HARMONIC ANALYSIS,THE TRACE FORMULA, AND SHIMURA VARIETIES

HARMONIC ANALYSIS,THE TRACE FORMULA, AND SHIMURA VARIETIESProceedings of the Clay Mathematics Institute2003 Summer School, The Fields InstituteToronto, Canada, June 227, 2003

James ArthurDavid EllwoodRobert KottwitzEditors

Clay Mathematics ProceedingsVolume 4

American Mathematical Society

Clay Mathematics Institute

2000 Mathematics Subject Classification. Primary 1102; Secondary 11F70, 11F72,11F85, 11G18, 14G35, 22E35, 22E50, 22E55.

Cover art developed with the assistance of Bill Casselman.

ISBN 0-8218-3844-X (alk. paper)

Copying and reprinting. Material in this book may be reproduced by any means for educa-tional and scientific purposes without fee or permission with the exception of reproduction by ser-vices that collect fees for delivery of documents and provided that the customary acknowledgmentof the source is given. This consent does not extend to other kinds of copying for general distribu-tion, for advertising or promotional purposes, or for resale. Requests for permission for commercialuse of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam-bridge, MA 02138, USA. Requests can also be made by e-mail to permissions@claymath.org.

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.)

c 2005 by the Clay Mathematics Institute. All rights reserved.Published by the American Mathematical Society, Providence, RI,

for the Clay Mathematics Institute, Cambridge, MA.Printed in the United States of America.

The Clay Mathematics Institute retains all rights

except those granted to the United States Government. 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/Visit the Clay Mathematics Institute home page at http://www.claymath.org/

10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05

Contents

Preface vii

An Introduction to the Trace Formula 1James Arthur

Introduction to Shimura Varieties 265J. S. Milne

Linear Algebraic Groups 379Fiona Murnaghan

Harmonic Analysis on Reductive p-adic Groups and Lie Algebras 393Robert E. Kottwitz

Homogeneity for Reductive p-adic Groups: An Introduction 523Stephen DeBacker

Compactifications and Cohomology of Modular Varieties 551Mark Goresky

Introduction to Shimura Varieties with Bad Reduction of Parahoric Type 583Thomas J. Haines

A Statement of the Fundamental Lemma 643Thomas C. Hales

Notes on the Generalized Ramanujan Conjectures 659Peter Sarnak

List of Participants 687

v

Preface

This volume is based on lectures given at the fourth Clay Mathematics Insti-tute Summer School entitled Harmonic Analysis, the Trace Formula, and ShimuraVarieties. It was held at the Fields Institute in Toronto, Canada, from June 2 toJune 27, 2003.

The main goal of the School was to introduce graduate students and youngmathematicians to three broad and interrelated areas in the theory of automorphicforms. Much of the volume is comprised of the articles of Arthur, Kottwitz, andMilne. Although these articles are based on lectures given at the school, the authorshave chosen to go well beyond what was discussed there, in order to provide both asense of the underlying structure of the subject and a working knowledge of someof its techniques. They were written to be self-contained in some places, and tobe used in conjunction with given references in others. We hope the volume willconvey the depth and beauty of this challenging field, in which there yet remainsso much to be discoveredperhaps some of it by you, the reader!

The theory of automorphic forms is formulated in terms of reductive algebraicgroups. This is sometimes a serious obstacle for mathematicians whose backgrounddoes not include Lie groups and Lie algebras. The monograph is by no means in-tended to exclude such mathematicians, even though the theory of reductive groupswas an informal prerequisite for the Summer School. Some modest familiarity withthe language of algebraic groups is often sufficient, at least to get started. Forthis reason, we have generally resisted the temptation to work with specific matrixgroups. The short article of Murnaghan contains a summary of some of the basicproperties of reductive algebraic groups that are used elsewhere in the monograph.

