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1、INTER-UNIVERSALTEICHMULLERTHEORYIV:¨LOG-VOLUMECOMPUTATIONSANDSET-THEORETICFOUNDATIONSShinichiMochizukiJuly2014Abstract.Thepresentpaperformsthefourthandfinalpaperinaseriesofpapersconcerning“inter-universalTeichm¨ullertheory”.Inthefirstthreepapersoftheseries,weintroducedan
2、dstudiedthetheorysurroundingthelog-theta-lattice,ahighlynon-commutativetwo-dimensionaldiagramof“miniaturemodelsofconventionalschemetheory”,calledΘ±ellNF-Hodgetheaters,thatwereassociated,inthefirstpaperoftheseries,tocertaindata,calledinitialΘ-data.Thisdataincludesanellip
3、ticcurveEFoveranumberfieldF,togetherwithaprimenumberl≥5.Considerationofvariouspropertiesofthelog-theta-latticelednaturallytotheestablishment,inthethirdpaperoftheseries,ofmultiradialalgorithmsforconstructing“splittingmonoidsofLGP-monoids”.Here,werecallthat“multiradialalg
4、orithms”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—a
6、lbeitfromanextremelynaive/non-expertpointofview!—thefoundational/set-theoreticissuessurroundingtheverticalandhorizontalarrowsofthelog-theta-latticebyintroducingandstudyingthebasicpropertiesofthenotionofa“species”,whichmaybethoughtofasasortofformalization,viaset-theoret
7、icformulas,oftheintuitivenotionofa“typeofmathematicalobject”.Thesefoundationalissuesarecloselyrelatedtothecentralroleplayedinthepresentseriesofpapersbyvariousresultsfromabsoluteanabeliangeometry,aswellastotheideaofgluingtogetherdistinctmodelsofconventionalschemetheory,
8、i.e.,inafashionthatliesoutsidetheframeworkofconventionalschemetheory.Moreover,itispreciselythesefoundationalissuessur
