Class invariants for quartic CM fields

arXiv:math/0404378v2 [math.NT] 14 May 2004EYALZ.GOREN&KRISTINE.LAUTER

Abstract.OnecandefineclassinvariantsforaquarticprimitiveCMfieldKasspecialvaluesofcertainSiegel(orHilbert)modularfunctionsatCMpointscorrespondingtoK.Suchcon-structionsweregivenin[DSG]and[Lau].Weprovideexplicitboundsontheprimesappearinginthedenominatorsofthesealgebraicnumbers.Thisallowsus,inparticular,toconstructS-unitsincertainabelianextensionsofK,whereSiseffectivelydeterminedbyK.

1.Introduction

OneofthemainproblemsofalgebraicnumbertheoryistheexplicitdescriptionofrayclassfieldsofanumberfieldK.Besidesthecaseofthefieldofrationalnumbers,thetheoryismostadvancedinthecasewhereKisacomplexmultiplication(CM)field.Effectiveconstructionsareavailableusingmodularfunctionsgeneralizingtheellipticmodularfunctionj;oneconstructsmodularfunctionsasquotientsoftwomodularformsonaSiegelupperhalfspaceandevaluatesatCMpointscorrespondingtoK.ThevalueslieinanexplicitlydeterminedextensionofthereflexfieldK∗ofK,thatdependsonthefieldoverwhichtheFouriercoefficientsofthemodularfunctionisdefined,onthelevelofthemodularfunction,andontheconductoroftheorderofKcorrespondingtotheCMpoint.Welooselycallmagnitudesconstructedthisway“classinvariants”ofK.TheterminologyisproposedbecausewhentheFouriercoefficientsarerationalandthelevelis1thevaluesofthemodularfunctionatCMpointslieinrayclassfieldsofK∗.Anoutstandingproblemistheeffectiveconstructionofunitsinabelianextensionsofnumberfields,eveninthecaseofcomplexmultiplication.AsolutionofthisproblemisexpectedtohavesignificantimpactonobtainingadditionalcasesofStark’sconjecture.Thecaseofcyclotomicunitsandellipticunitsiswelldeveloped,butinhigherdimensionalcaseslittlewasknown.Theessentialproblemisthatdivisorsofmodularfunctionscannotbesupportedattheboundaryofthemodulispace.ThepurposeofthispaperistoprovideexplicitboundsontheprimesappearinginthedenominatorsofclassinvariantsofaprimitivequarticCMfieldK.Thisyields,inparticular,anexplicitboundontheprimesdividingtheinvariantsu(a,b)constructedin[DSG],thusyieldingS-unitslyinginaspecificabelianextensionofK∗foranexplicitfinitesetofprimesS.ItalsoyieldsclasspolynomialsforprimitivequarticCMfieldswhosecoefficientsareS-integersasconjecturedin[Lau].

2.Elementsofsmallnorminadefinitequaternionalgebra

2.1.Simultaneousembeddings.LetB=Bp,∞be“the”quaternionalgebraoverQramifiedat{p,∞}.ConcretemodelsforBcanbefoundine.g.[Vig,p.98].LetTrandNbethe(reduced)traceandnormonBandx→

vj))=p2;

cf.[Piz,Prop.1.1].Further,usingthisbasiswemayidentifyB⊗RwithR4.Thebilinearform󰀞α,β󰀟=Tr(α

1991MathematicsSubjectClassification.Primary11G15,11G16Secondary11G18,11R27.

1

2EYALZ.GOREN&KRISTINE.LAUTER

matrixMwithevendiagonalentries,whichispositivedefiniteandsatisfiesdet(M)=p2.It󰀎definesaninnerproductonR4.Welet󰀠r󰀠=2N(r).Notethattheco-volumeofR(theabsolutevalue√ofthevolumeofafundamentalparallelepiped)isp.LetKi=Q(

Proof.Oneverifiesthatki=±mi√andmiisodd,and±mi

√

4

.

Di)/2ifDi≡1(mod4)

p/8(<

√

󰀎

−N(x)],Z[

p/8belongtoO1.Inparticular,foreveryconstantA<

A+2.

Corollary2.2.3.Letji,i=1,2,betwosingularj-invariants,jicorrespondingtoanellipticcurveEiwithcomplexmultiplicationbyanorderOiofconductormiinaquadraticimaginaryfieldKi⊂C.SupposethatK1=K2.Letpbeaprimeof

4

√

.

