CCllaayy MMaatthheemmaattiiccss PPrroocceeeeddiinnggss The modern theory of automorphic forms, embodied in 4 VVoolluummee 44 what has come to be known as the Langlands program, is an extraordinary unifying force in mathematics.It proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations.These “reciprocity laws”, conjectured by Langlands, are still largely unproved. a H However, their capacity to unite large areas of nt mathematics insures that they will be a central area dha er of study for years to come. S m hT o The goal of this volume is to provide an entry point into ir man this exciting and challenging field.It is directed on the ci u c one hand at graduate students and professional re mathematicians who would like to work in the area. a A F The longer articles in particular represent an attempt Von to enable a reader to master some of the more difficult ara techniques.On the other hand, the book will also be rimly Proceedings of the Clay Mathematics Institute eus useful to mathematicians who would like simply to 2003 Summer School,The Fields Institute tli understand something of the subject.They will be able ieas Toronto, Canada, June 2–27, 2003 ,, to consult the expository portions of the various articles. s The volume is centered around the trace formula and Shimura varieties.These areas are at the heart of the subject, but they have been especially difficult to learn because of a lack of expository material.The volume aims to rectify the problem.It is based on the courses given at the 2003 Clay Mathematics Institute Summer A School. However, many of the articles have been r t expanded into comprehensive introductions, either to h u the trace formula or the theory of Shimura varieties, or r , to some aspect of the interplay and application of the E two areas. ll w o o d James Arthur a n David Ellwood d K Robert Kottwitz o t Editors t w i t z , CMIP/4 E d it o r s www.ams.org American Mathematical Society AMS www.claymath.org CMI Clay Mathematics Institute 4-color process 704 pages • 1 5/16” spine HARMONIC ANALYSIS, THE TRACE FORMULA, AND SHIMURA VARIETIES Clay Mathematics Proceedings Volume 4 HARMONIC ANALYSIS, THE TRACE FORMULA, AND SHIMURA VARIETIES Proceedings of the Clay Mathematics Institute 2003 Summer School, The Fields Institute Toronto, Canada, June 2–27, 2003 James Arthur David Ellwood Robert Kottwitz Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 11–02;Secondary 11F70, 11F72, 11F85, 11G18, 14G35, 22E35, 22E50, 22E55. Cover art developedwith the assistance of Bill Casselman. ISBN 0-8218-3844-X (alk. paper) Copyingandreprinting. Materialinthisbookmaybereproducedbyanymeansforeduca- tionalandscientificpurposeswithoutfeeorpermissionwiththeexceptionofreproductionbyser- vicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledgment ofthesourceisgiven. Thisconsentdoesnotextendtootherkindsofcopyingforgeneraldistribu- tion,foradvertisingorpromotionalpurposes,orforresale. Requestsforpermissionforcommercial use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam- bridge,MA02138,[email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2005bytheClayMathematicsInstitute. Allrightsreserved. PublishedbytheAmericanMathematicalSociety,Providence,RI, fortheClayMathematicsInstitute,Cambridge,MA. PrintedintheUnitedStatesofAmerica. TheClayMathematicsInstituteretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 100908070605 Contents Preface vii An Introduction to the Trace Formula 1 James Arthur Introduction to Shimura Varieties 265 J. S. Milne Linear Algebraic Groups 379 Fiona Murnaghan Harmonic Analysis on Reductive p-adic Groups and Lie Algebras 393 Robert E. Kottwitz Homogeneity for Reductive p-adic Groups: An Introduction 523 Stephen DeBacker Compactifications and Cohomology of Modular Varieties 551 Mark Goresky Introduction to Shimura Varieties with Bad Reduction of Parahoric Type 583 Thomas J. Haines A Statement of the Fundamental Lemma 643 Thomas C. Hales Notes on the Generalized Ramanujan Conjectures 659 Peter Sarnak List of Participants 687 v Preface This volume is based on lectures given at the fourth Clay Mathematics Insti- tuteSummerSchoolentitled“HarmonicAnalysis,theTraceFormula,andShimura Varieties.” It was held at the Fields Institute in Toronto, Canada, from June 2 to June 27, 2003. The main goal of the School was to introduce graduate students and young mathematicianstothreebroadandinterrelatedareasinthetheoryofautomorphic forms. Much of the volume is comprised of the articles of Arthur, Kottwitz, and