the intrinsic hodge theory of p-adic hyperbolic curves (icm98)

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1、TheIntrinsicHodgeTheoryofp-adicHyperbolicCurvesbyShinichiMochizuki§1.Introduction(A.)TheFuchsianUniformizationAhyperboliccurveisanalgebraiccurveobtainedbyremovingrpointsfromasmooth,propercurveofgenusg,wheregandrarenonnegativeintegerssuchthat2g−2+r>0.IfXisahyperboliccurveoverthefieldofc

2、omplexnumbersC,thenXgivesriseinanaturalwaytoaRiemannsurfaceX.Asoneknowsfromcomplexanalysis,themostfundamentalfactconcerningsuchaRiemannsurface(duetoK¨obe)isthatitmaybeuniformizedbytheupperhalf-plane,i.e.,X∼=H/ΓdefwhereH={z∈C

3、Im(z)>0},andΓ∼=π1(X)(thetopologicalfundamentalgroupofX)isadi

4、scontinuousgroupactingonH.NotethattheactionofΓonHdefinesacanonicalrepresentationdefρX:π1(X)→PSL2(R)=SL2(R)/{±1}=AutHolomorphic(H)Thegoalofthepresentmanuscriptistosurveyvariouswork([Mzk1-5])devotedtogener-alizingK¨obe’suniformizationtothep-adiccase.First,weobservethatitisnotrealistictoe

5、xpectthathyperboliccurvesoverp-adicfieldsmaybeliterallyuniformizedbysomesortofp-adicupperhalf-planeinthefashionoftheK¨obeuniformization.Ofcourse,onehasthetheoryofMumford([Mumf]),butthistheoryfurnishesap-adicanaloguenotofK¨obe’sFuchsianuniformization(i.e.,uniformizationbyaFuchsiangroup)

6、,butratherofwhatinthecomplexcaseisknownastheSchottkyuniformization.Eveninthecomplexcase,theFuchsianandSchottkyuniformizationsarefundamentallydifferent:Forinstance,asthemoduliofthecurvevary,itsSchottkyperiodsvaryholomorphically,whereasitsFuchsianperiodsvaryonlyrealanalytically.Thisfacta

7、lreadysuggeststhattheFuchsianuniformizationisofamorearithmeticnaturethantheSchottkyuniformization,i.e.,itinvolvesrealanalyticstructures⇐⇒complexconjugation⇐⇒FrobeniusattheinfiniteprimeThus,sinceonecannotexpectap-adicanalogueintheformofaliteralglobaluniformiza-tionofthecurve,thefirstorde

8、rofbusinessistoreinterprettheFuchsianuniformizationinmoreabstracttermsthatgeneralizenaturallytothep-adicsetting.1991MathematicsSubjectClassification:14F30KeywordsandPhrases:hyperboliccurves,p-adic,fundamentalgroup,Hodgetheory1(B.)ThePhysicalInterpretationThefirstandmostobviousapproachis

9、toobs