a survey of the hodge-arakelov theory of elliptic curves i

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1、ASurveyoftheHodge-ArakelovTheoryofEllipticCurvesIShinichiMochizukiOctober2000Abstract:ThepurposeofthepresentmanuscriptistogiveasurveyoftheHodge-Arakelovtheoryofellipticcurves(cf.[Mzk1,2])—i.e.,asortof“Hodgetheoryofellipticcurves”analogoustotheclassicalcomplexandp-adicHodgetheories,bu

2、twhichexistsintheglobalarithmeticframeworkofArakelovtheory—asthistheoryex-istedatthetimeoftheworkshopon“GaloisActionsandGeometry”heldattheMathematicalSciencesResearchInstitute(MSRI)atBerkeley,USA,inOctober1999.Sincethen,variousfurtherimportantdevelopmentshaveoccurredinthistheory(cf.[

3、Mzk3,4,5],etc.),butweshallnotdiscussthesedevelopmentsindetailinthepresentmanuscript.Contents:§1.TheDiscretizationofLocalHodgeTheories§1.1.TheMainTheorem§1.2.TechnicalRoots§1.3.ConceptualRoots§1.4.TheArithmeticKodaira-SpencerMorphism§1.5.FutureDirections§2.TheThetaConvolution§2.1.Back

4、ground§2.2.StatementoftheMainTheoremTypesetbyAMS-TEX12SHINICHIMOCHIZUKISection1:TheDiscretizationofLocalHodgeTheories§1.1.TheMainTheoremThefundamentalresultoftheHodge-ArakelovtheoryofellipticcurvesisaComparisonTheorem(cf.TheoremAbelow)forellipticcurves,whichstatesroughlythat:Thespace

5、of“polynomialfunctions”ofdegree(roughly)

6、sticzero(cf.TheoremA).Forellipticcurvesinmixedcharacteristicanddegeneratingellipticcurves,thisstatementmaybemadeprecise(i.e.,therestrictionmapbecomesanisomorphism)ifonemodifiesthe“integralstructure”onthespaceofpolynomialfunctionsinanappropriatefashion(cf.TheoremA).Similarly,inthecaseo

7、fellipticcurvesoverthecomplexnumbers,onecanaskwhetherornotoneobtainsanisometryifoneputsnaturalHermitianmetricsonthespacesinvolved.In[Mzk1],wealsocomputewhatmodificationtothesemetricsisnecessarytoobtainanisometry(orsomethingveryclosetoanisometry).Incharacteristiczero,theuniversalextens

8、ionofanellip