MODULAR FORMS ELLIPTIC CURVES AND THE ABC CONJECTURE Dorian Goldfeld Dedicated to Alan - PDF document

MODULARFORMS,ELLIPTICCURVESANDTHEABCDorianGoldfeldDedicatedtoAlanBakerontheoccasionofhissixtiethbirthday.1.TheABCABC{conjecturewas¯rstformulatedbyDavidMasserandJosephOsterl¶e(see(see)in1985.Curiously,althoughthisconjecturecouldhavebeenformulatedinthelastcentury,itsdiscoverywasbasedonmodernresearchinthetheoryoffunction¯eldsandellipticcurves,whichsuggeststhatitisastatementaboutrami¯cationinarithmeticalgebraicgeometry.TheABC{conjectureseemsconnectedwithmanydiverseandwellknownproblemsinnumbertheoryandalwaysseemstolieontheboundaryofwhatisknownandwhatisunknown.Wehopetoelucidatethebeautifulconnectionsbetweenellipticcurves,modularformsandtheABCConjecture(ABCLetA;B;Cbenon{zero,pairwiserelativelyprime,rationalintegers=0ABCtobethesquarefreepartofABC.Thenforevery²�thereexistssuchthatmax(·AweakerversionoftheABC{conjecture(withthesamenotationasabove)maybegivenasfollows.Conjecture(ABC)(weak).Forevery²&#x-269;&#x 000;thereexistssuchthatABC ·Oesterl¶e,e,showedthatifwede¯ne)=infmax( N1+² SupportedinpartbyagrantfromtheNSF.TheauthorwouldliketothankIrisAnshelandShu-WuZhangformanyhelpfulconversations. )=Thebestresultinthisdirection,knowntodate,seemstobeinthepaperofStewartandTijdemanTijdeman.Theyprovethatforany¯xedpositivethereexistin¯nitelymanysolutionsof=0A;B)=1;NABCp�max(�N logN loglogIn1996AlanBakererproposedamorepreciseversionoftheABCConjecture(ABC(Baker)Forevery²�thereexistsaconstantsuchthatmax(·wheredenotesthenumberofdistinctprimefactorsofABCThisconjecturewouldgivethebestlowerboundsonecouldhopeforinthetheoryoflinearformsinlogarithms.InthesamepapererBakerattributestoGranvillethefollowingintriguingconjecture.Conjecture(ABC(Granville)Letdenotethenumberofintegerslessthanorequaltothatarecomposedonlyofprimefactorsof.Thenmax(NAtpresentthebestknownresultsinthedirectionoftheABC{conjectureareexponen-tialinsmallpowersofandareobtainedusingmachineryfromBaker'stheoryoflinearformsinlogarithms.The¯rstsuchresultwasobtainedbyStewartandTijdemanTijdemanin1986.Theorem1.LetA;B;CbepositiveintegerssatisfyingC;A;B)=1;C&#x-266;&#x 000;Thenthereexistsaconstant·&#x-266;&#x 000;(e®ectivelycomputable)suchthatCeThiswasimprovedin1990byStewartandYuutoTheorem2.LetA;B;CbepositiveintegerssatisfyingC;A;B)=1;C&#x-269;&#x 000;Thenthereexistsaconstant·&#x-269;&#x 000;(e®ectivelycomputable)suchthatCe 3+· loglogTheconstant hasrecentlybeenimprovedbyYuto 3:2 2.ApplicationsoftheABCInordertoshowtheprofoundimportanceoftheABC{conjectureinnumbertheory,weenumeratesomeremarkableconsequencesthatwouldfollowiftheABC{conjecturewereproven.Theorem3.AssumetheABC{conjecture.Fix²,and¯xnon{zerointegers®;¯;°.Thenthediophantineequation®x¯y°z=0hasonly¯nitelymanysolutionsinintegersx;y;z;r;s;txyz=0x;y)=(x;z)=(y;z)=1;r;s;t&#x 000; r+1 s+1 Moreover,thenumberofsuchsolutionscanbee®ectivelycomputedprovidedtheconstantintheABC{conjectureise®ective.Proof:®x;B¯y;C°zWithoutlossofgenerality,wemayassumethatisthemaximumofABC{conjecture(·)thenimpliesthat°z·¢j®¯°xyzj·jitimmediatelyfollowsthatj· ®¯¯1 ¢j j· ¯¯¯¯¯1 ¢j