Selected Titles in This Series

66 Robert S. Doran, Ze-Li D o u , and George T. Gilbert , Editors, Automorphic forms, automorphic representations, and arithmetic (Texas Christian University, Fort Worth, 1996)

65 M. Giaquinta, J. Shatah, and S. R. S. Varadhan, Editors, Differential equations: La Pietra 1996 (Villa La Pietra, Florence, Italy, 1996)

64 G. Ferreyra, R. Gardner, H. Hermes , and H. Sussmann, Editors, Differential geometry and control (University of Colorado, Boulder, 1997)

63 Alejandro A d e m , Jon Carlson, Stewart Priddy, and Peter Webb, Editors, Group representations: Cohomology, group actions and topology (University of Washington, Seattle, 1996)

62 Janos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, Algebraic geometry—Santa Cruz 1995 (University of California, Santa Cruz, July 1995)

61 T. N . Bai ley and A. W . Knapp, Editors, Representation theory and automorphic forms (International Centre for Mathematical Sciences, Edinburgh, Scotland, March 1996)

60 David Jerison, I. M. Singer, and Daniel W . Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994)

59 Wil l iam Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994)

58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992)

57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993)

56 Wil l iam J. Haboush and Brian J. Parshall , Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991)

55 U w e Jannsen, Steven L. Kle iman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991)

54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990)

53 James A. Carlson, C. Herbert Clemens , and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989)

52 Eric Bedford, John P. D'Ange lo , Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989)

51 Wil l iam B. Arveson and Ronald G. Douglas , Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988)

50 James Gl imm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988)

49 Robert C. Gunning and Leon Ehrenpreis , Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987)

48 R. O. Wells, Jr., Editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987)

47 Paul Fong, Editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986)

46 Spencer J. Bloch, Editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985)

45 Felix E. Browder, Editor, Nonlinear functional analysis and its applications (University of California, Berkeley, July 1983)

(Continued in the back of this publication)

http://dx.doi.org/10.1090/pspum/066.1

Automorphic Forms, Automorphic Representations, and Arithmetic

Professor Goro Shimura at the CBMS Conference "Euler Products and Eisenstein Series,"

held at Texas Christian University, May 1996

Proceedings of Symposia in

PURE MATHEMATICS

Volume 66, Part 1

Automorphic Forms, Automorphic Representations, and Arithmetic NSF-CBMS Regional Conference in Mathematics on Euler Products and Eisenstein Series May 20-24, 1996 Texas Christian University

Robert S. Doran Ze-Li Dou George T. Gilbert Editors

PROCEEDINGS OF AN NSF-CBMS REGIONAL CONFERENCE ON EULER PRODUCTS AND EISENSTEIN SERIES HELD AT TEXAS CHRISTIAN UNIVERSITY,

FORT WORTH, TEXAS, MAY 20-24, 1996

Any opinions, findings, and conclusions or recommendat ions expressed in this mater ia l are those of the au thors and do not necessarily reflect the view of the Nat ional Science Foundat ion .

1991 Mathematics Subject Classification. P r i m a r y l l F x x ; Secondary 14Gxx, 22Exx.

Library of Congress Cataloging-in-Publicat ion Data

NFS-CBMS Regional Conference in Mathematics on Euler Products and Eisenstein Series (1996 : Texas Christian University)

Automorphic forms, automorphic representations, and arithmetic : NFS-CBMS Regional Con­ference in Mathematics on Euler Products and Eisenstein Series, May 20-24, 1996, Texas Christian University / Robert S. Doran, Ze-Li Dou, George T. Gilbert, editors.

p. cm. — (Proceedings of symposia in pure mathematics, 0082-0717 ; v. 66, pts. 1-2) Includes bibliographical references. ISBN 0-8218-0659-9 (set) ISBN 0-8218-1050-2 (pt. 1) ISBN 0-8218-1051-0 (pt. 2) 1. Automorphic functions Congresses. 2. Automorphic forms Congresses. 3. Representa­

tions of groups Congresses. I. Doran, Robert S., 1937- . II. Dou, Ze Li, 1962- . III. Gilbert, George T., 1957- . IV. Title. V. Series. QA353.A9 N74 1996 511.3'3—dc21 99-28916

CIP

Copying and reprinting. Material in this book may be reproduced by any means for edu­cational 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 acknowledg­ment 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 Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org.

