Table Of ContentProgress in Mathematics
Volume258
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Eisenstein Series
and Applications
Wee Teck Gan
Stephen S. Kudla
Yuri Tschinkel
Editors
Birkha¨user
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
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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 work-
shop. 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
