Modular forms for the orthogonal group O(2,5) [Elektronische Ressource] / vorgelegt von Ingo Herbert Klöcker
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Modular forms for the orthogonal group O(2,5) [Elektronische Ressource] / vorgelegt von Ingo Herbert Klöcker

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ModularFormsfortheOrthogonalGroupO(2,5)VonderFakultätfürMathematik,InformatikundNaturwissenschaftenderRheinisch WestfälischenTechnischenHochschuleAachenzurErlangungdesakademischenGradeseinesDoktorsderNaturwissenschaftengenehmigteDissertationvorgelegtvonDiplom MathematikerIngoHerbertKlöckerausAachenBerichter: Univ. Prof. Dr.rer.nat. AloysKriegUniv. Prof. Dr.rer.nat. Nils PeterSkoruppaTagdermündlichenPrüfung: 30.November2005DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfügbar.ForOdo,Sandra,andMajaContentsIntroduction 10. BasicNotation 51. OrthogonalGroups 71.1. Latticesandorthogonalgroups . . . . . . . . . . . . . . . . . . . . . . . . 71.2. O(2,l+2)andtheattachedhalf space . . . . . . . . . . . . . . . . . . . . 91.3. Theorthogonalmodulargroup . . . . . . . . . . . . . . . . . . . . . . . . 101.4. Generatorsofcertainorthogonalmodulargroups . . . . . . . . . . . . . . 131.5. Thecommutatorsubgroupsofcertainorthogonalmodulargroups . . . . . . 211.6. Abeliancharactersoftheorthogonalmodulargroups . . . . . . . . . . . . 221.6.1. Thedeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.2. Theorthogonalcharacter(s) . . . . . . . . . . . . . . . . . . . . . 221.6.3. TheSiegelcharacter . . . . . . . . . . . . . . . . . . . . . . . . . 221.7. Parabolicsubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262. ModularForms 292.1. Orthogonalmodularforms . . . . . . . . . . . . . . . . . . . . . .

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Informations

Publié par
Publié le 01 janvier 2005
Nombre de lectures 37

Extrait

OrModularthogonalGrFormsoupforO(2,5)the

VonderFakultätfürMathematik,InformatikundNaturwissenschaften
derRheinisch-WestfälischenTechnischenHochschuleAachen
zurErlangungdesakademischenGradeseines
NaturwissenschaftenderDoktorsDissertationgenehmigte

vorgelegtvon

erDiplom-MathematikerKlöckHerbertIngoAachenaus

Berichter:UniUnivv.-Prof..-Prof.DrDr..rerrer..nat.nat.AloysNils-PeterKriegSkoruppa

TagdermündlichenPrüfung:30.November2005

DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfügbar.

orF

Odo,

Sandra,

and

Maja

Contents

oductionIntr

NotationBasic0.

oupsGrthogonalOr1.1.1.Latticesandorthogonalgroups........................
1.2.O(2,l+2)andtheattachedhalf-space....................
1.3.Theorthogonalmodulargroup........................
1.4.Generatorsofcertainorthogonalmodulargroups..............
1.5.Thecommutatorsubgroupsofcertainorthogonalmodulargroups......
1.6.Abeliancharactersoftheorthogonalmodulargroups............
1.6.1.Thedeterminant...........................
1.6.2.Theorthogonalcharacter(s).....................
1.6.3.TheSiegelcharacter.........................
1.7.Parabolicsubgroups..............................

FormsModular2.2.1.Orthogonalmodularforms..........................
2.2.Rankin-Cohentypedifferentialoperators...................
2.3.Jacobiforms..................................
2.4.Maaßspaces..................................
2.5.Restrictionsofmodularformstosubmanifolds................
2.5.1.Thegeneralcase...........................
2.5.2.RestrictionsofmodularformslivingonH............
D42.5.3.RestrictionsofmodularformslivingonH............
A32.5.4.RestrictionsofmodularformslivingonH(3)............
A12.6.Hermitianmodularformsofdegree2.....................
2.7.Quaternionicmodularformsofdegree2...................
2.8.Quaternionicthetaseries...........................

3.Vector-valuedModularForms
3.1.Themetaplecticgroup............................
3.2.Vector-valuedmodularforms.........................
3.3.TheWeilrepresentation............................
3.4.Adimensionformula.............................

