contemporary mathematicsasher ben-artzi and david soudry 13 gauss sum combinatorics and metaplectic...

CONTEMPORARYMATHEMATICS

488

Automorphic Forms and L-functions I. Global Aspects

A Workshop in Honor of Steve Gelbart on the Occasion of his Sixtieth Birthday

May 15–19, 2006 Rehovot and Tel Aviv, Israel

David Ginzburg Erez Lapid

David Soudry Editors

American Mathematical SocietyProvidence, Rhode Island

Bar-Ilan University Ramat Gam, Israel

Israel Mathematical Conference Proceedings


http://dx.doi.org/10.1090/conm/488

Steve Gelbart

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CONTEMPORARYMATHEMATICS

488


A Workshop in Honor of Steve Gelbart on the Occasion of His Sixtieth Birthday

May 15-19, 2006 Rehovot and Tel Aviv, Israel

David Ginzburg Erez Lapid

David Soudry Editors

Israel Mathematical Conference Proceedings

American Mathematical SocietyProvidence, Rhode Island

Bar-Ilan University Ramat Gam, Israel

Editorial Board of Contemporary Mathematics

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

Editorial Board of Israel Mathematical Conference Proceedings

Louis Rowen, Bar-Ilan University, managing editor

Z. Arad, Netanya Academic College M. Katz, Bar-Ilan UniversityJ. Bernstein, Tel-Aviv University B. Pinchuk, Netanya Academic CollegeH. Furstenberg, Hebrew University S. Shnider, Bar-Ilan UniversityS. Gelbart, Weizmann Institute L. Small, University of California

at San DiegoV. Gol′dshtein, Ben-Gurion University L. Zalcman, Bar-Ilan University

Miriam Beller, Technical Editor

2000 Mathematics Subject Classification. Primary 11F70, 11F67; Secondary 11F72,11F27, 11F33, 11F75, 11F80.

Photo courtesy of David Soudry.

Library of Congress Cataloging-in-Publication Data

Automorphic forms and L-functions : proceedings of a workshop in honor of Steve Gelbart onthe occasion of his sixtieth birthday : May 15–19, 2006, Rehovot and Tel Aviv, Israel / DavidGinzburg, Erez Lapid, David Soudry, editors.

v. cm. — (Contemporary mathematics ; v. 488–489) (Israel mathematical conference pro-ceedings)

Includes bibliographical references.Contents: 1. Global aspects — 2. Local aspects.ISBN 978-0-8218-4706-0 (alk. paper) — ISBN 978-0-8218-4708-4 (alk. paper)1. Automorphic functions—Congresses. 2. L-functions—Congresses. 3. Automorphic forms—

Congresses. I. Gelbart, Stephen S., 1946– II. Ginzburg, David, 1958– III. Lapid, Erez, 1971–IV. Soudry, David, 1956–

QA353.A9A928 2006515′.9—dc22 2009000412

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction byservices that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for generaldistribution, for advertising or promotional purposes, or for resale. Requests for permission forcommercial use of material should be addressed to the Managing Editor, IMCP, Department ofMathematics, Bar-Ilan University, Ramat-Gan, 52900 Israel. Requests can also be made by emailto [email protected].

Excluded from these provisions is material in articles for which the author holds copyright. Insuch cases, requests for permission to use or reprint should be addressed directly to the author(s).(Copyright ownership is indicated in the notice in the lower right-hand corner of the first page ofeach article.)

c© 2009 by Bar-Ilan University. Printed in the United States of America.©∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability.Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09

ContentsI: Global Aspects

II. Local Aspects vii

Preface ix

Bibliography of Stephen S. Gelbart xi

Conference Program xvii

List of Participants xix

Report on the Trace Formula

James Arthur 1

L-functions for Um ×RE/F GLn

(n ≤ [m

2

])Asher Ben-Artzi and David Soudry 13

Gauss Sum Combinatorics and Metaplectic Eisenstein Series

Ben Brubaker, Daniel Bump, and Solomon Friedberg 61

On Partial Poincare Series

J. W. Cogdell and I. I. Piatetski-Shapiro 83

Restrictions of Saito-Kurokawa Representations

Wee Teck Gan and Nadya Gurevich (with an Appendix by GordanSavin) 95

Models for Certain Residual Representations of Unitary Groups

David Ginzburg, Dihua Jiang, and Stephen Rallis 125

Crown Theory for the Upper Half Plane

Bernhard Krotz 147

Unitary Periods and Jacquet’s Relative Trace Formula

Omer Offen 183

Remarks on the Symmetric Powers of Cusp Forms on GL(2)

