inter-universal teichmuller theory ii

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1、INTER-UNIVERSALTEICHMULLERTHEORYII:¨HODGE-ARAKELOV-THEORETICEVALUATIONShinichiMochizukiDecember2015Abstract.Inthepresentpaper,whichisthesecondinaseriesoffourpa-pers,westudytheKummertheorysurroundingtheHodge-Arakelov-theoreticeval-uation—i.e.,evaluationinthest

2、yleofthescheme-theoreticHodge-Arakelovtheoryestablishedbytheauthorinpreviouspapers—ofthe[reciprocalofthel-throotofthe]thetafunctionatl-torsionpoints[strictlyspeaking,shiftedbyasuitable2-torsionpoint],forl≥5aprimenumber.Inthefirstpaperoftheseries,westudied“mini

3、aturemodelsofconventionalschemetheory”,whichwereferredtoasΘ±ellNF-Hodgetheaters,thatwereassociatedtocertaindata,calledinitialΘ-data,thatincludesanellipticcurveEFoveranumberfieldF,togetherwithaprimenum-berl≥5.TheunderlyingΘ-HodgetheatersoftheseΘ±ellNF-Hodgethea

4、tersweregluedtooneanotherbymeansof“Θ-links”,thatidentifythe[reciprocalofthel-throotofthe]thetafunctionatprimesofbadreductionofEFinoneΘ±ellNF-Hodgetheaterwith[2l-throotsof]theq-parameteratprimesofbadreductionofEFinan-otherΘ±ellNF-Hodgetheater.Thetheorydevelope

5、dinthepresentpaperallowsone×μtoconstructcertainnewversionsofthis“Θ-link”.OnesuchnewversionistheΘgau-link,whichissimilartotheΘ-link,butinvolvesthethetavaluesatl-torsionpoints,ratherthanthethetafunctionitself.Oneimportantaspectoftheconstructions×μthatunderlieth

6、eΘgau-linkisthestudyofmultiradialityproperties,i.e.,propertiesofthe“arithmeticholomorphicstructure”—or,moreconcretely,thering/schemestructure—arisingfromoneΘ±ellNF-Hodgetheaterthatmaybeformulatedinsuchawayastomakesensefromthepointofthearithmeticholomorphicstr

7、uctureofanotherΘ±ellNF-HodgetheaterwhichisrelatedtotheoriginalΘ±ellNF-Hodge×μtheaterbymeansofthe[non-scheme-theoretic!]Θgau-link.Forinstance,certainofthevariousrigiditypropertiesofthe´etalethetafunctionstudiedinanearlierpaperbytheauthormaybeintepretedasmultir

8、adialitypropertiesinthecontextofthetheoryofthepresentseriesofpapers.Anotherimportantaspectoftheconstructions×μthatunderlietheΘgau-linkisthestudyof“conjugatesynchronization”viatheF±-symme