Proof.If(j1−j2)∈pthenE1∼=E2(modp).LetE=E1(modp).SinceK1=K2,EissupersingularandafterfixinganisomorphismEnd(E)⊗Q∼=B,End(E)isamaximalorderofBcontainingO1,O2.󰀂

CLASSINVARIANTS3

2.3.Aremarkonsuccessiveminima.LetE,E′betwosupersingularellipticcurvesover

xMx)1/2.Geometryofnumbers,see[Sie,III§§3-4,X

§3],gives24(4!V)−1≤µ1·µ2·µ3·µ4≤24V−1,whereVisthevolumeoftheunitballwithrespecttof.SinceV=2π2/pwefindthat

2

t

(2.3)

1

π2

·p.

Proof.IfE=E′,thenµ1=1.Theembeddingisoptimalbecausetheelementofminimalnormindependentof1inanorderdeterminestheorder,cf.proofofLemma2.1.1.LetybeanelementforwhichthethirdsuccessiveminimumisobtainedandK2=Q(y).Bydefinitiony∈K1andwe󰀎√

N(x)areinthesituationof§2.1.WegetusingEquation(2.1),

N(xy)=µ2µ3.Wealsohaveµ24≥µ2µ3≥1.

Analysisoftheinequalitiesgivestheresult.󰀂ByEquation(2.3)

µ21

≤

√

2π

Proposition2.3.1.AssumethatE=E′.Letxbeanelementforwhichµ2isobtained,K1=

4

Q(x).ThenZ[x]isanorderofK1,optimallyembeddedinEnd(E).Wehaveµ22≤

√󰀅

1421/31/41/2

·p,·p1/2≤µ2µ3≤·p·p.≤µ≤4

2π31/3·π2/32

√E′

Fp.LetNbetheminimalinteger

over

2

√

p∼0.9004

suchthatforanysupersingular

√

p.NumericalevidenceshowsthatN≥0.7

d),

ford>0asquarefreeinteger.WriteK=d]atotallynegativeelement.√EveryquarticCMfieldcanbewrittenthiswayandthereismuchknownontheindexofZ[r]inOK.ThefollowingareequivalentforCMfieldsofdegree4:(i)Kisprimitive,i.e.,doesnotcontainaquadraticimaginaryfield;(ii)Kiseithernon-Galois,oracyclicGaloisextension;(iii)NQ(√

√K+(

∨:=Hom(E,E),F.Leta=Hom(E,E),ap2112󰀆󰀈

1aRi=End(Ei).ThenEnd(E1×E2)=R∨aR2.Theproductpolarizationinducedbythe

divisorE1×{0}+{0}×E2onE1×E2inducesaRosatiinvolutiondenotedby∨.Thisinvolution

4EYALZ.GOREN&KRISTINE.LAUTER

󰀂ab󰀁∨󰀂a∨c∨󰀁󰀂b󰀁

=b∨d∨,wherea∨,b∨etc.denotesthedualisogeny.Notethata∨=→isgivenbyacdcd

d.TheRosatiinvolutionisapositiveinvolution.Theembeddingproblem:Tofindaringembeddingι:OK֒→End(E1×E2)suchthattheRosatiinvolutioncomingfromtheproductpolarizationinducescomplexconjugationonOK.Asweshallseebelow,theproblemisintimatelyrelatedwithboundingprimesinthedenominatorsofclassinvariants.

Theorem3.0.4.Iftheembeddingproblemhasapositivesolutionthenp≤16·d2(Tr(r))2.Proof.√Assumesuchanembeddingιexists.Thenι(OK+)isfixedbytheRosatiinvolution,thus

d→M=

Letuswrite

󰀊

󰀋ab

,

b∨−a

√

a∈Z,b∈a,a2+bb∨=d.

√

TheconditionoftheRosatiinvolutioninducingcomplexconjugationisequivalenttoι(

√

r).So,ifι(

r=α+βd|.

r)=

󰀊

√

Afurtherconditionisobtainedfromι(

󰀋xy

,

−y∨w

00

x∈R1,w∈R2,y∈a.

