A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

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1、ASURVEYOFTHEHODGE-ARAKELOVTHEORYOFELLIPTICCURVESIISHINICHIMOCHIZUKIAbstract.ThepurposeofthepresentmanuscriptistocontinuethesurveyoftheHodge-Arakelovtheoryofellipticcurves(cf.[7,8,9,10,11])thatwasbegunin[12].Thistheoryisasortof“Hodgetheoryofellipticcurves”anal-ogo

2、ustotheclassicalcomplexandp-adicHodgetheories,butwhichexistsintheglobalarithmeticframeworkofArakelovtheory.Inparticular,inthepresentmanuscript,wefocusontheaspectsofthetheory(cf.[9,10,11])developedsubsequenttothosediscussedin[12],butpriortotheconference“AlgebraicG

3、eometry2000”heldinNagano,Japan,inJuly2000.Thesedevel-opmentscenteraroundthenaturalconnectionthatexistsonthepairconsistingoftheuniversalextensionofanellipticcurve,equippedwithanamplelinebundle.Thisconnectiongivesrisetoanaturalobject—whichwecallthecrys-tallinetheta

4、object—whichexhibitsmanyinterestingandunexpectedprop-erties.Thesepropertiesallowone,inparticular,tounderstandatarigorousmathematicallevelthe(hithertopurely“philosophical”)relationshipbetweentheclassicalKodaira-SpencermorphismandtheGalois-theoretic“arithmeticKodai

5、ra-Spencermorphism”ofHodge-Arakelovtheory.Theyalsoprovideamethod(undercertainconditions)for“eliminatingtheGaussianpoles,”whicharethemainobstructiontoapplyingHodge-Arakelovtheorytodiophantinegeometry.Finally,thesetechniquesallowonetogiveanewproofofthemainresultof[

6、7]usingcharacteristicpmethods.Itisthehopeoftheauthortosur-veymorerecentdevelopments(i.e.,developmentsthatoccurredsubsequentto“AlgebraicGeometry2000”)concerningtherelationshipbetweenHodge-Arakelovtheoryandanabeliangeometry(cf.[16])inasequeltothepresentmanuscript.1

7、2S.MOCHIZUKIContents1.GeneralIntroduction22.TheCrystallineThetaObject52.1.TheComplexAnalogue.2.2.TheCaseofOrdinaryp-adicEllipticCurves.2.3.IntegralStructuresandConnections.2.4.ComparisonIsomorphismatInfinity.2.5.TheAssociatedKodaira-SpencerMorphism.3.LagrangianGal

8、oisActions174.Hodge-ArakelovTheoryinPositiveCharacteristic23References231.GeneralIntroductionWebeginourgeneralintroductiontothetopicspresentedinthepresentmanus