The endomorphism rings of jacobians of cyclic cove

arXiv:math/0103203v2[math.AG]8Sep2002THEENDOMORPHISMRINGSOFJACOBIANSOFCYCLICCOVERSOFTHEPROJECTIVELINEBYYURIG.ZARHIN†DepartmentofMathematics,PennsylvaniaStateUniversity,UniversityPark,PA16802,USAe-mail:[e-mail]fieldofcharacteristiczero,Kaisitsalgebraicclosure,f(x)∈K[x]isanirreduciblepolynomialofdegreen≥5,whoseGaloisgroupcoincideseitherwiththefullsymmetricgroupSnorwiththealternatinggroupAn.Letpbeanoddprime,Z[ζp]theringofintegersinthepthcyclotomicfieldQ(ζp).SupposeCisthesmoothprojectivemodeloftheaffinecurveyp=f(x)andJ(C)isthejacobianofC.WeprovethattheringEnd(J(C))ofKa-endomorphismsofJ(C)iscanonicallyisomorphictoZ[ζp].1.IntroductionWewriteZ,Q,Cfortheringofintegers,thefieldofrationalnumbersandthefieldofcomplexnumbersrespectively.RecallthatanumberfieldiscalledaCM-fieldifitisapurelyimaginaryquadraticextensionofatotallyrealfield.Letpbeanoddprime,ζp∈Caprimitivepthrootofunity,Q(ζp)⊂CthepthcyclotomicfieldandZ[ζp]theringofintegersinQ(ζp).Itiswell-knownthatQ(ζp)isaCM-fieldofdegreep−1.WewriteFpforthefinitefieldconsistingofpelements.Letf(x)∈C[x]beapolynomialofdegreen≥4withoutmultipleroots.LetCf,pbeasmoothprojectivemodelofthesmoothaffinecurveyp=f(x).Itiswell-knownthatthegenusg(Cf,p)ofCf,pis(p−1)(n−1)/2ifpdoesnotdividenand(p−1)(n−2)/2ifitdoes.Themap(x,y)7→(x,ζpy)†PartiallysupportedbytheNSF.12YURIG.ZARHINgivesrisetoanon-trivialbirationalautomorphismδp:Cf,p→Cf,pofperiodp.ThejacobianJ(f,p):=J(Cf,p)ofCf,pisanabelianvarietyofdimensiong(Cf,p).WewriteEnd(J(f,p))fortheringofendomorphismsofJ(f,p)overC.ByAlbanesefunctoriality,δpinducesanautomorphismofJ(f,p)whichwestilldenotebyδp;itisknown([10,p.149],[11,p.458])thatδp−1p+···+δp+1=0inEnd(J(f,p)).ThisgivesusanembeddingZ[ζp]∼=Z[δp]⊂End(J(f,p))([10,p.149],[11,p.458]).Ourmainresultisthefollowingstatement.Theorem1.1.LetKbeasubfieldofCsuchthatallthecoefficientsoff(x)lieinK.Assumealsothatf(x)isanirreduciblepolynomialinK[x]ofdegreen≥5anditsGaloisgroupoverKiseitherthesymmetricgroupSnorthealternatinggroupAn.ThenEnd(J(f,p))=Z[δp]∼=Z[ζp].Inparticular,J(f,p)isasimplecomplexabelianvariety.Remark1.2.InthecasewhenpisaFermatprimetheassertionofTheorem1.1isprovenin[20].(Alsoin[20]theauthorprovedthatiftheconditionsofTheorem1.1holdtruethenZ[δp]isamaximalcommutativesubringinEnd(J(f,p))foralloddprimesp.See[21]forasimilarresultinpositivecharacteristicwhenp|nandn≥9.)The“opposite”casewhenJ(f,p)isanabelianvarietyofCM-typewasstudiedin[2].AnanalogueofTheorem1.1forhyperellipticjacobians(i.e.,thecaseofp=2)wasprovenin[17](seealso[18],[19]).Examples1.3.