Milne. Althoughthesearticlesarebasedonlecturesgivenattheschool,theauthors havechosentogowellbeyondwhatwasdiscussedthere,inordertoprovidebotha sense of the underlying structure of the subject and a working knowledge of some of its techniques. They were written to be self-contained in some places, and to be used in conjunction with given references in others. We hope the volume will convey the depth and beauty of this challenging field, in which there yet remains so much to be discovered—perhaps some of it by you, the reader! The theory of automorphic forms is formulated in terms of reductive algebraic groups. Thisissometimesaseriousobstacleformathematicianswhosebackground does not include Lie groups and Lie algebras. The monograph is by no means in- tendedtoexcludesuchmathematicians,eventhoughthetheoryofreductivegroups wasaninformalprerequisitefortheSummerSchool. Somemodestfamiliaritywith the language of algebraic groups is often sufficient, at least to get started. For this reason, we have generally resisted the temptation to work with specific matrix groups. The short article of Murnaghan contains a summary of some of the basic properties 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].) These problems are among the deepest questions in mathematics. It is premature to try to 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 be an essential part of the story. They have already been used to establish significant special cases. The traceformulahas perhapsbeen morecloselyidentified withthefirstprob- lem. Special cases of functoriality arise naturally from the conjectural theory of endoscopy, in which a comparison of trace formulas would be used to characterize the internal structure of the automorphic representations of a given group. (See [Sh]foradiscussionofthefirstcasetobeinvestigated.) Likewise,Shimuravarieties are usually associated with the second problem. As higher dimensional analogues of modular curves, they are attached by definition to certain reductive groups. In many cases, it has been possible to establish reciprocity laws between (cid:1)-adic Ga- lois representations on their cohomology groups and automorphic representations of 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 automorphic L-functions. (See [K] for a discussion of the rough form such a formula is expected to take. The word “rough” should be taken seriously, given the current limitations of our understanding.) vii viii PREFACE The work of Wiles that led to a proof of Fermat’s Last Theorem suggests that thetwoproblemsareinextricablylinked. Thisisalreadyapparentinthereciprocity laws that have been established for Shimura varieties. Indeed, the conjectural for- mula for the zeta function of a general Shimura variety requires the theory of endoscopy even to state. Moreover, the proof of these reciprocity laws requires a comparison of the (automorphic) trace formula with an ((cid:1)-adic) Lefschetz trace formula. Some of the most striking parts of the argument are in the comparison of the various terms in the two formulas. The most sophisticated Shimura vari- etiesforwhichtherearecompleteresultsaretheso-calledPicardmodularsurfaces. (See [LR], especially the summary on pp. 255–302.) Picard modular surfaces are attached 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 a classification of their automorphic representations [R]. There is some discussion of these problems in the articlesof Arthur and Milne. However,thearticlesofbothArthurandMilnereallyareintendedasintroductions, despite their length. The theory of endoscopy, and the automorphic description of zeta 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. They include the invariant orbital integrals and irreducible characters that are the basis forHarish-Chandra’stheoryoflocalharmonicanalysis. Theyalsoincludeweighted orbital integrals and weighted characters, objects that arose for the first time with the trace formula. The article of Kottwitz is devoted to the general study of these terms at p-adic places. It is a largely self-contained course, which covers many of Harish-Chandra’s basic results in invariant harmonic analysis, as well as their weighted, 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 of the basic theorems of p-adic harmonic analysis, such as Howe’s finiteness theorem and Harish-Chandra’s local character expansion. The article also explains how ho- mogeneity enters into Waldspurger’s analysis of stability for linear combinations of nilpotent orbital integrals. There are subtle questions concerning the terms in the trace formula