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《a survey of the hodge-arakelov theory of elliptic curves i》由會員上傳分享，免費在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。
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)ifonemodi?esthe“integralstructure”onthespaceofpolynomialfunctionsinanappropriatefashion(cf.TheoremA).Similarly,inthecaseo
7、fellipticcurvesoverthecomplexnumbers,onecanaskwhetherornotoneobtainsanisometryifoneputsnaturalHermitianmetricsonthespacesinvolved.In[Mzk1],wealsocomputewhatmodi?cationtothesemetricsisnecessarytoobtainanisometry(orsomethingveryclosetoanisometry).Incharacteristiczero,theuniversalextens
8、ionofanellip