Dinakar Ramakrishnan 237

The Cohomological Approach to Cuspidal Automorphic Representations

Joachim Schwermer 257

v

ContentsII: Local Aspects

Preface ix

Bibliography of Stephen S. Gelbart xi

Conference Program xvii

List of Participants xix

p-adic Interpolation of Triple L-functions: Analytic Aspects

Siegfried Bocherer and Alexei A. Panchishkin 1

Sur les Representations Modulo p de Groupes Reductifs p-adiques

Guy Henniart 41

Archimedean Rankin-Selberg Integrals

Herve Jacquet 57

On a Result of Venkatesh on Clozel’s Conjecture

Erez Lapid and Jonathan Rogawski 173

Paquets d’Arthur Discrets pour un Groupe Classique p-adique

Colette Mœglin 179

Complexity of Group Actions and Stability of Root Numbers

Freydoon Shahidi (with an Appendix by Wentang Kuo) 259

GL(n, C) and GL(n, R)

Marko Tadic 285

vii

Preface

The Workshop on Automorphic Forms and L-functions was held during the

beautiful spring days of May 15-19, 2006, in the Weizmann Institute of Science and

in Tel-Aviv University, Israel, in honor and celebration of the sixtieth birthday of

Steve Gelbart. This is the first of the two volumes which contain the proceedings

of this workshop.

The topics of the contributed articles represent leading themes of research in

automorphic forms today: the trace formula and its applications to functorial-

ity and representations of p-adic reductive groups; the relative trace formula and

periods of automorphic forms; Rankin-Selberg convolutions and L-functions; p-

adic L-functions. The articles in this volume concern mainly global aspects in

the study of automorphic forms, and those of the second volume (Contemporary

Mathematics, volume 489) concern mainly local aspects.

Steve Gelbart played a key role in the development of the theory of L-functions

of automorphic representations and its applications to the theta correspondence

and functoriality. The workshop and these two volumes are to acknowledge Steve

Gelbart’s achievements, impact and influence in automorphic forms and L-functions.

We are grateful to our sponsors for funding the workshop:

· The Israel Science Foundation (ISF)

· The Arthur and Rochelle Belfer Institute of Mathematics and Computer

Science

· The Maurice and Gabriella Goldschleger Conference Foundation at the

Weizmann Institute of Science

· Clay Mathematics Institute

· The European Research Network “Arithmetic Algebraic Geometry”

The Organizing Committee:

Vladimir Berkovich Erez Lapid

David Ellwood Peter Sarnak

David Ginzburg Freydoon Shahidi

Roger Howe David Soudry

ix

Bibliography of Stephen S. Gelbart

(1) Choquet, Gustave Lectures on Analysis. Vol. I: Integration and Topologi-cal Vector Spaces (edited by J. Marsden, T. Lance and S. Gelbart), W.A.

Benjamin, Inc., New York-Amsterdam, 1969 (with appendix).

(2) Choquet, Gustave Lectures on Analysis. Vol. II: Representation Theory(edited by J. Marsden, T. Lance and S. Gelbart), W. A. Benjamin, Inc.,

New York-Amsterdam, 1969.

(3) Choquet, Gustave, Lectures on Analysis. Vol. III: Infinite DimensionalMeasures and Problem Solutions (edited by J. Marsden, T. Lance and

S. Gelbart), W. A. Benjamin, Inc., New York-Amsterdam, 1969.

(4) Gelbart, Stephen S., Fourier analysis on GL(n, R), Proc. Nat. Acad.

Sci. U.S.A. 65 (1970), 14–18.

(5) Gelbart, Stephen S., Fourier Analysis on Matrix Space, Memoirs of the

American Mathematical Society, No. 108, American Mathematical Soci-

ety, Providence, R.I., 1971.

(6) Gelbart, Stephen S., Harmonics on Stiefel manifolds and generalizedHankel transforms, Bull. Amer. Math. Soc. 78 (1972), no. 3, 451–455.

