Siegel Modular Forms: A Classical and Representation-Theoretic Approach

142 Pages·2019·2.093 MB·English

by  Ameya Pitale

Lecture Notes in Mathematics 2240
Ameya Pitale
Siegel
Modular
Forms
A Classical and Representation-Theoretic
Approach
Lecture Notes in Mathematics
Volume 2240
Editor-in-Chiefs
Jean-MichelMorel,CMLA,ÉcoleNormaleSupérieuredeCachan,Cachan,France
Bernard Teissier, Equipe Géométrie et Dynamique, Institut Mathématique de
Jussieu-Paris Rive Gauche, Paris, France
Advisory Editor
Michel Brion, Institut Fourier, Université Grenoble Alpes Institut Fourier,
Grenoble, France
Camillo De Lellis, Institute for Advanced Study, Princeton, NJ, USA
AlessioFigalli,DepartmentofMathematics,SwissFederalInstituteofTechnology,
Zurich, Switzerland
Davar Khoshnevisan, Department of Mathematics, The University of Utah, Salt
Lake City, UT, USA
Ioannis Kontoyiannis, Department of Engineering, University of Cambridge,
Cambridge, UK
GáborLugosi,DepartmentofEconomicsandBusiness,UniversitatPompeuFabra,
Barcelona, Spain
MarkPodolskij,DepartmentofMathematics,AarhusUniversity,Aarhus,Denmark
Sylvia Serfaty, Courant Institute of Mathematics, New York University Courant
Institute of Mathematics, New York, NY, USA
Anna Wienhard, Mathematisches Institut, Universität Heidelberg, Heidelberg,
Germany
More information about this series at http://www.springer.com/series/304
Ameya Pitale
Siegel Modular Forms
A Classical and Representation-Theoretic
Approach
123
Ameya Pitale
Mathematics
University of Oklahoma
Norman, OK, USA
ISSN 0075-8434 ISSN 1617-9692 (electronic)
Lecture Notesin Mathematics
ISBN978-3-030-15674-9 ISBN978-3-030-15675-6 (eBook)
https://doi.org/10.1007/978-3-030-15675-6
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Contents
1 Introduction to Siegel Modular Forms . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Siegel Upper Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Siegel Modular Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Theta Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Saito–Kurokawa Lifts. . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Congruence Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Congruence Subgroups in Genus 2 . . . . . . . . . . . . . . . . 12
2.2.2 Siegel Modular Forms with Level. . . . . . . . . . . . . . . . . 13
3 Hecke Theory and L-Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 The Hecke Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Action of the Hecke Algebra on Siegel Modular Forms . . . . . . 17
3.3 Relation Between Fourier Coefficients and Hecke
Eigenvalues for Genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4.1 L-Function for Saito–Kurokawa Lifts . . . . . . . . . . . . . . 22
4 Nonvanishing of Fourier Coefficients and Applications. . . . . . . . . . 23
4.1 Generalized Ramanujan Conjecture . . . . . . . . . . . . . . . . . . . . . 23
4.2 Nonvanishing of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . 24
4.3 Application of the Nonvanishing Result. . . . . . . . . . . . . . . . . . 27
4.4 Böcherer’s Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Applications of Properties of L-Functions. . . . . . . . . . . . . . . . . . . . 31
5.1 Determining Cusp Forms by Size of Fourier Coefficients . . . . . 31
5.2 Sign Changes of Hecke Eigenvalues . . . . . . . . . . . . . . . . . . . . 33
5.3 Sign Changes of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . 35
v
vi Contents
6 Cuspidal Automorphic Representations Corresponding
to Siegel Modular Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1 Classical to Adelic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Hecke Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Satake Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 Spherical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 The Representation Associated to a Siegel Cusp Form . . . . . . . 47
7 Local Representation Theory of GSp4 (Qp). . . . . . . . . . . . . . . . . . . 49
7.1 Local Non-archimedean Representations for GSp . . . . . . . . . . 49
4
7.2 Generic Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.3 Iwahori-Spherical Representations . . . . . . . . . . . . . . . . . . . . . . 54
7.4 Paramodular Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 Bessel Models and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.1 The L-Function Lðs;…F (cid:2)¿Þ . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.2 Definition of Global Bessel Model. . . . . . . . . . . . . . . . . . . . . . 66
8.3 Local Bessel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.4 Explicit Formulas for Distinguished Vectors in Local
Bessel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
9 Analytic and Arithmetic Properties of GSp (cid:2) GL
4 2
L-Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.1 Functional Equation and Analytic Continuation . . . . . . . . . . . . 75
9.2 Transfer to GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4
9.3 Other Analytic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.4 Arithmetic Applications of the Integral Representation
of Lðs;…F (cid:2)¿Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10 Integral Representation of the Standard L-Function. . . . . . . . . . . . 83
10.1 The zeta integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 The Basic Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 The Local Integral Computation . . . . . . . . . . . . . . . . . . . . . . . 86
10.4 Global Integral Representation. . . . . . . . . . . . . . . . . . . . . . . . . 88
10.5 Classical Reformulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.6 Arithmetic Results in Genus 2. . . . . . . . . . . . . . . . . . . . . . . . . 89
Appendix A: GL Notes: Classical Modular Forms.... .... ..... .... 93
2
Appendix B: The p-Adic Fields Qp and the Ring of Adeles A of Q .... 97
Appendix C: GL Notes: Representation Theory.. .... .... ..... .... 99
2
Appendix D: Solutions to Exercises. .... .... .... .... .... ..... .... 103
