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1、INTER-UNIVERSALTEICHMULLERTHEORYII:¨HODGE-ARAKELOV-THEORETICEVALUATIONShinichiMochizukiJuly2014Abstract.Inthepresentpaper,whichisthesecondinaseriesoffourpa-pers,westudytheKummertheorysurroundingtheHodge-Arakelov-theoreticeval-uation—i.e.,evaluationinthestyleof
2、thescheme-theoreticHodge-Arakelovtheoryestablishedbytheauthorinpreviouspapers—ofthe[reciprocalofthel-throotofthe]thetafunctionatl-torsionpoints,forl≥5aprimenumber.Inthefirstpaperoftheseries,westudied“miniaturemodelsofconventionalschemetheory”,whichwereferredtoa
3、sΘ±ellNF-Hodgetheaters,thatwereassociatedtocertaindata,calledinitialΘ-data,thatincludesanellipticcurveEFoveranumberfieldF,togetherwithaprimenumberl≥5.TheseΘ±ellNF-Hodgetheatersweregluedtooneanotherbymeansof“Θ-links”,thatidentifythe[reciprocalofthel-throotofthe]
4、thetafunctionatprimesofbadreductionofEFinoneΘ±ellNF-Hodgetheaterwith[2l-throotsof]theq-parameteratprimesofbadreductionofEFinan-otherΘ±ellNF-Hodgetheater.Thetheorydevelopedinthepresentpaperallowsone×μtoconstructcertainnewversionsofthis“Θ-link”.Onesuchnewversion
5、istheΘgau-link,whichissimilartotheΘ-link,butinvolvesthethetavaluesatl-torsionpoints,ratherthanthethetafunctionitself.Oneimportantaspectoftheconstructions×μthatunderlietheΘgau-linkisthestudyofmultiradialityproperties,i.e.,propertiesofthe“arithmeticholomorphicst
6、ructure”—or,moreconcretely,thering/schemestructure—arisingfromoneΘ±ellNF-HodgetheaterthatmaybeformulatedinsuchawayastomakesensefromthepointofthearithmeticholomorphicstructureofanotherΘ±ellNF-HodgetheaterwhichisrelatedtotheoriginalΘ±ellNF-Hodge×μtheaterbymeanso
7、fthe[non-scheme-theoretic!]Θgau-link.Forinstance,certainofthevariousrigiditypropertiesofthe´etalethetafunctionstudiedinanearlierpaperbytheauthormaybeintepretedasmultiradialitypropertiesinthecontextofthetheoryofthepresentseriesofpapers.Anotherimportantaspectoft
8、heconstructions×μthatunderlietheΘgau-linkisthestudyof“conjugatesynchronization”viatheF±-symmetryofaΘ±ellNF-Hodgetheater.Conjugatesynchronizationreferstoalcertainsystemofisomorphis
