, n, = [F : K(x)], and w , is the so-called "ramification number" of F over K(x) which he defines to be the degree of the Dedekind different. Note that this definition covers the case of wild ramification which can appear in characteristic p. In contrast, Hensel-Landsberg [61] used a definition which looks simpler but is applicable in case of tame ramification only. It seems that F.K. Schmidt was aware of this situation and hence took care to choose the correct definition.

Although the definition (5) is not birationally invariant per se, F.K. Schmidt shows that the result of the expression on the right hand side of (5) does not depend on x. Hence g is indeed well defined as a birational invariant of the field. 33

3 2 ~ i t h o ~ t , however, using the terminology "separable" or LLseparating". He still uses the terminology "of the first kind" (von erster Ar t ) as introduced by Steinitz [105]. The term "separable" which is common today was introduced by van der Waerden in his textbook [I151 whose first edition appeared in 1930.

3 3 ~ . Witt, who had attended F.K. Schmidt's lectures on function fields in the winter semester 1933134 at Gottingen, presented in [I261 what he calls a simplification of this invariance proof. He explicitly refers to $4 of F.K. Schmidt's paper and proposes to replace that section by his (Witt's) proof. The "simplification" of Witt consists essentially of proving, in the algebraic setting including characteristic p, the well known explicit formula for the divisor of a differential, whereas F.K. Schmidt works with derivations only, not with differentials. - Independently of Witt and at the same time, Hasse [52] gave the same proof in the framework of his general theory of differentials. - F.K. Schmidt himself, in his later paper [loo] which appeared in 1936, proved the Riemann-Roch theorem for arbitrary function fields whose base field need not be perfect and, hence, there may not exist separating elements. In this

576 P. Roquette Class Field Theory i n Characteristic p, i ts Origin and Development 577

3. A certain difficulty arises for F.K. Schmidt because the base field K is not assumed to be algebraically closed. On several occasions he has to enlarge the base field K in order to be able to follow the lead of Dedekind-Weber. Then he has to show that those base changes do not disturb the arithmetic of the function field and are admissible for the respective problem. In doing this he relies heavily on the fact that K is perfect and, hence, algebraic extensions of the base field are separable. Today we would say that a function field over a perfect base field is conservative. But this notion did not exist at the time; in fact, F.K. Schmidt just proves it and uses the consequences. 34

4. After those preparations F.K. Schmidt is now ready for the proof of the Riemann-Roch theorem. Let C be a divisor class of FIK and C1 = W - C its dual class 35, where W denotes the differential class of FI K. Then the Riemann-Roch theorem says that

(6) dim C = deg C - g + 1 + dim(C1)

or, in symmetric form:

But F.K. Schmidt does not present the proof explicitly. He assumes the reader to be familiar with the book of Hensel-Landsberg and is content with saying that the Riemann-Roch theorem is important and can now be proved quite as in Hensel-Landsberg ( " . . . der wichtige Satz, der sich nunmehr ganz wie bei Hensel-Landsberg (S.301-304) beweisen la$t . . . '7.

In fact, this is correct: When checking the cited pages of Hensel- Landsberg the reader will find that all the notions and facts which are used in the proof there, had been transferred by F.K. Schmidt to the more general case of a perfect base field. The main fact is the construction of so-called "normal bases" which permit the explicit determination of the dimension of a divisor.

general situation he defined the genus as the constant which appears in the Riemann part of the Riemann-Roch theorem; this is a truly birationally invariant definition and is generally used today. Perhaps it is not without interest to add that this idea for the invariant definition of the genus arose directly from the correspondence of F.K. Schmidt with Hasse. In a letter to Hasse dated May 22, 1934 F.K. Schmidt outlined already the plan for his paper [loo].

3 4 ~ . ~ . Schmidt's proof of the Riemann-Roch theorem in [98] holds for any conservative function field, even if the base field is not perfect.

35 "Ergiinzung~klasse~ in the terminology of F.K. Schmidt.

To repeat: Concerning the Riemann-Roch theorem, the transition from the base field @ (Dedekind-Weber) to an arbitrary perfect base field and hence to finite base fields (F.K. Schmidt) is familiar to us and easily performed - but only thanks to F.K. Schmidt who did it first in his 1931 paper. His procedure was adequate, elegant and paved the way for further development. With this paper, F.K. Schmidt opened the general arithmetic theory of algebraic function fields.

4.3.3. Theory of the zeta function: But most important of all is his discovery that the Riemann-Roch theorem, in case the base field K is finite, is intimately connected with the properties of the zeta function [(s). In fact, in the second part of the paper he gives as almost immediate consequences of the Riemann-Roch theorem the following fundamental results.

Let q = IKI denote the order of the base field. The zeta function [ ( s ) of F is defined by the expansions (4) which converge if the real part of the complex variable s is %(s) > 1.

(i) [ ( s ) is a rational function of the variable t = q-". In particular it follows that [ ( s ) is analytically extendable to the whole complex plane as a periodic function with period 3. It admits essentially

1% 9 only two poles, of order 1, at the points s = 1 and s = 0 (and at those points which difler from these by an integral multiple of the period).

(ii) The residue of [ ( s ) at s = 1 is

where h is the class number of the field F , i.e., the number of the divisor classes of degree 0, and where g is the genus of F .

(iii) [ ( s ) satisfies the functional equation

or, i n symmetric form:

Let us add some comments: Ad (i) In the context of the proof of (i) F.K. Schmidt discovered

the important fact that every function field with finite base field admits a divisor of degree 1. This is today known as "F.K. Schmidt's theorem". It is remarkable that the proof of this algebraic statement was discovered

578 P. Roquette

and proved by analytic means. Later Witt provided an algebraic proof; we shall discuss this in section 8.1.

Ad (ii) The limit formula ( 8 ) looks somewhat different from the original limit formula (3). The reason is that now F.K. Schmidt uses a different zeta function, i.e., the birationally invariant one. In a separate section of his paper he explains the relation of the new zeta function [(s) with the former zeta function which now should be denoted by CX(s) since it refers to a given transcendental element x E F . 36 This is easily explained: Cx(s) is obtained from [(s) by multiplying with the finite product J&(l - Iql-") where q ranges over the poles of x. Hence the earlier limit formula (3) can be deduced from (8).

Ad (iii) In fact, the Riemann-Roch theorem is equivalent to the functional equation of [(s). To put this into evidence we may perhaps borrow from an idea of Witt which consists of rewriting the definition (4) of [(s) in such a way that the functional equation becomes obvious. 37

We use the variable u = qi-"; then the transformation s I-, 1 - s of the functional equation appears as u H u-l. Witt introduces the formal relation

which is interpreted in the following way: breaking up this relation at any index n into two partial sums, and adding the two rational functions in u which arise that way, this will always yield zero.

Now in the expansion on the right hand side of (4) we combine those divisors a which belong to the same divisor class C ; they have

d i m e - 1

the same degree deg C and their number is ' q-l . A straightforward

manipulation, adding a suitable multiple of (11) to the right hand side of (4) gives the following expansion:

where C ranges over all divisor classes of the function field FIK, including those of negative degree. (Note that dim C = 0 if deg C < O).) The Riemann-Roch theorem in its symmetric form (7) now says that

36 Note that, as said above already, our notation differs from F.K. Schmidt's.

3 7 ~ h i s idea has been recorded by Hasse in his survey [53]. - Witt used this idea in the paper [124] where he proved the functional equation for the zeta function of a simple algebra. We shall discuss that paper in section 6.2.


the coefficient qdimC-4 degC remains invariant under the substitution C H C'. On the other hand, this substitution transforms udegC-$ degW

into its inverse. In other words: the substitution C H C' (which is just a permutation of all divisor classes and hence leaves the right hand side of (12) invariant) is equivalent to u I-+ u-l. This is the functional equation in its symmetric form (10).

4.3.4. General comments: From what has been said above it is clear that F.K. Schmidt had conceived this paper with the explicit aim to transfer those tools of analytic number theory which are necessary to develop class field theory in the function field case. But the paper has exerted its influence much further than class field theory. The paper constitutes the first systematic presentation of the theory of algebraic function fields over arbitary base fields (or at least over perfect ones). It has served several generations of mathematicians as a basis for further research; in this sense it became a classic.

In this connection I would like to point out that F.K. Schmidt's paper appeared just in time in order to serve as a basis for Hasse's investigation of the Riemann hypothesis for function fields. As I have mentioned in another article [88], Hasse had been introduced by Daven- port to the problem of diophantine congruences. Hasse first met Dav- enport in the summer of 1931; at that time Hasse was already familiar with F.K. Schmidt's paper. Therefore he was able to realize at once that Davenport's problem was equivalent to the Riemann hypothesis for F.K. Schmidt's zeta functions.

In a later publication [50] Hasse presented the theory of F.K. Schmidt's zeta function in a form which he wished to use in further references. There he added some facts which were not explicitly mentioned in F.K. Schmidt's paper but which had been communicated to him by F.K. Schmidt in writing. One of those facts is the following representation of [(s) in terms of the variable t = q-" :

L(t) c(s) = (1 - t ) ( l - qt)

where L(t) is a polynomial with integer coefficients. Of course this is an immediate consequence of F.K. Schmidt's theorem (i) above, the poles t = 1 and t = q-l corresponding to s = 0 and s = 1 respectively. The numerator polynomial L(t) is known to play an important role in connection with the Riemann hypothesis for [(s). The degree of L(t) is 29, and this is given correctly by Hasse [50]. In F.K. Schmidt's paper [98] the formula (13) does not appear but if we would follow up F.K. Schmidt's arguments in [98] then we would obtain the degree 29 - 1.

580 P. Roquette

The reason is that F.K. Schmidt's formulas in [98] contain an "annoying misprint" as Hasse calls it (durchweg ein storender Druckfehler). 38 It seems that F.K. Schmidt had observed this "misprint" and informed Hasse about it, and at the same time he pointed out to Hasse the formula (13) with

which gives the correct degree of L ( t ) . (Here, N1 is the number of places of degree 1 in the given function field). By the way, a similar "misprint" occurs in the follow-up paper of F.K. Schmidt about class field theory [99] which we are going to discuss in section 5.

F.K. Schmidt's discovery that analytic properties of its zeta function are equivalent to the Riemann-Roch theorem of a function field, inspired several authors to look for an analogue of the Riemann-Roch theorem in a number field. One such analogue can be found in Tate's thesis [ I l l ] .

4.4. Summary A s a first step towards transferring class field theory, F.K. Schmidt transferred the necessary tools from analytic number theory. Thereby he generalized the analytic part of Artin's thesis and developed the theory of the zeta function of an arbitmry function field. Already in August 1926 he was able to report to Hasse the residue formula for the zeta function at s = 1. A preliminary announcement about his results was published i n November 1926. But in a note "Added in Proof" F.K. Schmidt changed his point of view and introduced his new, birationally invariant definition of the zeta function. F.K. Schmidt's final version of his paper, dealing with the birationally invariant zeta function, appeared in 1931 only, but the manuscript had been finished already in 1927 when he had submitted it to the Faculty in Erlangen for his "Habilitation~schrift'~. In its first part he developed the birationally invariant theory of divisors of function fields up to the Riemann-Roch theorem. His theory was modeled after the Dedekind- Weber paper (1880) on the classical theory of algebraic functions, and after the book of Hensel-Landsberg (1 902). F.K. Schmidt was able to transfer the

38This misprint occurs in the statement of the Riemann part of the Riemann-Roch theorem. It says that for a divisor class C we have dim C = deg C - g + 1, provided deg C > 29 - 2 . But F . K . Schmidt says in his paper that the condition degC 2 2 9 - 2 is already sufficient which is obviously not true.


methods of those classical sources to the case of arbitrary perfect base fields. I n its second part F.K. Schmidt developed the bimtionally invariant theory of the zeta function <(s ) of a function field. He had discovered that the main properties of the zeta function were closely connected with, and in fact immediate consequences of the Riemann-Roch theorem. This includes the rationality of the zeta function, the determination of the poles, their order and residues, and also the functional equation. Although the main aim of F.K. Schmidt was directed towards the establishment of class field theory in characteristic p, this paper obtained importance i n a much wider context. It constitutes the beginning of a systematic theory of algebraic function fields from the algebraic-arithmetic point of view. In addition, his theory of the zeta function proved to become the proper background for Hasse's investigations on the Riemann hypothesis i n characteristic p.