(7) Gelbart, Stephen ,Holomorphic discrete series for the real symplectic groupInvent. Math. 19 (1973), 49–58.

(8) Gelbart, Stephen, Bessel functions, representation theory, and automor-phic functions, Harmonic Analysis on Homogeneous Spaces (Proc. Sym-

pos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972),

Amer. Math. Soc., Providence, R.I., 1973, pp. 343–345.

(9) Gelbart, Stephen, An example in the theory of automorphic forms,Harmonic Analysis on Homogeneous Spaces (Proc. Sympos. Pure Math.,

Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Amer. Math.

Soc., Providence, R.I., 1973, pp. 437–439.

(10) Gelbart, Stephen, The decomposition of L2(Γ\G), Seminaire Choquet,

11e–12e annees (1971–1973), Initiation a l’analyse, Exp. No. 4, Secretariat

Mathmatique, Paris, 1973.

(11) Gelbart, Stephen, Introduction to the theory of group representations,Seminaire Choquet, 11e–12e annees (1971–1973), Initiation a l’analyse,

Exp. No. 3, Secretariat Mathmatique, Paris, 1973.

(12) Gelbart, Stephen S, A theory of Stiefel harmonics, Trans. Amer. Math.

Soc. 192 (1974), 29–50.

(13) Gelbart, Stephen S., Automorphic Forms on Adele Groups, Annals of

Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.,

University of Tokyo Press, Tokyo, 1975.

xi

xii BIBLIOGRAPHY OF STEPHEN S. GELBART

(14) Gelbart, Stephen; Sally, Paul, Intertwining operators and automorphicforms for the metaplectic group, Proc. Nat. Acad. Sci. U.S.A. 72 (1975),

1406–1410.

(15) Gelbart, Stephen, Automorphic Forms and Representations of Adele Groups,Lecture Notes in Representation Theory, Department of Mathematics,

University of Chicago, Chicago, Ill., 1975.

(16) Gelbart, Stephen; Jacquet, Herve, A relation between automorphic formson GL(2) and GL(3), Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 10,

3348–3350.

(17) Gelbart, Stephen S., Weil’s Representation and the Spectrum of the Meta-plectic Group, Lecture Notes in Mathematics, Vol. 530, Springer-Verlag,

Berlin-New York, 1976.

(18) Gelbart, Stephen, Elliptic curves and automorphic representations,Advances in Math. 21 (1976), no. 3, 235–292.

(19) Gelbart, Stephen, Automorphic forms and Artin’s conjecture, Modular

functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn.,

Bonn, 1976), Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977,

pp. 241–276.

(20) Gelbart, S.S.; Piatetski-Shapiro, I.I., Automorphic L-functions of half-integral weight, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 4, 1620–

1623.

(21) Gelbart, Stephen; Jacquet, Herve, A relation between automorphic rep-resentations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4) 11(1978), no. 4, 471–542.

(22) Gelbart, Stephen; Jacquet, Herve, Forms of GL(2) from the analytic pointof view, Automorphic Forms, Representations and L-functions (Proc. Sym-

pos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc.

Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979,

pp. 213–251.

(23) Gelbart, Stephen, Examples of dual reductive pairs, Automorphic Forms,

Representations and L-functions (Proc. Sympos. Pure Math., Oregon

State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math.,

XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 287–296.

(24) Gelbart, Stephen; Howe, Roger; Piatetski-Shapiro, Ilya, Uniqueness andexistence of Whittaker models for the metaplectic group, Israel J. Math.

34 (1979), no. 1-2, (1980), 21–37.

(25) Gelbart, Stephen; Piatetski-Shapiro, I.I., Distinguished representationsand modular forms of half-integral weight, Invent. Math. 59 (1980), no. 2,

145–188.

(26) Gelbart, S. S.; Knapp, A. W., Irreducible constituents of principal seriesof SLn(k), Duke Math. J. 48 (1981), no. 2, 313–326.

(27) Gelbart, S.; Piatetski-Shapiro, I., On Shimura’s correspondence for mod-ular forms of half-integral weight, Automorphic Forms, Representation

Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies

in Math. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 1–39.

(28) Gelbart, S.S.; Knapp, A.W., L-indistinguishability and R groups for thespecial linear group, Adv. in Math. 43 (1982), no. 2, 101–121.

