the galois-theoretic kodaira-spencer morphism of an elliptic curve

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1、TheGalois-TheoreticKodaira-SpencerMorphismofanEllipticCurveShinichiMochizukiJuly2000Contents:§0.Introduction§1.GaloisActionsontheTorsionPoints§2.LagrangianGaloisActions§2.1.DefinitionandConstruction§2.2.RelationtotheCrystallineThetaObject§3.GlobalMultiplicativeSubspaces§4.TheGroupTensorProduct

2、Section0:IntroductionThepurposeofthispaperistostudyingreaterdetailthearithmeticKodaira-Spencermorphismofanellipticcurveintroducedin[Mzk1],ChapterIX,inthegen-eralcontextoftheHodge-Arakelovtheoryofellipticcurves,developedin[Mzk1-3].Inparticular,aftercorrectingaminorerror(cf.Corollary1.6)intheco

3、nstructionofthisarithmeticKodaira-Spencermorphismin[Mzk1],ChapterIX,§3,wedefine(cf.§2.1)aslightlymodified“Lagrangian”versionofthisarithmeticKodaira-Spencermorphismwhichhasthefollowingremarkableproperties:(1)ThisLagrangianarithmeticKodaira-SpencermorphismisfreeofGaussianpoles(cf.Corollary2.5).(2

4、)AcertainportionofthereductionmodulopofthisLagrangianarithmeticKodaira-Spencermorphismmaybenaturallyiden-tifiedwiththeusualgeometricKodaira-Spencermorphism(cf.Corollary2.7).Werecallthatproperty(1)isofsubstantialinterestsinceitistheGaussianpolesthatarethemainobstructiontoapplyingtheHodge-Arakel

5、ovtheoryofellipticcurvestodiophantinegeometry(cf.thediscussionof[Mzk1],Introduction,§5.1,formoredetails).Ontheotherhand,property(2)isofsubstantialinterestinthatTypesetbyAMS-TEX12SHINICHIMOCHIZUKIitshowsquitedefinitivelythattheanalogyassertedin[Mzk1],ChapterIX,betweenthearithmeticKodaira-Spence

6、rmorphismoftheHodge-ArakelovtheoryofellipticcurvesandtheusualgeometricKodaira-Spencermorphismofafamilyofellipticcurvesisnotjustphilosophy,butrigorousmathematics!(cf.theRemarkfollowingCorollary2.7formoredetails).Infact,bothproperties(1)and(2)areessentiallyformalconsequencesofapropertythatweref

7、ertoasthe“crystallinenatureoftheLagrangianGaloisaction”(cf.Theorem2.4).Interestingly,thetheoryof§2ofthepresentpapermakesessentialusenotonlyofthetheoryof[Mzk1],butalsoof[Mzk2],[Mzk3].Unfortunately,however,thisLagrangianarithmeticKodaira-Spence