$5. Class field theory: the first step

5.1. General comments In his next paper [99] F.K. Schmidt started to deal with class field

theory proper. As with the foregoing paper on analytic number theory in characteristic p, this paper [99] appeared in 1931 but it had been completed in the summer of 1927 already, when F.K. Schmidt had used it as Part I1 of his Habilitationsschrift in Erlangen.

The title announces "Class field theory in the case of function fields with finite base fields". But a closer examination of the content of the paper shows that there are serious shortcomings and that this paper does not contain a full account of class field theory as announced in the title. The paper can be viewed only as a first approach to class field theory. It seems that F.K. Schmidt was well aware of this and had conceived the paper as a kind of a preliminary announcement, similarly as [96] which was published as a preliminary announcement of [98]. An indication for this is the fact that [99] appeared in the same not widely known journal as did [96], i.e., in the "Erlanger Berichte". Moreover, the presentation of the material is not as clear and final as it is in F.K. Schmidt's paper [98] on analytic number theory in function fields. While the latter has become a "classic" (we had mentioned this above already) this attribute cannot be given to the paper under discussion now.

The first serious shortcoming is stated already in the introduction of [99]. There the author says:

P. Roquette

Die vorliegende Darstellung beschrankt sich zunachst auf den Fall derjenigen A belschen Erweiterungen, deren Grad zur Charakteristik prim ist. Diejenigen Abelschen Erweit- erungen, deren Grad durch die Charakteristik teilbar ist, erfordern noch einige weitere Betrachtungen und sollen an anderer Stelle behandelt werden.

The present account is restricted to the case of abelian extensions whose degree is relatively prime to the characteristic. Those abelian extensions whose degree is divisible by the characterictic do require some further considerations and will be dealt with elsewhere.

Abelian extensions whose degree are divisible by the characteristic cannot be generated by radicals, not even after adjoining the proper roots of unity. Thus F.K. Schmidt excludes precisely those cases which cannot be dealt with by the classical methods employed by Takagi. Those new cases would require a new idea which is adapted to characteristic p particularly and cannot be obtained by transfer from characteristic 0.

Such an idea appeared in the same year 1927, namely in the paper by Artin and Schreier [8]. There it was shown that cyclic extensions of degree p in characteristic p are generated by the roots of (today) so- called Artin-Schreier equations: yp - y = a. Since F.K. Schmidt does not mention this result of Artin-Schreier we have to assume that he did not yet know about it; perhaps he had something different in mind when he mentioned "some further considerations" which he would deal with elsewhere. In any case, he never came back to this and it was Hasse in [51] who introduced Artin-Schreier theory into class field theory for function fields.

5.2. The main theorems of class field theory 1927 Now let us see what F.K. Schmidt did prove in [99]. We have said in the introduction already that class field theory in

characteristic p was developed parallel and in analogy to the characteristic 0 case. This can be well observed here, for the main theorems in characteristic p as formulated and proved in this paper, do reflect precisely the state of Takagi's class field theory in characteristic 0 in the year 1927. The source for F.K. Schmidt was Hasse's class field report [36] about which he had heard Hasse lecture at the Danzig meeting; Hasse later had sent him an offprint of the published version.

Let FIK be a function field with finite base field K. Given a positive divisor m, F.K. Schmidt defines the ray modulo m in the usual way: it consists of those principal divisors which can be generated by elements a with a = 1 (mod m). The full ray class group C, is defined to be


the factor group of the group of all divisors relatively prime to m, by the ray modulo m. Class field theory deals with subgroups H, c C,. Unlike in the number field case, however, C, is not finite in the function field case. Therefore, in the context of class field theory one has to add the additional requirement that the index h, = (C, : H,) is finite. F.K. Schmidt observes that this is satisfied if and only if H, contains at least one ray class of positive degree. For ray class groups belonging to different modules the following equivalence relation is introduced: H, Hml if and only if there exists m" 2 m, m' such that Hm and H,I have the same inverse image under the natural projections C,It + C, and C,II -t C,I respectively. If this is the case then, following Hasse [37] the groups H, and H,t are said to be "equal". In this way every H, c C, defines an equivalence class H of "equal" ray class groups. H is regarded as some kind of abstract ray class group which at m admits Hm as its "realization". m is called a "module of definition" (Erklarungsmodul) for H . The smallest module of definition for H is called the "conductor" (Fuhrer) of H , to be denoted by the letter f.

In modern terms, H indeed can be viewed as a group in the proper sense, namely as an open subgroup of the inverse limit

C, = lim C, t

which may be called the "universal ray class group" of F. The equivalence class corresponding to an open subgroup H C C, consists of all H, which have H as their inverse image under the natural map C, + C, . Thus it does not matter whether we talk about equivalence classes of "equal" ray class groups, or of open subgroups of C,.

But the notion of inverse limit of algebraic structures was not yet well established at the time when F.K. Schmidt wrote his paper. It was Chevalley who, at a later stage, realized C, as the idde class group of F and so simplified the conceptual framework of class field theory considerably [17]. Viewed from today, it does not matter whether we use the language of inverse limit of ray class groups, or avoid this and talk about equivalence classes of ray class groups; these are but two ways of describing the same object. F.K. Schmidt still used the old definition of Hasse [36] referring to equivalence classes of ray class groups.

Now let EIF be a Galois extension of degree n, say. Consider a module m in F and its ray class group C,. The norm map N : E + F yields a map of ray class groups whose image N, c C, is of finite index, say h,. Following Takagi, El F is called a class field defined modulo m if h, = n.

584 P. Roquette

This being said, the main theorems of class field theory as announced by F.K. Schmidt can be stated as follows:

I. Existence- a n d uniqueness theorem: Given any module m in F and any subgroup Hm c C, of finite index, there exists one and only one class field EIF defined over m which admits H, as its norm group, i.e., Hm = N,. If H,I is "equal" to H, in the sense as explained above then its class field coincides with E, and conversely.

11. Isomorphism theorem: EIF is abelian and its Galois group is isomorphic to the norm factor group C,/H,.

111. Discriminant-conductor theorem: The discriminant of E I F contains precisely those places which are contained in the conductor f of H.

IV. Decomposition theorem: If p is a prime of F not contained in m and if f denotes the order of p modulo H, then p splits in E into different primes of relative degree f .

V. Inversion theorem: Every abelian field extension EIF is a class field in the above sense.

In fact, these are the main theorems of class field theory which had been stated essentially in this form in Hasse's report [36]. We see that F.K. Schmidt is closely following Hasse's presentation indeed. He is going to prove those theorems under the additional hypotheses that the group index (in theorem I) and the field degree (in theorem V) are not divisible by the characteristic p.

Looking at the above list we observe the second serious shortcoming of this paper, namely that Artin's Reciprocity Law is completely missing. Artin had published his proof (in the case of number fields) in 1927 already, and according to Hasse it constituted a "progress of greatest importance" ( einen Fortschritt von der allergroflten Bedeutung ) [38]. F.K. Schmidt certainly must have heard of this by the time when his paper was sent to print (1930). So why didn't he attempt to prove Artin's reciprocity law in the function field case? Why didn't he even mention the reciprocity law? Again, we have only one explanation, namely that he had completed his paper in 1927 already (for his Habilitationsschrift) and at that time he did not yet know about Artin's result. Later, he did not change the text at all.

We shall see in section 6.1.1 that Hasse took up the problem and proved the reciprocity law in the function field case.

5.3. T h e L-series of F.K. Schmidt

Having stated the above 5 theorems F.K. Schmidt says:


Der Beweis dieser Satze vollzieht sich in entsprechenden Schritten wie in der Takagischen Theorie.

The proof of these theorems proceeds in analoguous steps as in Takagi's theory.

Accordingly he follows Hasse's report and starts with the proof of the so-called "first inequality" of class field theory. This inequality refers to the following situation: E J F is a Galois extension of finite degree n and m is a positive divisor in F. Let h, = (C, : N,(E)) denote the corresponding norm index. Then the first inequality says that

It is the proof of this inequality where F.K. Schmidt had to use analytic methods; the situation is just like in the number field case. More precisely, he had to use:

1. the theory of the zeta function; in particular the fact that [(s) has a pole of order 1 at s = 1;

2. the theory of L-series; in particular the fact that for any non- principal character x of C, of finite order, its L-series L(s, X ) assumes a finite value at s = 1.

F.K. Schmidt had dealt with item 1. in his former paper [98] which we have discussed above in section 4.3.3. In order to cover item 2., F.K. Schmidt introduces the L-series in the usual way:

where the dashes ' at the product sign and at the sum sign indicate that only those places p and divisors a are to be considered which are relatively prime to the given modulus m.

Using the Riemann-Roch theorem F.K. Schmidt is able to show that for every non-principal character x the series on the right hand side of (16) terminates, i.e. that L(s, X) is a polynomial in the variable t = q-'. Therefore, of course, L(l , X) is finite. 39 Thus the analytic theory in function fields turns out to be much simpler than in the number field case - thanks to the Riemann-Roch theorem.

Once having obtained item 2. above, F.K. Schmidt does not bother to present the proof of the first inequality (15) but he is content with saying:

3 9 ~ . ~ . Schmidt doesn't say anything about L(1, X ) # 0.

586 P. Roquette

Die L-Reihen des Funktionenkorpers verhalten sich bei Annaherung a n s = 1 genau ebenso wie die L-Reihen eines endlichen algebraischen Zahlkorpers. Man kann daher die bekannten zahlentheoretischen Schlujlweisen auf die L-Reihen des Funktionenkorpers ubertragen und ge- winnt so nach dem Vorbild von Hasse die Ungleichung . . .

Approaching s = 1, the L-series of the function field show the same behavior as the L-series of a finite algebraic number field. Therefore, it is possible to transfer the known number theoretic arguments to the L-series of the function field, and one obtains in this way, following Hasse, the inequality . . .

He is referring to Part Ia of Hasse's report [37] (the part where the proofs are presented). In other words: F.K. Schmidt assumes the reader to be familiar with Hasse's report including the proofs, and his arguments here are given only to the extent that the reader can do the transfer by himself: from characteristic 0 to characteristic p.

The paper does not contain any further systematic study of L-series in function fields. In particular the functional equation of the L-series is not discussed. This has been proved later by Witt; see section 7.4.

5.4. Further remarks

Takagi's original proof of the main theorems of class field theory for number fields is not straightforward. The structure of proof is a complicated net of back-and-forth arguments which finally yield the desired theorems but otherwise are quite unsatisfactory, in as much as they do not yield sufficient insight into the structure of the mathematical objects to be studied. 40

This initiated the search for simplification and re-organisation of class field theory. That process was quite under way in 1931 when F.K. Schmidt's paper [99] appeared. But we do not see any sign of that development reflected in this paper. The paper refers to Hasse's report Parts I and Ia only, and it closely follows the lines of arguments as given there.

In accordance with this, after having proved the first inequality F.K. Schmidt now switches to the proof of the inversion theorem for cyclic extensions EIF of prime degree n (where n $ 0 mod p). To this end he

4 0 ~ e e the remarks by Hasse which we have cited in section 2.2.3 from part I1 of his class field report.

Class Field Theory in Characteristic p , its Origin and Development 587

is performing a lengthy computation of group indices, including the so- called "Hauptgeschlechtssatz" which (from today's viewpoint) asserts the vanishing of a certain 1-cohomology group of ray classes. F.K. Schmidt puts into evidence that those computations (which were later much simplified by Herbrand [64]) can be carried out quite in the same manner as in the number field case. There are even certain simplifications due to the simple unit structure (every unit is a constant) and to the simple structure of cyclotomic fields (every cyclotomic extension is a base field extension and hence unramified) .

This being done, the rest of the paper is more or less hand waving. F.K. Schmidt seems to be in haste and therefore leaves all the rest to the reader, with the following comment:

Dabei wird m a n ganz von selbst auf einige leichte Ab- weichungen von den zahlentheoretischen Schlujlweisen gefuhrt, die aber durch das oben Gesagte bereits so nahe gelegt sind, dajl es sich eriibrigt, naher auf sie einzugehen.

One will be led automatically to some minor differences to the number theoretical arguments; but it does not seem necessary to discuss them in detail since they are suggested sufficiently by what has been said above already.

