the etale theta function and its frobenioid-theoretic manifestations

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1、THEETALETHETAFUNCTIONAND´ITSFROBENIOID-THEORETICMANIFESTATIONSShinichiMochizukiDecember2008WedevelopthetheoryofthetemperedanabelianandFrobenioid-theore-ticaspectsofthe“´etalethetafunction”,i.e.,theKummerclassoftheclassicalformalalgebraicthetafunctionassociatedtoaTatecurveoveranonarchimede

2、anmixed-characteristiclocalfield.Inparticular,weconsideracertainnatural“environment”forthestudyofthe´etalethetafunction,whichwerefertoasa“mono-thetaenviron-ment”—essentiallyaKummer-theoreticversionoftheclassicalthetatrivialization—andshowthatthismono-thetaenvironmentsatisfiescertainremarkablerigidit

3、ypropertiesinvolvingcyclotomes,discreteness,andconstantmultiples,allinafash-ionthatiscompatiblewiththetopologyofthetemperedfundamentalgroupandtheextensionstructureoftheassociatedtemperedFrobenioid.Contents:§0.NotationsandConventions§1.TheTemperedAnabelianRigidityoftheEtaleThetaFunction´§2.TheTheor

4、yofThetaEnvironments§3.TemperedFrobenioids§4.GeneralBi-KummerTheory§5.TheEtaleThetaFunctionviaTemperedFrobenioids´IntroductionThefundamentalgoalofthepresentpaperistostudythetemperedanabelian[cf.[Andr´e],[Mzk14]]andFrobenioid-theoretic[cf.[Mzk17],[Mzk18]]aspectsofthethetafunctionofaTatecurveoverano

5、narchimedeanmixed-characteristiclo-calfield.Themotivationforthisapproachtothethetafunctionarisesfromthelong-termgoalofovercomingvariousobstaclesthatoccurwhenoneattemptstoapplytheHodge-Arakelovtheoryofellipticcurves[cf.[Mzk4],[Mzk5];[Mzk6],[Mzk7],[Mzk8],[Mzk9],[Mzk10]]tothediophantinegeometryofellip

6、ticcurvesovernumberfields.Thatistosay,thetheoryofthepresentpaperismotivated2000MathematicalSubjectClassification:Primary14H42;Secondary14H30.TypesetbyAMS-TEX12SHINICHIMOCHIZUKIbytheexpectationthattheseobstaclesmaybeovercomebytranslatingthe[es-sentially]scheme-theoreticformulationofHodge-Arakelovtheo

7、ryintothelanguageofthegeometryofcategories[e.g.,the“temperoids”of[Mzk14],andthe“Frobe-nioids”of[Mzk17],[Mzk18]].Incertainrespects,thissituationisreminiscentofthewell-knownclassicalsolutiontotheproblemofrelatingth