Pluggingtheseboundsinto(21)andtakingtherootofbothsides,weobtain· r+1 s+1 ·wheretheimpliedconstantscanbee®ectivelycomputedanddependatmoston®;¯;°:Theinequality(22)plainlyimpliesthattherecanbeatmost¯nitelymanyintegerssatisfying(2Withoutlossofgenerality,wemaynowassumethatj·jItfollowsthatj· ®¯¯1 ¢j WritingtheABC{conjectureintheform¯y·¢j®¯°xyz andusingthepreviouslyprovedfactthatliesina¯niteset,itfollowsfrom(23)and4)that r+1 Thus,alsoliesina¯niteset.WritingtheABC{conjectureintheform¯x·¢j®¯°xyzandnotingthatmustbe2,itimmediatelyfollowsthatx sothatalsomustlieina¯niteset.Finally,weagainusetheABC{conjecturetowrite®x¯y°zx;y;zlieina¯niteset.Thus,r;s;talsomustlieina¯niteset.In1988Silvermanermanprovedthefollowingtheorem.Theorem4.AssumetheABC{conjecture.Thenthereexistin¯nitelymanyprimessuchthat1(modIn1991ElkiesElkiesprovedthattheABC{conjectureimpliestheMordellconjecture(thiswas¯rstprovedbyFaltingsaltings)whichstatesthateveryalgebraiccurveofgenusde¯nedoverhasonly¯nitelymanyrationalpoints.AnotherinterestingapplicationisduetoGranvilleville1998.Heprovedthefollowing.Theorem5.Letbeapolynomialwithintegercoe±cientswhichisnotdivisiblebythesquareofanotherpolynomial.Thenthereexistsaconstantsuchthat)issquarefree!1ThemostrecentapplicationofABCisduetoGranvilleandStarkStark(1999).TheyshowthataverystronguniformABC{conjecturefornumber¯eldsimpliestherearenoSiegelzerosforDirichletL{functionsassociatedtoimaginaryquadratic¯elds d0,andissquarefreewith1(4)orVojta((,1987)¯rstshowedhowtoformulatetheABC{conjecturefornumber¯elds.K=beanumber¯eldofdegreewithdiscriminant.Foreachprimeidealde¯neavaluationjjnormalizedsothat=NormK= Foreachembedding de¯neavaluationjjby ,andwherejjdenotestheordinaryabsolutevalueon.For;®;:::;®wede¯netheheight:;:::;®)=;:::;wheretheproductgoesoverallplaces(primeidealsandembeddings).Wealsode¯netheconductor:;:::;®)=denotesthesetofprimeidealssuchthat;:::;arenotallequal.Wecannowstate.ABCLet®;¯;°=0Thenforevery²�thereexistssuchthat®;¯;° ®;¯;°AssumingtheuniformABC{conjectureStarkandGranvilleobtainedthefollowinglowerboundfortheclassnumber)of ¡d):h(¡d)¸³¼ 3+o(1)´p d or0 3.EllipticcurvesoverQ(GlobalMinimalModels).Anellipticcurveovera¯eldisaprojectivenon{singularalgebraiccurveofgenusonede¯nedoverK;furnishedwitha{rationalpoint.EverysuchcurvehasageneralizedWeierstrassequationormodeloftheform:K;=16)with{rationalpoint(pointatin¯nity)giveninprojectivecoordinatesby(0Itwas¯rstprovedbyMordellMordell(forK=Q)andgeneralizedbyWeileiltoarbitrarythatthe{rationalpointson))forma¯nitelygeneratedabeliangroup(Mordell{Weilgroup).TherankoftheMordell{Weilgroupisde¯nedtobethenumberofgeneratorsofin¯niteorder. FollowingTate'sformulaireulaire,wede¯ne+4+2+4+4¢=+9where¢denotesthediscriminantofbeanotherellipticcurvede¯nedover.ThenE;Eareisomorphicifandonlyifthereisacoordinatechangeoftheformr;yr;s;twhichtransforms.Inthiscasewehavej;Foreachrationalprimenumber,considerthelocal¯eld.Letdenotethevaluationnormalizedsothat)=1denotestheringof{adicintegers.Fixarationalprime.Amongallisomorphicmodelsofagivenellipticcurvede¯nedover,wecan¯ndonewhereallcoe±cients,andthusiseasilyseenbythecoordinatechangex;ywhichsendseachChoosingtobeahighpowerofdoeswhatwewant.Sinceisdiscrete,wecanlookforanequationwith(¢)assmallaspossible.