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1、TheAbsoluteAnabelianGeometryofHyperbolicCurvesShinichiMochizukiContents:§0.NotationsandConventions§1.ReviewofAnabelianGeometry§1.1.TheAnabelianGeometryofNumberFields§1.2.TheAnabelianGeometryofp-adicLocalFields§1.3.TheAnabelianGeometryofHyperbolicCurves§2.ReconstructionoftheL
2、ogarithmicSpecialFiberAppendix:TerminologyofGraphTheoryIntroductionLetXKbeahyperboliccurve(cf.§0below)overafieldKofcharacteristic0.DenoteitsalgebraicfundamentalgroupbyΠXK.Thus,wehaveanaturalsurjectionΠXKGKofΠXKontotheabsoluteGaloisgroupGKofK.WhenKisafiniteextensionofQorQp,and
3、oneholdsGKfixed,thenitisknown(cf.[Tama1],[Mzk6];Theorem1.3.4ofthepresentmanuscript)thatonemayrecoverthecurveXKinafunctorialfashionfromΠXK.Thissortofresultmaybethoughtofasa“relativeresult”(i.e.,overGK).Thenthequestionnaturallyarises:Towhatextentarethe“absoluteanalogues”ofthisr
4、esultvalid—i.e.,whatifonedoesnotholdGKfixed?IfKisanumberfield,thenitisstillpossibletorecoverXKfromΠXK(cf.Corollary1.3.5),byapplyingthetheoremofNeukirch-Uchida(cf.Theorem1.1.3).Ontheother12SHINICHIMOCHIZUKIhand,whenKisap-adiclocalfield(i.e.,afiniteextensionofQp),theanalogueofthet
5、heoremofNeukirch-Uchidafailstohold,andindeed,itistheopinionoftheauthoratthetimeofwritingthatitisunlikely(inthep-adiclocalcase)thatonecanrecoverXKingeneral(i.e.,inthefashionofCorollary1.3.5)fromΠXK.Inthepresentmanuscript,webeginbyreviewing/surveyingin§1theanabeliangeometryofn
6、umberfields,p-adiclocalfields,andhyperboliccurvesfromthepointofviewofthegoalofunderstandingtowhatextenttheanabeliangeometryofhyperboliccurvesoverp-adiclocalfieldscanbemade“absolute”.Ourmainresult(Theorem2.7),givenin§2,statesthatwhenKisap-adiclocalfield,(althoughwemaybeunabletore
7、coverXKitself)onemayrecover(inafunctorialfashion)thespecialfiberofXK,togetherwithitsnaturallogstructure,inanabsolutefashion,i.e.,solelyfromtheisomorphismclassoftheprofinitegroupΠXK.Acknowledgements:IwouldliketothankA.Tamagawaforthetimethathesogen-erouslysharedwithmeinnumerouss
8、timulatingdiscussions,andespeciallyforthefollowing:(i)informingmeoftheargum
