00721 - Cohomology of Arithmetic Groups, L-Functions and Automorphic Forms.pdf - Books on Group Theory - BoxBooki

Proceedings of the International Conference on

Cohomology of Arithmetic Groups,

L-Functions and Automorphic Forms,

Mumbai 1998

TATA INSTITUTE OF FUNDAMENTAL RESEARCH

STUDIES IN MATHEMATICS

Proceedings of the International Conference on

Cohomology of Arithmetic Groups,

L-Functions and Automorphic Forms,

Mumbai 1998

Series Editor: S. RAMANAN

1. SEVERAL COMPLEX VARIABLES by M. Hew6

2. DIFFERENTIAL ANALYSIS

Proceedings of International Colloquium, 1964

3. IDEALS OF DIFFERENTIABLE FUNCTIONS by B. Malgrange

4. ALGEBRAIC GEOMETRY

Proceedings of International Colloquium, 1968

5. ABELIAN VARIETIES by D. Mumford

6. RADON MEASURES ON ARBITRARY TOPOLOGICAL

SPACES AND CYLINDRICAL MEASURES by L. Schwartz

Edited by

ToNo Venkataramana

7. DISCRETE SUBGROUPS OF LIE GROUPS AND

APPLICATIONS TO MODULI

Proceedings of International Colloquium, 1973

8. C.P. RAMANUJAM - A TRIBUTE

9. ADVANCED ANALYTIC NUMBER THEORY by C.L. Siege1

10. AUTOMORPHIC FORMS, REPRESENTATION THEORY AND

ARITHMETIC

Proceedings of International Colloquium, 1979

Published for the

Tata Institute of Fundamental Research

11. VECTOR BUNDLES ON ALGEBRAIC VARIETIES

Proceedings of International Colloquium, 1984

12. NUMBER THEORY AND RELATED TOPICS

Proceedings of International Colloquium, 1988

13. GEOMETRY AND ANALYSIS

Proceedings of International Colloquium, 1992

Narosa Publishing House

New Delhi Chennai Mumbai Kolkata

14. LIE GROUPS AND ERGODIC THEORY

Proceedings of International Colloquium, 1996

15. COHOMOLOGY OF ARITHMETIC GROUPS, L-FUNCTIONS

AND AUTOMORPHIC FORMS

Proceedings of International Conference, 1998

International distribution by

American Mathematical Society, USA

Contents

Converse Theorems for GL, and Their Application to Liftings

Cogdell and Piatetski-Shapiro .................................... 1

SERIES EDITOR

S. Ramanan

School of Mathematics

Tata Institute of Fundamental Research

Mumbai, INDIA

Congruences Between Base-Change and Non-Base-Change Hilbert

Modular Forms

Eknath Ghate ................................................... 35

Restriction Maps and L-values

Chandrashekhar Khare ........................................... 63

EDITOR

T.N. Venkataramana

School of Mathematics

Tata Institute of Fundamental Research

Mumbai, INDIA

On Hecke Theory for Jacobi Forms

M. Manickam ................................................... 89

The L2 Euler Characteristic of Arithmetic Quotients

Arvind N. Nair .................................................. 94

NAROSA PUBLISHING HOUSE

22 Daryaganj, Delhi Medical Association Road, New Delhi 110 002

35-36 Greams Road, Thousand Lights. Chennai 600 006

306 Shiv Centre, D.B.C. Secta 17, K.U. Bazar P.O., Navi Mumbai 400 705

2F2G Shivam Chambers, 53 Syed Amir Ali Avenue, Kolkata 700 019

The Space of Degenerate Whittaker Models for GL(4) over p-adic Fields

103

......................................

The Seigel Formula and Beyond

S. Raghavan ...............

retrieval system, or transmitted in any form a by any means, electronic, mechanical,

photocopying, recording or otherwise, witbout the prior permission of the publisher.

A Converse Theorem for Dirichlet Series with Poles

Raw' Raghunathan ............................................... .I27

Kirillov Theory for GL2 (V)

A. Raghuram .................................................... 143

All export rights for tbis book vest exclusively with Narosa Publishing House.

Unauthorised export is a violation of Copyright Law and is subject to Iegal action.

An Algebraic Chebotarev Density Theorem

C.S. Rajan ...................................................... 158

ISBN 81-7319-421-1

Theory of Newforms for the MadSpezialschar

B. Ramakrishnan ................................................ 170

Published by N.K. Mehra for Narosa Publishing House, 22 Daryaganj,

Delhi Medical Association Road, New Delhi 110 002 and printed at

Replika Press Pvt Ltd. Delhi 110 040 (India).

