absolute anabelian cuspidalizations

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1、ABSOLUTEANABELIANCUSPIDALIZATIONSOFPROPERHYPERBOLICCURVESShinichiMochizukiJune2007Inthispaper,wedevelopthetheoryof“cuspidalizations”ofthe´etalefundamentalgroupofaproperhyperboliccurveoverafiniteornonarchimedeanmixed-characteristiclocalfield.Theultimategoaloft

2、histheoryisthegroup-theoreticreconstructionofthe´etalefundamentalgroupofanarbitraryopensubschemeofthecurvefromthe´etalefundamentalgroupofthefullpropercurve.Wethenapplythistheorytoshowthatacertainabsolutep-adicversionoftheGrothendieckConjec-tureholdsforhyperboliccurv

3、es“ofBelyitype”.Thisincludes,inparticular,affinehyperboliccurvesoveranonarchimedeanmixed-characteristiclocalfieldwhicharedefinedoveranumberfieldandisogenoustoahyperboliccurveofgenuszero.Also,weapplythistheorytoprovetheanalogueforproperhyperboliccurvesoverfinitefieldsofthev

4、ersionoftheGrothendieckConjecturethatwasshownin[Tama].Contents:§0.NotationsandConventions§1.MaximalAbelianCuspidalizations§2.PointsandFunctions§3.MaximalPro-lCuspidalizationsAppendix:FreeLieAlgebrasIntroductionLetXbeaproperhyperboliccurveoverafieldkwhichiseitherfinite

5、ornonarchimedeanlocalofmixedcharacteristic;letU⊆XbeanopensubschemeofX.WriteΠXforthe´etalefundamentalgroupofX.Inthispaper,westudytheextenttowhichthe´etalefundamentalgroupofUmaybegroup-theoreticallyreconstructedfromΠX.In§1,weshowthattheabelianportionoftheextensionofΠX

6、determinedbythe´etalefundamentalgroupofUmaybegroup-theoreticallyreconstructedfromΠX[cf.Theorem1.16,(iii)],and,moreover,thatthisconstructionhascertainremarkablerigidityproperties[cf.Propositions1.15,(i);2.6,(i)].TypesetbyAMS-TEX12SHINICHIMOCHIZUKIIn§2,weshowthatthisa

7、belianportionoftheextensionissufficienttorecon-struct[inessence]themultiplicativegroupofthefunctionfieldofX[cf.Theorem2.5,(ii)].Inthecaseofnonarchimedean[mixed-characteristic]localfields,thisalreadyimpliesvariousinterestingconsequencesinthecontextoftheabsolutean-abelian

8、geometrystudiedin[Mzk5],[Mzk6],[Mzk8].Inparticular,itimpliesthattheabsolutep-adicversionoftheGrothendieckConjecture[i.e.,anabsoluteversion