This does not sound very convincing. In particular the transfer of the existence theorem of class field theory, which uses several delicate index

, computations, would remain doubtful unless it is presented explicitly - even if one restricts the discussion to the case where the characteristic p does not divide the relevant group index n. In fact, some years later in

L 1935 Witt, when presenting a simple proof of the existence theorem, did not say that F.K. Schmidt had already proved it for n $ 0 mod p, but

I that F.K. Schmidt "had already discussed the possibility of transferring L the proof" (hat die Moglichkeit einer ~ b e r t r a ~ u n ~ schon erortert). See

section 7.2 below. This paper is the last one by F.K. Schmidt about class field theory

in function fields. In the late twenties and thirties he had a number of other important papers on algebraic function fields and also on other topics, e.g., from the theory of local fields, some of them in cooperation with or inspired by Hasse. 41 I am planning in a separate publication to cover in more detail the results of his cooperation with Hasse. But since we are concerned with class field theory in function fields we have

or a list of publications of F.K. Schmidt see [75].

588 P. Roquette

now to turn to other authors who completed the work initiated by F.K. Schmidt.

5.5. Summary

A s a follow-up to his paper on analytic number theory in characteristic p, F.K. Schmidt published a second paper announcing class field theory i n characteristic p. A s with the former paper, the manuscript for this one too was finished already i n the summer of 1927. Notwithstanding its title the paper does not give a comprehensive presentation of all of class field theory. The following items are missing i n the function field case: 1. A belian extensions of degree divisible by the characteristic p, 2. Artin's Reciprocity Law, 3. Functional equation of the L-series. Thus the paper can be regarded as a first approach only to class field theory. The arguments of the paper follow closely the presentation of class field theory in Parts I and l a of Hasse7s class field report. The proofs are given only partially, and the reader is assumed to be able to transfer himself, mutatis mutandis, the proofs given by Hasse in his class field report.

56. The reciprocity law

6.1. Hasse's paper on cyclic function fields

After F.K. Schmidt's paper [99], the next one which contains a contribution to class field theory for function fields was Hasse's [51], published in 1934 with the title: Theorie der relativ-zyklischen algebraischen Funktionenkorper, insbesondere bei endlichem Konstantenkorper (The- ory of relatively cyclic algebraic function fields, in particular with finite base field).

This paper is a product of Hasse's cooperation with Davenport who, as said in section 4.3.4 already, had introduced him to the problem of diophantine congruences which is equivalent to the Riemann hypothesis for the zeta function of function fields. In January 1933 Hasse had succeeded in proving the Riemann hypothesis in the case of elliptic function fields [46]. 42 AS a step towards the general case, i.e., function fields of arbitrary genus, Davenport and Hasse investigated fields of the form

42~or more details on that story see [88].


K(x, y ) with defining relation of one of the following types:

where p is the characteristic and m, n are integers not divisible by p (the base field K is assumed to be finite and containing the m-th and the n-th roots of unity, respectively). In fact, in these cases Davenport and Hasse succeeded in proving the Riemann hypothesis since the roots of the zeta function can be interpreted by means of certain Gaussian sums and related expressions whose absolute value is known [19].

Now, from (17) we see that the field E = K(x, y ) can be regarded as a cyclic extension of F = K ( x ) (and also of F' = K(y)). Therefore Davenport and Hasse wished to write a preparatory paper for reference purposes, containing the necessary general facts from the theory of cyclic extensions of function fields and their corresponding Gseries.

The paper [51] under consideration was written for this purpose; it appeared in the same volume of Crelle's Journal as the Hasse-Davenport paper [19]. However, as it is often the case in Hasse's papers, he not only presented the facts which were necessary for the intended application but in addition he developed a comprehensive and systematic study of the objects under consideration, in this case the cyclic extensions EIF of function fields.

Thus Hasse's paper [51] was not written primarily with class field I

theory in mind. Class field theory is only one aspect of the theory of cyclic extensions of function fields, and Hasse deals with it only in

I passing. From the 18 pages of the paper, only 2 are concerned with class field theory proper (pages 45-46). Nevertheless these pages constitute

I L an important step in the development of class field theory for function

fields. For, Hasse proves the analogue of Artin7s Reciprocity Law in the f

I case of function fields. , ; In its proof Hasse uses quite new ideas when compared to the former

papers by F.K. Schmidt or by Artin. This reflects the state of the art in class field theory as of 1934: recently Hasse had introduced the theory of algebras into class field theory of number fields, following an idea

, of Emmy Noether [40], [43]. Now he uses algebras also in the case of function fields.

The general reciprocity law as conceived by Artin is concerned with abelian extensions. But it suffices to prove it for cyclic extensions only. For, as Hasse remarks, the general abelian case is reduced "immediately in a well known manner" to the cyclic case (ohne weiteres i n gelaufiger Weise). This is the justification for Hasse to include class field theory and the reciprocity law in a paper which is devoted to the study of cyclic function fields. In the following discussion we shall formulate the

590 P. Roquette

reciprocity law for arbitrary abelian extensions. Later, while discussing Hasse's proof we shall point out where and how Hasse uses the assump tion that the extension is cyclic. In section 6.1.4 we shall discuss what Hasse may have had in mind when he mentioned, without reference, the "immediate and well known" reduction to the cyclic case.

6.1.1. The reciprocity law: Theorems A , B and C: Let EIF be an abelian extension of function fields, with Galois group G . For each

unramified prime p of F , let (y) E G denote its Frobenius automor-

phism. 43 The map p I+ (y) extends uniquely to a homomorphism

a ++ ( ) of the group of unramified divisors (i.e., those divisors a

which are composed of unramified primes only) into G. This is the "Artin homomorphism". Artin's reciprocity law is concerned with this homomorphism and can be formulated as follows.

Theorem A. The kernel of the Artin homomorphism contains the ray modulo m where m denotes a sufficiently large positive divisor which contains all primes which are ramified i n E . The smallest m with this property is the conductor m = f of the extension EIF. (Definitions see below.)

Theorem B. Regarded as a homomorphism from the ray class group, the Artin homomorphism C, + G is surjective, and hence yields an isomorphism of the factor group Cm/Hm onto G , where Hm denotes the kernel of the Artin homomorphism. This kernel is called the Artin group of ElF modulo m . If m' is another module with the same properties and Hmt its Artin group then Hm and Hmt are "equal" in the sense as explained above in section 5.2.

Theorem C. The Artin group Hm equals the norm group Nm which consists of those ray classes modulo m which are norms from E, i.e., Hm = N,. The norm group is called the Takagi group of EIF (modulo m) . 44

As to the definition of a "ray" modulo m and the corresponding full ray class group Cm we refer to section 5.2.

The notion of "conductor" (Fuhrer) also had been defined in section 5.2 but there the definition is of group theoretic nature: it refers to a

4 3 ~ a s s e [51] calls it "Artin automorphism". 4 4 ~ h e terminology of "Artin group" and "Takagi group" has been intro-

duced by Chevalley. It is not used in Hasse's paper [51]. (But in his Marburg Lecture Notes [44] Hasse uses the terminology "Artin classes" for the residue classes modulo the Artin group.)

Class Field Theorg i n Characteristic p, its Origin and Development 591

ray class group of finite index. It is only after establishing the main theorems of class field theory that the conductor, as defined in section 5.2, can be associated to an abelian extension. In the present context, however, the conductor is defined a priori, without recourse to class field theory, for any finite abelian extension El F, namely by local norm conditions. For each prime p of F let Ep IFp denote the corresponding local extension. Then the p-component f p of f is defined to be minimal such that every a E Fp with a 1 (mod f p ) is a norm from Ep. 45

As to the existence of this conductor, Hasse refers to his paper [49] which had just appeared in the Science Journal of Tokyo University. There, Hasse discusses local numberfields only. But the paper [49] is based on the theory of local division algebras [40] which immediately can be transferred to the function field case; therefore, Hasse says, all the results of [49] hold also in the function field case. 46

Examples: If p is not ramified in E then f p is trivial. If p is ramified and [E : F] is not divisible by the characteristic p then f p = p (there is

1 tame ramification only). If [E : F] = p then E = F(y) with y* - y = a E F ; supposing that the pole order of a at p is m $ 0 (mod p) , the multiplicity of p in f is m+l . For the proof of these examples Hasse again refers to [49] but mentions that one could easily obtain them directly.

The above reciprocity law contains all the main theorems on class field theory which had been listed by F.K. Schmidt (see section 5.2) except the existence theorem (Theorem I in 5.2).

6.1.2. Hasse's proof of Theorem A , in the cyclic case, by means of algebras: Let A be a simple algebra over F . 47 For each prime p of F consider the p-adic completion Ap over Fp. Hasse refers to his former paper [43], published one year earlier in the Mathematische Annalen, where he had defined what today is called the Hasse invariant of AP which is a rational number modulo 1 . 48 In that former paper Hasse had worked with local number fields but, as said earlier, the local theory

45After establishing the main facts of class field theory it turns out that

b both notions of conductor become equivalent: the conductor of an abelian

i extension coincides with the group theoretical conductor of its Artin group. 46Witt has pointed out in [127] that the existence of the conductor is

i equivalent to the fact that the norm map is open in the topology of local i i fields, and that this is an easy consequence of Hensel's lemma.

! 471t is tacitly assumed that A is finite dimensional and that F is its center. 48That paper, dedicated to Emmy Noether, is the one where Hasse suc-

ceeded to prove the Artin reciprocity law in the number field case by means of the theory of algebras. The starting point for this was the fact that the local Hasse invariant (t) could be defined, following Chevalley [14], by purely local considerations whereas formerly, as in [41], Hasse could give the definition by

592 P. Roquette

can be transferred without problems to the case of local function fields (which are power series fields over finite base fields). In particular, the

p-adic Hasse invariant (:) is defined also in the function field case, as

a rational number modulo 1. The essential step in proving Theorem A is the proof of the sum

formula

(f) - 0 (mod 1)

where p ranges over all primes of F . For this, the proof in the number field case cannot be transferred directly to the function field case, because of the different behavior of cyclotomic fields in these two cases. Now in the function field case, Hasse proves (18) with the help of what today is known as Tsen's theorem.

In his Gottingen thesis [113], [I141 Ch.C. Tsen had proved, shortly before Hasse's paper, that there are no nontrivial simple algebras over a function field with algebraically closed base field. Tsen had studied with Emmy Noether as his advisor; in the preface of his thesis he mentions that he had also received valuable help from Artin. 49

Because of Tsen's theorem, for each algebra A over F there exists a finite base field extension L ( K such that A splits over FL. This is quite analogous to the fact, known from number theory, that every algebra admits a cyclotomic splitting field. But in the function field case the situation is much simpler because the splitting field F L is unramified over F . Moreover, F L is cyclic over F and its Galois group admits the Frobenius automorphism T of LIK as its generator.

means of the global class field product formula only. - As Auguste Dick [26] reports, Emmy Noether was extremely glad ("ganz besonders erfreut") about Hasse's results in this paper which confirmed her belief that non-commutative arithmetic can be profitably used to study commutative number fields.

4 9 ~ am indebted to Falko Lorenz for pointing out to me that also F.K. Schmidt is mentioned in Tsen's thesis, namely as his referee (Referent). This reflects the state of affairs at the Gottingen mathematical scene in 1933134. Emmy Noether had been dismissed from her university position in early 1933, due to the antisemitic policy of the National-Socialist regime in Germany since 1933. Also, many other mathematicians had left GWtingen; see the report by Schappacher and M. Kneser [go]. F.K. Schmidt had been called to Gottingen in the fall of 1933 as a temporary replacement of H. Weyl. In this position he took care of a number of students who had been advised by E. Noether, and in particular of Tsen. - F.K. Schmidt remained in Gottingen for one year; after that he received a position as a full professor at the University of Jena. For more biographical information about F.K. Schmidt see [75].


In this situation A is similar to a crossed product algebra: A - (p, FL, T) , defined as being generated by F L and an element u with the defining relations

where m = [L : K]. p is an element of F which is determined modulo norms from F L only. Now since F L l F is unramified the local Hasse invariants of such a crossed product are easily read off from their defi-

nition, namely (:) I deg(p:p (') (mod 1). (Here, up (P) is the order of

,B at p.) Hence the sum formula (18) is a consequence of the formula

which expresses the fact that every P # 0 admits as many poles as there are zeros. Having established the sum relation (18), the proof of Theorem A above is straightforward, once one has accepted Hasse's use of the theory of algebras in arithmetic:

At this point Hasse uses the assumption that EIF is cyclic. Accord- ingly let a be a fixed generator of the Galois group G. Let n = [E : F]. For each 0 # a E F consider the cyclic crossed product algebra

A = (a, E, a ) . Writing the p-adic Hasse invariant in the form A = 3 (J (mod 1) with rp E Z, Hasse defines the local norm symbol as follows: 50

We have (F) = 1 if and only if a is a norm from Ep 1 Fp .

