connections and related structures on the universal extension of an elliptic curve

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1、ConnectionsandRelatedIntegralStructuresontheUniversalExtensionofanEllipticCurvebyShinichiMochizukiMay2000Contents:§0.Introduction§1.TheEtaleIntegralStructureontheUniversalExtension´§2.TheEtaleIntegralStructureforanOrdinaryEllipticCurve´§2.1.Somep-adicFunctionTheory§2.2.

2、TheVerschiebungMorphism§3.CompactifiedHodgeTorsors§4.TheEtaleIntegralStructureontheHodgeTorsors´§4.1.NotationandSet-Up§4.2.DegeneratingEllipticCurves§4.3.OrdinaryEllipticCurves§4.4.TheGeneralCase§5.ConstructionoftheConnection§5.1.ComplexAnalogue§5.2.TheSchematicCase§6.Th

3、eSchottky-TheoreticHodge-ArakelovComparisonIsomorphism§7.CrystallineThetaExpansions§7.1.GeneralitiesonHigherp-Curvatures§7.2.ApplicationtoThetaExpansions§8.“GriffithsSemi-Transversality”§8.1.TheKodaira-SpencerMorphismoftheCrystallineThetaObject§8.2.CalculationoftheHigherp

4、-Curvatures§8.3.Hasse-typeInvariantsoftheCrystallineThetaObject§9.RelationtotheTheoryof[Mzk1]1§0.IntroductionInthispaper,wecontinueourstudyoftheHodge-Arakelovtheoryofellipticcurves,initiatedin[Mzk1],[Mzk2].Theessenceofthistheoryliesinthinkingofthecomparisonisomorphismof

5、the(complexorp-adic)HodgetheoryofanellipticcurveasarestrictionmorphismfromfunctionsonthedeRhamcohomologytofunctionsonsomesortof“torsionpoints”insidethedeRhamcohomology(cf.[Mzk1],Introduction).Thisfunction-theoreticpointofviewallowedusin[Mzk1]todiscretizetheusualcomplexa

6、ndp-adicHodgetheoriesofanellipticcurveintoaglobal,Arakelov-theoretic“Hodge-ArakelovComparisonIsomorphism”(cf.[Mzk1]).Thefirstmaingoalofthepresentpaperistoclearuptheconfusionsurroundingthediscussionofthe“´etaleintegralstructure”ontheuniversalextensionofanellipticcurve,whi

7、chwasintroducedin[Mzk1].Thisintegralstructuremaybedescribedasfollows.Overaformalneighborhoodofthepointatinfinityonthemodulistackofellipticcurves,thetautologicalellipticcurveElookslikeGm,whileitsuniversalextensionE†(roughlyspeaking:themodulispaceofdegreezerolinebundlesonE

8、equippedwithaconnection)maybedescribedastheproductG×A1ofGwiththeaffineline.Ifwewrite“T”mmforthestandardcoordinat