CONTEMPORARY MATHEMATICS 489 Israel Mathematical Conference Proceedings

Home , David Soudry, James Cogdell, Peter Sarnak, Zeev Rudnick

Automorphic Forms and L-functions II. Local 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 Society Providence, Rhode Island

Bar-Ilan University Ramat Gam, Israel http://dx.doi.org/10.1090/conm/489

Automorphic Forms and L-functions II. Local Aspects Photograph by David Soudry Photograph by David Steve Gelbart CONTEMPORARY MATHEMATICS

489

Israel Mathematical Conference Proceedings

Automorphic Forms and L-functions II. Local 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 Society Providence, 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 University J. Bernstein, Tel-Aviv University B. Pinchuk, Netanya Academic College H. Furstenberg, Hebrew University S. Shnider, Bar-Ilan University S. Gelbart, Weizmann Institute L. Small, University of California at San Diego V. Goldshtein, 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 on the occasion of his sixtieth birthday : May 15–19, 2006, Rehovot and Tel Aviv, Israel / David Ginzburg, Erez Lapid, David Soudry, editors. v. cm. — (Contemporary mathematics ; v. 488–489) (Israel mathematical conference proceedings) 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 2006 515.9—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 by services 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 general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Managing Editor, IMCP, Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900 Israel. Requests can also be made by email to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such 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 of each 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/ 10987654321 141312111009 Contents II: Local Aspects

I. Global Aspects vii 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 Repr´esentations Modulo p de Groupes R´eductifs p-adiques Guy Henniart 41 Archimedean Rankin-Selberg Integrals HerveJacquet´ 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)ˆ and GL(n, R)ˆ Marko Tadic´ 285

v Contents I: Global Aspects

Preface ix Bibliography of Stephen S. Gelbart xi Conference Program xvii List of participants xix Report on the Trace Formula James Arthur 1 × ≤ m L-functions for Um RE/F GLn n 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 Poincar´eSeries James W. Cogdell and Ilya I. Piatetski-Shapiro 83 Restriction of Saito-Kurokawa Representations Wee Teck Gan and Nadya Gurevich (with an Appendix by Gordan Savin) 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

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 second 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 functoriality 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 local aspects in the study of automorphic forms, and those of the ﬁrst volume (Contemporary Mathematics, volume 488) concern mainly global 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 Dimensional Measures 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 generalized Hankel transforms, Bull. Amer. Math. Soc. 78 (1972), no. 3, 451–455. (7) Gelbart, Stephen ,Holomorphic discrete series for the real symplectic group Invent. Math. 19 (1973), 49–58. (8) Gelbart, Stephen, Bessel functions, representation theory, and automorphic 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), Séminaire Choquet, 11e–12e années (1971–1973), Initiationa ` l’analyse, Exp. No. 4, Secrétariat Mathmatique, Paris, 1973. (11) Gelbart, Stephen, Introduction to the theory of group representations, Séminaire Choquet, 11e–12e années (1971–1973), Initiationa ` l’analyse, Exp. No. 3, Secrétariat 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 Adèle 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 automorphic forms 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, Hervé, A relation between automorphic forms on 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, Hervé, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole´ Norm. Sup. (4) 11 (1978), no. 4, 471–542. (22) Gelbart, Stephen; Jacquet, Hervé, Forms of GL(2) from the analytic point of 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 and existence 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 representations and 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 series of SLn(k), Duke Math. J. 48 (1981), no. 2, 313–326. (27) Gelbart, S.; Piatetski-Shapiro, I., On Shimura’s correspondence for modular 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 the special 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 forms and 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 vanishing of 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 his Sixtieth Birthday, Part I. Papers in Representation Theory,Papersfrom 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 representations and 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 functions: 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., Onperiodsofcuspformsandalge- braic cycles for U(3), Israel J. Math. 83 (1993), no. 1-2, 213–252. (44) Gelbart, Stephen; Rogawski, Jonathan; Soudry, David, Periods of cusp forms 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, Hervé; Rogawski, Jonathan, Représentations génériques du groupe unitaireatroisvariables ` (French) [Generic representations for the unitary group in three variables], C. R. Acad. Sci. Paris Sér. 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, Hervé; Rogawski, Jonathan, Generic representations 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), Birkhäuser Boston, Inc., Boston, MA, 2003. (54) Gelbart, Stephen S.; Miller, Stephen D., Riemann’s zeta function and beyond, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 1, 59–112 (electronic). (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, Anewmethodfor lower 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 the edge 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 the critical 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 Math´ematiques de Jussieu) Irreducibility and cuspidality of the Steinberg representation modulo 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 representations to 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 Shimura varieties

