complex multiplication

《complex multiplication》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ComplexMultiplicationJ.S.MilneApril7,2006Thesearepreliminarynotes1foramodernaccountofthetheoryofcomplexmultipli-cation.Ashortened(minimal)versionwillbeincludedinmybookonShimuravarieties,andacompletelongerversionmayonedaybepublishedseparately.v0.00April7,2006.Firstv

2、ersionpostedontheweb;113pages.Pleasesendcommentsandcorrectionstomeatmath@jmilne.org.Availableathttp://www.jmilne.org/mathCopyrightc2006.J.S.Milne.1Thisshouldbetakenseriously:thereareomissions,repetitions,clumsystatementsandproofs,andincon-sistenciesinnotation.Conte

3、ntsContents3IAnalytictheory91CM-algebrasandCM-types............................9Reviewofsemisimplealgebrasandtheirmodules9;CM-algebras11;CM-types.12;ThereflexfieldofaCM-pair14;Thereflexnorm.15;ClassificationoftheprimitiveCM-pairs19;PositiveinvolutionsandCM-algebras212C

4、omplexabelianvarieties.............................22Complextori22;Thecohomologyofcomplextori24;Hermitianformsandalternatingforms26;Riemannforms26;Abelianvarieties283Abelianvarietieswithcomplexmultiplication..................29DefinitionofCMabelianvarieties29;Therefl

5、exfieldofanabelianvarietywithcomplexmultiplication31;Classificationuptoisogeny31;Classificationuptoisomorphism324Mumford-Tategroups...............................33Reviewofalgebraicgroupsofmultiplicativetype33;CM-pairsandtori35;Thereflexnormintermsoftori36;Complexmulti

6、plicationintermsoftori36;Mumford-Tategroups37;Infinitytypes38;TheSerregroup42;AbelianvarietiesofCM-type465Motives......................................46TheHodgestructureofanabelianvariety46;Abelianmotives46;Hodgestructures46;CM-motives46IIThearithmetictheory496Abel

7、ianvarietiesandtheirgoodreductions...................49Complexabelianvarietiesandcomplextori49;Specializationofabelianvarieties51;Thegoodreductionofabelianvarieties517Abelianvarietieswithcomplexmultiplication..................55DefinitionofaCMabelianvariety55;Comple

8、xmultiplicationbyaQ-algebra55;Specialization57;Rigidity57;Goodreduction58;Thedegreesofisogenies58;a-multiplications(1)60;a-multiplications(2)64;a