(GlobalMinimalModel).LetbeanellipticcurveoverwithWeierstrassequationgivenbyisde¯nedtobeminimalat=1isminimal(amongallisomorphicmodelsoverWede¯netobeaglobalminimalmodelifisminimalateveryprime 4.ConjectureswhichareequivalenttoABCbeanellipticcurvede¯nedover(globalminimalmodel)withWeierstrassThenassociatedtowehavetwoimportantinvariants:Discriminant¢=+9)isnonsingular;)hasanodalsingularity;2+±;)hasacuspidalsingularity,with=0if=2Therecipefortheconductorwas¯rstshownbyOggOggin1967.AnalgorithmtocomputeinallcaseswasproposedbyTateinalettertoCassels(see(see).Anellipticcurvewhichneverhasbadreductionofcuspidaltypeissaidtobesemistable,andinthiscaseisalwaysthesquarefreepartof¢.ThisisthebridgebetweenthetheoryofellipticcurvesandtheABC(Szpiro,1981)Letbeanellipticcurveoverwhichisaglobalminimalmodelwithdiscriminantandconductor.Thenforevery²�,thereexistssuchthat·WeshowthatSzpiro'sconjectureaboveisequivalenttotheweakABC{conjecture.LetA;B;Cbecoprimeintegerssatisfying=0andABC=0ABCConsidertheFrey{HellegouarchcurveAminimalmodelforhasdiscriminant(ABCandconductorcertainabsolutelyboundedintegerss;t;(seeFreyrey).PluggingthisdataintoSzpiro'sconjectureimmediatelyshowstheequivalence.AnotherconjectureequivalenttoaversionoftheABC{conjectureisthedegreeconjec-ture.Let¡)denotethegroupofmatricesabcdSL)with0(modandset)tobethecompacti¯edRiemannsurfacerealizedasthequotientoftheupper{halfplaneby¡Anellipticcurvede¯nedoverissaidtobemodularifthereexistsanon{constantcoveringmapE;normalizedsothat)=0theoriginon.Itisnowknown(byworkofChristopheBreuil,BrianConrad,FredDiamond,RichardTaylor,andAndrewWiles)thatevery ellipticcurveoverismodular.Thedegreeconjectureconcernsthegrowthinofthetopologicaldegreeofthemap!1DegreeConjecture.(Frey1987)Forevery²�thereexistssuchthat·FreyreyprovedthatsomeboundforthedegreeimpliesaweakversionoftheABCconjecture.ItwasshownbyMai{Murtyy(1994)thattheABC{conjectureimpliesthedegreeconjectureforallFrey{HellegouarchcurvesandbyMurtyyin1996thatthedegreeconjectureimpliestheABC{conjecture.TheseresultsuseworkofWiles{Diamond[Wi],[D]aswellasworkofGoldfeld{Ho®stein{Liemann{Lockhart[G-H-L-L]onthenon{existenceofSiegelzeroson(3)whicharesymmetricsquareliftsfromABCconjectureisalsointimatelyrelatedtothesizeoftheperiodsoftheFrey{HellegouarchcurveBA:Thiscurvehastwoperiods:=2 p =2 p PeriodConjecture.(Goldfeld1988)LetbetheFrey{HellegouarchcurvewithA;BA;B)=1,andBA:Letdenotetheconductorof.Thenforevery²&#x-500;&#x 000;,thereexistssuchthat&#x-500;&#x 000;· ItwasshownininthattheperiodconjectureimpliestheweakABCThe¯nalconjectureweshalldiscuss(whichisequivalenttoABC)isaconjectureonthesizeoftheShafarevich{Tategroup[Sha],[T2])ofanellipticcurvede¯nedover.Itwasonlyrecently(see(see,[R2],[Kol1],[Kol2],[Kol3])thatwasproved¯niteforasingleellipticcurveandthisexplainswhytheABCconjectureissointractable.Weshallnowde¯nefrom¯rstprinciples.beaset.Wesayagroupactsonwithleftset{actionifforallG;xX;thebinaryoperation,andsatis¯es(forallg;gG;x)theidentities:istheidentityindenotesthegroupoperationin.Ifisanabeliangroupwithinternaloperation+,wesayactsonwithleft{groupactionisaleftset{actionwhichalsosatis¯es)=foralla;a