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1、INTER-UNIVERSALTEICHMULLERTHEORYIII:¨CANONICALSPLITTINGSOFTHELOG-THETA-LATTICEShinichiMochizukiJuly2014Abstract.Thepresentpaperconstitutesthethirdpaperinaseriesoffourpapersandmayberegardedastheculminationoftheabstractconceptualpor-tionofthetheorydevelopedintheseries.Inthepresentp
2、aper,westudythetheorysurroundingthelog-theta-lattice,ahighlynon-commutativetwo-dimensionaldia-gramof“miniaturemodelsofconventionalschemetheory”,calledΘ±ellNF-Hodgetheaters.Here,werecallthatΘ±ellNF-Hodgetheaterswereassociated,inthefirstpaperoftheseries,tocertaindata,calledinitialΘ-
3、data,thatincludesanellipticcurveEFoveranumberfieldF,togetherwithaprimenumberl≥5.Eachar-rowofthelog-theta-latticecorrespondstoacertaingluingoperationbetweentheΘ±ellNF-Hodgetheatersinthedomainandcodomainofthearrow.Thehorizontalarrowsofthelog-theta-latticearedefinedascertainversionsof
4、the“Θ-link”thatwasconstructed,inthesecondpaperoftheseries,byapplyingthetheoryofHodge-Arakelov-theoreticevaluation—i.e.,evaluationinthestyleofthescheme-theoreticHodge-Arakelovtheoryestablishedbytheauthorinpreviouspapers—ofthe[reciprocalofthel-throotofthe]thetafunctionatl-torsionpo
5、ints.Inthepresentpaper,wefocusonthetheorysurroundingthelog-linkbetweenΘ±ellNF-Hodgetheaters.Thelog-linkisobtained,roughlyspeaking,byapplying,ateach[say,forsimplicity,nonarchimedean]valuationofthenumberfieldunderconsider-ation,thelocalp-adiclogarithm.Thesignificanceofthelog-linklies
6、inthefactthatitallowsonetoconstructlog-shells,i.e.,roughlyspeaking,slightlyadjustedformsoftheimageofthelocalunitsatthevaluationunderconsiderationviathelocalp-adiclogarithm.Thetheoryoflog-shellswasstudiedextensivelyinaprevi-ouspaperbytheauthor.Theverticalarrowsofthelog-theta-latti
7、cearegivenbythelog-link.Considerationofvariouspropertiesofthelog-theta-latticeleadsnatu-rallytotheestablishmentofmultiradialalgorithmsforconstructing“splittingmonoidsoflogarithmicGaussianprocessionmonoids”.Here,werecallthat“multiradialalgorithms”arealgorithmsthatmakesensefromthep
8、ointofviewofan“alienarithmeticholomorphi