References.... .... .... .... ..... .... .... .... .... .... ..... .... 131
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 137
Introduction
The theory of modular forms is an important topic of research in number theory.
The modular forms are the centerpiece of several recent accomplishments such as
the proof of the Shimura–Taniyama–Weil conjecture due to Andrew Wiles and
others,whichledtotheresolutionofFermat’sLastTheorem.Siegelmodularforms
arethehigherdimensionalanaloguesofmodularformsandarethesubjectofthese
lecturenotes.Siegelmodularformshaveseentremendoussuccessrecentlyinboth
analytic and arithmetic aspects, with results toward Langlands functoriality and
Deligne’s conjectures.
These lecture notes are based on my workshop on Siegel modular forms at
IISER, Pune, India from August 8 to 18, 2017. The target audience for the
workshop, andfor thesenotes,aregraduatestudents andyoungresearchers. These
notes could also be used by senior researchers as reference material.
The main objective of the workshop was to introduce classical and representa-
tiontheoretictechniquesinmodularformsandtoexplicatetheinterconnectionsvia
current research and open problems. I believe that the classical and representation
theoreticmethodsareatwo-waystreetanditisabsolutelyessential for researchers
to gain expertise in going from one side to the other and back.
For this purpose, I decided to focus the workshop, and these lecture notes, on
Siegel modular forms. These are higher dimensional analogues of elliptic modular
forms and there is a long history of active research on this topic. The main
advantage is that one can approach Siegel modular forms from a purely classical
pointofviewasholomorphicfunctionsontheSiegelupperhalf-space.Inaddition,
these can be studied in the context of local and global representation theory of the
symplectic group.
ThesubjectofSiegelmodularformsisvastanditisimpossibletocoverallofit
in a 2-week workshop. This is my disclaimer that these notes are not comprehen-
sive. I have tried to give a detailed description of the basics of the classical theory
and representation theory of Siegel modular forms. But beyond the basics, my
emphasisismore onhow tousetheconceptstosolveseveralinterestingproblems
and to give an indication about the current open questions in the subject. I have
triedtoprovidereferencesforanythingthatisnotprovedinthelecturenotessothat
vii
viii Introduction
thereadercanaccessdetails.Ihopethatthiswillopencertainlockeddoorsforthe
readers,andgetthemexcitedenoughtopursuelearningorresearchingontheseand
related topics.
Some of the topics that have been omitted are the Galois representations asso-
ciated to Siegel modular forms, the paramodular conjecture, and vector-valued
Siegelmodularforms.ThefirsttwoomissionsarebychoicesinceIdonotworkon
problemsrelatedtothem,whilethelastoneisjusttosimplifythingsbyrestricting
to scalar-valued Siegel modular forms. Each of these aspects of Siegel modular
forms could be one or more chapters by themselves. Instead, I am just going to
provide some references to get the interested readers started on them. For Galois
representationslookat[57]and[105],fortheparamodularconjecturesee[16]and
for vector-valued Siegel modular forms refer to [71].
There are several good texts which explain details of the classical theory of
Siegel modular forms (see [4, 28, 46, 59]) and some research articles that provide
detailsoftherepresentationtheoryofthesymplecticgroup([6,81]).Thereisagap
in the literature when it comes to a good text or reference article giving the nec-
essaryin-depthinformationonthetwoapproachestoSiegelmodularformsaswell
as the interconnections between the two. I hope that these notes will fill that gap.
AnimportantfeatureofthenotesisthatIhavetriedtoprovideseveralexercises.
Theseplayadualrole.Theyallowmetointroducecertainresultswithouthavingto
gothroughtheirproofs.Butmoreimportantly,anyonewantingtoreallyunderstand
the material can only do so if they get their hands dirty doing these exercises. For
the convenience of the reader, I have provided solutions (or hints) for all the
exercises in an appendix.
In an ideal situation, the reader will already be familiar with the GLð2Þ theory.
ThiswillhelpinrealizinghowthetheoryandmethodsforSiegelmodularformsare
often a generalization of the elliptic modular forms theory. Nevertheless, I have
includedthreeappendicesrecallingtheclassicalconceptsandrepresentationtheory
for GLð2Þ and basics of p-adic numbers and the ring of adeles.
Thenotesareorganizedasfollows.Chapters1–3introducebasicinformationon
theclassical theoryofSiegel modularforms. Chapters4 and5introduceadvanced
topics,openconjecturesandrecentresultsthatusethetechniquesintroducedinthe
preceding Chapters. Chapter 6 is the transition from the classical to the represen-
tation theoretic. Chapter 7 provides much of the basic local representation theory.
Chapters 8–10 present current research and introduce the various techniques and
concepts required to understand them.
Acknowledgements:IwouldliketothankeveryoneatIISER,Pune,andespecially
BaskarBalasubramanyam,fortheirhospitalityandseamlessorganization.Iamalso
grateful to the participants of the workshop who spent close to 30 hours with me
over a period of 10 days, and countless many hours working on the exercises by
themselves. Their feedback was crucial in correcting mistakes and typos in earlier
versionsofthenotes.Iwouldalsoliketomentionthattheworkshopwaspartofthe
Global Initiative of Academic Networks (GIAN) proposal of the government of
India.
Introduction ix
IamthankfulforallthehelpandguidanceprovidedbyRalfSchmidtinmaking
sure that the lecture notes are accurate. I am also grateful to the referees whose
comments and suggestions were invaluable.
Finally, I want to thank my family—Swapna, Aayush, and Samay—for the
support and encouragement that makes all of it possible. These lecture notes are
dedicated to them.