BIBLIOGRAPHY OF STEPHEN S. GELBART xiii

(29) Gelbart, S.; Piatetski-Shapiro, I., Some remarks on metaplectic cusp formsand the correspondences of Shimura and Waldspurger, Israel J. Math. 44(1983), no. 2, 97–126.

(30) Gelbart, Stephen, An elementary introduction to the Langlands program,

Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219.

(31) Gelbart, Stephen; Piatetski-Shapiro, Ilya, Automorphic forms and L-functions for the unitary group, Lie Group Representations, II (College

Park, Md., 1982/1983), Lecture Notes in Math., 1041, Springer, Berlin,

1984, 141–184.

(32) Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen, Explicit Con-structions of Automorphic L-functions, Lecture Notes in Mathematics

1254 Springer-Verlag, Berlin, 1987.

(33) Gelbart, Stephen; Soudry, David, On Whittaker models and the vanishingof Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci.

97 (1987), no. 1-3, 67–74.

(34) Gelbart, Stephen; Shahidi, Freydoon, Analytic Properties of Automor-phic L-functions, Perspectives in Mathematics 6, Academic Press, Inc.,

Boston, MA, 1988.

(35) Gelbart, S., Recent results on automorphic L-functions, Number Theory,

Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston,

MA, 1989, pp. 265–280.

(36) Festschrift in Honor of I.I. Piatetski-Shapiro on the Occasion of hisSixtieth Birthday, Part I. Papers in Representation Theory, Papers from

the Workshop on L-Functions, Number Theory, and Harmonic Analysis

held at Tel-Aviv University, Ramat Aviv, May 14–19, 1989 (edited by

S. Gelbart, R. Howe and P. Sarnak), Israel Math. Conf. Proc. 2, Weiz-

mann Science Press of Israel, Jerusalem, 1990.

(37) Gelbart, Stephen S.; Rogawski, Jonathan D., Exceptional representationsand Shimura’s integral for the local unitary group U(3), Festschrift in

Honor of I.I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday,

Part I. (Ramat Aviv, 1989), Israel Math. Conf. Proc. 2, Weizmann

Science Press of Israel, Jerusalem, 1990, pp. 19–75.

(38) Festschrift in Honor of I.I. Piatetski-Shapiro on the Occasion of his Sixti-eth Birthday, Part II. Papers in Analysis, Number Theory and Automor-phic L-functions, Papers from the Workshop on L-Functions, Number

Theory, and Harmonic Analysis held at Tel-Aviv University, Ramat Aviv,

May 14–19, 1989 (edited by S. Gelbart, R. Howe and P. Sarnak), Israel

Math. Conf. Proc. 3, Weizmann Science Press of Israel, Jerusalem, 1990.

(39) Arthur, James; Gelbart, Stephen, Lectures on automorphic L-functions,L-functions and Arithmetic (Durham, 1989), London Math. Soc. Lecture

Note Ser. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 1–59.

(40) Gelbart, Stephen S.; Rogawski, Jonathan D., L-functions and Fourier-Jacobi coefficients for the unitary group U(3), Invent. Math. 105 (1991),

no. 3, 445–472.

(41) Gelbart, Stephen, Automorphic forms and Artin’s conjecture. II, Math-

ematische Wissenschaften gestern und heute, 300 Jahre Mathematische

Gesellschaft in Hamburg, Teil 4 (Hamburg, 1990), Mitt. Math. Ges. Ham-

burg 12 (1991), no. 4, 907–947 (1992).

xiv BIBLIOGRAPHY OF STEPHEN S. GELBART

(42) Gelbart, Stephen, On theta-series liftings for unitary groups, Theta func-

tions: from the classical to the modern, CRM Proc. Lecture Notes 1,

Amer. Math. Soc., Providence, RI, 1993, pp. 129–174.

(43) Gelbart, S.; Rogawski, J.; Soudry, D., On periods of cusp forms and alge-braic cycles for U(3), Israel J. Math. 83 (1993), no. 1-2, 213–252.

(44) Gelbart, Stephen; Rogawski, Jonathan; Soudry, David, Periods of cuspforms and L-packets, C. R. Acad. Sci. Paris Sr. I Math. 317 (1993),

no. 8, 717–722.

