The Hodge-Arakelov Theory of Elliptic Curves

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1、TheHodge-ArakelovTheoryofEllipticCurves:GlobalDiscretizationofLocalHodgeTheoriesbyShinichiMochizukiSeptember1999TableofContentsIntroduction§1.StatementoftheMainResults§2.TechnicalRoots:theWorkofMumfordandZhang§3.ConceptualRoots:theSearchforaGlobalHodgeTheory§3.1.From

2、AbsoluteDifferentiationtoComparisonIsomorphisms§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.DefinitionandFirstPropertiesofTorsors§3.SplittingswithBoundedDenominators§4.ExamplesfromGeom

4、etryChapterII:TheGaloisActiononTorsionPoints§0.Introduction§1.SomeElementaryGroupTheory§2.TheHeightofanEllipticCurve§3.TheGaloisActionontheTorsionofaTateCurve§4.AnEffectiveEstimateoftheImageofGalois1ChapterIII:TheUniversalExtensionofaLogEllipticCurve§0.Introduction§1.

5、DefinitionoftheUniversalExtension§2.CanonicalSplittingatInfinity§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.TheDefinitionoftheEvaluationMap§3.ExtensionoftheEtale-IntegralStructure§4.LinearRelationsAmongHigherSchottky-WeierstrassZetaFunctions§5.TheDeterminantoftheEvaluationMap§6.TheGeneric

8、CaseChapterVI:TheScheme-TheoreticComparisonTheorem§0.Introduction§1.DefinitionofaNewIntegralStructureatInfinity§2.CompatibilitywithBa