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1、COMMENTSONTHEMAINTHEOREMOFPOP-STIXShinichiMochizukiUpdatedNovember15,2010LetkbeafiniteextensionofQp,kanalgebraicclosureofk,andXaproperhyperboliccurveoverk.WriteΠXforthe´etalefundamentalgroupofX[relativetptosomebasepoint],ΠforthetemperedfundamentalgroupofX
2、[relativetosomeXbasepoint],anddeftpΠXGk=Gal(k/k),ΠXGkforthenaturalprojections.Also,wewriteKforthefunctionfieldofXandOkfortheringofintegersofk.Then:(1)DuringtheconferenceonGalois-Teichm¨ullertheoryheldatRIMS,KyotoUni-versity,inOctober2010,itwassuggestedb
3、yYvesAndr´ethattheMainTheoremof[PS]shouldimplythefollowingassertion:(A1)EverysectionofΠXGkarises,uptoΠX-conjugacy,fromasectiontpofΠXGk.Wereferto(7)belowforamoredetailedformulation.Oneconsequenceofthisassertion(A1)isthefollowingassertion:(A2)The“Profinit
4、ep-adicSectionConjecture”forhyperboliccurvesmaybereducedtothe“Temperedp-adicSectionConjecture”forhyperboliccurves.AsfarasIcansee,(A1)and(A2)maybeverifiedimmediately.(2)Inlightofthediscussionof(1),itisnaturaltoaskthequestion(cf.alsothediscussionof(8)below)
5、:tptp(Q1)Ifonestartswithasections:Gk→ΠXofΠXGk,thendoestheMainTheoremof[PS]tellyouanythingnewconcerningthesections?Inthiscontext,weobservethat,evenwithoutapplyingthetheoryof[PS],itfollowsimmediatelyfrom[Semi],Theorem5.4,(i)—cf.alsothediscussionof[Semi],T
6、ypesetbyAMS-TEX12SHINICHIMOCHIZUKIExample5.6;[Semi],Remark6.9.1—thatthesectionsdeterminesasystemofvertices...vi+1vi...ofanycofinalsystemoffinite´etaleconnectedGaloiscoveringsofXwithstablereduction...→Xi+1→Xi→...ofX—i.e.,eachviisanirreduciblecomponentoft
7、hespecialfiberofthestablemodelofXithatisfixedbythenaturalactionoftheimageIm(s)ofthesections.Here,irangesoverthepositiveintegers,and,afterpossiblypassingtoacofinalsubsystem,thenotation“”maybeinterpretedasthestatementthatoneofthefollowingtwo[mutuallyexclusiv
8、e]conditionsholds:(C1)Foreachpositiveintegeri,theirreduciblecomponentvi+1mapsquasi-finitelytotheirreduciblecomponentvi.(C2)Foreachpositiveintegeri,theirreduciblecomponentvi+1mapstoaclosedpointxioftheirreduciblecomponentvi.I