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.)

© 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. 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 URL: http: / /www.ams.org/

10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99

Dedicated to Goro Shimura

Table of Contents-Part 1

Foreword xi

On Period Torsors

DON BLASIUS and MIKHAIL BOROVOI 1

Near Holomorphy, Arithmeticity, and the Theta Correspondence

ANTONIA W. BLUHER 9

Torsion Points on Xo(N)

ROBERT COLEMAN, BRUCE KASKEL, and KENNETH A. RIBET 27

A Fundamental Group for Symbolic Dynamics I: Definitions

PAUL F E I T 51

Critical Values of Twisted Tensor //-Functions over CM-fields

EKNATH GHATE 87

Zeta Functions Formed with Modular Symbols

DORIAN GOLDFELD 111

Non-critical Values of Adjoint L-Functions for SL(2)

HARUZO HIDA 123

Bounds for Congruence Primes

M. RAM MURTY 177

Frobenius Distributions and Galois Representations

V. KUMAR MURTY 193

Mod 6 Representations of Elliptic Curves

KARL RUBIN and ALICE SILVERBERG 213

On Absolute CM-Periods

HIROYUKI YOSHIDA 221

IX

Table of Contents-Part 2

Foreword xi

Metaplect ic Genera t ing Functions and Shimura Integrals

DANIEL BUMP and SOLOMON FRIEDBERG 1

On the Plancherel Measure for t he Continuous Spec t rum of t h e Modular Group

BILL CASSELMAN 19

Average Values of Cubic L-Series

DAVID FARMER, JEFFREY HOFFSTEIN, and DANIEL LIEMAN 27

Euler Factorizat ion of Global Integrals

PAUL GARRETT 35

Cohomological Automorphic Forms on Uni ta ry Groups , I: Rat ional i ty of t h e T h e t a Correspondence

MICHAEL HARRIS 103

Pullbacks of Eisenstein Series, Hecke-Jacobi Theory and Automorphic L-Functions

BERNHARD E. HEIM 201

Omega Resul ts for Automorphic L-Series

JEFFREY HOFFSTEIN and PAUL LOCKHART 239

Representa t ions of SL(2) x G

KAMAL KHURI-MAKDISI 251

Trunca ted Integrals and the Shintani Zeta Function for t he Space of Binary Quar t ic Forms

JASON LEVY 277

On the Generalized Ramanujan Conjecture for GL(n)

WENZHI LUO, ZEEV RUDNICK, and P E T E R SARNAK 301

On t h e Global Root N u m b e r s of GL(n)x GL(ra)

DIPENDRA PRASAD and DINAKAR RAMAKRISHNAN 311

x

Foreword

Professor Goro Shimura was the principal speaker at the CBMS regional conference held in May, 1996 at Texas Christian University, entitled "Euler Products and Eisenstein Series." His ten lectures resulted in a monograph bearing the same title, which appeared as volume 93 of the CBMS Regional Conference Series in Mathematics. In these volumes we have collected papers based on most of the other talks presented at the conference, as well as many additional contributions.

Professor Shimura's stature as a mathematical giant, based on the plethora of seminal ideas he has introduced into number theory, on the numerous and varied theories that have matured in his hands, and on the high quality of the mathematicians he has cultivated and continues to influence, is certainly secure and will long endure. It is neither our place nor our intention to comment at length on Professor Shimura's achievements, but we believe that many aspects of Professor Shimura's work as well as that of his students are reflected in these volumes. It is therefore highly fitting that they be dedicated to Professor Shimura.

We have selected a rather general title, but an informed reader will find after a glance at the tables of contents that even this title does not adequately encompass the spectrum of the topics discussed in these volumes. We have also endeavored to place the papers which consider related topics and use similar techniques in the same volume. However, the division of the papers into two volumes cannot be based on a scientific principle. All in all, we must simply allow the mathematics to speak for itself.