beanabeliangroupwithinternaloperation+andletbeanothergroupwhichactsonwithleftgroup{action.Wede¯neG;A)tobethegroupofallfunctions(cocycles)whichsatisfythecocyclerelation)=)+denotesthegroupoperationin.ThesubgroupG;A)ofcoboundariesconsistsofallcocyclesoftheform.Wede¯nethe¯rstcohomologygroupG;A)tobethequotientgroupG;A)=G;AG;AFixanabeliangroupandanothergroupactingonwithaleftgroup{.AprincipalhomogeneousactionforG;A;isaleftset{actionwhichsatis¯estheidentityforalla;aWenowde¯neanequivalencerelationonthesetofprincipalhomogeneousactions.TwoprincipalhomogeneousactionsG;A;aresaidtobeequivalentforall,all,andsome¯xedWCG;A)denotethesetofequivalenceclassesofprincipalhomogeneousactionsfor(G;A;).WewillshowthatWCG;A)(Weil{Ch^ateletgroup)isinfactagroupbydemonstratingthatthereisabijection(ofsets)WCG;AG;AFirst,ifisaprincipalhomogeneousactionfor(G;A;)thenforsome¯xedwehavethat):=G;A)because)=()=)+Further,ifwereplacebyforanythenthecocyclechangesto)+whichisequivalentto)modG;AThuseachprincipalhomogeneousactiontoauniqueelementofG;AOnealsoeasilychecksthatequivalenthomogeneousactionsmaptothesameelementofG;AFinally,toshowthesurjectivity,letG;ADe¯nealeftactionby)+forallIfwechange)totheequivalentcocycle)+thenthisgivesrisetoanewgivenby)+.Clearlyareequivalentprincipalhomogeneousactions.TheidentityelementinthegroupWCG;Aistheequivalenceclassofallactionsequivalentto.Aprincipalhomogeneousactionisequivalenttoifandonlyifhasa¯xedpointundertheleftset{action,i.e.,ifandonlyifthereexistsforall(clearlytruebecauseisthezerococycle). Inordertoexplicitlyrealizeprincipalhomogeneousactions,itisoftenconvenienttoconsideraset)whereisabijection.Thebijectionleadstoatransitiverightset{actionof)andde¯nedby)forallandallInthissituation,theexistenceofaprincipalhomogeneousactionG;A;)givesrisetoaleftset{actionde¯nedbyforall,and.Onechecksthat=(forallThushasthepropertiesofaprincipalhomogeneousspace(see(see),i.e.,thereisarightset{actionofandaleftprincipalhomogeneousactionTode¯netheShafarevich{Tategroupforanellipticcurvede¯nedoverwe¯rstconsidertheWeil{Ch^ateletgroupWCG;E))where=Gal()whichactson),thegroupof{rationalpointson.ElementsofWCG;E))canberealizedascurvesofgenusone,denoted,de¯nedoverwhicharebirationallyequivalenttoovertogetherwithanappropriateactionNotethatacurveofgenusonede¯nedovermaynothaveapointin.Letbesuchabirationalequivalence.ThenforanythemapgÁisofthetype(see(see)a!a+c(g)witha2E(¹Q),c(g)2Z1(G;Eandadditionabovedenotingadditionontheellipticcurve.Therightactionof)on)isthengivenbytranslation(ontheellipticcurve)for;a;a).Theleftaction)isgivenby))with)whichisinducedfromthecocycle)associatedtothebirationalequivalence.TheTate{Shafarevichgroupoverisde¯nedtobethesubgroupofWCG;E))associatedtocurvesabovewhichhaveapointinandinevery{adic¯eld,orequivalently,theelementsWCG;E))whichhavetrivialimagesinWC;E)andWC;E),where=Gal()forall¯niteprimes,and=Gal().Ifhasapointinthenbytheremarkabove,theactionisintheidentityclassofprincipalhomogeneousactions.Thus,measurestheobstructiontotheHasseprinciple(Hasse'sprinciplestatesthatifacurvehaspointsinandinevery-adic¯eldthenithasapointininde¯nedthenotionofacompaniontoanellipticcurveasaofgenusonewhichisisomorphictoandoverforallprimes.TheShafarevich{Tategroupmaythenbede¯nedasthesetofisomorphismclassesoverofcompanionsof,eachendowed(asabove)withthestructureofaprincipalhomogeneousspace.ConjectureI.