topics in absolute anabelian geometry iii

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1、TOPICSINABSOLUTEANABELIANGEOMETRYIII:GLOBALRECONSTRUCTIONALGORITHMSShinichiMochizukiNovember2015Abstract.Inthepresentpaper,whichformsthethirdpartofathree-partseriesonanalgorithmicapproachtoabsoluteanabeliangeometry,weapplytheab-soluteanabeliantechniqueofBelyicuspidalizationdevelo

2、pedinthesecondpart,togetherwithcertainideascontainedinanearlierpaperoftheauthorconcerningthecategory-theoreticrepresentationofholomorphicstructuresviaeitherthetopologi-calgroupSL2(R)ortheuseof“parallelograms,rectangles,andsquares”,todevelopacertainglobalformalismforcertainhyperbo

3、licorbicurvesrelatedtoaonce-puncturedellipticcurveoveranumberfield.Thisformalismallowsonetoconstructcertaincanonicalrigidintegralstructures,whichwerefertoaslog-shells,thatareobtainedbyapplyingthelogarithmatvariousprimesofanumberfield.More-over,althougheachoftheselocallogarithmsis“f

4、arfrombeinganisomorphism”bothinthesensethatitfailstorespecttheringstructuresinvolvedandinthesense[cf.Frobeniusmorphismsinpositivecharacteristic!]thatithastheeffectofexhibitingthe“mass”representedbyitsdomainasa“somewhatsmallercollectionofmass”thanthe“mass”representedbyitscodomain,t

5、hisglobalformalismallowsonetotreatthelogarithmoperationasaglobaloperationonanumberfieldwhichsatisfiesthepropertyofbeingan“isomomorphismuptoanappropriaterenormal-izationoperation”,inafashionthatisreminiscentoftheisomorphisminducedondifferentialsbyaFrobeniuslifting,onceonedividesbyp.M

6、oregenerally,ifonethinksofnumberfieldsascorrespondingtopositivecharacteristichyperboliccurvesandofonce-puncturedellipticcurvesonanumberfieldascorrespondingtonilpotentordinaryindigenousbundlesonapositivecharacteristichyperboliccurve,thenmanyaspectsofthetheorydevelopedinthepresentpap

7、erarereminiscentof[thepositivecharacteristicportionof]p-adicTeichm¨ullertheory.Contents:Introduction§0.NotationsandConventions§1.Galois-theoreticReconstructionAlgorithms§2.ArchimedeanReconstructionAlgorithms§3.NonarchimedeanLog-FrobeniusCompatibility§4.ArchimedeanLog-FrobeniusCom

8、patibility§5.GlobalLog-FrobeniusCompatib