inter-universal%20teichmuller%20theory%20i[1]

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1、INTER-UNIVERSALTEICHMULLERTHEORYI:¨CONSTRUCTIONOFHODGETHEATERSShinichiMochizukiJuly2014Abstract.Thepresentpaperisthefirstinaseriesoffourpapers,thegoalofwhichistoestablishanarithmeticversionofTeichm¨ullertheoryfornumberfieldsequippedwithanellipticcurve—whichwerefertoas“inter-universalTei

2、chm¨ullertheory”—byapplyingthetheoryofsemi-graphsofanabelioids,Frobenioids,the´etalethetafunction,andlog-shellsdevelopedinearlierpapersbytheauthor.Webeginbyfixingwhatwecall“initialΘ-data”,whichconsistsofanellipticcurveEFoveranumberfieldF,andaprimenumberl≥5,aswellassomeothertechnicaldata

3、satisfyingcertaintechnicalproperties.Thisdatadeter-minesvarioushyperbolicorbicurvesthatarerelatedviafinite´etalecoveringstotheonce-puncturedellipticcurveXFdeterminedbyEF.Thesefinite´etalecoveringsadmitvarioussymmetrypropertiesarisingfromtheadditiveandmultiplicativestructuresontheringFl=

4、Z/lZactingonthel-torsionpointsoftheellipticcurve.Wethenconstruct“Θ±ellNF-Hodgetheaters”associatedtothegivenΘ-data.TheseΘ±ellNF-Hodgetheatersmaybethoughtofasminiaturemodelsofconventionalschemetheoryinwhichthetwounderlyingcombinatorialdimensionsofanumberfield—whichmaybethoughtofascorresp

5、ondingtotheadditiveandmultiplicativestructuresofaringor,alternatively,tothegroupofunitsandvaluegroupofalocalfieldassociatedtothenumberfield—are,insomesense,“dismantled”or“disentangled”fromoneanother.AllΘ±ellNF-Hodgetheatersareisomorphictooneanother,butmayalsoberelatedtooneanotherbymeans

6、ofa“Θ-link”,whichrelatescertainFrobenioid-theoreticportionsofoneΘ±ellNF-Hodgetheatertoanotherisafashionthatisnotcompatiblewiththerespectiveconven-tionalring/schemetheorystructures.Inparticular,itisahighlynontrivialproblemtorelatetheringstructuresoneithersideoftheΘ-linktooneanother.Thi

7、swillbeachieved,uptocertain“relativelymildindeterminacies”,infuturepapersintheseriesbyapplyingtheabsoluteanabeliangeometrydevelopedinearlierpapersbytheauthor.Theresultingdescriptionofan“alienringstructure”[asso-ciated,say,tothedomainoftheΘ-link]intermsofagivenringstructure[associated,

8、say,t