semi-graphs of anabelioids

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1、SEMI-GRAPHSOFANABELIOIDSShinichiMochizukiOctober2005Inthispaper,wediscussvarious“generalnonsense”aspectsofthegeometryofsemi-graphsofprofinitegroups[cf.[Mzk3],Appendix],byapplyingthelanguageofanabelioidsintroducedin[Mzk4].Afterprovingcertainbasi

2、cprop-ertiesconcerningvariouscommensuratorsassociatedtoasemi-graphofanabelioids,weshowthatthegeometryofasemi-graphofanabelioidsmayberecoveredfromthecategory-theoreticstructureofcertainnaturallyassociatedcategories—e.g.,“temperoids”[inessence,theanalogu

3、eofaGaloiscategoryforthe“temperedfunda-mentalgroups”of[Andr´e]]and“categoriesoflocalizations”.Finally,weapplythesetechniquestoobtaincertainresultsintheabsoluteanabeliangeometry[cf.[Mzk3],[Mzk8]]oftemperedfundamentalgroupsassociatedtohyperboliccurvesove

4、rp-adiclocalfields.§0.NotationsandConventions§1.Zariski’sMainTheoremforSemi-graphs§2.CommensurabilityProperties§3.TheTemperedFundamentalGroup§4.CategoriesofLocalizations§5.ArithmeticSemi-graphsofAnabelioids§6.TemperedAnabelianGeometryAppendix:Quasi-temp

5、eroidsIndexIntroductionInthispaper,wecontinuetopursuethethemeofcategoricalrepresentationofscheme-theoreticgeometries,whichplayedacentralrolein[Mzk6],[Mzk7],aswellasinthepreviousanabelianworkoftheauthor[e.g.,[Mzk2],[Mzk3],[Mzk5],[Mzk8]].Theoriginalmotiv

6、ationofthepresentworkliesintheproblemoffindinganappropriateandefficientwayofrepresenting,viacategories,thegeometryof2000MathematicalSubjectClassification.14H30,14H25.TypesetbyAMS-TEX12SHINICHIMOCHIZUKI“formallocalizations”ofhyperboliccurvesoverp-adiclocalfi

7、elds.Here,weusetheterm“formallocalizations”torefertothelocalizationsofthep-adicformalcomple-tionofastablelogcurveovertheringofintegersofap-adiclocalfieldobtainedbycompletingalongtheirreduciblecomponentsandnodesofthegeometriclogarithmicspecialfiberspecifie

8、dbysomesub-semi-graphofthe“dualsemi-graphwithcompactstructure”[cf.[Mzk3],Appendix]associatedtothisgeometriclogarithmicspecialfiber.Sincethegeometryofsuchformallocalizationsissubstantiallyreflectedinthegeometryoflocalizationsofthesemi-grap