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1、TheHodge-ArakelovTheoryofEllipticCurves:GlobalDiscretizationofLocalHodgeTheoriesbyShinichiMochizukiSeptember1999TableofContentsIntroduction§1.StatementoftheMainResults§2.TechnicalRoots:theWorkofMumfordandZhang§3.ConceptualRoots:theSearchforaGlobalHodgeTheory§3.1.From
2、AbsoluteDi?erentiationtoComparisonIsomorphisms§3.2.AFunction-TheoreticComparisonIsomorphism§3.3.TheMeaningofNonlinearity§3.4.HodgeTheoryatFiniteResolution§3.5.RelationshiptoOrdinaryFrobeniusLiftingsandAnabelianVarieties§4.GuidetotheText§5.FutureDirections§5.1.Gaussia
3、nPolesandtheThetaConvolution§5.2.HigherDimensionalAbelianVarietiesandHyperbolicCurvesChapterI:TorsorsinArakelovTheory§0.Introduction§1.ArakelovTheoryinGeometricDimensionZero§2.De?nitionandFirstPropertiesofTorsors§3.SplittingswithBoundedDenominators§4.ExamplesfromGeom
4、etryChapterII:TheGaloisActiononTorsionPoints§0.Introduction§1.SomeElementaryGroupTheory§2.TheHeightofanEllipticCurve§3.TheGaloisActionontheTorsionofaTateCurve§4.AnE?ectiveEstimateoftheImageofGalois1ChapterIII:TheUniversalExtensionofaLogEllipticCurve§0.Introduction§1.
5、De?nitionoftheUniversalExtension§2.CanonicalSplittingatIn?nity§3.CanonicalSplittingsintheComplexCase§4.Hodge-TheoreticInterpretationoftheUniversalExtension§5.AnalyticContinuationoftheCanonicalSplitting§6.HigherSchottky-WeierstrassZetaFunctions§7.CanonicalSchottky-Wei
6、erstrassZetaFunctionsChapterIV:ThetaGroupsandThetaFunctions§0.Introduction§1.Mumford’sAlgebraicThetaFunctions§2.ThetaActionsandtheSchottkyUniformization§3.TwistedSchottky-WeierstrassZetaFunctions§4.Zhang’sTheoryofMetrizedLineBundles§5.ThetaGroupsandMetrizedLineBundle
7、sChapterV:TheEvaluationMap§0.Introduction§1.ConstructionofCertainMetrizedLineBundles§2.TheDe?nitionoftheEvaluationMap§3.ExtensionoftheEtale-IntegralStructure§4.LinearRelationsAmongHigherSchottky-WeierstrassZetaFunctions§5.TheDeterminantoftheEvaluationMap§6.TheGeneric
8、CaseChapterVI:TheScheme-TheoreticComparisonTheorem§0.Introduction§1.De?nitionofaNewIntegralStructureatIn?nity§2.CompatibilitywithBa
