the local pro-p anabelian geometry of curves

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1、TheLocalPro-pAnabelianGeometryofCurvesbyShinichiMochizukiINTRODUCTIONTheAnabelianGeometryofGrothendieck:LetXbeaconnectedscheme.Thenonecanassociate(afterGrothendieck)toXitsalgebraicfundamentalgroupπ1(X).Thisgroupπ1(X)isaprofinitegroupwhichisuniquelydetermined(uptoinnerautomorphisms)bytheprope

2、rtythatthecategoryoffinite,discretesetsequippedwithacontinuousπ1(X)-actionisequivalenttothecategoryoffinite´etalecoveringsofX.Moreover,theassignmentX→π1(X)isafunctorfromthecategoryofconnectedschemes(andmorphismsofschemes)tothecategoryofprofinitetopologicalgroupsandcontinuousouterhomomorphisms

3、(i.e.,continuoushomomorphismsoftopologicalgroups,whereweidentifyanytwohomomorphismsthatcanbeobtainedfromoneanotherbycompositionwithaninnerautomorphism).NowletKbeafield.LetΓKbetheabsoluteGaloisgroupofK.Thenπ1(Spec(K))maybeidentifiedwithΓK.LetXKbeavariety(i.e.,ageometricallyintegralseparatedsch

4、emeoffinitetype)overK.ThenthestructuremorphismXK→Spec(K)definesanaturalaugmentationπ1(XK)→ΓK.Thekernelofthismorphismπ1(XK)→ΓKisaclosednormalsubgroupofπ1(XK)–calledthegeometricfundamentalgroupofXK–defwhichmaybeidentifiedwithπ1(XK)(whereXK=XK⊗KK).If,moreover,onefixesaprimenumberp,thenonecanformth

5、emaximalpro-pquotientπ(X)(p)ofπ(X).1K1KSincethequotientπ(X)→π(X)(p)ischaracteristic,itfollowsthatthekernelofthis1K1Kquotientis,infact,anormalsubgroupofπ1(XK).Thequotientofπ1(XK)bythisnormalsubgroupwillbedenotedΠXK.Thus,ΠXKinheritsanaturalaugmentationΠXK→ΓKfromthatofπ1(XK).Nowletusconsiderth

6、eassignmentπ1(−)K:{XK→Spec(K)}→{π1(XK)→ΓK}ThisassignmentdefinesafunctorfromthecategoryCKofK-varieties(whosemorphismsareK-linearmorphismsofvarieties)tothecategoryGKwhoseobjectsareprofinitetopo-logicalgroupsequippedwithanaugmentationtoΓK,andwhosemorphismsarecontinu-ousouterhomomorphismsoftopol

7、ogicalgroupsthatlieoverΓK.ItwastheintuitionofGrothendieck(see[Groth])that:1ForcertaintypesofK,ifonereplacesCKandGKby“certainappro-priate”subcategoriesCandG(suchthatπ(−)stillmapsCintoKK1KKG),thenπ(−)shouldbefullyfaithful.K1KHere,the“certainappropriate