CLASSINVARIANTS5

WeusethenotationN(s)=ss∨,N(y)=yy∨,etc..Notethatfors∈Rithisdefinitionofthenormistheusualoneand,inanycase,undertheinterpretationofelementsasendomor-phismsN(s)=deg(s)andsoN(st)=N(s)·N(t)whenitmakessense.Itfollowsfrom(⋆)that

(⋆⋆)

N(x)+N(y)=−(α+βa)N(w)+N(y)=−(α−βa).

Letϕ:E1→E2beanon-zeroisogenyofdegreeδ.Forf∈End(E1×E2)thecompositionofrationalisogenies

E1×E1−→E1×E2−→E1×E2−→E1×E1,

(1,ϕ)

f

(1,δ−1ϕ∨)

givesaringhomomorphismEnd0(E1×E2)−→End0(E1×E1)thatcanbewritteninmatrixformas

󰀊󰀋󰀊󰀋f11f12f11f12ϕf=→−1∨.

f21f22δϕf21δ−1ϕ∨f22ϕLetψbethecompositionK−→End0(E1×E2)−→End0(E1×E1).Thenψisanembeddingof

ringswiththeproperty

󰀊󰀋10

(3.4)ψ(OK)⊂M2(R1).

0δ

√󰀂ab󰀁∨r),Chooseϕ=y.Takingf=b∨−a(correspondingto

theembeddingψisdeterminedby

󰀋󰀊

√xδ

.r)=(3.5)ψ(

−1ywy∨/δWeconcludethat

S={by∨,yb∨,x,ywy∨}⊂R1.

Let

δ1

=min{−(α−βa),−(α+βa)}=|α|−|β|·|a|,

=|α|+|β|·|a|.

δ2=max{−(α−βa),−(α+βa)}Itfollowsfrom(3.1)and(⋆⋆)

N(by∨)=N(yb∨)≤dδ1,

N(x)≤δ2.

2,64δ2}.ThenN(by∨),N(yb∨)andN(x)areAssumethatp>16·d2(Tr(r))2≥max{64d2δ12√

allsmallerthan

6EYALZ.GOREN&KRISTINE.LAUTER

4.BadreductionofCMcurves

InthissectionwediscusstheconnectionbetweensolutionstotheembeddingproblemandbadreductionofcurvesofgenustwowhoseJacobianhascomplexmultiplication.WeshallassumeCMbythefullringofintegers,buttheargumentscaneasilybeadaptedtoCMbyanorder,atleastifavoidingprimesdividingtheconductoroftheorder.

4.1.Badreductionsolvestheembeddingproblem.FixaquarticprimitiveCMfieldK.√

r),r∈OK+,dapositiveinteger.LetCbeasmoothprojectivegenus2WriteK=Q(

curveoveranumberfieldL.WesaythatChasCM(byOK)ifJac(C)hasCMbyOK.BypassingtoafiniteextensionofLwemayassumethatChasastablemodeloverOLandthatalltheendomorphismsofJac(C)aredefinedoverL.SinceKisprimitive,Jac(C)isasimpleabelianvarietyandsoEnd0(Jac(C))=K.Inparticular,thenaturalpolarizationofJac(C),associatedtothethetadivisorC⊂Jac(C),preservesthefieldKandactsonitbycomplexconjugation.ItiswellknownthatJac(C)haseverywheregoodreduction.Itfollowsthatforeveryprimeidealp⊳OLeitherChasgoodreductionmoduloporisgeometricallyisomorphictotwoellipticcurvesE1,E2crossingtransverselyattheirorigins.Inthelattercasewehaveanisomorphismofprincipallypolarizedabelianvarietiesoverk(p)=OL/p,(Jac(C),C)∼=(E1×E2,E1×{0}+{0}×E2).SinceK֒→End(E1×E2)⊗QweseethatE1mustbeisogenoustoE2.Moreover,Eicannotbeordinary;thatimpliesthatK֒→M2(K1)forsomequadraticimaginaryfieldK1andoneconcludesthatK1֒→K,contradictingtheprimitivityofK.Weconclude

Lemma4.1.1.LetC/Lbeanon-singularprojectivecurveofgenus2withCMbyOK.AssumethatChasastablemodeloverOL.IfChasbadreductionmoduloaprimep|pofOLthentheembeddingproblemhasapositivesolutionfortheprimep.

ThefollowingtheoremnowfollowsimmediatelyusingTheorem3.0.4.