(45) The Schur Lectures (1992), Lectures delivered at Tel Aviv University, Tel

Aviv, May 1992 (edited by Ilya Piatetski-Shapiro and Stephen Gelbart),

Israel Math. Conf. Proc. 8, American Mathematical Society, Providence,

RI, 1995.

(46) Gelbart, Stephen, Lectures on the Arthur-Selberg Trace Formula, Univer-

sity Lecture Series 9, American Mathematical Society, Providence, RI,

1996.

(47) Gelbart, Stephen; Rogawski, Jonathan; Soudry, David, Endoscopy, theta-liftings, and period integrals for the unitary group in three variables, Ann.

of Math. (2) 145 (1997), no. 3, 419–476.

(48) Gelbart, Stephen, Three lectures on the modularity of ρE,3 and the Lang-lands reciprocity conjecture, Modular Forms and Fermat’s Last Theorem

(Boston, MA, 1995), Springer, New York, 1997, pp. 155–207.

(49) Friedberg, Solomon; Gelbart, Stephen; Jacquet, Herve; Rogawski,

Jonathan, Representations generiques du groupe unitaire a trois variables(French) [Generic representations for the unitary group in three variables],

C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 4, 255–260.

(50) Piatetski-Shapiro, Ilya, Selected Works of Ilya Piatetski-Shapiro (edited

and with commentaries by James Cogdell, Simon Gindikin, Peter Sar-

nak, Pierre Deligne, Stephen Gelbart, Roger Howe and Stephen Rallis),

American Mathematical Society, Providence, RI, 2000.

(51) Gelbart, Stephen; Shahidi, Freydoon, Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), no. 1, 79–107

(electronic).

(52) Gelbart, Stephen; Jacquet, Herve; Rogawski, Jonathan, Generic repre-sentations for the unitary group in three variables, Israel J. Math. 126(2001), 173–237.

(53) Bump, D.; Cogdell, J. W.; de Shalit, E.; Gaitsgory, D.; Kowalski, E.;

Kudla, S. S., An Introduction to the Langlands Program. Lectures Pre-sented at the Hebrew University of Jerusalem, Jerusalem, March 12–16, 2001 (edited by Joseph Bernstein and Stephen Gelbart), Birkhauser

Boston, Inc., Boston, MA, 2003.

(54) Gelbart, Stephen S.; Miller, Stephen D., Riemann’s zeta function andbeyond, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 1, 59–112 (elec-

tronic).

(55) Gelbart, Stephen, Joe Shalika and the Fine Hall days, 1968–1971, Con-

tributions to Automorphic Forms, Geometry, and Number Theory, ix–xi,

Johns Hopkins Univ. Press, Baltimore, MD, 2004.

BIBLIOGRAPHY OF STEPHEN S. GELBART xv

(56) Gelbart, Stephen S.; Lapid, Erez M.; Sarnak, Peter, A new method forlower bounds of L-functions, C. R. Math. Acad. Sci. Paris 339 (2004),

no. 2, 91–94.

(57) Gelbart, Stephen S.; Lapid, Erez, M. Lower bounds for L-functions at theedge of the critical strip, Amer. J. Math. 128 (2006), no. 3, 619–638.

(58) Gelbart, Stephen, When is an L-function non-vanishing in part of thecritical strip?, Harmonic Analysis, Group Representations, Automorphic

Forms and Invariant Theory, Lect. Notes Ser. Inst. Math. Sci. Natl.

Univ. Singap. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 73–87.

Conference Program

MONDAY, May 15 (Weizmann Institute)

09:00 - 10:00 Registration of participants

10:00 - 12:30 Morning session on the trace formula

James Arthur (University of Toronto)

14:30 - 15:30 Marie-France Vigneras (Institut Mathematiques de Jussieu)

Irreducibility and cuspidality of the Steinberg representationmodulo p

16:00 - 17:00 Prof. Chaim Leib Pekeris Memorial Lecture by

Peter Sarnak (Princeton University)

Equidistribution and primes

TUESDAY, May 16 (Tel-Aviv University)

10:00 - 12:30 Morning session on L-functions

Daniel Bump (Stanford University)

James Cogdell (Ohio State University)

14:00 - 15:00 Akshay Venkatesh (Courant Institute)

A spherical simple trace formula, and Weyls law for cusp forms

15:30 - 16:30 Wee Teck Gan (University of California, San Diego)

