elliptic curves - notes for math 679 - j. milne, u. michigan

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1、ELLIPTICCURVESJ.S.MILNEAugust21,1996;v1.01Abstract.ThesearethenotesforMath679,UniversityofMichigan,Winter1996,exactlyastheywerehandedoutduringthecourseexceptforsomeminorcorrections.Pleasesendcommentsandcorrectionstomeatjmilne@umich.eduusing“Math679”asthesubjec

2、t.ContentsIntroduction1FastfactorizationofintegersCongruentnumbersFermat’slasttheorem1.ReviewofPlaneCurves2AffineplanecurvesProjectiveplanecurves2.RationalPointsonPlaneCurves.6Hensel’slemmaAbriefintroductiontothep-adicnumbersSomehistory3.TheGroupLawonaCubicCurve

3、124.FunctionsonAlgebraicCurvesandtheRiemann-RochTheorem14RegularfunctionsonaffinecurvesRegularfunctionsonprojectivecurvesTheRiemann-RochtheoremThegrouplawrevisitedPerfectbasefields5.DefinitionofanEllipticCurve19Planeprojectivecubiccurveswitharationalinflectionpoint

4、GeneralplaneprojectivecurvesCompletenonsingularcurvesofgenus1Copyright1996J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.iiiJ.S.MILNEThecanonicalformoftheequationThegrouplawforthecanonicalform6.ReductionofanEllipticCurveModulop23Algebraicgroupsof

5、dimension1SingularcubiccurvesReductionofanellipticcurveSemistablereductionReductionmodulo2and3Otherfields7.EllipticCurvesoverQp298.TorsionPoints32FormulasSolutiontoExercise4.89.N´eronModels37WeierstrassminimalmodelsTheworkofKodairaThecompleteN´eronmodelSummary1

6、0.EllipticCurvesovertheComplexNumbers41LatticesandbasesQuotientsofCbylatticesDoublyperiodicfunctionsTheholomorphicmapsC=Λ!C=Λ0TheWeierstrass}functionEisensteinseriesThefieldofdoublyperiodicfunctionsTheellipticcurveE(Λ)ClassificationofellipticcurvesoverCTorsionpo

7、intsEndomorphismsAppendix:Resultants11.TheMordell-WeilTheorem:StatementandStrategy5412.Groupcohomology55CohomologyoffinitegroupsCohomologyofinfiniteGaloisgroups13.TheSelmerandTate-Shafarevichgroups59ELLIPTICCURVESiii14.TheFinitenessoftheSelmerGroup60Proofofthefin

8、itenessoftheSelmergroupinaspecialcaseProofofthefinitenessoftheSelmergroupinthegeneralcase15.Heights65HeightsonP1HeightsonE16.CompletionoftheProofoftheMordell-WeilTheorem,andFurtherR