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29293538424545495051535760

6565666870

vi

Contents

3.5.Examplesofvector-valuedmodularforms..................73
3.5.1.Eisensteinseries...........................73
3.5.2.Thetaseries..............................74

4.BorcherdsProducts77
4.1.WeylchambersandtheWeylvector.....................77
4.2.Quadraticdivisors...............................83
4.3.Borcherdsproducts..............................84
4.3.1.BorcherdsproductsforS=A...................87
3(3)4.3.2.BorcherdsproductsforS=A...................89
15.GradedRingsofOrthogonalModularForms91
5.1.ThegradedringforS=A3..........................91
(3)5.2.ThegradedringforS=A.........................98
1A.OrthogonalandSymplecticTransformations107
A.1.ThecaseS=D4...............................108
(3)A.2.ThecaseS=A...............................109
1A.3.ThecaseS=A...............................110
3B.OrthogonalandUnitaryTransformations111
(2)B.1.ThecaseS=A...............................112
1B.2.ThecaseS=A...............................113
2B.3.ThecaseS=S...............................114
2C.EichlerTransformations115

.DoupsGrDiscriminant

E.DimensionsofSpacesofVector-valuedModularForms

yliographBib

Notation

xInde

117

119

121

125

133

oductionIntr

Wtheecaseconsiderl=3.modularModularformsformsforforO(orthogonal2,3)groupscorrespondO(2,lto+Sie2)gelwithmodularparticularformsofemphasisdegreeon
2.Inthe1960sIgusa[Ig64]usedthetaconstantsinordertodescribethegradedringof
Sieminegelthemodulargradedringformsofofdesymmetricgree2.UsingHermitianIgusasmodularmethodformsFreitagofdegree[Fr67]2wovaserablethetoGaussiandeter-
√numberfieldQ(−1)whichcorrespondstothecaseofmodularformsforO(2,4).Na-
gaoka[Na96],Ibukiyama[Ib99b]andAoki[AI05]completedthedescriptionthegraded
O(ring2,in4)termswhereofdealtgeneratorswithbyandDernrelations.andKrieOtherg.ThecasesydeterminedcorrespondingthetogradedmodularringsformsofHerfor-
Q(mitian√−1),modularQ(√−2)formsandofQde(√−gree3)2(cf.including[De01],the[DK03],Abelian[DK04]).charactersInsteadfortheofusingnumberestima-fields
(cf.tionson[Bo98])thetainseriesorderastoinobtainIgusasHermitianapproachthemodularyappliedformsthewithknotheorywnofzeros.BorcherdsThenaproductssimilar
generalreductioncaseofprocessmasodulartheoneformsusedforbyO(2,Igusal+2)andwasFreitagstudiedyieldsbyFrtheireitagandstructureHermanntheorems.[FH00]The
fromageometricalpointofview.TheyderivedpartialresultsonmodularformsforO(2,5)
byembeddingsuitablelatticesintotheHurwitzquaternions.
UsingsimilarmethodsasDernandKriegwewillcharacterizethegradedringsofor-
positithogonalvedefinitemodularformssymmetricfortwmatrixoofmaximalrankl,discreteandletsubgroupsofO(2,5).LetSbeaneven
S0:=00−0S01,S1:=00S0001.
100100
ThecorrespondingbilinearformquadraticassociatedformistoqS0=is1(gi∙,v∙)en.byThe(a,b)0attached=taS0bhalf-spacefora,isb∈Rl+2andthe
002HS={w=u+iv∈Cl+2;v∈PS},
wherePS={v∈Rl+2;(v,v)0>0,(v,e)>0},e=t(1,0,...,0,1).Theorthogonal
groupO(S1;R)={M∈Mat(l+4;R);tMS1M=S1}
actsonHS∪(−HS)asgroupofbiholomorphicrationaltransformationsvia
tw→Mw=(−q0(w)b+Aw+c)j(M,w)−1forM=γbαtAdaδcβ∈O(S1;R),

