topics in absolute anabelian geometry i

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1、TOPICSINABSOLUTEANABELIANGEOMETRYI:GENERALITIESShinichiMochizukiMarch2012Abstract.Thispaperformsthefirstpartofathree-partseriesinwhichwetreatvarioustopicsinabsoluteanabeliangeometryfromthepointofviewofdevelop-ingabstractalgorithms,or“software”,thatmaybeappliedtoabstractprofinitegroupsthat“just

2、happen”toariseas[quotientsof]´etalefundamentalgroupsfromalgebraicgeometry.Onecentralthemeofthepresentpaperistheissueofunder-standingthegapbetweenrelative,“semi-absolute”,andabsoluteanabeliangeometry.Webeginbystudyingvariousabstractcombinatorialpropertiesofprofi-nitegroupsthattypicallyariseasa

3、bsoluteGaloisgroupsorarithmetic/geometricfundamentalgroupsintheanabeliangeometryofquitegeneralvarietiesinarbitrarydimensionovernumberfields,mixed-characteristiclocalfields,orfinitefields.Theseconsiderations,combinedwiththeclassicaltheoryofAlbanesevarieties,allowustoderiveanabsoluteanabelianalgor

4、ithmforconstructingthequotientofanarithmeticfundamentalgroupdeterminedbytheabsoluteGaloisgroupofthebasefieldinthecaseofquitegeneralvarietiesofarbitrarydimension.Next,wetakeamoredetailedlookatcertainp-adicHodge-theoreticaspectsoftheabsoluteGaloisgroupsofmixed-characteristiclocalfields.Thisallow

5、sus,forinstance,toderive,fromacertainresultcommunicatedorallytotheauthorbyA.Tamagawa,a“semi-absolute”Hom-version—whoseabsoluteanalogueisfalse!—ofthean-abelianconjectureforhyperboliccurvesovermixed-characteristiclocalfields.Finally,wegeneralizetothecaseofvarietiesofarbitrarydimensionoverarbitr

6、arysub-p-adicfieldscertaintechniquesdevelopedbytheauthorinpreviouspapersovermixed-characteristiclocalfieldsforapplyingrelativeanabelianresultstoobtain“semi-absolute”group-theoreticcontructionsofthe´etalefundamentalgroupofonehyperboliccurvefromthe´etalefundamentalgroupofanothercloselyrelatedhyp

7、erboliccurve.Contents:§0.NotationsandConventions§1.SomeProfiniteGroupTheory§2.Semi-absoluteAnabelianGeometry§3.AbsoluteOpenHomomorphismsofLocalGaloisGroups§4.ChainsofElementaryOperationsAppendix:TheTheoryofAlbaneseVarieties2000MathematicalSubjectCla