Theorem4.1.2.LetCbeanon-singularprojectivecurveofgenus2withCMbyOKandwithastablemodelovertheringofintegersOLofsomenumberfieldL.Letp|pbeaprimeidealofOL.Assumethatpisgreaterorequalto16·d2(Tr(r))2thenChasgoodreductionmodulop.4.2.Asolutiontotheembeddingproblemimpliesbadreduction.

Theorem4.2.1.Assumethattheembeddingproblemof§3hasasolutionwithrespecttoaprimitivequarticCMfieldK.ThenthereisasmoothprojectivecurveCofgenus2overanumberfieldLwithCMbyOK,whoseendomorphismsandstablemodelaredefinedoverOL,andaprimepofOLsuchthatChasbadreductionmodulop.

Ourstrategyforprovingthetheoremisthefollowing.Weconsideracertaininfinitesimaldefor-mationfunctorNforabeliansurfaceswithCMbyOK.WeshowthatNispro-representablebyaW(

Fp-pointxofSpec(Ru).WeprovethatRuisisomorphictothecompletedlocalringofapointonasuitableGrassmannvarietyanddeducethatRu⊗Q=0.Weconcludethatxcanbeliftedtocharacteristiczeroandfinishusingclassicalresultsinthetheoryofcomplexmultiplication.Beforebeginningtheproofproper,weneedsomepreliminariesaboutGrassmannvarieties.4.3.Grassmannschemes.ThefollowingappliestoanynumberfieldKwithaninvolution∗;wedenotethefixedfieldof∗byK+.Put[K:Q]=2g.

CLASSINVARIANTS7

4.3.1.

ConsiderthemoduleM0:=OK⊗ZW,W=W(

Fp-pointofit

liftstocharacteristiczero.ThatmeansthatforeverysubmoduleN1ofOK⊗Z

Fp=⊕p|pOK+/pep⊗Fp

are,respectively,freeOK+⊗ZWandOK+⊗Z

Fp,which

FpsubmoduleofB⊗W

Fp∼Fp,=

viz.(te).SincetheGrassmannschemeG′alwayshascharacteristiczerogeometricpointsandisprojective,aliftisprovidedby(any)characteristiczeropointofG′.

8EYALZ.GOREN&KRISTINE.LAUTER

InthesecondcasewehaveB⊗Wofrankeover

Fp[t]/(te)⊕

Fp

QpoverWandthateachliftingis

isotropicwhenconsideredasasubmoduleofB⊗WW′=A⊗WW′⊕A⊗WW′,whereW′isa“bigenough”extensionofW.Indeed,everygeometricpointoftheappropriateGrassmannscheme,beingproperoverSpec(W),extendstoanintegralpoint(definedoverafiniteintegralextensionW′/W).Suchageometricpointcorrespondstoachoiceof(ei)embeddingsA→

Fp).Thefollowingliftinglemma,thatholdsforanyCMfieldKandwhoseproofisgivenin§§4.4.1-4.4.4,isthekeypoint.

Lemma4.4.1.Let(A,λ,ι)beanabelianvarietywithCMover

Fp.Let(A′,λ′,ι′)beanabelianscheme

′1′overSwithCM.WeclaimthatH1dR(A/S)isafreeOK⊗ZS-moduleofrank1.SinceHdR(A/S)

isafreeS-moduleofrank2g,toverifythatitisafreeOK⊗ZS-moduleitisenoughtoprove

′Fp)isathatmodulothemaximalidealofS(cf.[DP,Rmq.2.8]),namely,thatH1dR(A⊗S

freeOK⊗ZFp/W)isafreeOK⊗ZW-module

4.4.2.Thepolarizationλinducesaperfectalternatingpairing󰀞·,·󰀟onthefreeOK⊗ZW-module1(A/W),whichweidentifywithM:=O⊗W.ThispairinginducescomplexconjugationHcrys0KZ

onOKandreducesmoduloptothepairinginducedbyλonH1dR(A/

)→H1dR(A/F

p

Qp(ϕ◦∗).consistingofachoiceofonesubspaceoutofeachpair

RecallthataCMtypeΦofKisasubsetofHom(K,C)(orofHom(K,

Qp=⊕{ϕ:K→

Qp(ϕ),

Qp)).