2

Introduction

wherej(M,w)=−γq0(w)+tdw+δ.Theorthogonalmodulargroupisgivenby
ΓS={M∈O(S1;R);MHS=HS,MΛ1=Λ1}.
Anorthogonalmodularformofweightk∈ZwithrespecttoanAbeliancharacterνofΓS
isaholomorphicfunctionf:HS→Csatisfying
(f|kM)(w):=j(M,w)−kf(Mw)=ν(M)f(w)forallw∈HS,M∈ΓS.
Thevectorspace[ΓS,k,ν]ofthosefunctionsisfinitedimensional.Iffj∈[ΓS,kj,νj],
j=1,2,thenf1f2∈[ΓS,k1+k2,ν1ν2].Thus
A(ΓS)=[ΓS,k,1]andA(ΓS)=[ΓS,k,ν],
k∈Zk∈Zν∈ΓSab
whereΓSisthecommutatorsubgroupofΓSandΓSabisthegroupofAbeliancharactersof
ΓS,formgradedrings.Ourmaingoalistheexplicitdescriptionofthosegradedringsin
forgeneratorsoftermsS=A3=012121210andS=A1(3)=002020200.
ItturnsoutthatinbothcasesthegradedringA(ΓS)isapolynomialringinsix(alge-
braicallyindependent)generatorswhilethegradedringsA(ΓS)arefreeR-modulesof
rank2and4,respectively,whereinbothcasesRisanextensionofdegreetwoofA(ΓS).
InthecaseofS=A3wecansimplytakecertainEisensteinseriesandBorcherdsproducts
asgenerators.Intheothercasewedeterminetheinvariantringofafiniterepresentation
whichisgivenbytheactionofasubgroupofthequaternionicsymplecticgrouponquater-
nionicthetaseries.TherestrictionsoftheprimaryinvariantsandsomeBorcherdsproducts
generatethegradedringsforS=A1(3).InbothcasesBorcherdsproductsplayanimpor-
tantrole.InafirststeptheexplicitlyknownzerosoftheBorcherdsproductsallowusto
reducetheproblemofdeterminingthegradedringA(ΓS)ofmodularformswithAbelian
characterstotheproblemofdeterminingthegradedringA(ΓS)ofmodularformsofeven
weightwithrespecttothetrivialcharacter.Inthenextstepweusethefactthatwealready
knowthegeneratorsofthegradedringsofmodularformslivingoncertainsubmanifolds
ofHSonwhichsuitableBorcherdsproductsvanishoffirstorder.InthecaseofS=A3we
canderiveour√resultsfromDernsresultforQ(√−3),andinthecaseofS=A1(3)weuse
theresultsforQ(−1).Asanapplicationofourresultswedescribetheattachedfieldsof
functions.modularorthogonalWenowbrieydescribethecontentofthisthesis:
Inthefirstchapterwecollectthenecessaryfactsandresultsaboutorthogonalgroups.In
particular,weexplicitlydeterminegeneratorsandAbeliancharactersofcertainorthogonal
modulargroupsΓS,andweintroducetheparamodularsubgroupofΓS.
Inthesecondchapterwedefinethemainobjectofourstudies,theorthogonalmodu-

Introduction

3

larforms,andstatesomefundamentalresults.Inparticular,weshowthat,unlikeelliptic
modularforms,orthogonalmodularformsautomaticallypossessanabsolutelyandlocally
uniformlyconvergentFourierseriesduetoKoechersprinciple.Moreover,weintroduce
thenotionofcuspformsandshowthat,asusual,thesubspaceofcuspformscanbechar-
acterizedbySiegelsΦ-operator.Thenweconsideracertaindifferentialoperatorwhich
allowsustoconstructnon-trivialorthogonalmodularformsfromanumberofalgebraically
independentorthogonalmodularforms.ThenexttwosectionsdealwithJacobiformsand
theMaaßspace.AnexplicitformulaforthedimensionofcertainMaaßspacesisderived
fromadimensionformulaforspacesofJacobiforms.Nextwetakealookatrestrictions
oforthogonalmodularformstosubmanifoldsandgiveabriefintroductionintoHermitian
andquaternionicmodularformsofdegree2.Wetranslatetheresultsaboutgradedringsof
Hermitianmodularformsofdegree2fromthesymplecticpointofviewtoourterminol-
ogy,andwedefineorthogonalEisensteinseriesforS=A3andS=A1(3)asrestrictionsof
quaternionicEisensteinseries.Finally,weconsidera5-dimensionalfiniterepresentationof
ΓA1(3),determineitsinvariantringusingtheMAGMAandgetfivealgebraicallyindependent
modularformsforΓA(3)whoserestrictionstoacertainsubmanifoldgeneratethegraded
ringoforthogonal1modularformsofevenweightandtrivialcharactercorrespondingto
HermitianmodularformsovertheGaussiannumberfield.
Inthethirdchapterwerecallfundamentalfactsaboutvector-valuedellipticmodular
formsforthemetaplecticgroupMp(2;Z).Wefocusonholomorphicvector-valuedmodu-
larformswithrespecttotheWeilrepresentationρSattachedtoacertainquadraticmodule
(Λ/Λ,qS)associatedtoS.Adimensionformulaforspacesofholomorphicvector-valued
modularformsisgiven,andtwoclassesofvector-valuedmodularformswhoseFourier
expansionscanbeexp

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