By means of the definition (20) the sum formula (18) is translated into the product formula

Now if a e 1 (mod f) then for every ramified p we have by definition that a is a local norm from Ep, hence the corresponding algebra A splits

at p and therefore (y) = 1. On the other hand, for unramified p

5 0 ~ h e minus sign in front of the exponent r p on the right hand side is for normalizing purposes only and is not important for the following argument.

594 P. Roquette

it follows from the definition that (F) is a power of the Frobenius

automorphism, namely (F) = ( ) and hence the product

formula (21) yields for the principal divisor a = (a):

--up (ff

unramified p all p

which shows that, indeed, a = (a) is in the kernel of the Artin homomorphism.

6.1.3. The proof of Theorems B and C: For the proof of Theorem B Hasse uses the fact, proved by F.K. Schmidt, that the zeta function of every function field has a pole of order 1 at the point s = 1. He argues as follows: Let G' be the image of the Artin homomorphism and E' the subfield of E corresponding to G' by Galois theory; put n' = [El : F]. Then every prime p of F which is not contained in m splits completely in E', i.e., it has precisely n' extensions in El. It follows from the product representation of the zeta function of E' that CE/ ( s ) is the n'-th power of CF(s) - except perhaps for finitely many Euler factors belonging to the primes of m. In any case, the zeta function of E' has a pole of order n' at s = 1. Thus n' = 1 and E' = F.

We see that for Theorem B, Hasse used the following lemma which he proved with the help of F.K. Schmidt's zeta function:

Lemma 1. Let E I F be an abelian field extension such that almost every prime 51 p of F splits completely i n E. Then E = F.

For the proof of Theorem C Hasse uses the "first inequality" h, 5 n of (15) which had been proved by F.K. Schmidt by means of L-series. Here, h, is the index of the Takagi group. Hasse remarks that this part of F.K. Schmidt's paper [99] in which he proved the first inequality, is generally valid and does not depend on the assumption, otherwise imposed in [99], that the field degree n $ 0 (mod p).

According to Theorems A and B, the field degree n equals the index of the Artin group. Since the Takagi group is contained in the Artin group it follows h, = n and both groups coincide.

6.1.4. Further remarks: We have discussed Hasse's proof in such detail in order to put into evidence that his idea of using algebras in class field theory did contribute essentially to simplify and systematize

' l ~ h i s means every prime but finitely many.


the proofs. There are two comments of Hasse on his proof which perhaps need some further attention. The first of these comments, found on page 142, we have mentioned above already:

Der obige Klassenkorperhauptsatz iibertragt sich ohne weiteres i n gelaufiger Weise auf beliebige separable abelsche Erweiterungskorper.

The above main theorem of class field theory can be extended immediately and in a well known manner to the case of arbitrary separable abelian extension fields.

By "main theorem" Hasse means the union of what we have called The- orems A, B and C . His proof, as presented above, covers only cyclic extensions. In order to obtain class field theory in its full extent one has to reduce the general abelian case to the cyclic case. Hasse does not give any reference, nor does he explain what he means by an "immediate" and "well known" method to carry out this reduction. A closer look may perhaps reveal what he had in mind. Let us explain the situation:

Our above presentation puts into evidence that the proofs of Theo- rems B and C are generally valid, and it is only Hasse's proof of Theorem A where the cyclic property of the extension EIF is used.

Let EIF be an arbitrary abelian extension with group G. Consider the cyclic subextensions Ei)F of E ) F , with conductors fEi. Let the divisor a of F be unramified in E and (T) E G its image under the

Artin map. When restricted to Ei this gives (y). Hence, if a is contained in the ray modulo fEl then, by the cyclic case of Theorem A, the restriction of (T) to Ei is trivial; if this is true for all i then

(T) = 1 in G. Now by the very definition of the conductor, we have

That is, the ray modulo the conductor f E is contained in the intersection of the rays modulo the conductors fEi of its cyclic subextensions. It follows that the Artin homomorphism vanishes on the ray modulo fE.

Thus indeed, this "immediate" argument shows that Theorem A holds for any abelian extension EIF - except for the fact that the conductor fE is the smallest divisor with the property as stated in the theorem. For the proof of this additional contention one has to use that equality holds in (22):

(23) f ~ = m a x

596 P. Roquette

Since conductors are defined locally, this is a purely local affair, concerning the local abelian extensions Ep IFp for each prime p of F. It is clear that this formula is a consequence of local class field theory.

Already in 1930 Hasse and F.K. Schmidt had developed local class field theory [39], [97] in the number field case. But in their first approach the local theory was based on the global theory, because it was not possible, at that time, to define the norm residue symbol on purely local terms. It was Chevalley who in his famous thesis [15] had developed the local class field theory directly, without using global arguments.

But, considering the year of publication (1934), where Chevalley's thesis had just appeared, is it conceivable that Hasse would refer to it by naming it "immediate and well known", without mentioning explicitly what he has in mind? And without giving any particular reference?

In looking for further evidence we discover that two pages earlier, on page 140, Hasse gives a reference to another of his papers [49], on the norm residue theory of Galois fields with applications to conductor and discriminant of abelian fields. (We have mentioned this earlier already.) That paper appeared right after Chevalley's thesis in the same Japanese journal. 52 It contains a detailed study of the norm map when compared with the higher ramification groups. Although it is concerned with local number fields, it is clear from the context and mentioned explicitly by Hasse that the local theory can be transferred directly to the function field case. 53 For US it is of interest that this paper [49] contains an explicit formula for the conductor of an abelian extension, from which (23) can be deduced. In proving that formula, Hasse had to use certain facts from local class field theory, and he said about it:

Was den Beweis . . . angeht, so bildet der Spezialfall, wo die Erweiterung zylclisch ist, den einen Hauptpunkt . . . Fur diesen zylclischen Spezialfall hat Herbrand einen sehr eleganten Beweis gegeben. (Eine Darstellung dieses Be- weises siehe i n der Th&e von C. Chevalley, die dieser Arbeit unmittelbar vorangeht . . . ) Den berga an^ zum allgemein abelschen Fall kann man entweder unter voller

52See also Hasse's Comptes Rendus Notes [47],[48] where he announced the results of [49].

53~asse 's paper [49] became widely known because it contains the Hasse part of the "Theorem of Hasse-Arf" on the ramification numbers of local abelian extensions. The Hasse part is concerned with local fields whose residue field is finite. Hasse conjectured that the same result would hold for arbitrary perfect residue fields and he gave this problem to his student Cahit Arf who solved it in his thesis [2].


Ausnutzung der Hauptsatze der Klassenkorpertheorie i m Grossen durch Entwiclclung der Theorie des Normenrest- symbols vollziehen - das ist aber methodisch unschon - oder aber auch direlct durch methodisch i n die Klassen- lcorpertheorie i m Kleinen gehorende Betrachtungen aus- fuhren. (Eine Ausfuhrung dieses Beweises siehe eben- falls i n der Chevalleyschen Th$se.) - Die Ausfuhrungen meiner vorliegenden Arbeit erganzen den Herbrandschen Beweis fur den zyklischen Fall und den Chevalleyschen ubergang zum allgemein-abelschen Fall eben i n der Weise, dafl sie die genaue Bestimmung des p-Fuhrers liefern . . .

Concerning the proof . . . , the main point is the special case where the extension is cyclic . . . For this cyclic case a very elegant proof has been given by Herbrand. (For an exposition of this proof see Chevalley's thesis which immediately preceeds this paper . . . ) The transition to the general abelian case can be given either with full use of the main theorems of global class field theory by develop ing the norm residue symbol - but that is not desirable from a methodical point of view - or else directly, using arguments which methodically belong to local class field theory. (For an exposition of this see Chevalley's thesis again.) - The discussion in my present paper supplement Herbrand's proof in the cyclic case and Chevalley's transition to the general abelian case in such a way that they yield the exact determination of the p-conductor.

Thus here in [49] we find what we have missed in [51], namely an explanation of how Hasse envisages the transition from the cyclic to the abelian case, i.e. the proof of (23). From today's viewpoint, since local class field theory is well known nowadays, the proof of (23) is indeed "immediate and well known", but it seems doubtful whether this could be said in 1934 already. In 1934, the reader of [51] would perhaps have preferred a more detailed explanation of what Hasse had in mind.

In any case, as said earlier already, Hasse says explicitly that the results of [49] which were stated and proved there for local number fields, remain valid for local function fields.

By the way, in 1933 Hasse had already accepted a paper by Chevalley for Crelle's Journal [14], where the latter also presented his method how to prove (23). It is quite apparent that his method is of cohomological

598 P. Roquette

nature, as the "crossed products" (verschrankte Produkte) of E. Noether are used to compare the norm groups of different cyclic extensions. -

Now we quote the second comment of Hasse on page 142 in [51], on his proof of Artin's reciprocity law:

Damit ist eine dem heutigen Stande angepaflte Begriin- dung der von F.K. Schmidt entwickelten Klassenkorper- theorie gegeben und insbesondere die dortige Beschrankung auf zur Charakteristik p prime Grade beseitigt.

Herewith we have given a presentation, adapted to the present state of knowledge, of the class field theory which had first been developed by F.K. Schmidt; in particular we have eliminated the restriction to those degrees which are relatively prime to the characteristic. 54

Hasse's wording that his presentation corresponds to the "present state" of knowledge may reflect that he did not consider it as final; he leaves it open that further simplifications are to come in due time. Had he envisaged already the penetration of cohomology into class field theory ?

If we review the above proof of Theorem A we see that simple algebras are used mostly in a formal way: as crossed products which, in the cyclic case, reflect the norm class structure for the splitting field. 55 Nev- ertheless it seems unlikely that Hasse was contemplating to substitute, as regards class field theory, the theory of algebras by a more formal calculus of cohomological nature. In fact, he has always propagated Emmy Noether's dictum: Use non-commutative arithmetic to get results i n the commutative case ! And later in the forties and fifties, when cohomology indeed had found its place in class field theory due to the works of Hochschild, Nakayama, Artin and Tate [9], [112], then Hasse did never

54~asse ' s citation list includes all 4 papers by F.K. Schmidt which we have discussed above: [95], [96], [98], [99]. He gives full credit to F.K. Schmidt for having developed the general theory of function fields, in particular with finite base fields. (Side remark: Erroneously Hasse cites [95] as F.K. Schmidt's thesis in Erlangen but as we have mentioned above, this thesis was written at the university of Freiburg. This seems quite curious since Hasse had sent the proof sheets of his paper to F.K. Schmidt and asked for his comments. It seems that F.K. Schmidt himself did not discover this error.)

5 5 ~ t is only in the local case that Hasse had to regard algebras not only through the crossed product formalism but in fact with their arithmetic structure: In order to prove that a local division algebra admits an unramified splitting field Hasse extended the canonical valuation of the center to the division algebra and studied its arithmetic properties [40].


take up those ideas in his own work - although of course he himself had started all this by using crossed products in class field theory. It seems that he did not wish to part from Emmy Noether's idea about the role of non-commutative algebras incommutative number theory. 56

More likely, Hasse may have had in mind to free class field theory from analytic methods and to find proofs which are based solely on algebra and arithmetic. Such tendency was spreading in the thirties, with the intention to gain more insight into the structures connected with class field theory.

Hasse7s proof of Theorem A is certainly of algebraic-arithmetic nature. But Theorems B and C still rested on analytic arguments. We shall see in section 8 how it became possible to replace these analytic arguments by algebraic ones.

6.2. Witt: Riemann-Roch theorem and zeta function for algebras

In the year 1934, at about the same time when Hasse's paper [51] was published, there appeared a paper by Witt [I241 which also contained important contributions to class field theory for function fields. This was Witt's Gottingen thesis of 1933. The aim of Witt's thesis was to transfer the theory of Kathe Hey to the function field case.

In the year 1929 Kathe Hey had completed her thesis [65] in Ham- burg, with E. Artin as her advisor. She had considered simple algebras over a number field and developed analytic number theory in this setting; in particular the zeta function was defined and investigated in the non- commutative case, in analogy to the Dedekind zeta function of a number field. Hey's thesis has never been published 57 but it was well known at that time in the context of algebraic and analytic number theory. It contained also a new analytic foundation of the main theorems in class field theory; according to Deuring [22] "die starkste Zusammenfassung der analytischen Hilfsmittel zur Erreichung des Zieles" (the strongest concentration of analytic tools in order to reach the goal).

