| Chapter1 Ellipticfunctions 1.1 Introduction 1.2 Doublyperiodicfunctions 1.3 Fundamentalpairsofperiods 1.4 Ellipticfunctions 1.5 Constructionofellipticfunctions 1.6 TheWeierstrassfunction 1.7 TheLaurentexpansionofganeartheorigin 1.8 Differentialequationsatisfiedbyξ 1.9 TheEisensteinseriesandtheinvariantsg2andg3 1.10 Thenumberse1，e2，e3 1.11 ThediscriminantA 1.12 KleinsmodularfunctionJ（τ） 1.13 InvarianceofJunderunimodulartransformations 1.14 TheFourierexpansionsofg2（τ）andg3（τ） 1.15 TheFourierexpansionsof△（τ）andJ（τ） ExercisesforChapter1 Chapter2 TheModulargroupandmodularfunctions 2.1 M6biustransformations 2.2 Themodulargroup 2.3 Fundamentalregions 2.4 Modularfunctions 2.5 Specialvaluesof 2.6 Modularfunctionsasrationalfunctionsof 2.7 Mappingpropertiesof 2.8 ApplicationtotheinversionproblemforEisensteinseries 2.9 ApplicationtoPicardstheorem ExercisesforChapter2 Chapter3 TheDedekindetafunction 3.1 Introduction 3.2 SiegeisproofofTheorem3.1 3.3 Infiniteproductrepresentationfor△（τ） 3.4 Thegeneralfunctionalequationforη（τ） 3.5 Isekistransformationformula 3.6 DeductionofDedekindsfunctionalequationfromIsekisformula 3.7 PropertiesofDedekindsums 3.8 ThereciprocitylawforDedekindsums 3.9 CongruencepropertiesofDedekindsums 3.1 0TheEisensteinseriesG2（τ） ExercisesforChapter3 Chapter4 Congruencesforthecoefficientsofthemodularfunctionj 4.1 Introduction 4.2 ThesubgroupFo（q） 4.3 FundamentalregionofFo（p） 4.4 FunctionsautomorphicunderthesubgroupFo（p） 4.5 ConstructionoffunctionsbelongingtoFo（p） 4.6 Thebehavioroffpunderthegeneratorsofг 4.7 Thefunction（τ）=△（qτ）/△（τ） 4.8 Theunivalentfunctionφ（τ） 4.9 Invarianceofφ（τ）undertransformationsofг0（q） 4.1 0Thefunctionjpexpressedasapolynomialinφ ExercisesforChapter4 Chapter5 Rademachersseriesforthepartitionfunction 5.1 Introduction 5.2 Theplanoftheproof 5.3 DedekindsfunctionalequationexpressedintermsofF 5.4 Fareyfractions 5.5 Fordcircles 5.6 Rademacherspathofintegration 5.7 Rademachersconvergentseriesforp（n） ExercisesforChapter5 Chapter6 Modularformswithmultiplicativecoefficients 6.1 Introduction 6.2 Modularformsofweightk 6.3 Theweightformulaforzerosofanentiremodularform 6.4 RepresentationofentireformsintermsofG4andG6 6.5 ThelinearspaceMkandthesubspaceMk.o 6.6 Classificationofentireformsintermsoftheirzeros 6.7 TheHeckeoperatorsTn 6.8 Transformationsofordern 6.9 BehaviorofTnfunderthemodulargroup 6.10 MultiplicativepropertyofHeckeoperators 6.11 EigenfunctionsofHeckeoperators 6.12 Propertiesofsimultaneouseigenforms 6.13 Examplesofnormalizedsimultaneouseigenforms 6.14 RemarksonexistenceofsimultaneouseigenformsinM2k.0 6.15 EstimatesfortheFouriercoefficientsofentireforms 6.16 ModularformsandDirichletseries Exerci |
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