The regularized Siegel-Weil formula for exceptional groups

17:00 - 18:00 Jiu-Kang Yu (Purdue University)

Construction of tame types

WEDNESDAY, May 17 (Weizmann Institute)

10:00 - 12:30 Morning session on theta correspondence

Roger Howe (Yale University)

Stephen Kudla (University of Maryland)

xvii

xviii CONFERENCE PROGRAM

THURSDAY, May 18 (Weizmann Institute)

10:00 - 11:00 Birgit Speh (Cornell University)

The restriction of cohomologically induced representationsto subgroups

11:30 - 12:30 Guy Henniart (University of Paris-Sud)

Explicit local Langlands correspondence for GL(N): the tame case

14:30 - 17:00 Afternoon session on arithmetic applications

Peter Sarnak (Princeton University)

Freydoon Shahidi (Purdue University)

FRIDAY, May 19 (Weizmann Institute)

10:00 - 11:00 Laurent Clozel (University of Paris-Sud)

Equidistribution of adelic tori and of CM-points for Shimuravarieties

11:30 - 12:30 Dinakar Ramakrishnan (Caltech, Pasadena)

Rational cusp forms and Calabi-Yau varieties

14:00 - 15:00 Marko Tadic (University of Zagreb)

On irreducible unitary representations of classical p-adic groups

15:15 - 16:15 Werner Muller (University of Bonn)

The trace formula and the cuspidal spectrum of GL(n)

List of Participants

James Arthur

University of Toronto, Canada

Moshe Baruch

Technion, Israel

Gal Binyamini

Weizmann Institute of Science, Israel

Vladimir Berkovich


Joseph Bernstein

Tel-Aviv University, Israel

Amnon Besser

Ben Gurion University of the Negev,

Israel

Siegfried Bocherer

University of Mannheim, Germany

Eliot Brenner


Israel

Daniel Bump

Stanford University, U.S.A.

CheeWhye Chin

National University of Singapore,

Singapore

Laurent Clozel

Universite Paris-Sud, France

James Cogdell

Ohio State University, U.S.A.

Ehud de Shalit

Hebrew University of Jerusalem, Israel

Tobias Finis

University of Leipzig, U.S.A.

Solomon Friedberg

Boston College, U.S.A.

Wee Teck Gan

University of California, San Diego,

U.S.A.

Steve Gelbart


David Ginzburg


Nadya Gurevich


Israel

Guy Henniart

Universite Paris-Sud, France

Roger Howe

Yale University, U.S.A.

Joseph Hundley

Pennsylvania State University, U.S.A.

Anthony Joseph


Martin Karel

Rutgers University, U.S.A.

David Kazhdan


Bernhard Krotz

Max Planck Institute, Bonn, Germany

Stephen Kudla

University of Maryland, U.S.A.

Erez Lapid


xix

xx PARTICIPANTS

Alex Lubotzky


Zhengyu Mao

Rutgers University, U.S.A.

Kimball Martin

Columbia University, U.S.A.

Werner Muller

University of Bonn, Germany

Omer Offen


Ilya Piatetski-Shapiro

Yale University, U.S.A.

Dipendra Prasad

Tata Institute, India

Dinakar Ramakrishnan

Caltech, Pasadena, U.S.A.

Andre Reznikov

Bar-Ilan University, Israel

Zeev Rudnick


Peter Sarnak

Princeton University, U.S.A.

Eitan Sayag


Freydoon Shahidi

Purdue University, U.S.A.

Lior Silberman

IAS, Princeton, U.S.A.

Andy Sinton

Hebrew University of Jerusalem,Israel

David Soudry


Birgit Speh

Cornell University, U.S.A.

Marko Tadic

University of Zagreb, Croatia

Jimi Lee Truelsen

University of Aarhus, Denmark

Yakov Varshavsky


Akshay Venkatesh

Courant Institute, U.S.A.

Marie-France Vigneras

Institut de Mathematiques de Jussieu,

France

David Whitehouse

IHES, Bures-sur-Yvette, France

Jiu-Kang Yu

Purdue University, U.S.A.