Now, after F.K. Schmidt had succeeded in transferring the theory of the Dedekind zeta function to characteristic p there arose the question whether Hey's theory of the zeta function for division algebras could be transferred too. E. Noether had posed this question to Witt and he answered it in his thesis.

5 6 ~ e e the last words of Hasse in his paper on the history of class field theory [56].

5 7 ~ h e thesis contained some errors which, however, could be corrected. See e.g., Zorn [133], Deuring [22] chap.VI1, Fj8.


E. Witt had studied one year in Freiburg (since 1929) and then 3 years in Gottingen. As he himself recalls [131]:

Tief beeindruckt haben mich 1932 die beriihmten 3 Vor- trage von Artin uber Klassenkorpertheorie. Die ansch- liessenden Ferien verbrachte ich i n Hamburg, u m dort die Klassenkorpertheorie intensiv zu studieren. I n den folgen- den Jahren war es mein Ziel, diese Klassenkorpertheorie auf Funktionenkorper zu ubertragen.

In the year 1932 I was deeply impressed by the famous three lectures of Artin on class field theory. 58 In the next academic vacations I went to Hamburg for an intensive study of class field theory for number fields. In the following years it was my aim to transfer class field theory to function fields.

Witt was 21 when he decided to complete what F.K. Schmidt had started. From the above we see that Witt in his work was much influenced by E. Artin. Other people who influenced Witt were Emmy Noether, his thesis advisor, and H. Hasse whose assistant in Gottingen he became in 1934. 59

The title of Witt's thesis is: "Riemann-Rochscher Satz und <-Funk- tion im Hyperkomplexen" (Riemann-Roch theorem and <-function in the hypercomplex domain). Witt cites F.K. Schmidt's already classical paper [98]. In fact, Witt's proof of the Riemann-Roch theorem in the non-commutative case copies F.K. Schmidt's proof very closely; he says that F.K. Schmidt's proof served him as a model ("nach dem Vorbild von F.K. Schmidt"). After establishing the proper notions of "divisor" etc. of a simple algebra, Witt showed that the Riemann-Roch theorem can be formulated and proved precisely as in the commutative case, with the exception that the genus of the algebra may be negative. In fact, the treatment by Witt puts into evidence that the Riemann-Roch theorem essentially belongs to linear algebra, hence the non-commutativity of the multiplication does not disturb the general picture.

Thus this paper continues the historical line which had been started 1880 by Dedekind-Weber [21], which was followed 1902 by Hensel- Landsberg [61] and had been taken up 1927 by F.K. Schmidt [98]. Witt

5 8 ~ h e r e were notes taken by Olga Taussky from Artin's lectures. A copy is preserved in the library of the Mathematics Institute in Gottingen [7]. I am indebted to F. Lemmermeyer for pointing out to me that an English translation of these lecture notes has been included as an appendix to H. Cohn's "Classical Invitation to Algebraic Numbers and Class Fields'' [18].

5 9 ~ o r e biographical information about Witt can be obtained from [74].

seems to have been fully aware of this background; he says that his construction of a normal basis follows the usual way ( " in der ublichen Weise"). But in contrast to F.K. Schmidt [98], Witt presented fully the algebraic proof of the Riemann-Roch theorem. (Recall that F.K. Schmidt had been content to wave his hands and just said that the Riemann-Roch theorem can be proved in quite the same way as in Hensel-Landsberg ; see section 4.3.2).

Similarly as in the case of fields, in the case of division algebras the Riemann-Roch theorem leads to a birational invariant zeta function; this is the function field analogue to Hey's zeta function. Comparison of Witt's new zeta function of the division algebra with F.K. Schmidt's zeta function of its center field leads to the following conclusion (as in Hey's thesis for number fields):

Every non-trivial division algebra (or, more generally, simple algebra) over a function field F admits at least two places where the algebra does not split. In other words: If a simple algebra splits locally for all but possibly one place then it splits globally.

Based on this local-global principle Witt presents an alternative proof of the sum formula (18) which Hasse had used for the proof of Artin's reciprocity law in function fields.

Thus in this paper, Witt's result concerning class field theory for function fields overlaps widely with Hasse's [51]. But the methods are different: whereas Hasse's proof of the sum formula (18), based on Tsen's theorem, is essentially of algebraic nature, Witt's proof of (18) is based very much on analytic properties of the zeta function of algebras. It constitutes, to use Deuring's words once more, "the strongest concentration of analytic tools i n order to reach the goal". But the trend in the further development was more towards the algebraic direction. Witt's analytic proof of the reciprocity law is not widely known today, and his paper [I241 is known mainly for the Riemann-Roch theorem for algebras, i.e., as a contribution to non-commutative algebraic geometry. 60

By the way, in this paper Witt also gives a complete description of the Brauer group of algebras over function fields. The result is of the same type as Hasse had found a year ago in the number field case [43].

6.3. Summary

I n 1934 Hasse published a paper on cyclic extensions of func- t ion fields. His original motivation came from his joint work with Davenport on the Riemann hypothesis for certain function

'Osee e.g., the Remarks by Giinter Tamme in [132], page 60.


fields of higher genus. But Hasse's paper contained also important contributions to class field theory for function fields. Its main achievement regarding class field theory was Hasse's proof of Artin's general reciprocity law in the function field case. The methods used i n that proof belong to the arithmetic theory of algebras and their splitting behavior; these methods had recently been successfully used by Hasse in the number field case (responding to a question of E m m y Noether) and were now transferred to the function field case.

0 Hasse's proof i n the function field case relied heavily on the theorem which Tsen had just obtained i n his Gottingen thesis (with Emmy Noether as his main advisor). In 1934, parallel to Hasse7s paper, there appeared Wit t 's Gottin- gen thesis (again with E m m y Noether as thesis advisor). This paper was concerned with the transfer of Kathe Hey's theory to the function field case; i.e., developing the theory of zeta functions for simple algebras over function fields. To this end Wit t proved the Riemann-Roch theorem for simple algebras over function fields, in generalization of F.K. Schmidt's work. Witt's theory of zeta functions for division algebras leads to a local-global principle for algebras over function fields and, consequently, to a new proof of the Artin reciprocity law for function fields. Hence, Witt 's results overlap with those of Hasse but the methods used are diflerent. Witt's paper was conceived as the first of a series in which Witt planned to complete class field theory for function fields, which had been started by F.K. Schmidt.

57. The final steps

7.1. H.L. Schmid: Explicit reciprocity formulas In the case when the ground field is a rational function field, F =

K(x ) , Hasse in his 1934 paper [51] provided a second proof of Artin's reciprocity law, not depending on Tsen's theorem and being of "elementary" nature in the sense that only elementary manipulations of polynomials and rational functions are used. In doing this he distinguished two different cases, depending on the degree n = [E : F], namely n f 0 (mod p) and n = p. (Recall that p denotes the characteristic.) In the case n $ 0 (mod p) Hasse observed that his arguments are essentially identical to those which lead to the power reciprocity law (1) in F.K. Schmidt's thesis which we have discussed in section 3.2.2.

But the arguments in the case n = p, where Artin-Schreier theory had to be used, were new. Hasse's computations in this case involved

logarithmic derivatives of rational functions. He pointed out that these are the precise analogues of Kummer's logarithmic derivatives which appear in the explicit reciprocity formulas in the number theory case. 61

There arose the question whether those computations could be generalized to arbitrary function fields F, not necessarily rational. Hasse had put this question to his student H.L. Schmid. 62

H.L. Schmid solved Hasse's question in his 1934 Marburg thesis which appeared in print one year later [91]. The paper has the title ~ b e r das Reziprozitiitsgesetz i n relativ-zyklischen Funktionenkorpern mi t endlichem Konstantenkorper (On the reciprocity law in relatively cyclic function fields with finite fields of constants). It is conceived as a follow- up to Hasse's paper [51], with the aim of supplementing it by giving explicit formulas for the local norm symbols.

Let EIF be cyclic of degree n. Following Hasse, H.L. Schmid deals separately with the two cases n $ 0 (mod p) and n = p.

The most interesting is the case n = p. Then El F admits an Artin- Schreier generation

with p E F . For any a E FX and any prime p of F consider the

local norm symbol (7) as defined above in (20). For computational

purposes it is convenient to replace this symbol, which is an element in the Galois group G, by another symbol which is an element in the prime

field Zip. Namely, if the automorphism (7) is applied to y then the

result is y + c with c in the prime field Zip. This c is then denoted by

{ ); in other words, the defining relation for the new symbol is

This symbol is multiplicative in the first variable a and additive in the second variable 0. 63

' l ~ a s s e had discussed and generalized Kummer's formulas in Part I1 of his class field report [38].

6 2 ~ o t to be confused with F.K. Schmidt. For biographical information about H.L. Schmid see the obituary [55], written by Hasse in 1958.

6 3 ~ n the literature there is no unique notation for this symbol. Here we use H.L. Schmid's notation. Note that this symbol is asymmetric; there is no formula for exchanging the arguments a and P. See e .g . , the notation used by Witt in [127].

604 P. Roquette

Now, H.L. Schmid gives the following explicit formula for the computation of this symbol:

Here, resp (. . . ) denotes the residue at p of the differential in question; this is an element in the residue field Kp . 64 And Sp, : Kp -+ K is the trace function ("Spur") to K, whereas 6 : K -+ Z / p is the absolute trace from K to its prime field.

The formula (24) contains the logarithmic differential % which again puts into evidence the analogy to Kummer7s formulas in number theory - this time for an arbitrary function field F instead of the rational field as in Hasse7s paper. The importance of the formula lies in the following:

Firstly, in view of the theorem of the residues in function fields:

it follows immediately from (24) that

which is equivalent to the sum formula (18) for the algebra A = ( a , E, a), i.e., for all algebras A over F which admit a cyclic splitting field EIF of degree p. This proof of (18) does not need the theorem of Tsen. In this way it is possible to prove Artin7s reciprocity law for cyclic extensions of degree p without Tsen's theorem, using the theorem of the residues instead.

Secondly, the formula (24) immediately gives the multiplicity of p in the conductor of EIF: it is m + 1 if m $ 0 (mod p) is the pole order of p at p. (Note that by definition, a is a norm from Ep if and only

if (7) = 0.) Whereas Hasse [51] had to rely on the theory of higher

ramification groups and its connection to conductors, as developed in [49], the formula (24) shows this result immediately.

Thirdly, the formula (24) gives rise to a formalism about p-algebras over arbitrary fields of characteristic p; this has later been observed and used, e.g., by Witt [I271 (see section 7.2).

6 4 ~ . L . Schrnid defines Sppres,(. . . ) to be the residue of the differential.


Let us add some remarks concerning the case n $ 0 (mod p). It is assumed that the n-th roots of unity are in F. Then E(F is a cyclic Kummer extension:

E = F ( Y ) , yn = P with ,f3 E F. Again for any prime p of F , H.L. Schmid is concerned with

the local norm symbol (y) which is an element of the Galois group

G. The corresponding numerical symbol ( Y ) ~ is now to be defined

multiplicatively, in the form:

This time the symbol ( Y ) ~ is an n-th root of unity; it is multiplicative

in both variables a , P. 65 NOW H.L. Schmid arrives at the following explicit formula. For simplicity let us write a = up (a) and b = up (P) .

where Np is the norm function from the residue field to K. (For any function f E F we denote by f (p) its image in the residue field Kp .)

H.L. Schmid points out that the formula (27) is the multiplicative analogue to (24). But, he says, while (24) leads to a new proof of Hasse's sum formula (18) (via the theorem of the residues) and hence to Artin's reciprocity law in the case n = p, the formula (27) does not so in the case n f 0 mod p.

Hasse, when reporting about H.L. Schmid's work in 1958, also says that a multiplicative analogue to the residue theorem has not been found in this connection [55].

It seems that both H.L. Schmid and Hasse had overlooked the relation of the n-th norm symbol (?Irn to the universal symbol in arbitrary

.. conservative function fields, over any base field. That universal symbol

( ) is defined by the same formula (27) but without the exponent

on the right hand side. It is well known that for this universal symbol the product formula

6 5 ~ n the literature this symbol is called the Hilbert symbol.