AchoiceofliftoftheHodgefiltrationprovidesuswithCMtypeΦ.LetK∗bethereflexfielddefinedbyΦ.Weseethat,infact,aliftoftheHodgefiltrationisdefinedoverΛ,whereΛisthecompositumofWwiththevaluationringofthep-adicreflexfieldassociatedtoΦ.

4.4.3.LetV=OK⊗ZC-acomplexvectorspaceonwhichOKacts.ChooseaZ-basise1,...,e2gforOKandconsiderfΦ(x

OK∗

→Sets

CLASSINVARIANTS9

󰀇

whereA/SisanabelianschemewithCManddet(eixi,Lie).Thatis,thetriple(A/S,λ,ι:OK֒→EndS(A))satisfiestheKottwitzcondition[Kot,§5]uniquelydeterminedbyΦ.

Forthegivenpointx=(A,λ,ι)∈M(

FptothecategorySets

Fp

)→H1dR(A/

Lemma4.4.1,wemaylift(A/Fp,λ0,ι0)definedoversomep-adicfieldK1

andso,byLefschetzprinciple,definedoverC.Bythetheoryofcomplexmultiplication(A0,λ0,ι0)isdefinedoversomenumberfieldK2.SincetheCMfieldKisprimitive,A0issimpleandprinci-pallypolarized.ByatheoremofWeil[Wei]thepolarizationisdefinedbyanonsingularprojectivegenus2curveCanditfollowsthatA0∼=Jac(C)aspolarizedabelianvarieties.Furthermore,CisdefinedoveranumberfieldK3(thatisatmostaquadraticextensionofK2).BypassingtoafiniteextensionLofK3,wegetastablemodel.

5.Applications

5.1.Ageneralprinciple.Thefollowinglemmaisfolkloreandeasytoprove:

Lemma5.1.1.Letπ:S→RbeaproperschemeoveraDedekinddomainRwithquotientfieldH.LetL→SbealinebundleonSandf,g:S→Lsections.Letx∈S(H′)beapoint,whereH′isafinitefieldextensionofH.Letu=(f/g)(x)∈H′.LetpbeaprimeofR′,theintegralclosureofRinH′.Letx¯betheR′-pointcorrespondingtox.Thenvalp(u)<0impliesthatx¯intersectsthedivisorofginthefiberofSoverp.Corollary5.1.2.LetA2→Spec(Z)bethemodulispaceofprincipallypolarizedabeliansurfacesandlet(5.1)

Θ(τ)=

1

Fp,λ,ι)=(E1×E2/

Fp(E1

×E2)).By

10EYALZ.GOREN&KRISTINE.LAUTER

theringofintegersofanumberfieldL,andthattheweightoffisoftheform10k,kapositiveinteger.

LetτbeapointonSp4(Z)\\H2correspondingtoasmoothgenus2curveCwithCMbythefullringofintegersofaprimitiveCMfieldK.Then(f/Θk)(τ)isanalgebraicnumberlyinginthecompositumHK∗LoftheHilbertclassfieldofK∗andL.Ifaprimepdividesthedenominatorof(f/Θk)(τ)thenChasbadreductionmodulop.

Proof.Theargumentisessentiallythatof[DSG,§4.4]:Igusa[Igu2]provedthatΘisamodularformonSp4(Z)\\H2(see[Igu2,Thm.3],Θisdenotedthereχ10).Itiswellknowntohaveweight10andacomputationshowsthatitsFouriercoefficientsareintegersandhaveg.c.d.1.Theq-expansionprinciple[FC,Ch.V,Prop.1.8]showsthatfandΘkaresectionsofasuitablelinebundleofthemodulischemeA2⊗ZOL.Thevalue(f/Θk)(τ)liesinHK∗Lbythetheoryofcomplexmultiplication.

