An Elementary Introduction to Elliptic Curves, II

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1、CCRExpositoryReport34AnElementaryIntroductiontoEllipticCurvesIILeonardS.CharlapRaymondColeyJuly1990AbstractThispaperisthesecondinaserieswhosepurposeistoprovideel-ementaryproofsofthemainresultsonellipticcurveswithemphasisoncurvesoverniteelds.Themainresultsprovedinthispartconcernrati

2、onalmapsbetweenellipticcurvesandWeilreciprocity.WeprovethatallisogeniesarehomomorphismsandtheLowerStarTheorem",aswellasgeneralizedWeilreciprocity.IntroductionThesenotesareacontinuationof[5].Ourgoalin[5]wastoprovideelementaryproofsfortheresultsusedbySchoofin[12].Indoingthisweleftoutt

3、hosepartsofthebasicmaterialonellipticcurvesthatwereunnecessaryforthispurpose.Inthesenoteswewouldliketollinsomeofthosegaps.Ourmaintopicconcernsrationalmapsbetweenellipticcurves,atopicomit-tedin[5].AsecondarytopicisWeilreciprocitywhichweusetofurtherthestudyoftheWeilpairing.Someofthere

4、sultsaboutrationalmapsbetweenellipticcurvessimplycarryoverfromsimilarresultsprovedin[5]aboutrationalmapsfromanellipticcurvetoitself.Manyoftheotherresultswepresentareusuallyprovedusinglargeamountsofcommutativealgebra.Wehavebeenabletondelementaryproofsmuchinthestyleof[5].Acharacterist

5、icoftheseproofsisthatalthoughthetheoremsholdformoregeneralvarieties,ourelementaryproofsonlyworkinthecaseofellipticcurves.UnfortunatelythesituationisdierentinthecaseofWeilreciprocity.Wehavenotbeenabletondaproofthatavoidssomemoreorlessadvancednotions,i.e.,Dedekinddomainsorvaluationth

6、eory.Wehavechosentopresenttherequiredresultwithoutproofinthebodyofthenotesandtorelegatethemoredicultmaterial(andtheproof)toaratherlongAppendix.InthisAppendixwedevelopthenecessarytheorystartingfromaboutthesamelevelrequiredintherestofthenotessoagainyoushouldnothavetoconsultalotofother

7、books.Onesomewhatunsatisfactoryresultofthistreatmentisthatthe1theoremsoftheAppendixholdinsuchgeneralitythattheyincludesomeoftheearlierresults.WewouldliketothankDavidRobbinsforinsightfulremarks,abundantusefulcriticism,andunstintingencouragement.WewouldalsoliketothankNoamElkiesforanumb

8、erofincisive