MATIMY´ASMATEMATIKAJournaloftheMathematicalSocietyofthePhilippines

ISSN0115-6926Vol.48No.1(2025)pp.1-5

Anobservationconcerningtherankofy2=x3+2nx

P.G.WalshUniversityofOttawaOttawa,Canadagwalsh@uottawa.ca

Abstract

InrecentworkofCorpuzandDimabayao,theauthorsdescribeaclevermethodtoderiveafamilyofellipticcurvesoftheformy2=x3+2nx,forwhichallbutfinitelymanyhaveMordell-Weilrankatleast2.Thesetofintegersnforwhichthisholdsappearstobequitethincomparedtothesetofallintegersnforwhichcurvesoftheaboveformhavealowerboundof2ontherank.Thepurposeofthispaperistodescribeadifferentconstruction,whichcombinedwitharesultonrootnumbersofellipticcurves,impliestheexistenceofaconsiderablylargerfamilyofcurveswiththeabovepropertiesundertheassumptionoftheParityConjecture.

Keywords:ellipticcurve,rank,torsionsubgroup2020MSC:11G05

1Introduction

Numerousarticleshaveappearedintheliteraturewhichinvestigatetherankofcurveswhichtaketheformy2=x3+nx(seeforexample[1],[2],[3],and[8]).AveryusefultoolforthispurposeistheParityConjecture(see[1],[4]and[6]),whichremainsamongthemostimportantopenproblemsinNumberTheoryandArithmeticGeometry.Ourresearchwasmotivatedbytherecentworkin[2],inwhichSilverman’sSpecializationtheoremwasusedonasubfamilyofcurveswhichtaketheformy2=x3+2nx.ThisisthesubjectofCorollary2.2below,inwhichafamilyofcurvesof(conjectural)rankatleast2isgiveninwhichnisfarsmallerthanthecurvesin[2],albeitunderthehypothesisoftheparityconjecture.Alongthewaytoarrivingatthatresult,itbecameapparentthattherootnumberofsuchcurveshaveinterestinguniformity,whichweestablishusingtheworkofRohrlich[7]andHalberstadt[5]onlocalrootnumbers.

2MainResults

Beforeproceedingtothemainresults,weprovidesomebackgroundontherootnumberofanellipticcurve.TheinterestedreadercanrefertoSection2of[6]formoredetailsonthenotionswesetoutbelow.LetEdenoteanellipticcurve,givenbytheequation

y2+a1xy+a3y=x3+a2x2+a4x+a6,

2P.G.Walsh

witha1,a2,a3,a4,a6∈Q.Assumethatthecurveisnon-singularandthismodelisminimal.Forarationalprimep,thetraceapofEatpisthequantityap=p+1|E(Fp)|.UsingthetraceonecanthendefinetheEulerfactoratpasfollows.Fors∈C,define

Lp(E,s)=



(1apps+p12s)1

ifEhasgoodreductionatp

ifEhassplitmultiplicativereductionatpifEhasnon-splitmultiplicativereductionatp

(1ps)1

(1+ps)1

WiththesefactorsonecandefinetheL-functionofEby

L(E,s)=

Y

Lp(E,s).

p

Thesefunctionsarewidelystudiedandknowntosatisfymanyextraordinaryproperties.Ourinterestinthemliesexclusivelyonhowtheycanbeusedtodeterminetherankofthecorrespondingellipticcurve.