We thank Professor Shimura for making the conference a success, and for making these volumes possible. We also acknowledge the hard work and patience of our authors and referees. We are especially grateful to Professor Hiroyuki Yoshida for his advice and for his lending a helping hand on several occasions. We are indebted to Shirley Dor an, our technical secretary, for a thoroughly professional job. Finally, our heartfelt gratitude is reserved for our families, for their cheerful long-suffering.

Robert S. Doran Ze-Li Dou George T. Gilbert February 1999

XI

Selected Titles in This Series (Continued from the front of this publication)

44 Wil l iam K. Allard and Frederick J. Almgren, Jr., Editors, Geometric measure theory and the calculus of variations (Humboldt State University, Areata, California, July/August 1984)

43 Frangois Treves, Editor, Pseudodifferential operators and applications (University of

Notre Dame, Notre Dame, Indiana, April 1984)

42 Anil Nerode and Richard A. Shore, Editors, Recursion theory (Cornell University,

Ithaca, New York, June/July 1982)

41 Yum-Tong Siu, Editor, Complex analysis of several variables (Madison, Wisconsin, April 1982)

40 Peter Orlik, Editor, Singularities (Humboldt State University, Areata, California, July/August 1981)

39 Felix E. Browder, Editor, The mathematical heritage of Henri Poincare (Indiana

University, Bloomington, April 1980)

38 Richard V. Kadison, Editor, Operator algebras and applications (Queens University,

Kingston, Ontario, July/August 1980)

37 Bruce Cooperste in and Geoffrey Mason, Editors, The Santa Cruz conference on

finite groups (University of California, Santa Cruz, June/July 1979)

36 Robert Osserman and Alan Weinste in , Editors, Geometry of the Laplace operator

(University of Hawaii, Honolulu, March 1979)

35 Guido Weiss and Stephen Wainger, Editors, Harmonic analysis in Euclidean spaces

(Williams College, Williamstown, Massachusetts, July 1978)

34 D . K. Ray-Chaudhuri , Editor, Relations between combinatorics and other parts of

mathematics (Ohio State University, Columbus, March 1978)

33 A. Borel and W . Casselman, Editors, Automorphic forms, representations and

L-functions (Oregon State University, Corvallis, July/August 1977)

32 R. James Milgram, Editor, Algebraic and geometric topology (Stanford University,

Stanford, California, August 1976)

31 Joseph L. D o o b , Editor, Probability (University of Illinois at Urbana-Champaign,

Urbana, March 1976)

30 R. O. Wells , Jr., Editor, Several complex variables (Williams College, Williamstown,

Massachusetts, July/August 1975)

29 Robin Hartshorne, Editor, Algebraic geometry - Areata 1974 (Humboldt State

University, Areata, California, July/August 1974)

28 Felix E. Browder, Editor, Mathematical developments arising from Hilbert problems

(Northern Illinois University, Dekalb, May 1974)

27 S. S. Chern and R. Osserman, Editors, Differential geometry (Stanford University,

Stanford, California, July/August 1973)

26 Calvin C. Moore , Editor, Harmonic analysis on homogeneous spaces (Williams College,

Williamstown, Massachusetts, July/August 1972)

25 Leon Henkin, John Addison, C. C. Chang, Wil l iam Craig, D a n a Scott , and Robert Vaught, Editors, Proceedings of the Tarski symposium (University of

California, Berkeley, June 1971)

24 Harold G. Diamond, Editor, Analytic number theory (St. Louis University, St. Louis,

Missouri, March 1972)

23 D . C. Spencer, Editor, Partial differential equations (University of California, Berkeley,

August 1971)

22 Arunas Liulevicius, Editor, Algebraic topology (University of Wisconsin, Madison,

June/July 1970)

21 Irving Reiner, Editor, Representation theory of finite groups and related topics (University of Wisconsin, Madison, April 1970)

(See the AMS catalog for earlier titles)