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1、INTER-UNIVERSALTEICHMULLERTHEORYI:¨CONSTRUCTIONOFHODGETHEATERSShinichiMochizukiJuly2014Abstract.Thepresentpaperisthefirstinaseriesoffourpapers,thegoalofwhichistoestablishanarithmeticversionofTeichm¨ullertheoryfornumberfieldsequippedwithanellipticcurve—whichwerefertoas“in
2、ter-universalTeichm¨ullertheory”—byapplyingthetheoryofsemi-graphsofanabelioids,Frobenioids,the´etalethetafunction,andlog-shellsdevelopedinearlierpapersbytheauthor.Webeginbyfixingwhatwecall“initialΘ-data”,whichconsistsofanellipticcurveEFoveranumberfieldF,andaprimenumberl≥
3、5,aswellassomeothertechnicaldatasatisfyingcertaintechnicalproperties.Thisdatadeter-minesvarioushyperbolicorbicurvesthatarerelatedviafinite´etalecoveringstotheonce-puncturedellipticcurveXFdeterminedbyEF.Thesefinite´etalecoveringsadmitvarioussymmetrypropertiesarisingfromth
4、eadditiveandmultiplicativestructuresontheringFl=Z/lZactingonthel-torsionpointsoftheellipticcurve.Wethenconstruct“Θ±ellNF-Hodgetheaters”associatedtothegivenΘ-data.TheseΘ±ellNF-Hodgetheatersmaybethoughtofasminiaturemodelsofconventionalschemetheoryinwhichthetwounderlyingc
5、ombinatorialdimensionsofanumberfield—whichmaybethoughtofascorrespondingtotheadditiveandmultiplicativestructuresofaringor,alternatively,tothegroupofunitsandvaluegroupofalocalfieldassociatedtothenumberfield—are,insomesense,“dismantled”or“disentangled”fromoneanother.AllΘ±ell
6、NF-Hodgetheatersareisomorphictooneanother,butmayalsoberelatedtooneanotherbymeansofa“Θ-link”,whichrelatescertainFrobenioid-theoreticportionsofoneΘ±ellNF-Hodgetheatertoanotherisafashionthatisnotcompatiblewiththerespectiveconven-tionalring/schemetheorystructures.Inparticu
7、lar,itisahighlynontrivialproblemtorelatetheringstructuresoneithersideoftheΘ-linktooneanother.Thiswillbeachieved,uptocertain“relativelymildindeterminacies”,infuturepapersintheseriesbyapplyingtheabsoluteanabeliangeometrydevelopedinearlierpapersbytheauthor.Theresultingdes
8、criptionofan“alienringstructure”[asso-ciated,say,tothedomainoftheΘ-link]intermsofagivenringstructure[associated,say,t