(BoundforLetbeanellipticcurvede¯nedoverofconductorwithShafarevich{Tategroup.Thenforevery²�,thereexistssuchthat· !1 OneofthemostremarkableconjecturesinnumbertheoryistheBirch{Swinnerton{DyererwhichrelatestherankoftheMordell{WeilgroupofanellipticcurveandtheShafarevich{Tategroupoftothespecialvalueat=1oftheHasse{WeilL{functionassociatedtotoforthede¯nitionoftheHasse{WeilL{function).ItwasshowninGoldfeld{Szpiro(1995)(1995)thatassumingtheB{S{D(forrank0curvesonly),theaboveconjecturedboundforimpliesthefollowingversionoftheABCABC NIfonefurtherassumesthegeneralizedRiemannhypothesis(fortheRankin{Selbergzetafunctionassociatedtotheweight cuspformcomingfromtheShintani{Shimuralift)thenitwasalsoshownininthattheaboveconjecturedboundfor(forrank0curvesonly)impliestheweakABCABC NActually,similarimplicationscanbeobtainedfromthefollowingweakerconjecture.ConjectureII.(AverageBoundforLetbeanellipticcurveofconductora;b.Forasquare{freeinteger,de¯nethetwistedcurvewithMordell{WeilrankandShafarevich{TategroupThenthereexistsaconstantc&#x-287;&#x 000;andforevery²&#x-287;&#x 000;thereexistsaconstantsuchthatqN· !1WenowsketchtheproofthatConjectureIIplusB-S-DimpliesaversionoftheABCconjecture.TheB-S-DconjecturestatesthattheHasse{WeilL{function)ofanellipticcurvede¯nedoverhasazerooforder=rankoftheMordell-Weilgroupof)andthattheTaylorseriesof)about=1isgivenby)=¢jj¢vol( Here­iseithertherealperiodortwicetherealperiodof(dependingonwhetherornot)isconnected),istheorderoftheTate{ShafarevichgroupofE=,vol(isthevolumeoftheMordell{WeilgroupfortheN¶eron{Tatebilinearpairing,istheorderofthetorsionsubgroupofE=,and=1unlessbadreductionatinwhichcaseistheorder)(Here)isthesetofpointsreducingtonon-singularpointsof)(see(see).Itisknownthat256(Mazur1977 andthatvol())=1if=0Sointherank=0situation,alowerboundfor(1)togetherwithanupperboundfortheorderofwouldimplyalowerboundfortheperiod­.IfthelowerboundfortheperiodwerestrongenoughtogivetheperiodconjecturewewouldgetaversionofABC.Itisenoughtodothisforonetwistedcurvesincetheperiodchangesby .Now,byatheoremofWaldspurger(see[Wa],[Koh]onecan¯ndenoughtwists(qNc&#x-290;&#x 000;&#x-290;&#x 000;1)ofwithMordell{Weilrankzerowheretodowhatwewant.Inthecase0c1itisnecessarytousethegeneralizedRiemannhypothesis.ConjectureIcanbeprovedforCMellipticcurveswith=01728(weactuallygetbetterbounds).Thiswas¯rstdoneinGoldfeld{Lieman(1996)(1996)(seeTheorem6below).ForCMellipticcurvesde¯nedoverweexpect.LetbeaCMellipticcurvede¯nedoverwithShafarevich{Tategroup.ThenN =01728)N =0)N =1728)Theconstantdependsonlyonandise®ectivelycomputable.Theorem6.LetbeaCMellipticcurvede¯nedoverwithMordell{WeilrankandShafarevich{Tategroup.ThenN =01728)N =0)N =1728)Theconstantdependsonlyonandise®ectivelycomputable.ThisresultusesthedeepworkofK.RubinRubin(wheretheB-S-DconjectureisprovedforCMellipticcurvesoverofMordell{Weilrank0),togetherwiththeupperboundsforspecialvaluesof{functionsobtainedbyDuke{Friedlander{Iwaniecaniec.x5.LargeShafarevich{Tategroups.groups.in1964showedthattheTate{Shafarevichgroupofanellipticcurveovercanbearbitrarilylarge.Casselsmethodactuallyshowsthatthereexista¯xedconstantc�0andin¯nitelymanyintegersforwhichthereexistanellipticcurveofconductor,de¯nedover,with��N loglog Thisresultwasobtainedbyadi®erentmethodbyKramerKramerin1983.AssumingtheBirch{Swinnerton{Dyerconjecture,Mai{Murtyyshowedin1994thattherearein¯nitelymanyellipticcurves,de¯nedoverforwhich��N ThiswasimprovedbyDeWegerin1996[We]whoshowedthat��N in¯nitelyoften,undertheassumptionofboththegeneralizedRiemannhypothesisandtheBirch{Swinnerton{Dyerconjecture.TheconnectionbetweentheABC{conjectureandthegrowthofallowsonetocon-structellipticcurveswithlargeShafarevich{TategroupsfrombadABCexamples.B.DeWeger(1997)[We]hasfound11examplesofcurveswith N:Cremona(1993)(1993)(byothermethods)hadalsofoundseveralsuchcurves.ThebestknownexampleofaFrey{Hellegouarchcurvewithlarge+2)comingfromtheABC=3;B=2;C=23duetoReyssatwith=15042Inthiscase: p =06.ModularSymbols.)=¼inzbeaholomorphicHeckenewformofweighttwofor¡normalizedsothat(1)=1For)wede¯nethemodularsymbol°;f&#x-269;&#x 000;¼i°¿whichisindependentofofQ[fShimura(1973)(1973)showedthatthemodularsymbolisahomomorphismof¡)intotheperiodlatticeassociatedwithspeci¯cally,ifthecoe±cients)alllieinthenthehomomorphismisintotheperiodlatticeofanellipticcurve,i.e.,°;f&#x-290;&#x 000;;mm;­1;­2]:Forabcd,de¯netheheightof)tobethemaximumofModularSymbolConjecture.(Goldfeld1988)Let°;f&#x-290;&#x 000;asabove.;mhaveatmostapolynomialgrowthin Itisnothardtoshowthatthereexists·�0suchthat°;f&#x-269;&#x 000;islargerthanwithheightNTheaboveconjecturethenimpliesalowerboundfortheperiodswhichcanbeused(viatheperiodconjecture)toproveaversionoftheABCconjecture.Alternatively,thespecialvalue(1)(attheB-S-Dpoint)canbeexpressedasalinearcombinationofmodularsymbolswhichalsoprovidesabridgetothegrowthofInordertostudythegrowthpropertiesofmodularsymbols,wehaveintroducedanewtypeofEisensteinseriestwistedbymodularsymbols,whichisde¯nedasfollows:z;s)=°;f&#x-269;&#x 000;Im(°zNowisnotanautomorphicform,butitsatis¯es(forall))thefollowingautomorphicrelation°z;s)=z;s°;f&#x-269;&#x 000;Ez;sz;s)=Im(°zistheclassicalEisensteinseries.WehaveshownwnthatE¤(z;s)hasameromorphiccontinuationtotheentirecomplex{planewithonlyonesimplepoleat=1withresiduegivenby ¼N1+ )=2¼idw:Asaconsequence,itfollows(see(see)thatfor¯xedM;N!1°;f&#x-269;&#x 000;e x»3 ¼N ThisresultwasrecentlyimprovedbyO'SullivananwhoexplicitlyevaluatedtheerrortermasafunctionofM;Nandfoundexponentialdecayin.Anintriguingpossibilityistochoosesothat)ispreciselytherealperiodoftheassociatedellipticcurve.Theproblemisthatthereisalotofcancellationinthemodularsymbolssothattheasymptoticrelation(61)givesnoinformationinthedirectionofthemodularsymbolsconjecture.Itwouldbeofgreatinteresttotrytoconstructothersuchserieswhichhavepositivecoe±cientsandhaveasimplepoleat=1withresiduegivenbytheperiodofanellipticcurve.Iftheperiodweretoosmall,suchserieswouldhavetohaveaSiegelzero. 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