L-Functions and Automorphic Forms: LAF, Heidelberg, February 22-26, 2016

367 Pages·2017·5.131 MB·English

by  Jan Hendrik Bruinier,Winfried Kohnen (eds.)

Contributions in Mathematical and Computational
Sciences 10
Jan Hendrik Bruinier
Winfried Kohnen Editors
L-Functions and
Automorphic
Forms
LAF, Heidelberg, February 22-26, 2016
Contributions in Mathematical and
Computational Sciences
Volume 10
Serieseditors
HansGeorgBock
WilliJäger
HansKnüpfer
OtmarVenjakob
Moreinformationaboutthisseriesathttp://www.springer.com/series/8861
Jan Hendrik Bruinier (cid:129) Winfried Kohnen
Editors
L-Functions and
Automorphic Forms
LAF, Heidelberg, February 22-26, 2016
123
Editors
JanHendrikBruinier WinfriedKohnen
FachbereichMathematik MathematischesInstitut
TechnischeUniversitätDarmstadt UniversitätHeidelberg
Darmstadt,Germany Heidelberg,Germany
ISSN2191-303X ISSN2191-3048 (electronic)
ContributionsinMathematicalandComputationalSciences
ISBN978-3-319-69711-6 ISBN978-3-319-69712-3 (eBook)
https://doi.org/10.1007/978-3-319-69712-3
LibraryofCongressControlNumber:2017961817
MathematicsSubjectClassification(2010):11-XX,11MXX,11FXX,11GXX,14CXX,14GXX,
32MXX,32NXX
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Preface
This volume presents the proceedings of the conference L-Functions and Auto-
morphic Forms that took place from February 22 to 26, 2016 at the University
of Heidelberg, Germany. The theory of automorphic forms and their associated
L-functionsis one of the central research areas in modern number theory,linking
numbertheory,arithmeticalgebraic geometry,representationtheory,and complex
analysis in many profound ways. It is an area of enduring interest to a wide
class of mathematicians. The present volume contains carefully refereed articles
byleadingexpertsin the field,givingnewand originalresults. Thetopicsinclude
automorphicL-functionsandtheirspecialvalues,p-adicmodularforms,Eisenstein
series,Borcherdsproducts,automorphicperiods,andmanymore.
In addition,four preparatorycoursestook place in the week before the confer-
ence, with the aim to introduce PhD students to basic and advanced concepts of
the theory of automorphicforms. Lecture notes of three of those courses are also
included.
The conference took place in the Internationales Wissenschaftsforum Heidel-
berg,whichisacenterforscholarlyexchangeinallareasofscienceandacademic
researchlocated in the old town of Heidelberg.We are gratefulfor the hospitality
andthatwehadtheopportunitytousethisexcellentvenue.
WethankClaudiaAlfes-NeumannandEricHofmannfortheirhelpinpreparing
theconferenceprogram,thewebpage,andthepreparatorycourses.Specialthanks
are also due to Nicole Umlas and David Obermayr for their support in preparing
and running the conference. We are grateful for the generous financial support
from the DFG-Forschergruppe 1920 Heidelberg/Darmstadt Symmetry, Geometry,
andArithmeticandtheMathematicsCenterHeidelberg(MATCH).Finally,wewish
toextendoursincerethankstoallcontributorstothisvolumeandallspeakersofthe
conference.
Darmstadt,Germany JanHendrikBruinier
Heidelberg,Germany WinfriedKohnen
December2017
v
vi Preface
Grouppictureoftheconference
Contents
Sturm-LikeBoundforSquare-FreeFourierCoefficients ................... 1
PramathAnambyandSoumyaDas
ImagesofMaass-PoincaréSeriesintheLowerHalf-Plane.................. 9
NickolasAndersen,KathrinBringmann,andLarryRolen
OnDenominatorsofValuesofCertainL-FunctionsWhenTwisted
byCharacters..................................................................... 25
SiegfriedBöcherer