Asisnowwellknown,L(E,s)hasananalyticcontinuationtotheentirecomplexplane,andsatisfiesthefunctionalequation

(2π)s(s)Ns/2L(E,s)=W(E)(2π)s2(2s)N(2s)/2L(E,2s),

whereNistheconductorofEandW(E)istherootnumberofE,(W(E)=±1).ThefunctionL(E,s)canbewrittenasaTaylorseriesats=1,andcorrespondingly,theindexrAofitsfirstnon-zerocoefficientiscalledtheanalyticrank.Inotherwords,rA≥0istheorderofvanishingofL(E,s)ats=1.ItisnotdifficulttoverifyfromtheequationabovethattherootnumberofEsatisfiesW(E)=(1)rA.ThefamousconjectureofBirchandSwinnerton-DyerstatesthattheanalyticrankofEisequaltotheMordell-Weilrank.Therefore,undertheassumptionofthisconjectureitfollowsthatW(E)=(1)r,andthisiswhatisknownastheParityConjecture.

Theorem2.1.Letndenoteanoddcube-freeinteger.Writenasn=ab2,witha,bsquarefreeandpairwisecoprime.ThentherootnumberW(E)ofthecurve

E:y2=x3+2nx

is(1)(bsgn(n))/2.

TheproofofTheorem2.1willbegiveninthenextsection,althoughitisworthnot-ingthatthisveryproblemwasexploredinconsiderabledetailintheworkofBirchandStephens[1].Beforeproceedingtotheproof,westateacorollarywhichwasactuallythedrivingforcebehindthiswork.

Unlikethecurvesin[2]whichhavetwoparametricpoints,theprofoundnatureoftheparityconjectureisonfulldisplayhere,asthefollowingfamilyofcurveshaveanexplicitpointofinfiniteorder,whiletheparityconjectureindicatesthattherankofeachcurveiseven.Therefore,asecondgeneratoroftheMordell-WeilgroupofthecurveoverQmustexist,andevenamodestcomputationindicatesthatitsheightisquiteerratic.

Anobservationconcerningtherank...

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Corollary2.2.Fori≥1definetheellipticcurveE±iby

E±i:y2=x3+2(2i4±4i2+1)x.

ThenthepointPi=(2i2,2i(2i2±1))isapointofinfiniteorderonE±i.Consequently,ifn=2i4±4i2+1issquarefree,thenundertheparityconjecture,therankofE±iisatleast

2.

TheproofofCorollary2.2amountstonumericallyverifyingthatthepointgivenisonthecurve,andshowingthatitisnotatorsionpoint.Theformeristrivial,whilethelatterisasimpleapplicationoftheNagell-Lutzcriterion.ThestatementofCorollary2.2couldbeextendedtoincludesquaredfactors,solongasthenumberofprimesinthefactorizationofnwhichare3modulo4iseven.

Wewillassumeforsimplicitythatnispositive,astheproofforthecasethatnisnegativeisidentical.Asinthestatementofthetheorem,assumethatn=ab2,witha,bodd,squarefree,andcoprime.Wewillassumethatbothaandbarenotdivisibleby3,astheargumentabovegoesthroughifaorbismultipliedby3.Forprimespdividinga,byProposition2(v)of[7],thelocalrootnumberatpis(2/p),whileforprimespdividingb,thevalueofeinProposition2(v)of[7]ise=2,andsothelocalrootnumberis(1/p).Asfortheprimep=2,thelocalrootnumberisgiveninTable1of[5]onthelinewithν(c4)=5,ν(c6)≥9andν(δ)=9.Inparticular,thelocalrootnumberatp=2is1ifandonlyiftheoddpartofc4,whichis3ab2,is1or3modulo8.Inotherwords,ifandonlyifn≡5,7(mod8).Thisshowsthattheglobalrootnumbercontributionfrom2aisequalto1,whichiscancelledbythelocalfactoratinfinity.Thus,theglobalrootnumbercontributionfromb2isexactlythenumberofprimesdividingbwhichare3modulo4,givingthedesiredresult.

Acknowledgements.Theauthorisgratefultotwoanonymousrefereesforprovidinginsightfulcommentsandimprovingthepresentationofthiswork.

References

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Topology5(1966),295–299.

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4P.G.Walsh

[6]J.R.Love,Rootnumbersofafamilyofellipticcurvesandtwoapplications,Indag.

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