ItisclassicalthatthedivisorofΘoverC,sayDgen,isthelocusofthereduciblepolarizedabeliansurfaces–thosethatareaproductofellipticcurveswiththeproductpolarization.The

clofDZariskiclosureDgengeninA2iscontainedinthedivisorDarithofΘ,viewedasasectionofa

cl=DlinebundleoverA2,andthereforeDgenarith,becausebytheq-expansionprincipleDarithhasclalsoparameterizesreduciblepolarizedabeliansurfaces,itno“verticalcomponents”.SinceDgen

followsthatDarithparameterizesreduciblepolarizedabeliansurfaces.(Furthermore,itiseasytoseebyliftingthateveryreduciblepolarizedabeliansurfaceisparameterizedbyDarith.)TheCorollarythusfollowsfromLemma5.1.1.󰀂Corollary5.1.3.(f/Θk)(τ)isanS-integer,whereSisthesetofprimesoflyingoverrationalprimesplessthan16·d2(Tr(r))2andsuchthatpdecomposesinacertainfashioninanormalclosureofKasimposedbysuperspecialreduction[Gor,Thms.1,2](forexample,ifKisacyclicGaloisextensionthenpiseitherramifiedordecomposesasp1p2inK).

5.2.Classinvariants.Igusa[Igu,p.620]definedinvariantsA(u),B(u),C(u),D(u)ofasexticu0X6+u1X5+···+u6,󰀉withrootsα1,...,α6,ascertainsymmetricfunctionsoftheroots.

210Forexample,D(u)=u0iafieldofcharacteristicdifferentfrom2,thecomplementofD=0inProjk[A,B,C,D],whereA,B,C,Dareofweights2,4,6,10respectively,isthecoarsemodulispaceforhyperellipticcurvesofgenus2.Moreover,theringofrationalfunctionsisgeneratedbythe“absoluteinvariants”B/A2,C/A3,D/A5(see[Igu1,p.177],[Igu,p.638]).Onecanchooseothergeneratorsofcourse,andforourpurposesitmakessensetochoosegeneratorswithdenominatorapowerofD.Choosethenasin[Wam,p.313]thegenerators

i1=A5/D,

i2=A3B/D,

i3=A2C/D.

Oneshouldnotethoughthattheseinvariantsarenotknowna-prioritobevalidincharacteristic2,sinceWeierstrasspoints“donotreducewell”modulo2.TheinvariantsincanbeexpressedintermsofSiegelmodularformsthus:

65

i1=2·35χ−10χ12,

See[Igu1,pp.189,195]forthedefinitions;ψiareEisensteinseriesofweighti,−22χ10isourΘ.Anotherinterestingapproachtothedefinitionofinvariantsisthefollowing:LetI2=h12/h10,I4=h4,I6=h16/h10,I10=h10bethemodularformsofweight2,4,6,10,respectively,asin[Lau].Theappealofthisconstructionisthateachhnisasimplepolynomial󰀇expressioninRie-ǫ

mannthetafunctionswithintegralevencharacteristics[ǫ′];forexample,h4=10(Θ[ǫǫ′](0,τ))4,

43

i2=2−3·33ψ4χ−10χ12,32−432

i3=2−5·3ψ6χ−10χ12+2·3ψ4χ10χ12.

CLASSINVARIANTS11

h10=212Θ.Itisnothardtoprovethattheg.c.d.oftheFouriercoefficientsofΘ[ǫǫ′](0,τ),for[ǫǫ′]anintegralevencharacteristic,is1ifǫ∈Z2(thathappensfor4evencharacteristics)and2ifǫ∈Z2(thathappensfor6evencharacteristics).UsingthatandwritingIn=∗/Θ,onefindsthatthenumeratorofInhasanintegralFourierexpansion.Onethenlets

5−12

j1:=I2/2I10,

3

j2:=I2I4/2−12I10,

2

j3:=I2I6/2−12I10.

Thesearemodularfunctionsoftheformf/Θk,suchthatthenumeratorhasintegralFourier

coefficients.Slightlymodifyingthedefinitionof[Lau](thereoneusesji:=2−12ji),weput

󰀃

(5.2)Hi(X)=(X−ji(τ)),i=1,2,3,

τ

wheretheproductistakenoverallτ∈Sp(4,Z)\\H2suchthattheassociatedabelianvarietyhasCMbyOK(thusallpolarizationsandCMtypesappear).Weremarkthatj1=i1,j2=i2;thiscanbeverifiedusingtheformulasgivenin[Igu3,p.848].