FirstOrderp-AdicDeformationsofWeightOneNewforms................ 39
HenriDarmon,AlanLauder,andVictorRotger
ComputingInvariantsoftheWeilRepresentation........................... 81
StephanEhlenandNils-PeterSkoruppa
The Metaplectic Tensor Product asan Instance of Langlands
Functoriality...................................................................... 97
WeeTeckGan
OnScatteringConstantsofCongruenceSubgroups......................... 115
MiguelGradosandAnna-MariavonPippich
The Bruinier–Funke PairingandtheOrthogonalComplement
ofUnaryThetaFunctions....................................................... 139
BenKaneandSiuHangMan
Bounds for Fourier-JacobiCoefficientsof Siegel Cusp Forms
ofDegreeTwo..................................................................... 159
WinfriedKohnenandJyotiSengupta
HarmonicEisensteinSeriesofWeightOne ................................... 171
YingkunLi
vii
viii Contents
ANoteontheGrowthofNearlyHolomorphicVector-ValuedSiegel
ModularForms................................................................... 185
AmeyaPitale,AbhishekSaha,andRalfSchmidt
CriticalValues of L-Functions for GL (cid:2)GL over a Totally
3 1
RealField ......................................................................... 195
A.RaghuramandGunjaSachdeva
IndecomposableHarish-ChandraModulesforJacobiGroups............. 231
MartinRaum
MultiplicityOneforCertainParamodularFormsofGenusTwo .......... 251
MirkoRösnerandRainerWeissauer
RestrictionofHeckeEigenformstoHorocycles .............................. 265
HoChungSiuandKannanSoundararajan
OntheTripleProductFormula:RealLocalCalculations................... 275
MichaelWoodbury
AnIntroductiontotheTheoryofHarmonicMaassForms ................. 299
ClaudiaAlfes-Neumann
ElementaryIntroductiontop-AdicSiegelModularForms ................. 317
SiegfriedBöcherer
LiftingsandBorcherdsProducts............................................... 333
EricHofmann
Sturm-Like Bound for Square-Free
Fourier Coefficients
PramathAnambyandSoumyaDas
Abstract Inthisshortarticle,weshowtheexistenceofananalogueoftheclassical
Sturm’sboundinthecontextofthesquare-freeFouriercoefficientsforcuspforms
of square-free levels. This number is a cut-off to determine a cusp form from its
initialfewsquare-freeFouriercoefficients.Wealsomentionsomequestionsinthis
regard.
1 Introduction
Thetheoryofmodularformsbynowoccupiesacentralplaceinnumbertheory,and
its wide ranging applications in various branches of mathematics is well known.
Onepleasant,andcomputationallyimportantfeatureofthese objectsisthatiff is
such a form in Mk.(cid:2)/ ((cid:2) (cid:3) SL2.Z/ is a congruencesubgroupand k (cid:4) 0) with a
Fourierexpansion,say
X1
f.(cid:3)/D a.f;n/e2(cid:4)in(cid:3); .(cid:3) 2H Dfz2Cj=.z/>0g/
nD0
(1.1)
thenthereexistsanumberA > 0dependingonthespacesuchthatifa.f;n/ D 0
foralln (cid:5) A,thenf D 0.ThesmallestsuchboundingeneralisknownasSturm’s
boundintheliterature.Letusdenoteitby(cid:5).k;(cid:2)/andrecallthat
(cid:5).k;(cid:2)/WD kŒSL2.Z/W(cid:2)(cid:6): (1.2)
12
InfactSturm’sboundisknownforvariouskindsofmodularforms,e.g.,half-integer
weightforms,Siegelmodularformsetc.Inthispaperwewilldiscussthefollowing
P.Anamby(cid:129)S.Das((cid:2))
DepartmentofMathematics,IndianInstituteofScience,Bangalore560012,India
e-mail:[email protected];[email protected];[email protected];
[email protected]
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J.H.Bruinier,W.Kohnen(eds.),L-FunctionsandAutomorphicForms,
ContributionsinMathematicalandComputationalSciences10,
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