; TeX output 2000.06.19:2238 sXO2Dt G cmr17The eldgeneratedbzAythep ointsofsmallpEYprime orderonanellipticcurvzAe4do+- cmcsc10LoK*cMerel K`y cmr10andWilliamA.SteinE"V cmbx10In9troQductionO Letkb> cmmi10pbGeaprimenumber.Letqƍ<:kQybekanalgebraicclosureofQ.DenotebyQ( 0er cmmi7pR)the cyclotomic̢subeldofqƍg:Q<ϫgeneratedbythepthroGotsofunity*.LetE`/bGeanellipticcurveoverUUQ(pR),suchthatthepGointsoforderpofE (qƍ:Q)areallQ(pR)-rational.TheoremUU.q|$ ': cmti10Onehasp>1000,p<6orp=13. W*efnotethatthecasep3=7fwastreatedbyEmmanuelHalbGerstadt.,Thepartofthetheoremthatconcernsthecasep!", cmsy103 (moGd4)isgivenin[2].W*epropGosetogivethedetailsthatpGermitourtreatingthemoredicultcaseinwhichp1 (mod4).QW*etreatthisUUlastcasewiththeaidofPropGosition2below,whichisnotpresentinlo}'c.cit. W*eUUdidnotthoroughlystudythecasep=13.1.pW eTrecalltheresultsof[2] Denote5bySٓR cmr72|s( 0(p))5thespaceofcuspformsofweight2forthecongruencesubgroup 0|s(p).DenotembyTthesubringofEnd"ñS2( 0(p))generatedbytheHeckeopGerators.Letf2sS2|s( 0(p))}haveq[٫-expansionu cmex10PލJO! cmsy71%Jn=1anq~q^nW.?WhenisaDirichletcharacter,GdenotebyL(f V;;s)UUtheentirefunctionwhichextendstheDirichletseriesPލ 㐺1% 㐴n=1anq~(n)=n^sF:. LetS1%bGethesetofisomorphismclassesofsupersingularellipticcurvesincharacter-isticNCp.olDenotebyS Jthegroupformedbythedivisorsofdegree0withsuppGortonS .olItisequippGedwithastructureofT-module(deduced,forexample,fromtheactionoftheHeckeUUcorrespGondencesontheberatpoftheregularminimalmodelofX0|s(p)overUUZ). Letٱj2qƍw+:XFp s8JS,ºwhereJS wdenotesthesetofsupGersingularmodularinvqariants.xTW*edenotebyj(SthehomomorphismofgroupsS !qƍ֫:˟FpthatassoGciatestoP EnEm[E ]thequantityUUP 㐟 EOnEm=(jk 8j (E )),UUwherej(E )denotesthemoGdularinvqariantUUofE. OnesaysthatanelementjY2Fp`isanomalousifthereexistsanellipticcurveoverFpwith)moGdularinvqariant)j!thatpossessesanFpR-rationalpointoforderp(thennecessarilyjv=Y2gJS).PropositionL1.|Supp}'ose'thatpiscongruentto1modulo4.xUSupposethatforal lanomalous$j^2/FpYandal lnon-quadr}'aticDirichletcharacters:ի(Z=pZ)^e !/C,Hthere ;1 * sexistsȱtF52Tand'2S VsuchthatL(f V;;1)6=0foreverynewformfڧ2tS2|s( 0(p))and j6(t`)6=0. Thenforal lsub}'groupsCofor}'derpofE (qƍ:Q),thereexistsanel lipticcurveEC overQ(pUW fe ;p ]W)ee}'quippedwithaQ(pUW fe ;p ]W)-rationalsubgroupDC N?oforderp,andthepairs(E ;C )and(ECڱ;DC)ar}'eqƍe:Q7r-isomorphic.Pr}'oof.|!W*eindicatehowthisisdeducedfrom[2].ThehypGothesisj6(t`)6=0!forcest I=2C=pTand,aWfortiori,t6=0;inaddition,thenon-vqanishinghypGothesisontheL-seriesforcesUUthehypGothesisHpR()oflo}'c.cit,introGduction. AccordingtoCorollary3ofPropGosition6oflo}'c.ncit,E\haspotentiallygoodreductionat QtheprimeidealPΫofZ[pR]thatliesabGove QponceweknowthathypGothesisHpR()issatised6wforallnon-quadraticDirichletcharactersofconductorp(thisisthecasebyhypGothesis). DenoteAbyjcthemoGdularinvqariantofthebGeratPUoftheNGeronmoGdelofE .AccordingtoUUthecorollaryofPropGosition15oflo}'c.cit.,jisanomalous. LetNCjbGeasubgroupofE (qƍ:Q)oforderp. !ByassumptionEx۫isanellipticcurveover>'Q(pR)whosepGointsoforderpareallQ(p)-rational,x\sothepair(E ;C )denesaQ(pR)-rationalUUpGointPofthemodularcurveX0|s(p). Consider5themorphismF5=t O \ cmmi5(seelo}'c.cit.g section1.3).Whenj6(t`)6=0,;xthis5isaformalimmersionatthepGointP:=f$ cmbx7F puQ,İaccordingtolo}'c.b$cit.,PropGosition4.5ThehypothesisthatcL(f V;;1)6=0foreverynewformfڧ2tS2|s( 0(p)),translatesintoL(tJ0(p);;1)6=0,which|inturnimpliesthatthe-isotypicalcompGonentoftJ0|s(p)(Q(pR))isnite(thisisKato's+theorem,seethediscussioninlo}'c.dcit.Vdsection1.5).W*ecanthenapplyCorollary1ofPPropGosition6oflo}'c.4cit.pFThisprovesPthatPbisQ(pUW fe ;p ]W)-rational;RSthistranslatesintotheconclusionUUofPropGosition1.V2.pATlemmaabQoutellipticcurv9es{PropositionU2.Rr|L}'et=mpbeaprimenumberthatiscongruentto1modulo4.|LetEbeanel lipticcurveoverqƍm:Q3.q,Ther}'eexistsacyclicsubgroupCoforderpofE (qƍ:Q)[p],2suchthatforal l)ellipticcurvesE ^0woverQ(pUW fe ;p ]W)%Re}'quippedwithaQ(pUW fe ;p ]W)!)-rationalsubgroupC ^0U,6thepairs(E ;C )and(E^0aƱ;C ^0U)ar}'enotqƍe:Q7r-isomorphic.Pr}'oof.p|R@W*eproGcedebycontradiction.pLetE0γbGeanellipticcurveoverQ(pUW fe ;p ]W)thatisqƍ$:Qisomorphic toEY(itexistsbyhypGothesis).BW*erstshowthatthesubgroupGalf(qƍ:Q=Q(pUW fe ;p ]W) )actsUUbyscalarsontheFpR-vectorspaceE0|s(qƍ:Q)[p]. DenotebyX (p)thealgebraiccurveoverQthatclassies(nelysincep>2)ofgeneralizedlellipticcurvesequippGedwithanembGedding'E:](Z=pZ)^2Z $!E [p].Consider ^rthejEmorphism(ofalgebraicvqarietiesoverjEQ):X (p) !X0|s(p)^PrZ cmr51 (F p2Ԯ):thatto(E ;[٫)assoGciatesQ t2P 1 (F p2Ԯ)/(E ;[٫(t))./DenotebyXɫ(p)theimageof.Thecovering(ofalgebraic }vcurvesoverQ)^0T:RX (p)` !`Xɫ(p)isGaloiswithGaloisgroupisomorphictoF^፴pm(theactionUUbGeingdeducedfromthescalaractionofF^፴ponE [p]). Letsf0٫bGeanembedding(Z=pZ)^2C `!E0|s[p].&xDenotebyPtheqƍE+:Q-rationalpointofX (p)deduced>from(E0|s;0).IIts>imagebyisQ(pUW fe ;p ]W)-rationalbyhypGothesis.IW*ehavethenacharacter5:݇Gal}(qƍ:Q=Q(pUW fe ;p ]W)) =X! F^፴p*suchthat[٫(Pc) = z():Pī(|2Gal(qƍ:Q=Q(pUW fe ;p ]W))).gInotherUUwords,GalW(qƍ:Q=Q(pUW fe ;p ]W))actsbyscalarsonE0|s(qƍ:Q)[p]viathecharacter z. ;2 M s BecauseGoftheW*eilpairing,r z^2coincideswiththecyclotomiccharactermoGdulop, anditfactorsthroughGal5(Q(pR)=Q(pUW fe ;p ]W)).3KBut,%whenp1 (moGd4),thegroupGal(Q(pR)=Q(pUW fe ;p ]W) )gisofevenorder,andthecharactersmoGdulopformagroupgener-atedz3bythereductionmoGdulopofthecyclotomiccharacter,kwhich,therefore,canz3notbGeaUUsquare.Ut3.pV ericationToftheh9ypQothesisofProposition1 LetS̱pbGeaprimenumber.qDInSthissectionweexplainhowtouseacomputertoverifythat sthehypGothesisofProposition1aresatised.XW*ehave scarriedoutthisvericationforp=11UUand13