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CONTEMPORARY MATHEMATICS 488 Israel Mathematical Conference Proceedings 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 Society Providence, Rhode Island Bar-Ilan University Ramat Gam, Israel http://dx.doi.org/10.1090/conm/488 Automorphic Forms and L-functions I. Global Aspects Photograph by David Soudry Photograph by David Steve Gelbart CONTEMPORARY MATHEMATICS 488 Israel Mathematical Conference Proceedings 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 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 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 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 I: 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 × ≤ 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 J. W. Cogdell and I. I. Piatetski-Shapiro 83 Restrictions 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 v Contents II: 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 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 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 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 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), S´eminaire Choquet, 11e–12e ann´ees (1971–1973), Initiationa ` l’analyse, Exp. No. 4, Secr´etariat Mathmatique, Paris, 1973. (11) Gelbart, Stephen, Introduction to the theory of group representations, S´eminaire Choquet, 11e–12e ann´ees (1971–1973), Initiationa ` l’analyse, Exp. No. 3, Secr´etariat 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 Ad`ele 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),
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  • The Computation of Local Exterior Square L-Functions for Glm Via Bernstein-Zelevinsky Derivatives

    The Computation of Local Exterior Square L-Functions for Glm Via Bernstein-Zelevinsky Derivatives

    The Computation of Local Exterior Square L-functions for GLm via Bernstein-Zelevinsky Derivatives Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Yeongseong Jo, B.S., M.S. Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: James Cogdell, Advisor Roman Holowinsky Wenzhi Luo Cynthia Clopper c Copyright by Yeongseong Jo 2018 Abstract Let π be an irreducible admissible representation of GLm(F ), where F is a non- archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square L-function L(s; π; ^2) in terms of L-functions of supercuspidal representa- tions via an integral representation established by Jacquet and Shalika in 1990. We analyze the local exterior square L-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic L-functions L(s; π; ^2) via integral integral representations for the irreducible admis- 2 sible representation π for GLm(F ) and the local arithmetic L-functions L(s; ^ (φ(π))) of its Langlands parameter φ(π) via local Langlands correspondence. ii This is dedicated to my parents and especially my lovely sister who will beat cancer. iii Acknowledgments First and foremost, I would like to express my sincere gratitude to my advisor, Professor Cogdell, for his encouragement, invaluable comments, countless hours of discussions, and guidances at every stage of a graduate students far beyond this dissertation.