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Progress in Mathematics Volume258 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein Eisenstein Series and Applications Wee Teck Gan Stephen S. Kudla Yuri Tschinkel Editors Birkhaüser Boston • Basel • Berlin WeeTeckGan StephenS.Kudla DepartmentofMathematics DepartmentofMathematics UniversityofCalifornia,SanDiego UniversityofToronto 9500GilmanDrive 40St.GeorgeStreet LaJolla,CA92093 Toronto,OntarioM5S2E4 U.S.A. Canada [email protected] [email protected] YuriTschinkel CourantInstituteofMathematicalSciences NewYorkUniversity 251MercerStreet NewYork,NY10012 U.S.A. [email protected] ISBN-13:978-0-8176-4496-3 e-ISBN-13:978-0-8176–4639-4 DOI:10.1007/978-0-8176-4639-4 LibraryofCongressControlNumber:2007937323 MathematicsSubjectClassification(2000):11F70,22E55,11F67,32N15 (cid:1)c2008BirkhaüserBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMediaLLC,233SpringStreet,NewYork, NY10013,USA)andtheauthor,exceptforbriefexcerptsinconnectionwithreviewsorscholarly analysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedis forbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 www.birkhauser.com To Robert Langlands, on the occasion of his seventieth birthday Preface The theory of Eisenstein series, in the general form given to it by Robert Langlands some forty years ago,has been an important and incredibly useful tool in the fields of automorphicforms, representationtheory,number theory andarithmeticgeometry.Forexample,thetheoryofautomorphicL-functions arises out of the calculation of the constant terms of Eisenstein series along parabolicsubgroups.Notsurprisingly,thetwoprimaryapproachestotheana- lytic properties of automorphic L-functions, namely the Langlands–Shahidi methodandtheRankin–Selbergmethod,bothrelyonthetheoryofEisenstein series. In representation theory, Eisenstein series were originally studied by Langlands in order to give the spectral decomposition of the space of L2- functions oflocally symmetricspacesattachedto adelic groups.This spectral theory has been used to prove the unitarity of certain local representations. Finally,onthemorearithmeticside,theFouriercoefficientsofEisensteinseries containawealthofarithmeticinformationwhichisfarfrombeingcompletely understood.Thep-divisibilitypropertiesofthesecoefficients,forexample,are instrumental in the construction of p-adic L-functions. In short, the theory of Eisenstein series seems to have, hidden within it, an inexhaustible number of treasures waiting to be discovered and mined. With such diverse applications, it is not easy even for the conscientious researcher to keep abreast of current developments. Indeed, different users of Eisenstein series often focus on different aspects of the theory. With this in mind, the workshop “Eisenstein Series and Applications” was held at the American Institute of Mathematics (Palo Alto) from August 15 to 19, 2005. ThegoaloftheworkshopwastobringtogetherusersofEisensteinseriesfrom different areas who do not normally interact with each other, with the hope that such a juxtaposition of perspectives would provide deeper insight into the arithmetic of Eisenstein series and foster fruitful new collaborations. This volume contains a collection of articles related to the theme of the workshop.Some,butnotallofthem,arebasedonlecturesgiveninthe workshop. We hope that the articles assembled here will be useful to a diverse audience and especially to students who are just entering the field. viii Preface Wewouldliketo takethis opportunitytothank alltheparticipantsofthe workshopfortheirenthusiasticparticipation,andtheauthorswhocontributed articles to this volume for their efforts and timely submissions, as well as, all the referees who gave the articles their thoughtful considerations. We are grateful to the American Institute of Mathematics and the National Science Foundation for providing generous support, and especially to Brian Conrey, David Farmer and Helen Moore of AIM for their invaluable assistance in the organizationof the workshop. We find it appropriate to dedicate this volume to Robert Langlands, who started it all, on the occasion of his seventieth birthday. New York August 2007 Wee Teck Gan, Stephen Kudla and Yuri Tschinkel Contents Preface ........................................................ vii Twisted Weyl Group Multiple Dirichlet Series: The Stable Case Ben Brubaker, Daniel Bump, and Solomon Friedberg ................. 1 A Topological Model for Some Summand of the Eisenstein Cohomology of Congruence Subgroups Jens Franke ..................................................... 27 The Saito–Kurokawa Space of PGSp and Its Transfer to 4 Inner Forms Wee Teck Gan................................................... 87 Values of Archimedean Zeta Integrals for Unitary Groups Paul Garrett ....................................................125 A Simple Proof of Rationality of Siegel–Weil Eisenstein Series Michael Harris ..................................................149 Residues of Eisenstein Series and Related Problems Dihua Jiang.....................................................187 Some Extensions of the Siegel–Weil Formula Stephen S. Kudla ................................................205 A Remark on Eisenstein Series Erez M. Lapid ...................................................239 Arithmetic Aspects of the Theta Correspondence and Periods of Modular Forms Kartik Prasanna .................................................251 x Contents Functoriality and Special Values of L-Functions A. Raghuram and Freydoon Shahidi ................................271 Bounds for Matrix Coefficients and Arithmetic Applications Ramin Takloo-Bighash............................................295 Twisted Weyl Group Multiple Dirichlet Series: The Stable Case Ben Brubaker1, Daniel Bump2, and Solomon Friedberg3 1 Department of Mathematics, MIT, Cambridge, MA 02139-4307, U.S.A. 2 Department of Mathematics, Stanford University,Stanford, CA 94305-2125, U.S.A. 3 Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, U.S.A. Summary. Weyl group multiple Dirichlet series were associated with a root sys- temΦandanumberfieldF containingthen-throotsofunitybyBrubaker,Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflectthecombinatorics of theroot system.Conjecturally,thesefunctionscoincide with Whittakercoefficients of metaplectic Eisenstein series, buttheyare studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. “Twisted”DirichetserieswereintroducedinBrubaker,Bump,Friedberg,andHoff- stein [5] without the stability assumption, but only for root systems of type Ar. Theirdescriptionisgivendifferently,intermsofGausssumsassociatedtoGelfand– Tsetlin patterns.Inthispaper,wereimposethestability assumptionandstudythe twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ=Ar we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for thestable case. 1 Introduction ThepresentpapercontinuesthestudyofWeylgroupmultipleDirichletseries: familiesofDirichletseriesinseveralcomplexvariablesassociatedtoareduced rootsystemΦ, a positive integer n,and a number field F containing the n-th roots of unity. These series take the form (cid:1) Z (s,m)= HΨ(c;m)N(c)−2s (1) Ψ c where the ingredients in the above formula are defined as follows. Letting r denotetherankoftherootsystemΦ,s=(s ,...,s )∈Cr,m=(m ,...,m ) 1 r 1 r