the intrinsic hodge theory of p-adic hyperbolic curves (nara 1998)

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1、TheIntrinsicHodgeTheoryofp-adicHyperbolicCurvesbyShinichiMochizukiContents:§1.UniformizationTheoryasaHodgeTheoryatArithmeticPrimes(A.)UniformizationasaCatalogueofRationalPoints(B.)“Intrinsic”HodgeTheories(C.)CompletionatArithmeticPrimes§2.ThePhysicalAspect:EmbeddingbyAutom

2、orphicForms(A.)TheComplexCase(B.)TheArithmeticFundamentalGroup(C.)TheMainTheorem(D.)ComparisonwiththeCaseofAbelianVarieties§3.TheModularAspect:CanonicalFrobeniusActions(A.)TheComplexCase(B.)Teichm¨ullerTheoryinCharacteristicp(C.)Canonicalp-adicLiftings§1.UniformizationTheo

3、ryasaHodgeTheoryatArithmeticPrimes(A.)UniformizationasaCatalogueofRationalPointsWebeginourdiscussionbyposingthefollowingelementaryproblemconcerningalge-braicvarietiesoverthecomplexnumbers(where,roughlyspeaking,an“algebraicvarietyoverthecomplexnumbers”isageometricobjectdefin

4、edbypolynomialequationswithcoefficientswhicharecomplexnumbers):Problem:GivenanalgebraicvarietyZoverC,itispossibletogivesomesortofnaturalexplicitcatalogueoftherationalpointsZ(C)ofZ?1Togainasenseofwhatismeantbytheexpression“anaturalexplicitcatalogue,”itisusefultobeginbythinkin

5、gaboutsomebasicexamples.Perhapsthesimplestnontrivialexamplesofalgebraicvarietiesareplanecurves,i.e.,subvarietiesofA2(two-dimensionalCaffinespaceoverC)definedbyasinglepolynomialequationf(X,Y)=0intwovariables.Inthiscase,thesetofrationalpointsZ(C)ofthecorrespondingvarietyZisgive

6、nbyZ(C)={(x,y)∈C2

7、f(x,y)=0}Moreover,wecanclassifyplanecurvesbythedegreeofthedefiningequationf(X,Y).WethenseethattheresultingsetsZ(C)maybeexplicitlydescribedasfollows:(1.)TheLinearCase(deg(f)=1):Uptocoordinatetransformations,thisisthecasegivenbytheequationf(X,Y)=X.Inthiscase

8、,wethenobtainanexplicitcatalogueoftherationalpointsby:∼(0,?):C→Z(C)(i.e.,mappingz∈Cto(0,z)∈Z(C)).(2.)TheQuadraticCase(deg(f)=2):Uptocoordinatetransformations(andrulingoutdegeneratecases),weseethatthisisessentiallythecasewheretheequationf(X,Y)=X·Y−1.Inthiscase,anexplicitcat

9、alogueisgivenbytheexponentialmap:exp:C→Z(C)=C×(Infact,themapmaybedefinedintrinsically,with