怀尔斯的费马大定理证明

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1、AnnalsofMathematics,141(1995),443-552ModularellipticcurvesandFermat’sLastTheoremByAndrewJohnWiles*ForNada,Claire,KateandOliviaPierredeFermatAndrewJohnWilesCubumauteminduoscubos,autquadratoquadratuminduosquadra-toquadratos,etgeneraliternullamininfinitumultraquadratumpotestatuminduosejus

2、demnominisfasestdividere:cujesreidemonstrationemmirabilemsanedetexi.Hancmarginisexiguitasnoncaperet.-PierredeFermat∼1637Abstract.WhenAndrewJohnWileswas10yearsold,hereadEricTempleBell’sTheLastProblemandwassoimpressedbyitthathedecidedthathewouldbethefirstpersontoproveFermat’sLastTheorem.

3、Thistheoremstatesthattherearenononzerointegersa,b,c,nwithn>2suchthatan+bn=cn.Thisobjectofthispaperistoprovethatallsemistableellipticcurvesoverthesetofrationalnumbersaremodular.Fermat’sLastTheoremfollowsasacorollarybyvirtueofworkbyFrey,SerreandRibet.IntroductionAnellipticcurveoverQissa

4、idtobemodularifithasafinitecoveringbyamodularcurveoftheformX0(N).AnysuchellipticcurvehasthepropertythatitsHasse-Weilzetafunctionhasananalyticcontinuationandsatisfiesafunctionalequationofthestandardtype.IfanellipticcurveoverQwithagivenj-invariantismodularthenitiseasytoseethatallellipticc

5、urveswiththesamej-invariantaremodular(inwhichcasewesaythatthej-invariantismodular).Awell-knownconjecturewhichgrewoutoftheworkofShimuraandTaniyamainthe1950’sand1960’sassertsthateveryellipticcurveoverQismodular.However,itonlybecamewidelyknownthroughitspublicationinapaperofWeilin1967[We]

6、(asanexercisefortheinterestedreader!),inwhich,moreover,Weilgaveconceptualevidencefortheconjecture.Althoughithadbeennumericallyverifiedinmanycases,priortotheresultsdescribedinthispaperithadonlybeenknownthatfinitelymanyj-invariantsweremodular.In1985Freymadetheremarkableobservationthatthis

7、conjectureshouldimplyFermat’sLastTheorem.TheprecisemechanismrelatingthetwowasformulatedbySerreastheε-conjectureandthiswasthenprovedbyRibetinthesummerof1986.Ribet’sresultonlyrequiresonetoprovetheconjectureforsemistableellipticcurvesinordertodeduceFermat’sLastTheorem.*Theworkonthispaper

8、wassu