inter-universal teichmuller theory i

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1、INTER-UNIVERSALTEICHMULLERTHEORYI:¨CONSTRUCTIONOFHODGETHEATERSShinichiMochizukiFebruary2016Abstract.Thepresentpaperisthefirstinaseriesoffourpapers,thegoalofwhichistoestablishanarithmeticversionofTeichm¨ullertheoryfornumberfieldsequippedwithanellipticcurve—whichwerefertoas“inter-universalTeichm¨ulle

2、rtheory”—byapplyingthetheoryofsemi-graphsofanabelioids,Frobenioids,the´etalethetafunction,andlog-shellsdevelopedinearlierpapersbytheauthor.Webeginbyfixingwhatwecall“initialΘ-data”,whichconsistsofanellipticcurveEFoveranumberfieldF,andaprimenumberl≥5,aswellassomeothertechnicaldatasatisfyingcertaintec

3、hnicalproperties.Thisdatadeter-minesvarioushyperbolicorbicurvesthatarerelatedviafinite´etalecoveringstotheonce-puncturedellipticcurveXFdeterminedbyEF.Thesefinite´etalecoveringsadmitvarioussymmetrypropertiesarisingfromtheadditiveandmultiplicativestructuresontheringFl=Z/lZactingonthel-torsionpointsof

4、theellipticcurve.Wethenconstruct“Θ±ellNF-Hodgetheaters”associatedtothegivenΘ-data.TheseΘ±ellNF-Hodgetheatersmaybethoughtofasminiaturemodelsofconventionalschemetheoryinwhichthetwounderlyingcombinatorialdimensionsofanumberfield—whichmaybethoughtofascorrespondingtotheadditiveandmultiplicativestructur

5、esofaringor,alternatively,tothegroupofunitsandvaluegroupofalocalfieldassociatedtothenumberfield—are,insomesense,“dismantled”or“disentangled”fromoneanother.AllΘ±ellNF-Hodgetheatersareisomorphictooneanother,butmayalsoberelatedtooneanotherbymeansofa“Θ-link”,whichrelatescertainFrobenioid-theoreticporti

6、onsofoneΘ±ellNF-Hodgetheatertoanotherinafashionthatisnotcompatiblewiththerespectiveconven-tionalring/schemetheorystructures.Inparticular,itisahighlynontrivialproblemtorelatetheringstructuresoneithersideoftheΘ-linktooneanother.Thiswillbeachieved,uptocertain“relativelymildindeterminacies”,infuturep

7、apersintheseriesbyapplyingtheabsoluteanabeliangeometrydevelopedinearlierpapersbytheauthor.Theresultingdescriptionofan“alienringstructure”[asso-ciated,say,tothedomainoftheΘ-link]intermsofagivenringstructure[associated,s