ThepolynomialsappearinginEquation(5.2)haverationalcoefficientsthataresymmetricfunctionsinmodularinvariants,viz.thevaluesofthefunctionsjiassociatedtoCMpoints.Assuch,itisnaturaltoaskfortheprimefactorizationofthesecoefficients.Forexample,theresultsof[GZ]givethefactorizationofthediscriminantoftheHilbertclasspolynomialinthecaseofimaginaryquadraticfieldsandsoprovideaboundontheprimeswhichcanappear.In[Lau],itwasconjecturedthatprimesdividingthedenominatorsofthecoefficientsofHi(X)areboundedbythediscriminantofK(notethattheonlydifferencebetweenthecurrentdefinitionandloc.cit.ispowersof2).Wededucefromtheprecedingresultsthefollowing:

Corollary5.2.1.ThecoefficientsoftherationalpolynomialsHi(X)areS-integerswhereSisthesetofprimessmallerthan16·d2(Tr(r))2andsatisfyingacertaindecompositionpropertyinanormalclosureofKasimposedbysuperspecialreduction.Weprovidesomenumericaldatain§§6.1-6.2.

Remark5.2.2.Theorem4.2.1givesapartialconversetothiscorollary.

5.3.Units.LetKbeaprimitivequarticCMfieldasbefore.In[DSG],DeShalitandthefirstnamedauthorconstructedclassinvariantsu(Φ;a),u(Φ;a,b)associatedtocertainidealsofKandaCMtypeΦ.TheconstructionessentiallyinvolvestheevaluationofΘatvariousCMpointsassociatedtoK.Thoughtheconstructionisgeneral,werecallitonlyfortheu(Φ;a)andunderveryspecialconditions.Forthegeneralcase,refertoloc.cit.

Example5.3.1.AssumethatKisacyclicCMfieldwithoddclassnumberhK,hK+=1.LetΦbeaCMtypeofKandassumethatthedifferentidealDK/Q=(δ)witha=(a).Theform󰀞f,g󰀟=TrK/Q(

∆(Φ(OK))

,u(Φ;a,b)=

u(Φ;ab)

12EYALZ.GOREN&KRISTINE.LAUTER

Corollary5.3.2.Theinvariantsu(Φ;a,b)areS-unitsforSthesetofprimesofHK∗thatlieoverrationalprimespsmallerthan16·d2(Tr(r))2suchthatpdecomposesinacertainfashioninanormalclosureofKasimposedbysuperspecialreduction.

6.Appendix:Numericaldata

6.1.Classinvariants.LetK=󰀎Q[x]/(x4+50x2+93)bethenon-normalquarticCMfield

√

ofclassnumber4generatedbyi133overitstotallyrealsubfieldK0=Q(

√

13−3

37·2312·13112

,i2=

2·5·113·533·67193·7229·301133

y2=−70399443x6+36128207x5+262678342x4−48855486x3−112312588x2+36312676x.Thereductionofagenus2curveataprimecanbecalculatedusing[Liu,Thm1,p.204].Fortheseexamplesweactuallycalculatedthereductionusingthegenus2reductionprogramwrittenbyLiu.Theoutputoftheprogramshowsthatattheprimesp=2,3,23,131,thecurve

34·238·1318

,hasanaffinemodel

CLASSINVARIANTS13

haspotentialstablereductionequaltotheunionoftwosupersingularellipticcurvesE1andE2intersectingtransversallyatonepoint.

1055

Thesecondcurvehasinvariantsequaltoi1=2·7·11·21059,i3=33·238

2·76·112·210592·71347·739363

Fp.Becauseofrunningtimeandmemoryrestrictionswedidonlysamplecalculations.Forp=10007,theTotalComputationTimewas22688.710seconds,TotalMemoryUsagewas1213.97MB.TheprogramranonanIntelPentium4,2.53GHz,1GBmemory.

p101

18

307

34

503

50

701

68

907

84

2003

250

4001

418

6007

584

8009

750

10007

10089

59

0.700

78

56

0.674

63

46

0.654

45

34

0.698

30

20

0.667

26

18

0.633

22

14

0.654

18

12

0.682

√[10

N

0.6009

0.611

p

14EYALZ.GOREN&KRISTINE.LAUTER

Acknowledgments:WewouldliketothankPaymanKassaeiforvaluablediscussionsandEhudDeShalitforsomeusefulcomments.Thefirstnamedauthor’sresearchwaspartiallysupportedbyNSERC;hewouldliketothankMicrosoftResearchforitshospitalityduringavisitwherethisprojecttookshape.ThesecondnamedauthorthanksTonghaiYangformanystimulatingdiscussionsandMcGillUniversityforitshospitality.

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