Automorphic Forms, Representation Theory and Arithmetic: Papers presented at the Bombay Colloquium 1979

358 Pages·1981·9.03 MB·English

by  Gelbart Harder Iwasawa

AUTOMORPHIC FORMS,
REPRESENTATION THEORY
AND ARITHMETIC
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
STUDIES IN MATHEMATICS
General Editor: K. G. RAMANATHAN
1. M. Herve: SEVERAL COMPLEX VARIABLFS
2. M. F. Atiyah and others: DIFFERENTIAL ANALYSIS
3. B. Malgrange: IDEALS OF DIFFERENTIABLE FUNCTIONS
4. S. S. Abhyankar and others: ALGEBRAIC GEOMETRY
5. D. Mumford: ABELIAN VA RmTIES
6. L. Schwartz: RADoN MEASURES ON ARBITRARY TOPOLOGICAL
SPACFS AND CYLINDRICAL MEASURES
7. W. L. Baily, Jr., and others: DISCRETE SUBGROUPS OF Lm
GROUPS AND APPLICATIONS TO MODULI
8. C. P. RAMANUJAM: A TRIBUTE
9. C. L. Siegel: ADVANCED ANALYTIC NUMBER THEORY
10. S. Gelbart and others: AUTOMORPHIC FORMS, REpRESENTATION
THEoRY AND ARITHMETIC
AUTOMORPHIC FORMS,
REPRESENTATION THEORY
AND ARITHMETIC
Papers presented at the Bombay Colloquium 1979, by
GELBART HARDER IWASAWA
JACQUET KATZ PIATETSKI-SHAPIRO
RAGHAVAN SHINTANI STARK. ZAGIER
PrIbIisIml/or lite
TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1981
© SPRINGER-VERLAG BERLIN HEIDELBERG 1981
ORIGINALLY PUBLISHED BY SPRINGER-VERLAG BERLIN HEIDELBERG NEW YORK IN 1981
ISBN 978-3-540-10697-5 ISBN 978-3-662-00734-1 (eBook)
DOI 10.1007/978-3-662-00734-1
No part of this book may be reproduced in any form by print,
microfilm or any other means without written permission
from the Tata Institute of Fundamental Research,
Bombay 400005
INTERNATIONAL COLLOQUIUM
ON AUTOMORPHIC FORMS
REPRESENTATION THEORY
AND ARITHMETIC
BOMBAY, 8-15 JANUARY 1979
REPORT
AN INTERNATIONAL COLLOQUIUM on Automorphic forms, Representation
theory and Arithmetic was held at the Tata Institute of Fundamental
Research, Bombay, from 8 to 15 January 1979. The purpose of the
Colloquium was to discuss recent achievements in the theory of auto
morphic forms of one and several variables, representation theory with
special reference to the interplay between these and number theory,
e.g. arithmetic automorphic forms, Hecke theory, Representation of
GLz and GLn in general, class fields, L-functions, p-adic automorphic
forms and p-adic L-functions.
The Colloquium was jointly sponsored by the International Mathe
matical Union and the Tata Institute of Fundamental Research, and was
financially supported by them and the Sir Dorabji Tata Trust.
An Organizing Committee consisting of Professors P. Deligne,
M. Kneser, M.S. Narasimhan, S. Raghavan, M.S. Raghunathan and
C.S. Seshadri was in charge of the scientific programme. Professors
P. Deligne and M. Kneser acted as representatives of the International
Mathematical Union on the Organising Committee.
The following mathematicians gave invited addresses at the
Colloquium: W. Casselman, P. Deligne, S. Gelbart, G. Harder,
K. Iwasawa, H. Jacquet, N.M. Katz, I. Piatetski-Shapiro, S. Raghavan,
T. Shintani, H.M. Stark and D. Zagier.
Professor R. Howe was unable to attend the Colloquium but has
sent a paper for publication in the Proceedings.
Professors A. Borel and M. Kneser who accepted our invitation, were
unable to attend the Colloquium.
The invited lectures were of fifty minutes' duration. These were followed
by discussions. In addition to the programme of invited addresses, there
were expository and survey lectures by some invited speakers giving
more details of their work. Besides the mathematicians at the Tata
Institute, there were also mathematicians from other universities in India
who were invitees to the Colloquium.
The social programme during the Colloquium included a Tea Party on
8 January; a programme of Weste m music on 9 January; a programme
oflnstrumental music on lO January; a dinner at the Institute to meet the
members of the School of Mathematics on 11 January; a performance of
classical Indian Dances (Bharata Naty am) on 12 January; a visit to
Elephanta on 13 January; a programme of Vocal music on 13 January
and a dinner at the Institute on 14 January.
CONTENTS
GELBART, S. and I. PIATETSKI-SHAPIRO: On Shimura's corres
pondence for modular forms of half-integral weight.
HARDER, G.: Period integrals of cohomology classes which are
represented by Eisenstein series 41
HOWE, ROGER: Wave front sets of representations of Lie groups 117
IWASAWA, KENKICHI: On p-adic representations associated with
Zp- extensions . 141
JACQUET, HERvE: Dirichlet series for the group GL(n) 155
KATZ, NICHOLAS M: Crystalline cohomology, Dieudonne'modules
and Jacobi sums . 165
RAGHAVAN, S: Estimates of coefficients of modular forms and
generalized modular relations . 247
SHINTANI, TAKURO: A remark on zeta functions of algebraic
number fields 255
STARK, H. M.: Derivatives of L-series at s = 0 261
ZAGIER, D.: Eisenstein series and the Riemann zeta function 275
ZAGIER, D. : Eisenstein series and the Selberg trace formula I 303
AUTOMORPHIC FORMS,
REPRESENTATION THEORY
AND ARITHMETIC
ON SHIMURA'S CORRESPONDENCE
FOR MODULAR FORMS OF
HALF-INTEGRAL WEIGHT*
By S. GELBART AND I. PIATETSKI-SHAPIRO
Table of Contents
Chapter I. Local Theory
§ 1. The Metaplectic Group
§ 2. Admissible Representations
§ 3. Whittaker Models
§ 4. The Representations r x
§ 5. A Functional Equation of Shimura Type
§ 6. Land E - factors
§ 7. A Local Shimura Correspondence
Chapter II. Global Theory
§ 8. The Metaplectic Group
§ 9. Automorphic Representations of Half-integral Weight
§ 10. Fourier Expansions
§ 11. Theta-Representations
- Chapter III. A Generalized Shimura Correspondence
§ 12. A Rankin-Selberg-Shimura type Zeta-integral
§ 13. An Euler Product Expansion
§ 14. The Global Shimura Correspondence
§ 15. The Main Theorem
§ 16. Applications and Concluding Remarks
Introduction
G. Shimura has shown how to attach to each holomorphic cusp form
of half-integral weight a modular form of even integral weight. More
precisely, suppose f(z) is a cusp form of weight k/2, level N, and character
*Talk presented by S.G.