; TeX output 2003.04.25:1528 y ? ㎍1ݛDt G cmr17The eldgeneratedbzAythep ointsofsmallpZlprime orderonanellipticcurvzAe6 1- cmcsc10LoK*cMerelK`y cmr10andWilliamA.SteinH $ E"V cmbx10In9troQduction 3 ELetqƍ*:XQUubGeXanalgebraicclosureofQ,andforanyprimenumbGer b> cmmi10p,denoteby $ EQ( 0er cmmi7pR)UUthecyclotomicsubeldofqƍ':QN5generatedbythepthroGotsofunity*.$ ETheorem..| ': cmti10L}'et[ٵpbeaprime.kIfthereexistsanel lipticcurveEfoverQ(pR)$ Esuchthatthep}'ointsoforderpofE (qƍ:Q)areal lQ(pR)-rational,thenp_\=2;3;5;13$ Eorp>1000.3 ETheJcasep=7JwastreatedbyEmmanuelHalbGerstadt.SnThepartofthetheorem$ Ethatconcernsthecasep !", cmsy103(moGd4)isgivenin[3].InthispapGer,Nwegivethe$ Edetails0thatpGermitourtreatingthemoredicultcaseinwhichp4}1(mod4).$ EW*eUtreatthislastcasewiththeaidofPropGosition2below,UwhichUisnotpresentin$ Elo}'c.(cit..~TheYcasep8=13YiscurrentlyunderinvestigationbyMarusiaRebGolledo,Zas$ EpartUUofherPh.D.thesis.$ E1.pCoun9terexamplesTdenepQointsonXٓR cmr70|s(p)(Q(pUW fe ;p ]W))3 EFirstZ$werecallsomeoftheresultsandnotationof[3].4LetS2|s( 0(p))Z$denote$ Etheaspaceofcuspformsofweighta2forthecongruencesubgroup 0|s(p).PDenote$ Eby|TthesubringofEndDҵS2|s( 0(p))|generatedbytheHeckeopGeratorsTn forall$ EintegersoHn.Letf2WS2|s( 0(p))oHhaveq[ٲ-expansion u cmex10Pލ O! cmsy71% n=19anq~q^nW.WhenisaDirichlet$ Echaracter,denote}byL(f V;;s)theentirefunctionwhichextendstheDirichletseries$ EPލ.1%.n=1?ꩵanq~(n)=n^sF:.3 ELetSEbGethesetofisomorphismclassesofsupersingularellipticcurvesincharac-$ Eteristicp.^RDenotebySthegroupformedbythedivisorsofdegree0withsuppGort$ EonVٵS .vSItisequippGedwithastructureofT-module(induced,W:forexample,fromthe$ Eaction7OoftheHecke7OcorrespGondencesontheberatpoftheregularminimalmodel$ EofUUX0|s(p)overUUZ).3 ELetjVݸ2qƍ≲:RFp JS,BwhereJS QdenotesthesetofsupGersingularmodularinvqari-$ Eants.PlW*eJ7denotebyjthehomomorphismofgroupsS *@ F!qƍ}p:_9Fp1thatassoGciates$ EtoP0ӟ EnEm[E ]thequantityP0ӟ EnEm=(j j (E )),wherej(E)denotesthemoGdular$ EinvqariantUUofE . g1 *y ? 3 EOnesaysthatanelementj12bFpvnisanomalousifthereexistsanellipticcurve $ Eoverb Fp \withmoGdularinvqariantb jthatpossessesanFpR-rationalpointoforderp$ E(thenUUnecessarilyjv=Y2gJS).3 ELet~GpbGeaprimethatiscongruentto1modulo4.Inthefollowingproposition$ Ewe¬prove,underahypGothesisonp,thatifEV9isanellipticcurveoverQ(pR)allof$ EwhosetorsionisQ(pR)-rational,HthenforeachsubgroupC̸!E (qƍ:Q)oforderp,the$ EpGointn(E ;C )onX0|s(p)isdenedoverQ(pUW fe ;p ]W).AswewillseeinPropGosition2,uthis$ EQ(pUW fe ;p ]W)-rationality&conclusioniscontrarytofact,/fromwhichweconcludethatsuch$ Eelliptic/|curvesE donotexistwhenthehypGothesisonpissatised.e*InSection3we$ EverifyUUthishypGothesisforp=11UUand13