Some Remarks on the Riemann Hypothesis

M. Ram Murty .................................................. 180

Dipendra Pmsad ...........

Contents

International Conference

Cohomology of Arithmetic Groups, L-functions

and Autpmorphic Forms

On the Restriction of Cuspidal Representations to Unipotent Elements

..............................

Dipendm Prasad and Nilabh Sanat

197

Nonvanishing of Symmetric Square L-functions of Cusp Forms Inside

the Critical Strip

....................................

Mumbai, December 1998 - January 1999.

W.Kohnen and J. Sengupta

.202

Symmetric Cube for GL2

.............................

205

Henry H. Kim and hydoon Shahzdi

L-functions and Modular Forms in Finite Characteristic

...............................................

Danesh S. Thakur

214

This volume consists of theproceedings of an International Conference

on Automorphic Forms, L-functions and Cohomology of Arith-

metic Groups, held at the School of Mathematics, Tata Institute of Fun-

damental Research, during December 1998-January 1999. The conference

was part of the 'Special Year' at the Tata Institute, devoted to the above

topics.

Automorphic Forms for Siege1 and Jacobi Modular Groups

................................................

T.C. Vasudevan

.229

Restriction Maps Between Cohomology of Locally Symmetric Varieties

............................................

T.N. Venkatammana

.237

The Organizing Committee consisted of Prof. M.S. Raghunathan, Dr.

E. Ghate, Dr. C. Khare, Dr. Arvind Nair, Prof. D. Prasad, Dr. C.S. Rajan

and Prof. T.N. Venkataramana.

Professors J. Cogdell, M. Ram Murty, F. Shahidi and D.S. Thakur,

respectively from Oklahoma State University, Queens University, Purdue

University and University of Arizona, took part in the Conference and

kindly agreed to have the expositions of their latest research work pub-

lished here. From India, besides the members of the Institute, Professors

M. Manickam, D. Prasad, S. Raghavan, B. Ramakrishnan T.C. Vasudevan

and N. Sanat gave invited talks at the conference.

Mr. V. Nandagopal carried out the difficult task of converting into one

format, the manuscripts which were typeset in different styles and software.

Mr. D.B. Sawant and his colleagues at the School of Mathematics office

helped in the organization of the Conference with their customary efficiency.

Converse Theorems for GL, and Their

Application to Liftings*

J.W. Cogdell and 1.1. Piatetski-Shapiro

L(M, s) = n, L(Mv,s) where for each finite place v, L(Mv,s) encodes

Diophantine information about M at the prime v and is the inverse of a

pol~nomialin p, whose degree for almost all v is independent of v. The

product converges in some right half plane. Each M usually has a dual

object M with its own L-function L(M,s). If there is a natural tensor

product structure on the M this translates into a multiplicative convolution

(or twisting) of the L-functions. Conjecturally, these L-functions should

all enjoy nice analytic properties. In particular, they should have at least

meromorphic continuation to the whole complex plane with a finite number

of poles (entire for irreducible objects), be bounded in vertical strips (away

from any poles), and satisfy a functional equation of the form L(M, s) =

E(M, S)L(M,1 - S) with E(M,S) of the form E(M, S) = AeBs. (For a brief

exposition in terms of mixed motives, see [ll].)

There is another class of objects which also have complex analytic in-

variants enjoying similar analytic properties, namely modular forms f or

automorphic representations T and their L-functions. These L-functions are

also Euler products with a convolution structure (Rankin-Selberg convolu-

tions) and they can be shown to be nice in the sense of having meromorphic

continuation to functions bounded in vertical strips and having a functional

equation (see Section 3 below).

The most common way of establishing the analytic properties of the

L-functions of arithmetic objects L(M,s) is to associate to each M what

Siege1 referred to as an "analytic invariant", that is, a modular form or

automorphic representation T such that L(T, s) = L(M, s). This is what

*The first author was supported in part by the NSA. The second author was supported

in part by the NSF.

Since Riemann [57] number theorists have found it fruitful to attach

to an arithmetic object M a complex analytic invariant L(M,s). usually

called a zeta function or L-function. These are all Dirichlet series having

similar properties. These L-functions are usually given by an Euler product

00721 - Cohomology of Arithmetic Groups, L-Functions and Automorphic Forms.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: