C ONTEMPORARY M ATHEMATICS 489 Automorphic Forms and L-functions I I. 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 Israel Mathematical Conference Proceedings
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CONTEMPORARYMATHEMATICS
489
Automorphic Forms and L-functions I I.
Local AspectsA 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
Automorphic Forms and L-functions II.
Local Aspects
http://dx.doi.org/10.1090/conm/489
Steve Gelbart
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CONTEMPORARYMATHEMATICS
489
Automorphic Forms and L-functions II.
Local AspectsA 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
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.)
(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,
(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.
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
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
Weizmann Institute of Science, Israel
Joseph Bernstein
Tel-Aviv University, Israel
Amnon Besser
Ben Gurion University of the Negev,
Israel
Siegfried Bocherer
University of Mannheim, Germany
Eliot Brenner
Ben Gurion University of the Negev,
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
Weizmann Institute of Science, Israel
David Ginzburg
Tel-Aviv University, Israel
Nadya Gurevich
Ben Gurion University of the Negev,
Israel
Guy Henniart
Universite Paris-Sud, France
Roger Howe
Yale University, U.S.A.
Joseph Hundley
Pennsylvania State University, U.S.A.
Anthony Joseph
Weizmann Institute of Science, Israel
Martin Karel
Rutgers University, U.S.A.
David Kazhdan
Hebrew University of Jerusalem, Israel
Bernhard Krotz
Max Planck Institute, Bonn, Germany
Stephen Kudla
University of Maryland, U.S.A.
Erez Lapid
Hebrew University of Jerusalem, Israel
xix
xx PARTICIPANTS
Alex Lubotzky
Hebrew University of Jerusalem, Israel
Zhengyu Mao
Rutgers University, U.S.A.
Kimball Martin
Columbia University, U.S.A.
Werner Muller
University of Bonn, Germany
Omer Offen
Weizmann Institute of Science, Israel
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
Yakov Varshavsky
Hebrew University of Jerusalem, Israel
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 orms 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
f
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 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 �-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 488) is devoted to global aspects of auto-morphic forms.