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1、BOGOMOLOV’SPROOFOFTHEGEOMETRICVERSIONOFTHESZPIROCONJECTUREFROMTHEPOINTOFVIEWOFINTER-UNIVERSALTEICHMULLERTHEORY¨ShinichiMochizukiJanuary2016Abstract.Thepurposeofthepresentpaperistoexpose,insubstantialdetail,certainremarkablesimilaritiesbetweeninter-universalTeichm¨ullertheoryandthetheorysurroundingB
2、ogomolov’sproofofthegeometricversionoftheSzpiroConjecture.Thesesimilaritiesare,insomesense,consequencesofthefactthatboththeoriesarecloselyrelatedtothehyperbolicgeometryoftheclassicalupperhalf-plane.Wealsodiscussvariousdifferencesbetweenthetheories,whicharecloselyrelatedtotheconspicuousabsenceinBogom
3、olov’sproofofGaussiandistributionsandthetafunctions,i.e.,whichplayacentralroleininter-universalTeichm¨ullertheory.Contents:Introduction§1.TheGeometrySurroundingBogomolov’sProof§2.FundamentalGroupsinBogomolov’sProof§3.EstimatesofDisplacementsSubjecttoIndeterminacies§4.SimilaritiesBetweentheTwoTheori
4、es§5.DifferencesBetweentheTwoTheoriesIntroductionCertainaspectsoftheinter-universalTeichm¨ullertheorydevelopedin[IUTchI],[IUTchII],[IUTchIII],[IUTchIV]—namely,(IU1)thegeometryofΘ±ellNF-Hodgetheaters[cf.[IUTchI],Definition6.13;[IUTchI],Remark6.12.3],(IU2)thepreciserelationshipbetweenarithmeticdegrees—
5、i.e.,ofq-pilot×μandΘ-pilotobjects—givenbytheΘ-link[cf.[IUTchIII],DefinitionLGP3.8,(i),(ii);[IUTchIII],Remark3.10.1,(ii)],and(IU3)theestimatesoflog-volumesofcertainsubsetsoflog-shellsthatgiverisetodiophantineinequalities[cf.[IUTchIV],§1,§2;[IUTchIII],Re-mark3.10.1,(iii)]suchastheSzpiroConjecture—ares
6、ubstantiallyreminiscentofthetheorysurroundingBogomolov’sproofofthegeometricversionoftheSzpiroConjecture,asdiscussedin[ABKP],TypesetbyAMS-TEX12SHINICHIMOCHIZUKI[Zh].Putanotherway,theseaspectsofinter-universalTeichm¨ullertheorymaybethoughtofasarithmeticanaloguesofthegeometrictheorysurroundingBo-gomol
7、ov’sproof.Alternatively,Bogomolov’sproofmaybethoughtofasasortofusefulelementaryguide,orblueprint[perhapsevenasortofRosettastone!],forunderstandingsubstantialportionsofinter-universalTeichm¨ullertheory.Theau