the profinite grothendieck conjecture for closed hyperbolic curves over number fields

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1、TheProfiniteGrothendieckConjectureforClosedHyperbolicCurvesoverNumberFieldsbyShinichiMochizukiSection0:IntroductionIn[Tama],aproofoftheGrothendieckConjecture(reviewedbelow)wasgivenforsmoothaffinehyperboliccurvesoverfinitefields(andovernumberfields).Thepurposeofthispaperistoshowhowonecanderivet

2、heGrothendieckConjectureforarbitrary(i.e.,notnecessarilyaffine)smoothhyperboliccurvesovernumberfieldsfromtheresultsof[Tama]foraffinehyperboliccurvesoverfinitefields.Weobtainthreetypesofresults:oneovernumberfields,oneoverfinitefields,andoneoverlocalfields.Weremarkherethatwhenthispaperwasfirstwritten(

3、October1995),TheoremsAandCbelowwerethestrongestknownresultsoftheirrespectivekinds.Sincethen,theauthorwrote[Mzk2](November1995),whichgivesrisetomuchstrongerresultsthanTheoremsAorCofthepresentpaper.Moreover,theproofsof[Mzk2]arecompletelydifferentfrom(and,inparticular,donotrelyon)theproofsof

4、thepresentpaper.Nevertheless,itseemstotheauthorthatthepresentpaperstillhassomemarginalinterest,partlybecausemostofthepresentpaperisdevotedtotheproofofTheoremBbelow(whichisnotimpliedbyanyresultof[Mzk2]),andpartlybecauseitisinsomesenseofinteresttoseehowTheoremsAorCcanbederivedwithinthecont

5、extofthetheoryof[Tama].Ourmainresultovernumberfields(Theorem10.2inthetext)isasfollows:TheoremA:LetKbeafiniteextensionofQ;letKbeanalgebraicclosureofK.LetX→Spec(K)andX→Spec(K)besmooth,geometricallyconnected,propercurvesKKoverK,ofgenus≥2.LetΔXK(respectively,ΔX)bethegeometricfundamentalgroup

6、KofX(respectively,X).ThenthenaturalmapKKIsomK(XK,XK)→Outρ(ΔXK,ΔX)Kisbijective.Here,“Outρ”referstoouterisomorphismsthatrespectthenaturalouterrepresentationsofGal(K/K)onΔXKandΔX.KThestatementofthisTheoremiscommonlyreferredtoas“theGrothendieckConjecture.”In[Tama],atheoremsimilartoTheore

7、mA,exceptthatXisreplacedbyahyperbolicaffinecurve,isproven.Itisasimpleexercisetoderivetheaffinecasefromtheproper1case.Ontheotherhand,toderivethepropercasefromtheaffinecaseisbynomeansstraightforward;inthispaper,wederivethepropercaseovernumberfieldsfromtheaffinecaseoverfinitefields.Inf