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1、TheScheme-TheoreticThetaConvolutionbyShinichiMochizukiSeptember1999Contents:§0.Introduction§1.TheScheme-TheoreticFourierTransform§2.DiscreteGaussiansandGaussSums§3.ReviewofDegreeComputationsin[HAT]§4.TheFourierTransformofanAlgebraicThetaFunction:TheCaseof
2、anEtaleLagrangianSubgroup´§5.TheFourierTransformofanAlgebraicThetaFunction:TheCaseofaLagrangianSubgroupwithNontrivialMultiplicativePart§6.GenericPropertiesoftheNorm§7.SomeElementaryComputationalLemmas§8.TheVariousContributionsatInfinity§9.TheMainTheorem§10
3、.TheThetaConvolution§0.IntroductionInthispaper,wecontinueourdevelopmentofthetheoryoftheHodge-ArakelovCom-parisonIsomorphismof[HAT].OurmainresultconcernstheinvertibilityofthecoefficientsoftheFouriertransformofanalgebraicthetafunction.Usingthisresult,weobtain
4、amodi-fiedversionoftheHodge-ArakelovComparisonIsomorphismof[HAT],whichwerefertoastheTheta-ConvolutedComparisonIsomorphism.Thesignificanceofthismodifiedversionisthattheprincipleobstructiontotheapplicationofthetheoryof[HAT]todiophantinegeometry—namely,theGauss
5、ianpoles—partiallyvanishesinthetheta-convolutedcon-text.Thus,theresultsofthispaperbringthetheoryof[HAT]onestepclosertopossibleapplicationtodiophantinegeometry.Perhapsthesimplestwaytoexplainthemainideaofthepresentpaperisthefollowing:Thetheoryof[HAT]maybeth
6、oughtofasasortofdiscrete,scheme-theoreticversionof12thetheoryoftheclassicalGaussiane−x(ontherealline)anditsderivatives(cf.[HAT],Introduction,§2).Thereasonfortheappearanceofthe“Gaussianpoles”—which,asremarkedabove,constitutetheprincipleobstructiontotheappl
7、icationofthetheoryof[HAT]todiophantinegeometry—isthat,if,forinstance,P(−)isapolynomialwithconstantcoefficients,then,atitsmostfundamentalcombinatoriallevel,the“comparisonisomorphism”of[HAT]maybethoughtofasthemapping∂−x2−x2P(−)→P()e=P(−2x)·e∂xfromacertainspa
8、ceofpolynomials—whichconstitutesthedeRhamsideofthecom-parisonisomorphism—toacertainspaceofset-theoreticfunctions—whichconstitutesthe´etalesideofthecomparisonisomorphism.Roughlyspeaking,inordertomakethis2morphisminto
