combinatorial grothendieck conjecture

《combinatorial grothendieck conjecture》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ACOMBINATORIALVERSIONOFTHEGROTHENDIECKCONJECTUREShinichiMochizukiApril2007Westudythe“combinatorialanabeliangeometry”thatgovernstherela-tionshipbetweenthedualsemi-graphofapointedstablecurveandvariousassociatedprofinitefundamentalgroupsofthepointedstablecurve.Althoughmanyresu

2、ltsofthistypehavebeenobtainedpreviouslyinvariousparticularsituationsofinterestunderunnecessarilystronghypotheses,thegoalofthepresentpaperistostepbackfromsuch“typicalsituationsofinterest”andinsteadtoconsiderthistopicintheabstract—apointofviewwhichallowsonetoproveresultsofthistypeinm

3、uchgreatergeneralityunderveryweakhypotheses.IntroductionInthispaper,weapplythelanguageofanabelioids[cf.[Mzk5],[Mzk7]]tostudythe“profinitecombinatorialgrouptheory”arisingfromtherelationshipbetweenthesemi-graphofanabelioidsassociatedtoapointedstablecurve[i.e.,a“semi-graphofanabelioids

4、ofPSC-type”—cf.Definition1.1,(i),belowformoredetails]andacertainassociatedprofinitefundamentalgroup[cf.Definition1.1,(ii)].Inparticular,weshowthat:(i)Thecuspidalportionofthesemi-graphmayberecoveredgroup-theoretical-lyfromtheassociatedprofinitefundamentalgroup,togetherwithcertainnumeric

5、alinformation[roughlyspeaking,thenumberofcuspsofthevar-iousfinite´etalecoveringsofthegivensemi-graphofanabelioids]—cf.Theorem1.6,(i).(ii)Theentire“semi-graphofanabelioidsstructure”mayberecoveredgroup-theoreticallyfromtheassociatedprofinitefundamentalgroup,togetherwithacertainfiltratio

6、n[arisingfromthis“semi-graphofanabelioidsstruc-ture”]oftheabelianizationsofthevariousfinite´etalecoveringsofthegivensemi-graphofanabelioids—cf.Theorem1.6,(ii).2000MathematicalSubjectClassification.Primary14H30;Secondary14H25.PartlysupportedbytheGrant-in-AidforYoungScientists(B),TheMi

7、nistryofEducation,Culture,Sports,ScienceandTechnology,Japan.TypesetbyAMS-TEX12SHINICHIMOCHIZUKIMoreover,weshowthatfromthepointofviewof“weights”[i.e.,logarithmsofabsolutevaluesofeigenvaluesoftheactionoftheFrobeniuselementoftheGaloisgroupofafinitefield],thedatanecessaryfor(i)(respectiv

8、ely,(ii))abovemayberecover