606 P. Roquette

holds. See, e.g., the treatment of those symbols in Serre's book on algebraic groups and class fields [103], or in [87]. This product formula,

when taken into its G - t h power, yields the product formula for ( Y ) ~ and hence formula (21), thus it gives a new proof of Artin's reciprocity law, independent of Tsen's theorem, also in the case n $ 0 mod p - at least if the n-th roots of unity are contained in K.

7.2. The existence theorem

As we have said above already, Artin's general reciprocity law does not cover the existence theorem of class field theory as formulated in statement I in section 5.2. Witt takes up the challenge in his second paper [I271 of his series on class field theory, which appeared 1935 in Crelle's Journal with the title "Der Existenzsatz fur abelsche Funktio- nenkorper" (The existence theorem for abelian function fields).

The existence theorem is a major ingredient in general class field theory. In the case of number fields, the existence theorem had been part of the results of Takagi [108]. The proof had been included in Hasse's class field report but was later much simplified by Herbrand and in Chevalley's thesis [15].

For function fields, F.K. Schmidt [99] had claimed to have a proof in the case when the index is not divisible by p. As I have said already in section 5.4 his claim was not too convincingly substantiated since F.K. Schmidt did not go into the details of proof which would involve delicate index computations. And for subgroups of index divisible by p, before Witt's paper there had been no hint of how to approach this problem.

Witt's paper constitutes a major advance in the development of class field theory for function fields. It is a masterpiece not only because of its results but also because of its concise and precise style which became the characteristic of Witt's papers. Witt's reputation as a first rate and very original mathematician was fully established with this paper.

The existence theorem can now be formulated as follows:

Theorem D. Given a module m i n a function field F and a subgroup Hm of finite index i n the ray class group C,, there exists a unique abelian extension E ( F such that (i) every prime p of F which does not appear i n m is unramified in E ; (ii) H, is the kernel of the Artin homomorphism from C, to the Galois group G of EIF.

Consequently the factor group C,/H, is isomorphic to G in view of part B of Artin's reciprocity law (see section 6.1.1). Moreover, H,,, = N, in view of part C .


Let n denote the exponent of the factor group Cm/Hm. Witt discusses separately the two cases n $ 0 mod p and n = p ; the general case is then treated by induction.

It suffices to prove the existence theorem for the smallest ray class group in C, whose factor group is of exponent n, i.e., for the group C:. For, if there exists a class field El F for this group, then the subgroups Hm between C: and Cm correspond, via the isomorphism of Artin's reciprocity law and Galois theory, 1 - 1 to the intermediate fields between E and F ; it is immediate that for each such subgroup (ii) holds with respect to the corresponding field.

Witt cites Hasse's class field results in [51] and says that, by Hasse, every abelian field extension EIF is a class field for some ray class group. So he is going, for given m and n, to construct a certain field extension EIF bythe usual algebraic procedures, namely Kummer extension in case n $ 0 mod p (after adjoining the n-th roots of unity) and Artin-Schreier extension in case n = p ; then he verifies that because of his careful construction of EIF its Artin group is precisely C:. This requires some rather straightforward index computations.

In case n $ 0 mod p there will be tame ramification only and m can be assumed to be "square free", which means that every prime occurring in m has multiplicity 1 in m. Witt says that his proof in this case is just a copy of "Herbrand's proof". For this he cites Hasse's Marburg lecture notes [44] where Herbrand's computations are presented. (As it is to be expected, the computations in the function field case require some modifications.) It seems strange that Witt does not cite Chevalley's thesis [15]. It is also strange that Witt does not cite the paper by Chevalley and Nehrkorn [16] which appeared at about the same time as Witt's. In that paper the existence theorem is discussed (in the number field case) from the point of view of arithmetic-algebraic proofs (see section 8). Neither do Chevalley-Nehrkorn cite Witt, and hence it seems that none of the two parties knew about the work of the other party before it was too late to insert a reference. "

6 6 ~ n November 1934, Chevalley informed Hasse about the results of his paper with Nehrkorn. At the same time Chevalley announced that he was working on a proof of the existence theorem in characteristic p. Hasse replied that just recently Witt had submitted to him a paper containing the proof of the existence theorem, and he added a sketch of Witt's proof. He also informed Witt about Chevalley's letter. It appears that Chevalley's ideas were quite similar to Witt's on this matter. Thus Witt and Chevalley were informed about the work of the other through Hasse.

608 P. Roquette

As a side result, Witt develops the theory of arbitrary abelian Kum- mer extensions (not necessarily cyclic) of a given exponent n $ 0 mod p. This is the form which today is usually given in algebra textbooks. 67

The case n = p was new and Witt could not rely on analogues in number fields; Herbrand and Chevalley did not cover this case. Witt relied, however, on H.L. Schmid's paper [91] and the formula (24); it permits to estimate in advance the conductor of an abelian extension of exponent p. Actually, Witt generalized H.L. Schmid's formula in the following way: For a + 0 and /? in F Witt defines the algebra (a, /?I over F given by generators u, y with the defining relations:

Let p be a prime of the function field F and Fp its completion. Over Fp one can perform a similar construction; the corresponding algebra

is denoted by (?I. (Witt's notation is (a , /?I since he considers p as

being fixed.) Now Witt states and uses the formula

where - denotes the equivalence of algebras and t is a uniformizing variable at p. In this formula, like in H.L. Schmid's formula (24), there appears a logarithmic derivative. Witt mentions H.L. Schmid's paper 68

and refers to the proof there, although the formula (24) is not quite the same as Witt's (29): for (24) concerns Hasse invariants of algebras, whereas (29) holds for the algebras themselves. Accordingly Witt's formula is more general: it holds over arbitrary perfect base fields while H.L. Schmid's formula (24) makes sense only if the base field is finite. But his formula, Witt says, is proved by the same methods as H.L. Schmid's. It is straightforward to extract H.L. Schmid's formula from Witt's.

Again as a side result, Witt develops the theory of abelian extensions of exponent p, not necessarily cyclic, thereby generalizing Artin-Schreier

PI. It turns out that the computations in case n = p are easier than

those in case n $ 0 mod p. Let us cite Witt:

6 7 ~ e e e.g., Lorenz [76]. 6 8 ~ h i s had not yet appeared when Witt wrote his manuscript; so he

referred to [91] by saying: "erscheint demnachst i n der Math. Zeitschr." (will appear soon in the Mathematische Zeitschrift).


Ein Analogon der Theollle der Kummerschen Korper erhalten wir i m Falle n = p, indem wir die Produkte ad- ditiv schreiben. Es ist bemerkenswert, dafi der Existenz- beweis i m vorliegenden Falle vie1 einfacher gefuhrt werden lcann. Durch direlctes Schliefien mit Hilfe des Riemann- Rochschen Satzes werden lunge Indexrechnungen vennei- den.

In case n = p we obtain an analogue to the theory of Kum- mer fields by writing all products in an additive manner. It is remarkable that in this case the existence proof can be given in a much easier way. By direct recourse to the Riemann-Roch theorem one can avoid long index compu- t ations.

With Artin's reciprocity law and the existence theorem, the foundation of general class field theory was now achieved in the function field case. But we have still to mention two other items which concern class field theory for function fields: Explicit reciprocity formula for cyclic extensions of p-power degree, and the functional equation for F.K. Schmidt's L-series. The relevant papers for these were published by H.L. Schmid and Witt who, both being assistants to Hasse in Gottingen, seem to have worked closely together.

7.3. Cyclic field extensions of degree pn

In 1937, as a result of the legendary Gottinger Arbeitsgemeinschaft (workshop) headed by E. Witt, there appeared his great paper [I301 where he introduced what is now known as Witt vectors. The construc-

, tion of Witt vectors "is of fundamental importance for modern algebra and some of the most recent developments i n arithmetical algebraic ge-

l ometry" (G. Harder in [132], page 165). I It seems not to be widely known that Witt vectors were discovered , in connection with a problem belonging to class field theory in function

fields. The problem was to generalize H.L. Schmid's explicit formula for I

the norm symbol (24) (section 7.1) to cyclic extensions of p-power degree, I not just of degree p. This was not necessary for the proof of Artin's

reciprocity law or of the existence theorem since for those purposes one had an argument using induction with respect to the degree. But the problem was of importance in order to fully transfer class field theory, including the explicit reciprocity formulas, to the function field case.

Witt had transformed H.L. Schmid's formula (24) into (29) which concerned algebras of rank p ; now the problem was to arrive at similar formulas for cyclic algebras of p-power rank.

610 P. Roquette

In order to attack this problem one first had to generalize the Artin- Schreier generation of cyclic fields of degree p to cyclic fields of p-power. This had been done in 1936 by H.L. Schmid [92]. He had found that an earlier solution of the problem, given by Albert [I], was not suited for the intended arithmetical application. Instead, he had discovered that a cyclic field extension EIF of degree pn in characteristic p can be generated in the form

where the Witt vector y = (yo, y1, . . . , yn-1) of length n has components in E and satisfies an equation of the form

with a vector ,B = (Po,&, . . . over F . In formula (30) one has to interpret yp as the vector with the components yr (as it is usual with Witt vectors), and the minus sign is to be interpreted in the sense of the additive group of Witt vectors.

H.L. Schmid, however, had not yet the formalism of Witt vectors at his disposal. Recall that every Witt vector y = (yo, yl, . . . ) is also given by its "ghost components" (Nebenkomponenten) y = (y(0), y( l ) , . . . ) ; the algebraic operations are given componentwise in the ghost components which yield polynomially defined operations for the main components

69 ?4i.

Those polynomials are quite complicated to work with explicitly. It was a high accomplishment that H.L. Schmid was able to get through with the very complicated polynomial computations, proving associativ- ity, distributivity etc. for those operations and, moreover, using this to study the arithmetic notions like Artin-symbol, norm symbol etc. in this situation.

H.L. Schmid had reported about his results in Witt7s workshop. It was again a high accomplishment, this time by Witt, to see through this jungle of polynomial identities and to find out that it could be reduced to simple operations on the ghost components. This then was the birth of the Witt vector calculus, soon to be amended by Teichmiiller's multiplication and so providing a solid foundation for the structure theory of complete unramified local fields.

6 9 ~ h e connection between ghost components and main components is defined in characteristic 0. Hence if we talk about ghost components of vectors over a field of characteristic p then we tacitly assume that the given field has been represented as the reduction mod p of some integral domain in characteristic 0, and the ghost components belong to foreimages of those vectors.


Here we are interested in that part of Witt's paper [I301 which concerns the arithmetic of function fields.

Given an element a # 0 in F and a Witt vector p = (Po, . . . , over F of length n, Witt considers the algebra ( a I P ] defined by generators u, yo,. . . , yn-1 where the yi are commuting with each other and the following defining relations hold, with y = (yo, . . . , Yn- 1) considered as a Witt vector:

Here, 1 = (1,0, . . . ,0) denotes the unit element of the ring of Witt vectors, and uyu-' means (uyou-l , . . . , uyn- lu-l ) . These relations define a simple algebra with center F ; it has the cyclic splitting field E = F(y), and this is of precise degree pn if Po is not of the form b p - b with b E F . The symbol (a1 P] , when considered as an element in the Brauer group over F, is multiplicative in the first variable a and additive in the second Witt vector variable P.

Note that the algebra (a I PI, defined with Witt vectors, is a direct generalization of (a , P] which is defined by field elements (28). There arises the question whether for these new algebras the formula (29) can be generalized, and how the generalization looks like. Witt gives the following solution.

If p is a prime of F then one can consider the same algebra over the

corresponding p-adic completion Fp. This is to be denoted by (y 1 . Consider the p-adic completion Fp as power series field over the

residue field Kp of p. Consider the ghost components /3(') as power series and let the operation resp p be defined ghost-componentwise. There results a Witt vector which Witt calls the "residue vector" and denotes by (a, P)p. The components of this Witt vector are contained in the residue field Kp . Now the analogue of (29) is as follows:

where, again, t is a uniformizing variable at p. For the computation of the Hasse invariant of this algebra similar formulas are available, which generalize H.L. Schmid's formula (24).

Without proof Witt mentions the relation

7 0 ~ i t t writes (a 1 ,L?] and regards p as fixed.

612 P. Roquette

which is a generalization of the residue theorem, now for residues of Witt vectors. (Sp denotes the trace from the p-adic residue field Kp to K , extended to Witt vectors.)

