Coates-Wiles towers for CM abelian varieties

Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the

COATES-WILESTOWERSFORCMABELIANVARIETIES

arXiv:math/0404022v1 [math.NT] 2 Apr 2004CHRISTOPHERM.ROWEAbstract.Theaimofthispaperistocomputecongruencerelationsonunitsin eldsgeneratedbyadjoiningtorsionpointsofaCMabelianvarietytoanumber eld.Forellipticcurves,congruencerelationsoftheformwecomputewereanimportantingredientintheearlyproofsoftheCoates-Wilestheorem.Ingeneral,wecomputecongruencerelationsonexteriorproductsofunitsinpision elds,whichmorenatu-rally tintotheframeworkofRubin’sgeneralizationofStark’sconjecture.IntroductionLetEbeanellipticcurvede nedoverFwithcomplexmultiplicationbytheringofin-tegersofanimaginaryquadraticextensionKofQ,whereFiseitherKorQ.InCoatesandWilesprovedthatifE(K)haspositiverankthentheHasse-WeilL-seriesL(E/F,s)vanishesats=1.(InRubin’simportantworkonTate-Shafarevichgroups,hedeterminedboundsonSelmergroups,whichyieldanotherproofoftheCoates-Wilestheorem[Rub87].)CoatesandWilesusedformalgroupsandIwasawa-theoretictechniquestorelateellipticunitswithspecialvaluesofL(E/F,s).Usingmoreclassicaltechniques,StarkandGuptawereabletogiveaproofoftheCoates-Wilestheoremforellipticcurvesde nedoverQGup85].Theproofsutilizedi erenttechniques,butbothincludemanyofthesameideas.Speci cally,letEbeanellipticcurvede nedoverQwithcomplexmultiplication(CM)bytheringofintegersofanimaginaryquadraticextensionKofQ(sonecessarilyofclassnumberone).Letp=(π)beoneofin nitelymanysuitablychosenprimesofK,andKnthe eldofπn-pisionvaluesofE(i.e.,the niteextensionofKobtainedbyadjoiningalltheπn-torsionofEtoK).ThenthereexistsauniqueprimepnofKnlyingoverp.FixapointQ∈E(Q)\πE(Q)ofin niteorderandletLnbethe eldof

pn-pisionvaluesofQ,i.e.,Ln=Kn(1

2000MathematicsSubjectClassi cation.Primary:11G10;Primary:11G15;Secondary:11R27.

1

Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the

2C.ROWE

KwithcomplexmultiplicationbytheringofintegersofK(withthedegree2goverQ).FurthermoreletpbeaprimeofKlyingovertheoddrationalprimep,andassumethatbothpsplitscompletelyinKandAhasgoodreductionatallprimeslyingoverp.ThenwedescribecongruencerelationsonunitsofKn,theextensionofKgeneratedbythepn-torsionpointsofA.Forabeliansurfaces,

Grant

described

congruence

relationsonunitssimilartothoseofGupta(see[Gra88]),butadditionalprogresswasstymiedbythelackofunderstandingandconstructionof“abelianunits”toparallelthetheoryofellipticunits.Moreover,itseemstobeadi cultproblemtocomeupwithageneraltheoryofabelianunits.

Sincewerequirethatpsplitscompletely,the eldextensionsKn/Kcanbeshowntohavedegreepn 1(p 1)andtobetotallyrami edathalfoftheprimesofKlyingabovepandunrami edattheotherhalf(whichprimesramifydependsuponpandthe“CMtype”ofA).We xapointQ∈A(K)\pA(K)ofin niteorderandconstruct eldextensionsLn=Kn(1

wF γ∈Gχ(γ)log|γ( 1)|P1.

WhenF=QoranimaginaryquadraticextensionofQ,Starkwasabletouseproper-tiesofcyclotomicandellipticunits,respectively,toprovehisre nedconjecture[Sta80].

Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the

COATES-WILESTOWERSFORCMABELIANVARIETIES3

Forr≥1,Rubingaveageneralizedre nedStark’sconjecture,whichconjecturedare-lationbetweenexteriorproductsofS-unitsandtheleadtermintheTaylorexpansion

anentryinalatticeofQ rofL(s,χ)ats=0[Rub96].Rubin’sconjecturerelatesUS,Ttotherthderivativeats=0ofL(s,χ),whereUS,Tisaspeci csubgroupofthegroup

ofS-units,dependinguponanauxiliary, nitesetofprimesT.

Rubin’sconjectureappliedtoKe/Kshouldproduceexteriorproductsof“abelianS-units”inKe.OutsideofRubin’soriginalpaper,theonlydirectevidenceforRubin’sconjectureisgivenin[Gra99],whichlooksatexteriorproductsofunitsarisingfrom5-torsionontheJacobianofy2=x5+1/4.However,forg=1,Starkfurtherre nedhisconjecturesothattheS-unitinKnrelatingtotheL-seriesisactuallyaunit,anditwascongruencesonunits,notthecorrespondingS-units,thatwereemployedintheproofoftheCoates-Wilestheorem.Theseresultsledustoconsiderwhethertheexistenceofapointofin niteorderinA(K)shouldforcecongruenceconditionsonexteriorproductsofunitsinKe;theresultofwhichisTheorem2.

The rstsectionofthispaper xesnotationandassumptionsaboutourabelianvarietyAandnumber eldK.Inthesecondsection,wecollecttheinformationweneedaboutformalgroupsattachedtoabelianvarieties.Thenweusepropertiesofformalgroupsattachedtoabelianvarietiesinsections3and4todescribepropertiesofthe eldextensionsKn/KandLn/Kn,respectively.WeproveTheorem1insection6andTheorem2insection7.

Mostoftheresultsofthispaperwerecontainedintheauthor’sPh.D.thesis,andtheauthorwouldberemissifhedidnotthankhisadvisor,DavidGrant,forhisinvaluableassistance.Also,theauthorwouldliketothankbothWolfgangSchmidtandthePaci cInstitutefortheMathematicalSciencesfortheirsupportduringthewritingofthispaper.

Notation.LetF/Ebenumber elds,andqaprimeofE.Weletf(F/E)andD(F/E)denotetheconductoranddiscriminantofF/Erespectively.Welet

Let A be a CM abelian variety defined over a number field K. We compute congruence relations on units in fields generated by adjoining torsion points of A to K. For elliptic curves, congruence relations of the type we compute were used in the proof of the

4C.ROWE

ACMabelianvarietywillbeapair(A,i).Wesaythat(A,i)isde nedoveranumber eldKifbothAandeveryelementofEnd(A)arede nedoverK.Let

K)→A(

K/K).WeletA[α]=ker[α]denotethe

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