categories of hyperbolic riemann surfaces

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1、CONFORMALANDQUASICONFORMALCATEGORICALREPRESENTATIONOFHYPERBOLICRIEMANNSURFACESShinichiMochizukiAugust2006Inthispaper,weconsidervariouscategoriesofhyperbolicRiemannsur-facesandshow,invariouscases,thattheconformalorquasiconformalstructureoftheRi

2、emannsurfacemaybereconstructed,uptopossibleconfusionbetweenholo-morphicandanti-holomorphicstructures,inanaturalwayfromsuchacategory.Thetheoryexposedinthepresentpaperismotivatedpartlybyaclassicalresultconcern-ingthecategoricalrepresentationofsobertopolo

3、gicalspaces,partlybypreviousworkoftheauthorconcerningthecategoricalrepresentationofarithmeticlogschemes,andpartlybyacertainanalogywithp-adicanabeliangeometry—ananalogywhichthetheoryofthepresentpaperservestorendermoreexplicit.Contents:Introduction§0.Not

4、ationsandConventions§1.ReconstructionviatheUpperHalf-PlaneUniformization§2.CategoriesofParallelograms,Rectangles,andSquaresAppendix:QuasiconformalLinearAlgebraIntroductionInthispaper,wecontinueourstudy[cf.,[Mzk2],[Mzk10]]ofthetopicofrepresentingvarious

5、objectsthatappearinconventionalarithmeticgeometrybymeansofcategories.Asdiscussedin[Mzk2],[Mzk10],thispointofviewispartiallymotivatedbytheanabelianphilosophyofGrothendieck[cf.,e.g.,[Mzk3],[Mzk4],[Mzk5]],and,inparticular,bythemorerecentworkoftheauthorona

6、bsoluteanabeliangeometry[cf.[Mzk6],[Mzk7],[Mzk8],[Mzk9],[Mzk11],[Mzk12]].Onewaytothinkaboutanabeliangeometryisthatitconcernstheissueofrepresentingschemesbymeansofcategories[i.e.,Galoiscategories]thatcapturecertainaspectsofthe[´etale]topologyofthescheme

7、[i.e.,itsfundamentalgroup].Fromthispointofview,anotherimportant,albeitelementary,exampleoftheissue2000MathematicalSubjectClassification.14H55,30F60.TypesetbyAMS-TEX12SHINICHIMOCHIZUKIofrepresentinga“space”bymeansofa“categoryoftopologicalorigin”isthewell

8、-knownexampleofthecategoryofopensubsetsofasobertopologicalspace[cf.,e.g.,[Mzk2],Theorem1.4;[Mzk10],Proposition4.1].Insomesense,thisexampleistheexamplethatmotivatedtheconstructionofthecategoriesappearinginthepresentpaper.Themainresultsof