In the same volume of Crelle's Journal as Witt's paper [I301 there appeared another paper of H.L. Schmid on the arithmetic of cyclic fields of ppower degree [93]. There, building on the now established theory of Witt vectors he continues his investigation of [92]. Given a cyclic extension E = F(y) of degree pn with Witt vector generation yp- y = ,B, H.L. Schmid establishes formulas for the conductor, the discriminant and the genus of E in terms of ,B. These formulas are very useful in various arithmetic and geometric applications. 71

7.4. T h e functional equation for t h e L-series

In the preface to his paper [I271 on the existence theorem, Witt mentioned the proof of the functional equation of the L-series for function fields as a further desideratum. According to his own testimony [I311 he had completed the proof one year later in 1936. But he abstained from publication because he was asked by Artin to do so; Artin had a doctoral student who was working on the same subject. 72

Let x be a non-trivial ray class character in F with conductor f. Thus ~ ( a ) is defined for divisors which are relatively prime to f, and ~ ( a ) = 1 if a = (a) with a - 1 mod f. For divisors a which are not relatively prime to f we may put ~ ( a ) = 0. Then F.K. Schmidt's L- series is

The product ranges over all primes p of F and the sum over all positive divisors; hence the dashes at n and C which appear in formula (16) are not necessary here.

The functional equation establishes a relation between L(s , X) and L ( l - s, X) where 15 denotes the conjugate complex character; X has the same conductor f as does X. Let d = deg f; then the functional equation can be written in the following form:

7 1 ~ . ~ . Schmid's paper [93] and Witt's [I301 were two papers out of seven which all arose in the Gottingen workshop and which all appeared in a single fascicle of Crelle's Journal, together with a paper by Hasse.

72 This was J. Weissinger; his w roof of the functional equation appeared 1938 in [122]. - Later, Weissinger went to applied mathematics.


with ) E ( x ) ~ = 1. Although not publishing this result, Witt presented his proof in the

Gottingen seminar, and so in the course of time it became known in wider circles. We know about the proof from various sources, namely:

In a letter from Hasse to Davenport dated April 30, 1936, Hasse gave a three page outline of Witt's proof. 73

In a letter from Hasse to A. Weil dated July 12, 1936, Hasse informed Weil about Witt's proof (among other number theoretic news) and included a sketch of it. 74

In the year 1943 the Hamburger Abhandlungen accepted a paper by H.L. Schmid and 0. Teichmiiller for volume 15, which contained essentially a presentation of Witt's proof as seen by those authors [94]. 75

The recently published Collected Papers of E. Witt [I321 contain a note, written by Schulze-Pillot, where Witt's proof is sketched

7 3 ~ h i s letter is contained among the Davenport papers at the archive of Trinity College, Cambridge. - Somewhat later Davenport himself gave another proof which, as Hasse said in a letter to Weil (Feb 4, 1939), proceeds in a more computational way ("auf mehr rechnerische Art") .

7 4 ~ e i l seemed to have forgotten about it, for on Jan 20, 1939 he informed Hasse that he had a proof of the functional equation. In his reply Hasse mentioned Witt's proof again and sent Weil a new, more detailed exposition; but he also mentioned Weissinger's and Davenport's proof. Upon this Weil wrote to Hasse that he had checked Weissinger's proof which had already appeared in [122], and he found that his (Weil's) proof was essentially the same as Weissinger's. He also apologized to Hasse that he had forgotten Hasse's former information about Witt's proof in 1936. -

As a side remark it may be mentioned that in this letter Hasse informs Weil also about other news, one of them being Deuring's algebraic theory of correspondences of algebraic function fields (published later 1937 and 1941 in [23],[24]). Hasse explains to Weil that this theory will open the way to the proof of the Riemann hypothesis for function fields of arbitrary genus: one would have to prove that the algebraic analogue of the hermitian form from the period matrix is positive definite. In his reply (dated July 17, 1936) Weil appreciated Deuring's promising idea ( " . . . es ist sehr schon, dass durch die Idee von Deuring nunmehr die Losung dieses Problems i n Aussicht gestellt wird."). And he adds a reference to Severi's "Trattato" [104]. It seems remarkable that Deuring's idea turned out to be precisely the same which A.Weil several years later used when he indeed arrived at the proof of the Riemann hypothesis [120].

7 5 ~ u e to war time difficulties of publication, volume 15 was completed in 1947 only.

614 P. Roquette

after Witt's own handwritten notes (which are not dated, however).

It seems remarkable that Witt had conceived his proof as an analogue to the classical proof by Hecke [58] who worked with theta functions and the theta transformation formulas. Although in the function field case the analogues to these are purely algebraic identities and hence of quite another type, Witt named those algebraic lemmas in the same way as their classical counterparts - in order to stress the analogy between both. This analogy is not so transparent, however, in the presentation given by H.L. Schmid and Teichmiiller [94].

Any known proof so far is based on a generalization of the Riemann- Roch theorem, much the same way as the functional equation of the ordinary zeta function by F.K. Schmidt is based on the ordinary Riemann Roch theorem. Perhaps it is not without interest to cite from the first paragraph of Hasse's letter to Davenport 1936 where he gave an outline of Witt's proof. (This letter is handwritten in English.)

The main source for Riemann-Roch's theorem and generalizations to character classes is, according to Witt , the following theorem:

Let k be a n arbitrary field and K the field of all power series CFVo a,tY with a , i n k and t an indeterminate; furthermore:

R1 the ring of all polynomials i n f over k , R2 the ring of all integral power series z:=o a,tV over

k , both subrings of K . Let M be a matrix with determinant # 0 , consisting of elements i n K . Then there are a matrix Al over R1 and a matrix A2 over R2, both with determinant a unit (element # 0 in k ) such that

We observe that this theorem embodies the classical method of so-called "normal bases" which had been used by Dedekind-Weber and Hensel- Landsberg in proving the Riemann-Roch theorem, taken up by F.K. Schmidt (see section 4.3.2), and also used by Witt himself in his proof of the Riemann-Roch theorem in non-commutative algebras (see section 6.2). This same theorem appears here again in connection with Witt's proof of the functional equation. And this time it is formulated as a separate result, independent of the intended application.


Hasse was convinced that this theorem is an important result which should be included as a separate lemma (or theorem) in his textbook "Zahlentheorie" which he had finished in 1938. And the name "Witt's lemma" seemed to him justified because Witt had fully seen its importance, had used it in various different situations, and now had formulated it as a separate statement. Hasse was fully aware of the role of this lemma in the historical development connected to the Riemann-Roch theorem. In his letter to Weil on March 7, 1939 he says:

Den Inhalt von $1 habe ich ubrigens i n meinem Buch ver- arbeitet, indem ich von diesem Hilfssatz aus - der ja i m Wesentlichen der Satz von der Existenz der Normalba- sis ist - direkt zum Riemann-Rochschen Satz vorstosse, also ohne Einfuhrung des Wittschen Formalismus mit der Theta-Funktion. 76

Geyer [33], p.125 has pointed out, however, that this "Witt's lemma" had been formulated already in various other situations in the course of history.

7.5. Summary

A student of Hasse, H.L. Schmid, gave i n his thesis (1934) a n amendment to Hasse's paper on cyclic extensions of function fields (see section 6.1). H.L. Schmid's main achievement was an explicit formula, involving logarithmic diflerentials, for the p- th norm symbol when p is the characteristic. A s a side result, this yields another proof of Artin's reciprocity law, without referring to Tsen's theorem, for cyclic extensions of degree p, namely with the help of the residue theorem i n function fields. H.L. Schmid found also a n explicit formula for the n- th norm symbol i n case n $ 0 mod p. But he failed to see that this too could have been used to give another proof of Artin's reciprocity law, i n

76~asse 's "Zahlentheorze" appeared in 1949 only, with another publisher as originally planned. 11 years earlier (more precisely: in a letter dated Nov

I 28, 1938) the original publisher had rejected the book because it had become larger than originally planned. When Weil heard of this situation he wrote that he was highly indignant ("aufs hochste emport") about this situation and he offered to try to have Hasse's book published in France. But Hasse did not consider this possibility. Later, on June 8, 1939, the publisher accepted a recommendation by C.L. Siege1 and, reversing his former decision, agreed to publish Hasse's book. But due to the outbreak of the second world war this could not be realized.

616 P. Roquette

case n $ 0 mod p, namely with the help of the product formula for the universal symbol. I n 1935 Wit t published a proof of the class field existence theorem for function fields. In the case of exponent n $ 0 mod p his proof supersedes that of F.K. Schmidt, and he follows the methods of index computations as given earlier by Herbrand and Chevalley for number fields. In the case of exponent p he relies o n H.L. Schmid's formulas for the norm symbol i n order to have a n estimate for the conductor of an abelian extension of exponent p. His methods are original and quite new. - This paper was the second in Witt's planned series devoted to class field theory for function fields. It provided the last missing stone for the building of general class field theory i n function fields. I n 1936 there appeared Witt's great paper where he introduces what is now called Wit t vectors. The discovery of Wi t t vectors was intimately connected with problems from class field theory for function fields, namely the search for explicit formulas for the norm symbol in function fields. H.L. Schmid had done this in case of degree p, and now this became possible also i n the cyclic case of p-power degree, thanks to the calculus of Witt vectors. I n 1936 Wit t arrived at a proof of the functional equation for F.K. Schmidt's L-series with ray class characters for function fields. The proof presents, in the function field case, the algebraic analogues to the analytic tools which Hecke had used i n the number field case. Wi t t never published his proof; i t is preserved as handwritten note only, by h im and by other mathematicians who had heard him lecture on this. The main ingredient is still another variant of the Riemann-Roch theorem, whose proof .is based on the classical method of "normal bases". 77 There is also a paper by H.L. Schmid and Teichmuller i n which they present Witt's proof as they saw it.

$8. Algebraizat ion

Let us recall the main steps in the foundation of class field theory for function fields which we have discussed above.

1. F.K. Schmidt's theory of the zeta function and the L-functions: 1927-31 (sections 4 and 5)

2. Hasse's proof of the Artin reciprocity law: 1934 (section 6)

77 It seems not to be widely known in the mathematical public that Witt, in proving this Riemann-Roch theorem, preceeded Rosenlicht [89] by 16 years.


3. Witt's existence theorem for function fields: 1935 (section 7)

In the thirties there arose the question whether the use of analysis was really necessary for the foundation of class field theory. Would it be possible to prove the Artin reciprocity law and the existence theorem without F.K. Schmidt's analytic theory?

Looking more closely into the matter we see that only little "analysis" was involved. For, F.K. Schmidt had proved that his zeta functions and L-series were rational functions and polynomials, respectively, in the variable t = q - l . Hence what was considered an "analytic" argument turns out, from this point of view, to be "algebraic" after all. But this is algebra over the field of complex numbers, not over the given function field. Hence the search for algebraic proofs in the function field case did not so much care about the relation between algebra and analysis: it was the search for intrinsic, structural proofs which yield more insight into the relevant structures of function fields.

But the terminology was not quite clear. Some authors spoke of "algebraic" proofs, and some of "algebraic-arithmetic" proofs. It is not quite clear what should have been the difference between both. Some authors used "arithmetic" in the sense that the proof works for function fields over finite base fields: these are global function fields which resem- ble the global number fields most. But other authors used "arithmetic" also for function fields over arbitrary base fields; in such case the methods used were valuation theoretic or ideal theoretic and, in this sense the methods came from the study of ordinary arithmetic of algebraic

I numbers. I

I Let us here use the terminology of "algebraic proof" for a proof

which avoids the use of zeta functions and L-functions and works with

1 structures inside the function field only. In this sense, it turned out that the algebraization of class field theory for function fields was indeed possible. In the following we shall report on the work in this direction, and the results.

8.1. F.K. Schmidt's theorem

I I The first instance where the use of analysis had been found to be

unnecessary was F.K. Schmidt's theorem. As explained in section 4.3.3, this theorem asserts that every function field FIK with finite base field admits a divisor of degree 1. F.K. Schmidt was quite aware of the curious

618 P. Roquette

fact that his original proof used analytic arguments. In a letter to Hasse dated January 21, 1933 he comments on this: 78

Bekannte Tatsache i m Fall der algebraischen Funktionen mit bel. komplexen Zahlkoefizienten! Aber hier, wo der Konstantenkorper ein Galoisfeld ist, keineswegs trivial, ja bisher nicht einmal rein algebraisch, sondern nur mit transzendenten Methoden beweisbar.

Known fact in case of algebraic functions with arbitrary complex numbers as coefficients! But here, where the field of constants is a Galois field, it is by no means trivial, up to now it is not even provable purely algebraically, but with transcendental met hods only.