Titles in This Subseries

Israel Mathematical Conference Proceedings (IMCP) is a publication, part of the ContemporaryMathematics Series, devoted to the proceedings of conferences, symposia and seminars.Collections of papers focusing on a certain subject will also be published. Prospective volumesmay be submitted to any member of the editorial board. Each volume has an editor (or editors)responsible for its preparation. In order to ensure inexpensive and timely distribution, authorsare requested to submit to the Editor of the volume an electronic TEX file of their manuscript inAMS-LATEX, using the Contemporary Mathematics style file which can be downloaded athttp://www.ams.org/tex/author-info.html. For further information, contact the ManagingEditor, IMCP, Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel;e-mail: [email protected].

489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms andL-functions II. Local aspects, 2009

488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms andL-functions I. Global aspects, 2009

455 Mark Agranovsky, Daoud Bshouty, Lavi Karp, Simeon Reich, David Shoikhet,and Lawrence Zalcman, Editors, Complex analysis and dynamical systems III, 2008

433 Pavel Etingof, Shlomo Gelaki, and Steven Shnider, Editors, Quantum groups, 2007

404 Alexander Borichev, Hakan Hedenmalm, and Kehe Zhu, Editors, Bergman spacesand related topics in complex analysis, 2006

402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi,Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006

387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry,spectral theory, groups, and dynamics, 2005

382 Mark Agranovsky, Lavi Karp, and David Shoikhet, Editors, Complex analysis anddynamical systems II, 2005

364 Mark Agranovsky, Lavi Karp, David Shoikhet, and Lawrence Zalcman, Editors,Complex analysis and dynamical systems, 2004

Published Earlier as IMCP

15 Vitali Milman, Iossif Ostrovskii, Mikhail Sodin, Vadim Tkachenko,and Lawrence Zalcman, Editors, Entire functions in modern analysis: Boris Levinmemorial conference, 2001

14 Robert Brooks and Mikhail Sodin, Editors, Lectures in memory of Lars Ahlfors, 2000

13 Yuri Brudnyi, Michael Cwikel, and Yoram Sagher, Editors, Function spaces,

interpolation spaces, and related topics, 1999

12 Mina Teicher, Editor, The heritage of Emmy Noether, 1999

11 Lawrence Zalcman, Editor, Proceedings of the Ashkelon workshop on complex functiontheory (May 1996), 1997

10 Jean-Pierre Fouque, Kenneth J. Hochberg, and Ely Merzbach, Editors,Stochastic analysis: random fields and measure-valued processes, 1995

9 Mina Teicher, Editor, Proceedings of the Hirzebruch 65 conference on algebraicgeometry, 1995

8 Ilya Piatetski-Shapiro and Stephen Gelbart, Editors, The Schur lectures (1992),1995

7 Anthony Joseph and Steven Shnider, Editors, Quantum deformations of algebrasand their representations, 1993

6 Haim Judah, Editor, Set theory of the reals, 1992

5 Michael Cwikel, Mario Milman, and Richard Rochberg, Editors, Interpolationspaces and related topics, 1992

TITLES IN THIS SUBSERIES

4 Simson Baron and Dany Leviatan, Editors, Approximation interpolation andsummability, in honor of Amnon Jakimovski, 1991

3 Stephen Gelbart, Roger Howe, and Peter Sarnak, Editors, Festschrift in honor of I.I. Piatetski-Shapiro, part II: Papers in analysis, number theory and automorphicL-Functions, 1990

2 Stephen Gelbart, Roger Howe, and Peter Sarnak, Editors, Festschrift in honor of I.I. Piatetski-Shapiro, part I: Papers in representation theory, 1990

1 Louis Rowen, Editor, Ring theory, in honor of S. A. Amitsur, 1989

This book is the first of two volumes which represent leading themes of current research in automorphic forms and representation theory of reductive groups over local fields. Articles in this volume mainly represent global aspects of automorphic forms. Among the topics are the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin–Selberg convolutions and �-functions; and ��-adic �-functions. The articles are written by leading researchers in the field, and bring the reader, advanced graduate students and researchers alike, to the frontline of the vigorous research in these deep, vital topics. The companion volume (Contemporary Mathematics, Volume 489) is devoted to local aspects of automorphic forms.

CONM/488www.ams.orgAMS on the Webwww.ams.org

contemporary mathematicsasher ben-artzi and david soudry 13 gauss sum combinatorics and metaplectic...

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