that go beyondthosetreatedbyKottwitzandDeBacker. Themostbasicoftheseisknown asthefundamentallemma,eventhoughitisstilllargelyconjectural.1 Thearticleby Hales contains a precise statement of the conjecture and some remarks on progress toward a general proof. The fundamental lemma occupies a unique place in the theory. It is a critical ingredient in the comparison of trace formulas that is part of the theory of endoscopy. It has an equally indispensable role in the comparison of (automorphic and (cid:1)-adic) trace formulas needed to establish reciprocity laws for Shimura varieties. SomeShimuravarietiesareprojective,whichistosaythattheyarecompactas complexvarieties. TheycorrespondtoreductivegroupsoverQthatareanisotropic. Thetraceformulainthiscasesimplifiesconsiderably. ItreducestotheSelbergtrace formula for compact quotient. On the other hand, the arithmeticgeometry of such varieties is still very rich. In particular, the comparison of individual terms in the 1Moreover,thetermlemma isultimatelyagrossunderstatement. PREFACE ix two kinds of trace formulas is of major interest. There is a great deal left to be done, but it is in this case that there has been the most progress. If the Shimura variety is not projective, the comparison is more sophisticated. IthastobebasedontherelationshipbetweenL2-cohomologyandintersectioncoho- mology,conjecturedbyZucker,andestablishedbySaperandStern,andLooijenga. The article of Goresky describes several compactifications of open Shimura vari- eties and their relations with associated cohomology groups. Goresky’s article also serves as an introduction to work of Goresky and MacPherson, in which weighted cohomology complexes on the reductive Borel-Serre compactification are used to obtaina Lefschetz formula for the intersectioncohomologyof the Baily-Borel com- pactification. According to Zucker’s conjecture, this last formula is equivalent to the relevant form of the automorphic trace formula. There remains the important open problem of establishing a corresponding (cid:1)-adic Lefschetz formula that can be compared with either one of these two formulas. ThereciprocitylawsprovedforPicardmodularsurfacesin[LR]applytoplaces ofgoodreduction. Thesamerestrictionhasbeenimplicitinourdiscussionofother Shimura varieties. In the final analysis, one would like to establish reciprocity laws between (cid:1)-adic Galois representations and automorphic representations that apply to all places. The theory of Shimura varieties at places of bad reduction is considerablylessdeveloped,althoughtherehascertainlybeenprogress. Thearticle of Haines is a survey of recent work in this direction, concentrating on the case of level structures of parahoric type. It also touches upon the problem of comparing the automorphic trace formula with the Lefschetz formula, now in the context of bad reduction. ThearticleofSarnakconcernstheclassicalRamanujanconjectureformodular forms and its higher dimensional analogues. Langlands has shown that the gen- eralized Ramanujan conjecture is a consequence of the principle of functoriality. Conversely, it is possible that the generalized Ramanujan conjecture could play a critical role in the study of those cases of functoriality that are not part of the the- oryofendoscopy. Sarnakdescribesthepresentstateoftheconjectureanddiscusses various techniques that have been successfully applied to special cases. Wehavetriedtopresentthecontentsofthemonographfromaunifiedperspec- tive. Ourdescriptionhasbeencenteredaroundtwofundamentalproblemsthatare the essential expression of the Langlands program. The two problems ought to be treated as signposts, which give direction to current work, but which point to destinationsthatwillnotbereachedintheforeseeablefuture. Thereaderisfreeto drawwhateverinspirationfromthemhisorhertemperamentpermits. Inanycase, many of the questions discussed in the various articles here are of great interest in their own right. In point of fact, there is probably too much in the monograph for anyone to learn in a limited period of time. Perhaps the best strategy for a beginner would be to start with one or two articles of special interest, and try to master them. As we have mentioned, participants were encouraged to bring a prior under- standingofthebasicpropertiesofalgebraicgroups. Thetheoryofreductivegroups isrootedinthestructureofcomplexsemisimpleLiealgebras,forwhich[Se]and[H] aregoodreferences. Asfor algebraicgroupsthemselves, a familiaritywithmanyof the topics in [B] or [Sp] is certainly desirable, though perhaps not essential.