But some months later, on August 7, 1933, he reported to Hasse on a postcard:

Wit t schrieb mir vorige Tage, er konne n u n bei algebr. Fkt. mit einem Galoisfeld als Konstantenkorper rein arithmetisch einen Divisor von der Ordnung 1 nachweisen. Sein Beweis sei allerdings langer als mein analytischer. Leider teilte er mi r aber seinen Beweis nicht mit.

Witt wrote me some days ago that he was able, in algebraic function fields with a Galois field as its field of constants, to construct a divisor of degree 1. However his proof was longer than my analytic proof. Unfortunately he did not convey his proof to me.

Witt included his proof in the first section of his paper [I251 which appeared in 1934. Its title is " ober ein Gegenbeispiel zum Normensatt" (On a counter example to the norm theorem). This title does not give any hint that the paper also contains a new algebraic proof of F.K. Schmidt's theorem; this is perhaps the reason why Witt's proof did not become widely known at the time and was rediscovered several times. It seems that Witt included that proof because of the similarity of methods used in both cases: for F.K. Schmidt's theorem and for the discussion of the norm theorem for function fields. In the introduction to this paper Witt writes:

Fur den Satz "In einem Funktionenkorper iiber einem Ga- loisfeld gibt es Divisoren jeder Ordnung " hat F. K. Schmidt

7 8 ~ t r i ~ t l y speaking, in this letter F.K. Schmidt did not directly refer to his theorem but to the following result which is a consequence of his theorem: Every function field of genus zero is rational.

Class Field Theory i n Characteristic p, its Origin and Development

einen sehr kurzen und eleganten analytischen Beweis gegeben. Vom algebraischen Standpunkt ist es wohl nicht unnutz, wenn bei dieser Gelegenheit ein gruppentheoretis- cher Beweis mitgeteilt wird.

The theorem " I n a function field over a Galois field there exist divisors of every degree" has been given by F.K. Schmidt with a very short and elegant analytical proof. From the algebraic point of view it may not be super- fluous if on this occasion we present a group theoretical proof.

The "group theoretical" proof which Witt mentions is of cohomological nature. Of course, Witt does not explicitly use the modern notions and notations of algebraic cohomology; they did not yet exist at the time. But in fact, Witt's computations can be interpreted as determining the Galois cohomology of the divisor group and related groups, with respect to the Galois group G of a finite base field extension FLIL. Here, L is chosen as the field whose degree [L : K] equals the smallest positive divisor degree of F ( K . This choice implies that every prime p of FIK splits completely in the extension FL. Hence the divisor group, as a G-module, is cohomologically trivial and from this Witt deduces that [ L : K] = 1.

In his computations Witt uses a technique very similar to what today is known as "Herbrand's lemma" in cohomology; note that G is cyclic and hence Herbrand's lemma is applicable. With today's cohomological formalism it is possible to rewrite Witt's algebraic proof such that it does not appear longer than F.K. Schmidt's. In fact, with only minor changes 79 Witt's proof yields the following more general

Lemma 2. If EIF is a cyclic extension such that every prime p of F splits completely in E then E = F .

7 9 ~ h e changes are as follows: Witt uses the fact that the multiplicative group L X of any finite extension L of K is cohomologically trivial, with respect to the action of the Galois group. Now with respect to any cyclic group action the cohomology of L X may not be trivial, but since L X is finite both cohomology groups H'(L' ) and H1 (LX ) have the same order. That is what is actually needed. ( H O is to be understood in the sense of Tate's modified cohomology. )

620 P. Roquette

This is almost the statement of the Lemma 1 (see section 6.1.3) which Hasse had used in the proof of Theorem B. The differences are that, firstly, here we deal with cyclic extensions whereas Lemma 1 refers to arbitrary abelian extensions. But this is not essential: if Lemma 1 holds for cyclic extensions (or only for cyclic extensions of prime degree) then trivially it holds for arbitrary abelian extensions. The second difference seems to be more essential: whereas in Lemma 1 it is assumed that almost all primes are completely split, in Lemma 2 this is required for all primes.

Now it has been shown by Chevalley and Nehrkorn [16] how to reduce Lemma 1 to Lemma 2. They show (in case EIF is cyclic of prime degree) that if almost all primes p of F split completely in E then there exists a field F' containing F, linearly disjoint to EIF, such that indeed all primes p' of F' split completely in the composite field EF'; hence (using Lemma 2) EF' = F' and so E = F.

Chevalley and Nehrkorn, however, discuss only number fields; their construction uses radicals, i.e., Kummer theory, and this is not always applicable in the function field case, not if the field degree equals the characteristic p. It has been observed by Moriya [80], [81] that the Chevalley-Nehrkorn construction works also in the case of degree p if Kummer theory is replaced by Artin-Schreier theory.

Accordingly, Lemma 1 can be reduced to Lemma 2, also in the function field case. Now we have said above already that Lemma 2 had been proved algebraically by Witt; more precisely, it could have been proved with the same cohomological arguments as are used in Witt's proof 11251 of F.K. Schmidt's theorem. In 1937 Moriya [80] published a proof along similar lines as Witt7s proof. He does not seem to have known Witt's paper because he says:

Ich vermeide es, diese Tatsache [dug es einen Divisor 1- ten Grades gibt] zu benutzen, weil man, soweit ich wei,P, zum Beweis die Kongruenzzetafunktion zu Hilfe nehmen mufl.

I avoid to use this fact [that there exists a divisor of degree 11 because, as far as I know, for its proof it is necessary to use the congruence zeta function.

Moriya's paper carries the title: "Rein arithmetisch-algebraischer Aufbau der Klassenkorpertheorie uber algebraischen Funktionenkorpern einer

8 0 ~ e use the notations as introduced earlier: Theorems A,B and C are stated in section 6.1.1; they concern Artin's reciprocity law. Theorem D is the existence theorem in 7.2.


Unbestimmten mit endlichem Konstantenkorper7' (Purely arithmetic- algebraic foundation of the class field theory for algebraic function fields in one indeterminate with finite field of constants). Thus his aim is precisely to eliminate the use of analytic arguments from class field theory for function fields. In the course of this he discusses Hasse's Theorem B and, as we have said above, reduces it to Lemma 2 and then presents a proof of Lemma 2.

We have seen: Hasse's 1934 proof of Theorem A [51] was of algebraic nature. His proof of Theorem B was not, but Moriya [81] gave an algebraic proof in 1938. The methods are of cohomological nature and very similar to those which Witt used 1934 in his proof of F.K. Schmidt's theorem [125].

8.2. The new face of class field theory

Witt says in the introduction to his 1935 paper [I271 on the existence theorem:

Die Voranstellung des Artinschen Reziprozitatsgesetzes hat eine grofle Wandlung mi t sich gebracht. Die friihere Klassenkorpertheorie ist heute einer Theorie der abelschen Korper gewichen. Die friiher an die Spitze gestellte Ta- kagische Definition des Klassenkorpers hat heute eine andere Bedeutung. Sie dient nur noch zur Gewinnung eines handlichen Kriteriums fur abelsche Korper. Ein solches Kriterium wird namlich fur den vollstandigen Beweis des Existenzsatzes benotigt.

Putting Artin's Reciprocity Law first has brought great changes. Today the former class field theory has given way to a theory of abelian fields. Takagi's definition of a class field, which formerly had been the starting point, is today regarded from a different perspective. It is considered as a convenient criterion for abelian fields only. For, such a criterion is necessary for the complete proof of the existence theorem.

What does this mean? What kind of changes did Witt have in mind? Witt distinguishes between "former class field theory" and "theory

of abelian fields". Takagi's definition of a class field, he says, is not fundamental in the body of the "theory of abelian fields" but of secondary importance only.

This seems to indicate that Witt proposes to include into the body of his "theory of abelian fields7' the union of Theorems A, B and D only

622 P. Roquette

while Theorem C , referring to the Takagi groups, is separated and not regarded any more to be fundamental.

Note that Theorems A and B are those which we just have listed as having been proved algebraically. Is Witt's proof of the existence theorem D also algebraic ?

Witt himself does not discuss this question. But since he is separating Theorem C (which was not yet proved algebraically) from the body of the other theorems, he seems to have been aware of the problem. In his proof of the existence theorem he says that he accepts the full result of Hasse's paper which is partly based on analytic properties of L-series. But if one looks more closely into his proofs then it turns out that in fact, from Hasse he uses only Theorems A and B in order to prove D. Consequently, Witt's proof yields an algebraic foundation of what he calls "theory of abelian fields" in the function field case.

As to Theorem C , the case was discussed carefully by Chevalley- Nehrkorn in their 1935 paper 1161 (which Witt seemed not to know). They presented a method how to reinterpret Witt's proof in a purely algebraic manner, such that at the same time it also yields C . This is a nice idea and is worthwhile to be discussed a little bit further. To be sure, Chevalley and Nehrkorn did not discuss function fields; they were concerned with class field theory in number fields (and apparently did not know Witt's paper). But the same idea applies to the function field case, and this was explicitly pointed out by Moriya [81].

Let El F be an abelian extension of degree n, and H, its Artin group for a suitable module m (e.g., we can take for m the conductor of E IF). By Theorems A and B we see that the index (C, : H, ) = n. The Takagi group, or norm group N, is a subgroup of H,, as follows immediately from the definitions. Hence for the index h , = (C, : N,) of the Takagi group it follows

This is the "second" inequality of classical class field theory. Usually, in classical class field theory one proves first the "first" inequality h , 5 n, namely by analytic means. The "second" inequality then shows h , = n and hence H, = N,.

Thus in the new algebraic setting, the second inequality is proved before the first inequality! This has led several authors to rename those inequalities: what formerly was the first was now named second, and vice versa. As could be expected, this produced a certain amount of uncertainty. Anyone reading the literature of the time should be aware that the terminology in this respect is not uniform.


Using the inequality (32), be it called "first" or "second", Chevalley and Nehrkorn proved the following result which we state as a lemma.

Lemma 3. Let F c E' c E be a tower of abelian fields over F , of degree n', n over F respectively. Let m contain the conductor of E I F . If h , = n then also h', = n'. In other words: If El F is a class field i n Takagi 's sense then E ' J F is so too.

This being said, Witt's existence proof can now be interpreted as being purely algebraic, also yielding C , as follows: We use the same notations as in our discussion of Witt's proof in section 7.2. Given m and n, Witt's construction yields an abelian field EIF whose Artin group is precisely C: and coincides with its Takagi group. Then for all intermediate groups H, between C: and C, there exists, by Artin's reciprocity isomorphism and Galois theory, an intermediate field between F and E whose Artin group is H,; using Lemma 3 we conclude that H, coincides with the Takagi group of that field.

A slight difficulty arises when n $ 0 mod p and the n-th roots of unity are not contained in F . Then in order to construct the field E Witt has first to adjoin the n-th roots of unity and then perform his construction via Kummer theory. One has to be sure that the constructed field is abelian over the original field F. For this Witt uses Takagi's theorem C as a "convenient criterion for abelian fields", as he had announced in the introduction (see above). He sketches a new proof of this criterion 81 and informs us that it is based on an idea of Iyanaga.

8.3. Summary

I n the thirties we observe a tendency to eliminate analytic arguments from the foundations of class field theory, i n particular from class field theory for function fields. The motivation was to arrive at a better understanding of the underlying structures of class field theory. The first theorem which was freed from analytical proofs was F.K. Schmidt's theorem on the existence of a divisor of degree 1. Wi t t discovered an algebraic proof i n 1933. He included his proof i n his paper on a counterexample of the norm theorem, but it seems that it did not become widely known at the time.

he criterion had been stated and proved already by Hasse in $5 of Part I1 of his class field report [38].

624 P. Roquette

Chevalley and Nehrkorn 1935 supplied useful ideas for the algebraization of the proofs i n class field theory. They discussed number fields only but Moriya 1937 showed that their results could be transferred to the function field case. Consequently it became possible to give algebraic proofs of all main theorems of class field theory i n the function field case. The main ingredients are (i) Hasse's algebraic proof of the sum relation for the local invariants of a simple algebra, and (ii) Witt's proof of the existence theorem. The latter rests on an idea of Herbrand in the case when the index is not divisible by the characteristic, and otherwise on H.L. Schmid's explicit formulas for the norm residue symbol. A n exposition of the algebraic foundation of class field theory in function fields was given in Moriya's paper 1937.

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