CONTEMPORARY MATHEMATICS

15

Advanced Analytic Number Theory

Part l: Ramification

Theoretic Methods

Carlos J. Moreno

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http://dx.doi.org/10.1090/conm/015

Advanced Analytic Number Theory

Part 1: Ramification

Theoretic Methods

CoNTEMPORARY MATHEMATICS

15

Advanced Analytic Number Theory

Part 1: Ramification

Theoretic Methods

Carlos J. Moreno

American Mathematical Society Providence, Rhode Island

1980 Mathematics Subject Classification. Primary 10015, 12A65, 12A67, 12B10, 12B25, 12B27, 14G10, 10H99.

Key words and phrases. Ramification theory, analytic number theory, algebraic number theory, algebraic geometry, automorphic forms.

Library of Congress Cataloging-in-Publication Data

Moreno, Carlos J., 1946-Advanced analytic number theory. (Contemporary mathematics, ISSN 0271-4132; v. 15-) Bibliography: p. Includes index. Contents: pt. 1. Ramification theoretic methods. 1. Numbers, Theory of. 2. Fields, Algebraic. I. Title. II. Series.

QA241.M64 1983 5121.7 ISBN 0-8218-5015-6

82-22620

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Copyright @1983 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted

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The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.@

This volume was printed directly from author-prepared copy.

111098765432 989796959493

CONTENTS

Preface ChapteP 0. IntPoduction

§1, A historical perspective §2, The explicit formulas: motivation

ChapteP I. GaLois Theory foP Infinite Extensions

§1. Fundamental Theorem of Galois Theory for infinite extensions §2, Topological properties of Gal(K'/K) §3. Example

ChapteP II. FTojective Limits

§1. Basic definitions §2. Examples §3. Homomorphisms

ChapteP III. ELementary Theory of L-adic Integration

§1. Distributions and measures §2. Measures on groups §3. Riemann sums §4. !-adic Haar measures §5. Examples

ChapteP IV. Ramification Theory

§1. The v-function §2. Transitivity properties of the v-function §3. The different §4. Transitivity properties of the different §5. Complementary basis §6. Fundamental inequality and the C-invariant §7. Quotients of Galois groups and Herbrand's Theorem

ChapteP V. MULtipLicative vePsus Additive Reduction

§1. The case of a local field §2. The module structure of the unit group §3. The case of an elliptic curve

v

1

25

31

45

55

vi CONTENTS

Chapter VI. Ramification of Abelian Extensions

§1. Index computations §2. Main Theorem

Chapter VII. The Weil Groups of a Loaal Field

§1. The Frobenius automorphism §2. The relative Wei! group of a non-archimedean local field §3. The absolute Wei! group §4. The Wei! group of an archimedean local field §5. The Weil-Deligne group

Appendix: The non-abelian local reciprocity law

Chapter VIII. Shafarevitah's Theorem

§1. Representations of a simple algebra §2. Factor sets §3. Subalgebras of central simple algebras §4. Cyclic algebras §5. Algebras and group extensions §6. Powers of central simple algebras §7. Generators of the Brauer group §8. Structure of the relative Wei! group §9. Shafarevitch's Theorem

Appendix: I. The Brauer group of central 'simple algebras over K 2 X II. The group H (G,Ks)

Chapter IX. The Herbrand Distribution

§1. The Herbrand distribution: definition and basic facts §2, The support of the Herbrand distribution §3. Determination of the Herbrand distribution on Gal(Kab/K) §4. Functoriality properties of the Herbrand distribution §5. The Herbrand distribution and measures on the Wei! group §6. The integral formula for the Herbrand distribution §7. Conductors, the Artin representation, and the Herbrand distribution

Bib Ziogr>aphy

Index

95

109

133

157

PREFACE

These notes are intended as an introduction to those aspects of analytic

number theory which depend on and have applications to the theory of algebraic

numbers. As is well known the central problem of the theory is the general

algebraic formulation of ·the following distribution results: prime number

theorem, Dirichlet's theorem on primes in arithmetic progressions, Cebotarev's

density theorem on the distribution of Frobenius conjugacy classes in Galois

groups, Heeke's density theorem on the distribution of the arguments of quasi-

characters of idele class groups, Sato-Tate conjectural densities for the value

distribution of the traces of Frobenius elements, strong multiplicity one

theorem for automorphic representations of the general linear group, etc.

These are all variants of the following fundamental distr-ibution pPoblem. Let

G be the Galois group of the separable closure of the rationals; let Fp be

the Frobenius conjugacy class associated with the rational prime p; let P : G 4 GL(V) be a representation of G on some (l-adic) vector space V.

The main problem then is the study of weighted sums of the form

I x(F Jw , (x = tP(rJJ, p p p

where p runs over finite sets of primes. The basic analytic tools needed for

the study of such sums are the so called explicit formulas of prime number

theory which relate sums over primes with sums over zeros of 1-functions. The

principal aim of these notes is to develop these formulas and to give some of

their applications.

Part I: Ramification Theoretic Methods treats all the necessary local

aspects of algebraic number theory that appear in the formal description of

the structure of the general explicit formulas, notably the inte,gral represent-

ation of the Herbrand distribution. We believe that Part I in itself is of

independent interest and indeed chapters I, II, IV and V could be used as an introduction to local class field theory.

Part II: 1-functions treats certain global aspects of algebraic number

theory, e.g. relative Weil groups of number fields and their finite dimensional

representations, functional equations of 1-functions, distribution of zeros,

theorems of Hadamard-De La Vallee Poussin, explicit formulas, generalized prime

vii

viii PREFACE

number theorems, lower bounds for conductors, etc. For a precise description

of the new results see Part I, Chapter 0, §2.

We are indebted to Weil's Basic Number Theory for the influence it has had

on our presentation. Undoubtedly in our mind this book has paved the way from

Heeke's classical approach to algebraic number theory and modular forms to the

recent advanced developments in the theory of automorphic forms. Our discus-

sion of the ramification theory of abelian local extensions, Shafarevitch's

theorem and the Herbrand distribution follows rather closely that of the third

edition.

These notes were originally prepared to serve the author as a dictionary

for future reference. It now appears that the elementary and informal nature

of our presentation may also serve other number theorists and algebraists as an

introduction to an area of mathematics which traditionally has been considered

difficult and inaccessible. The author hopes that they are more pleasant to

read than they were to write.

Urbana June 1982

CARLOS J. MORENO

ISBN 0-8218-5015-6

9 780821 850152