椭圆函数（第2版）

图书标准信息

• 作者 S.Lang 编
• 出版社 世界图书出版公司
• 出版时间 2003-11
• 版次 2
• ISBN 9787506265508
• 定价 45.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 326页
• 正文语种 英语

【内容简介】

Ellipticfunctionsparametrizeellipticcurves,andtheinterminglingoftheanalyticandalgebraic-arithmetictheoryhasbeenatthecenterofmathematicssincetheearlypartofthenineteenthcentury.Somenewtechniquesandoutlookshaverecentlyappearedontheseoldsubjects,continuinginthetraditionofKronecker,Weber,Fricke,Hasse,Deuring.Shimura'sbookIntroductiontothearithmetictheoryofautomorphicfunctionsisasplendidmodernreference,whichIfoundveryhelpfulmyselftolearnsomeaspectsofellipticcurves.ItemphasizesthedirectionoftheHasse-Weilzetafunction,Heckeoperators,andthegeneralizationsduetohimtothehigherdimensionalcase（abelianvarieties,curvesofhighergenuscomingfromanarithmeticgroupoperatingontheupperhalfplane,boundedsymmetricdomainswithadiscretearithmeticgroupwhosequotientisalgebraic）.Irefertheinterestedreadertohisbookandthebibliographytherein.

【目录】

PARTONEccGENERALTHEORY
Chapter1ElliptiFunctions
1ThecLiouvilleTheorems
2TheWeierstrassFunction
3TheAdditioncTheorem
4IsomorphismClassescofEllipticCurves
5EndomorphismsandAutomorphisms

Chapter2Homomorphisms
1PointsofFiniteOrder
2Isogenies
3TheInvolution

Chapter3hecModularFunction
1TheModularGroup
2AutomorphicFunctionsofDegree2k
3TheModularFunctionj

Chapter4FourierExpansions
1ExpansionforGk,cg2,cg3,c△candcj
2ExpansionfortheWeierstrassFunction
3BernoulliNumbers

Chapter5TheModularEquation
1IntegralMatriceswithPositiveDeterminant
2TheModularEquation
3RelationswithIsogenies

Chapter6HigherLevels
1CongruenceSubgroups
2TheFieldofModularFunctionsOvercC
3TheFieldofModularFunctionsOvercQ
4SubfieldsoftheModularFunctionField

Chapter7AutomorphismsoftheModularFunctionField
1RationalAdelesofGL
2OperationoftheRationalAdelescontheModularFunctionField
3TheShimuraExactSequence
PARTcTWOccCOMPLEXcMULTIPLICATIONELLIPTICcCURVEScWITHcSINGULARcINVARIANTS

Chapter8ResultsfromAlgebraicNumberTheory
1LatticescinQuadraticFields
2Completions
3TheDecompositionGroupandFrobeniusAutomorphism
4SummaryofClassFieldTheory

Chapter9ReductionofEllipticCurves
1Non-degenerateReduction,GeneralCase
2RedutionofHomomorphisms
3CoveringsofLevelcN
4ReductionofDifferentialForms

Chapter10ComplexMultiplication
1GenerationofClassFields,Deuring'sApproach
2IdelicFormulationforArbitraryLattices
3GenerationofClassFieldsbySingularValuesofModularFunctions
4TheFrobeniusEndomorphism
AppendixARelationofKronecker

Chapter11Shimura'sReciprocityLaw
IRelationBetweenGenericandSpecialExtensions
2ApplicationtoQuotientscofModularForms

Chapter12TheFuntion△（at）/△（t）
1BehaviorUndertheArtinAutomorphism
2PrimeFactorizationofitsValues
3AnalytiProoffortheCongruenceRelationcofj

Chapterc13Thel-adicandp-adicRepresentationsofDeuring
1Thecl-adicSpaces
2RepresentationsinCharacteristip
3RepresentationsandIsogenies
4ReductioncofcthecRingcofcEndomorphisms
5TheDeuringLiftingTheorem

Chapter14Ihara'sTheory
1.DeuringRepresentatives
2TheGenericSituation
3SpecialSituations
PARTTHREEELLIPTICCURVEScWITHNON-INTEGRALINVARIANT

Chapter15TheTateParametrization
1EllipticCurveswithNon-integralInvariants
2ElliptiCurvesOveraCompleteLocalRing

Chapter16TheIsogenyTheorems
1TheGaloisp-adicRepresentations
2ResultsofKummerTheory
3TheLocalIsogenyTheorems
4SupersingularRedution
5TheGlobalIsogenyTheorems

Chapter17DivisionPointsOverNumberFields
1AcTheoremofShafarevic
2TheIrreducibilityTheorem
3TheHorizontalGaloisGroup
4TheVerticalGaloisGroup
5EndoftheProof
PARTcFOURccTHETAcFUNCTIONScANDcKRONECKERcLIMITFORMULA

Chapter18ProductExpansions
1TheSigmaandZetaFunction
AppendixTheSkewSymmetricPairing
2ANormalizationandtheq-productforthea-function
3q-expansionsAgain
4Theq-productforcA
5TheEtaFunctionofDedekind
6ModularFunctionsofLevelc2

Chapter19TheSiegelFunctionsandKleinForms
1TheKleinForms
2TheSiegelFunctions
3SpecialValuesoftheSiegelFunctions

Chapter20TheKroneckerLimitFormulas
1ThePoissonSummationFormula
2Examples
3TheFunctioncKs（x）
4TheKroneckerFirstLimitFormula
5TheKroneckerSecondLimitcFormula

Chapter21TheFirstLimitFormulaandL-series
1RelationwithL-series
2TheFrobeniusDeterminant
3ApplicationtothecL-series

Chapter22TheSecondLimitFormulaandL-series
1GaussSums
2AnExpressionfortheL-series
APPENDICESELLIPTICCURVESINCHARACTERISTICp
Appendixc1AlgebraicFormulasinArbitraryCharateristicBYcJ.cTATE
1GeneralizedWeierstrassForm
2CanonicalForms
3ExpansionNearO;TheFormalGroup
Appendix2TheTracecofFrobeniusandtheDifferentialofFirstcKind
1TheTraceofFrobenius
2Duality
3TheTateTrace
4TheCartierOperator
5TheHasseInvariant
Bibliography
Index