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1、INTER-UNIVERSALTEICHMULLERTHEORYIV:¨LOG-VOLUMECOMPUTATIONSANDSET-THEORETICFOUNDATIONSShinichiMochizukiDecember2015Abstract.Thepresentpaperformsthefourthandfinalpaperinaseriesofpapersconcerning“inter-universalTeichm¨ullertheory”.Inthefirstthreepapersoftheseries,weintroduce
2、dandstudiedthetheorysurroundingthelog-theta-lattice,ahighlynon-commutativetwo-dimensionaldiagramof“miniaturemodelsofconventionalschemetheory”,calledΘ±ellNF-Hodgetheaters,thatwereassociated,inthefirstpaperoftheseries,tocertaindata,calledinitialΘ-data.Thisdataincludesanell
3、ipticcurveEFoveranumberfieldF,togetherwithaprimenumberl≥5.Considerationofvariouspropertiesofthelog-theta-latticelednaturallytotheestablishment,inthethirdpaperoftheseries,ofmultiradialalgorithmsforconstructing“splittingmonoidsofLGP-monoids”.Here,werecallthat“multiradialal
4、gorithms”arealgorithmsthatmakesensefromthepointofviewofan“alienarithmeticholomorphicstructure”,i.e.,thering/schemestructureofaΘ±ellNF-HodgetheaterrelatedtoagivenΘ±ellNF-Hodgetheaterbymeansofanon-ring/scheme-theoretichorizontalarrowofthelog-theta-lattice.Inthepresentpape
5、r,estimatesarisingfromthesemultiradialalgorithmsforsplittingmonoidsofLGP-monoidsareappliedtoverifyvariousdiophantineresultswhichimply,forinstance,theso-calledVojtaConjectureforhyperboliccurves,theABCConjecture,andtheSzpiroConjectureforellipticcurves.Finally,weexamine—al
6、beitfromanextremelynaive/non-expertpointofview!—thefoundational/set-theoreticissuessurroundingtheverticalandhorizontalarrowsofthelog-theta-latticebyintroducingandstudyingthebasicpropertiesofthenotionofa“species”,whichmaybethoughtofasasortofformalization,viaset-theoretic
7、formulas,oftheintuitivenotionofa“typeofmathematicalobject”.Thesefoundationalissuesarecloselyrelatedtothecentralroleplayedinthepresentseriesofpapersbyvariousresultsfromabsoluteanabeliangeometry,aswellastotheideaofgluingtogetherdistinctmodelsofconventionalschemetheory,i.e
8、.,inafashionthatliesoutsidetheframeworkofconventionalschemetheory.Moreover,itispreciselythesefoundationalissue