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 Solomon Friedberg University of Toronto, Canada Boston College, U.S.A. Moshe Baruch Wee Teck Gan Technion, Israel University of California, San Diego, U.S.A. Gal Binyamini Weizmann Institute of Science, Israel Steve Gelbart Weizmann Institute of Science, Israel Vladimir Berkovich Weizmann Institute of Science, Israel David Ginzburg Joseph Bernstein Tel-Aviv University, Israel Tel-Aviv University, Israel Nadya Gurevich Amnon Besser Ben Gurion University of the Negev, Ben Gurion University of the Negev, Israel Israel Guy Henniart Siegfried Böcherer Université Paris-Sud, France University of Mannheim, Germany Roger Howe Eliot Brenner Yale University, U.S.A. Ben Gurion University of the Negev, Joseph Hundley Israel Pennsylvania State University, U.S.A. Daniel Bump Stanford University, U.S.A. Anthony Joseph Weizmann Institute of Science, Israel CheeWhye Chin National University of Singapore, Martin Karel Singapore Rutgers University, U.S.A. Laurent Clozel David Kazhdan Université Paris-Sud, France Hebrew University of Jerusalem, Israel James Cogdell Bernhard Krötz Ohio State University, U.S.A. Max Planck Institute, Bonn, Germany Ehud de Shalit Stephen Kudla Hebrew University of Jerusalem, Israel University of Maryland, U.S.A. Tobias Finis Erez Lapid University of Leipzig, U.S.A. Hebrew University of Jerusalem, Israel

xix xx PARTICIPANTS

Alex Lubotzky Yakov Varshavsky Hebrew University of Jerusalem, Israel Hebrew University of Jerusalem, Israel Zhengyu Mao Akshay Venkatesh Rutgers University, U.S.A. Courant Institute, U.S.A. Kimball Martin Marie-France Vigneras Columbia University, U.S.A. Institut de Mathématiques de Jussieu, France Werner Müller University of Bonn, Germany David Whitehouse IHES, Bures-sur-Yvette, France Omer Offen Weizmann Institute of Science, Israel Jiu-Kang Yu Purdue University, U.S.A. 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 Tel-Aviv University, Israel Peter Sarnak Princeton University, U.S.A. Eitan Sayag Weizmann Institute of Science, Israel Freydoon Shahidi Purdue University, U.S.A. Lior Silberman IAS, Princeton, U.S.A. Andy Sinton Hebrew University of Jerusalem,Israel David Soudry Tel-Aviv University, Israel Birgit Speh Cornell University, U.S.A. Marko Tadic University of Zagreb, Croatia Jimi Lee Truelsen University of Aarhus, Denmark Titles in This Subseries

Israel Mathematical Conference Proceedings (IMCP) is a publication, part of the Contemporary Mathematics Series, devoted to the proceedings of conferences, symposia and seminars. Collections of papers focusing on a certain subject will also be published. Prospective volumes may 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, authors are requested to submit to the Editor of the volume an electronic TEX ﬁle of their manuscript in AMS-LATEX, using the Contemporary Mathematics style ﬁle which can be downloaded at http://www.ams.org/tex/author-info.html. For further information, contact the Managing Editor, IMCP, Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel; e-mail: [email protected]. 489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphicf orms and L-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and L-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, H˚akan Hedenmalm, and Kehe Zhu, Editors, Bergman spaces and 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 and dynamical 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 Levin memorial 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 function theory (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 algebraic geometry, 1995 8 Ilya Piatetski-Shapiro and Stephen Gelbart, Editors, The Schur lectures (1992), 1995 7 Anthony Joseph and Steven Shnider, Editors, Quantum deformations of algebras and their representations, 1993 6 Haim Judah, Editor, Set theory of the reals, 1992 5 Michael Cwikel, Mario Milman, and Richard Rochberg, Editors, Interpolation spaces and related topics, 1992 TITLES IN THIS SUBSERIES

4 Simson Baron and Dany Leviatan, Editors, Approximation interpolation and summability, 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 automorphic L-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 second 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 L-functions; and p-adic L-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 488) is devoted to global aspects of automorphic forms.

CONM/489 AMS on the Web www.ams.orgwww.ams.org