(1)thepolynomialxn−x−1∈Q[x]hasGaloisgroupSnoverQ([13,p.42]).Thereforetheendomorphismring(overC)ofthejacobianJ(C)ofthecurveC:yp=xn−x−1isZ[ζp]ifn≥5.(2)theGaloisgroupofthe“truncatedexponential”expn(x):=1+x+x22+x36+···+xnn!∈Q[x]CYCLICCOVERS,JACOBIANSANDENDOMORPHISMRINGS3iseitherSnorAn[12].Thereforetheendomorphismring(overC)ofthejacobianJ(C)ofthecurveC:yp=expn(x)isZ[ζp]ifn≥5.Remark1.4.Iff(x)∈K[x]thenthecurveCf,panditsjacobianJ(f,p)aredefinedoverK.LetKa⊂CbethealgebraicclosureofK.Clearly,allendomorphismsofJ(f,p)aredefinedoverKa.ThisimpliesthatinordertoproveTheorem1.1,itsufficestocheckthatZ[δp]coincideswiththeringofallKa-endomorphismsofJ(f,p)orequivalently,thatQ[δp]coincideswiththeQ-algebraofKa-endomorphismsofJ(f,p).Thepaperisorganizedasfollows.Section2containsauxiliaryresultsaboutendomorphismalgebrasofcomplexabelianvarieties.WeusetheminSection3inordertostudyendomorphismsofJ(f,p).InSection4weprovethemainresult.TheshortlastSectioncontainscorrigendumto[20].Theauthorwouldliketothanktherefereeforusefulcomments.2.ComplexabelianvarietiesThroughoutthissectionweassumethatZisacomplexabelianvarietyofpositivedimension.Asusual,wewriteEnd0(Z)forthesemisimplefinite-dimensionalQ-algebraEnd(Z)⊗Q.WewriteCZforthecenterofEnd0(Z).Itiswell-knownthatCZisadirectproductoffinitelymanynumberfields.AllthefieldsinvolvedareeithertotallyrealnumberfieldsorCM-fields.LetH1(Z,Q)bethefirstrationalhomologygroupofZ;itisa2dim(Z)-dimensionalQ-vectorspace.ByfunctorialityEnd0(Z)actsonH1(Z,Q);hencewehaveanembeddingEnd0(Z)֒→EndQ(H1(Z,Q))(whichsends1to1).SupposeEisasubfieldofEnd0(Z)thatcontainstheidentitymap.ThenH1(Z,Q)becomesanE-vectorspaceofdimensiond=2dim(Z)[E:Q].WewriteTrE:EndE(H1(Z,Q))→EforthecorrespondingtracemapontheE-algebraofE-linearoperatorsinH1(Z,Q).4YURIG.ZARHINExtendingbyC-linearitytheactionofEnd0(Z)andofEonthecomplexcoho-mologygroupH1(Z,Q)⊗QC=H1(Z,C)ofZwegettheembeddingsE⊗QC⊂End0(Z)⊗QC֒→EndC(H1(Z,C))whichprovideH1(Z,C)withanaturalstructureoffreeEC:=E⊗QC-moduleofrankd.IfΣEisthesetofembeddingsofσ:E֒→Cthenitiswell-knownthatEC=E⊗QC=Yσ∈ΣEE⊗E,σC=Yσ∈ΣECσwhereCσ=E⊗E,σC=C.SinceH1(Z,C)isafreeEC-moduleofrankd,thereisthecorrespondingtracemapTrEC:EndEC(H1(Z,C))→ECwhichcoincidesonECwithmultiplicationbydandwithTrEonEndE(H1(Z,Q)).WewriteLie(Z)forthetangentspaceofZ;itisadim(Z)-dimensionalC-vectorspace.Byfunctoriality,End0(Z)andthereforeEactsonLie(Z).ThisprovidesLie(Z)withanaturalstructureofE⊗QC-module.WehaveLie(Z)=Mσ∈ΣECσLie(Z)=⊕σ∈ΣELie(Z)σwhereLie(Z)σ=CσLie(Z)={x∈Lie(Z)|ex=σ(e)x∀e∈E}.Letusputnσ=nσ(Z,E)=dimCσLie(Z)σ=dimCLi