Much of the modern theory of automorphic forms is governed by two funda-mental problems that are at the heart of the Langlands program. One is Lang-lands principle of functoriality. The other is the general analogue of the Shimura-Taniyama-Weil conjecture on modular elliptic curves. (See [A] and [L, 2].) Theseproblems are among the deepest questions in mathematics. It is premature to tryto guess what various techniques will play a role in their ultimate resolution. How-ever, the trace formula and the theory of Shimura varieties are both likely to bean essential part of the story. They have already been used to establish significantspecial cases.

The trace formula has perhaps been more closely identified with the first prob-lem. Special cases of functoriality arise naturally from the conjectural theory ofendoscopy, in which a comparison of trace formulas would be used to characterizethe internal structure of the automorphic representations of a given group. (See[Sh] for a discussion of the first case to be investigated.) Likewise, Shimura varietiesare usually associated with the second problem. As higher dimensional analoguesof modular curves, they are attached by definition to certain reductive groups. Inmany cases, it has been possible to establish reciprocity laws between -adic Ga-lois representations on their cohomology groups and automorphic representationsof the corresponding reductive groups. These laws can be formulated as an ex-plicit formula for the zeta function of a Shimura variety in terms of automorphicL-functions. (See [K] for a discussion of the rough form such a formula is expectedto take. The word rough should be taken seriously, given the current limitationsof our understanding.)

vii

viii PREFACE

The work of Wiles that led to a proof of Fermats Last Theorem suggests thatthe two problems are inextricably linked. This is already apparent in the reciprocitylaws that have been established for Shimura varieties. Indeed, the conjectural for-mula for the zeta function of a general Shimura variety requires the theory ofendoscopy even to state. Moreover, the proof of these reciprocity laws requiresa comparison of the (automorphic) trace formula with an (-adic) Lefschetz traceformula. Some of the most striking parts of the argument are in the comparisonof the various terms in the two formulas. The most sophisticated Shimura vari-eties for which there are complete results are the so-called Picard modular surfaces.(See [LR], especially the summary on pp. 255302.) Picard modular surfaces areattached to unitary groups in three variables. It is no coincidence that the the-ory of endoscopy has also been established for these groups, thereby yielding aclassification of their automorphic representations [R].

There is some discussion of these problems in the articles of Arthur and Milne.However, the articles of both Arthur and Milne really are intended as introductions,despite their length. The theory of endoscopy, and the automorphic description ofzeta functions of Shimura varieties, are at the forefront of present day research.They are for the most part beyond the scope of this monograph.

The local terms in the trace formula are essentially analytic objects. Theyinclude the invariant orbital integrals and irreducible characters that are the basisfor Harish-Chandras theory of local harmonic analysis. They also include weightedorbital integrals and weighted characters, objects that arose for the first time withthe trace formula. The article of Kottwitz is devoted to the general study of theseterms at p-adic places. It is a largely self-contained course, which covers manyof Harish-Chandras basic results in invariant harmonic analysis, as well as theirweighted, noninvariant analogues.

The article of DeBacker focuses on the phenomenon of homogeneity in invari-ant harmonic analysis at p-adic places. It concerns quantitative forms of some ofthe basic theorems of p-adic harmonic analysis, such as Howes finiteness theoremand Harish-Chandras local character expansion. The article also explains how ho-mogeneity enters into Waldspurgers analysis of stability for linear combinations ofnilpotent orbital integrals.

There are subtle questions concerning the terms in the trace formula that gobeyond those treated by Kottwitz and DeBacker. The most basic of these is knownas the fundamental lemma, even though it is still largely conjectural.1 The article byHales contains a precise statement of the conjecture and some remarks on progresstoward a general proof. The fundamental lemma occupies a unique place in thetheory. It is a critical ingredient in the comparison of trace formulas that is partof the theory of endoscopy. It has an equally indispensable role in the comparisonof (automorphic and -adic) trace formulas needed to establish reciprocity laws forShimura varieties.

Some Shimura varieties are projective, which is to say that they are compact ascomplex varieties. They correspond to reductive groups over Q that are anisotropic.The trace formula in this case simplifies considerably. It reduces to the Selberg traceformula for compact quotient. On t