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1、APanoramicOverviewofInter-universalTeichm¨ullerTheoryByShinichiMochizuki∗AbstractInter-universalTeichm¨ullertheorymaybedescribedasasortofarithmeticversionofTeichm¨ullertheorythatconcernsacertaintypeofcanonicaldeformationassociatedtoanellipticcurveoveranumberfieldandaprime
2、numberl≥5.Webeginoursurveyofinter-universalTeichm¨ullertheorywithareviewofthetechnicaldifficultiesthatariseinapplyingscheme-theoreticHodge-Arakelovtheorytodiophantinegeometry.Itispreciselythegoalofovercomingthesetechnicaldifficultiesthatmotivatedtheauthortoconstructthenon-sc
3、heme-theoreticdeformationsthatformthecontentofinter-universalTeichm¨ullertheory.Next,wediscussgeneralitiesconcerning“Teichm¨uller-theoreticdeformations”ofvariousfamiliargeometricandarithmeticobjectswhichatfirstglanceappearone-dimensional,butinfacthavetwounderlyingdimensio
4、ns.Wethenproceedtodiscussinsomedetailthevariouscomponentsofthelog-theta-lattice,whichformsthecentralstageforthevariousconstruc-tionsofinter-universalTeichm¨ullertheory.Manyoftheseconstructionsmaybeunderstoodtoacertainextentbyconsideringtheanalogyoftheseconstructionswiths
5、uchclassicalresultsasJacobi’sidentityforthethetafunctionandtheintegraloftheGaussiandistribu-tionovertherealline.Wethendiscussthe“inter-universal”aspectsofthetheory,whichleadnaturallytotheintroductionofanabeliantechniques.Finally,wesummarizethemainabstracttheoreticanddiop
6、hantineconsequencesofinter-universalTeichm¨ullertheory,whichincludeaverificationoftheABC/SzpiroConjecture.Contents§1.Hodge-Arakelov-theoreticMotivation§2.Teichm¨uller-theoreticDeformations§3.TheLog-theta-lattice§4.Inter-universalityandAnabelianGeometry2000MathematicsSubje
7、ctClassification(s):Primary:14H25;Secondary:14H30.KeyWords:ellipticcurve,numberfield,thetafunction,hyperboliccurve,anabeliangeometry,ABCConjecture,SzpiroConjecture.∗RIMS,KyotoUniversity,Kyoto606-8502,Japan.e-mail:motizuki@kurims.kyoto-u.ac.jp2ShinichiMochizuki§1.Hodge-Arak
8、elov-theoreticMotivationTheinter-universalTeichm¨ullertheorydevelopedin[IUTchI],[IUTchII],[IUTchIII],[I
