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SATO-TATEDISTRIBUTIONSANDREWV.
SUTHERLANDABSTRACT.
InthisexpositoryarticleweexploretherelationshipbetweenGaloisrepresentations,motivicL-functions,Mumford-Tategroups,andSato-Tategroups,andwegiveanexplicitformulationoftheSato-TateconjectureforabelianvarietiesasanequidistributionstatementrelativetotheSato-Tategroup.
WethendiscusstheclassicationofSato-Tategroupsofabelianvarietiesofdimensiong≤3andcomputesomeofthecorrespondingtracedistributions.
Thisarticleisbasedonaseriesoflecturespresentedatthe2016ArizonaWinterSchoolheldattheSouthwestCenterforArithmeticGeometry.
1.
ANINTRODUCTIONTOSATO-TATEDISTRIBUTIONSBeforediscussingtheSato-TateconjectureandSato-Tatedistributionsinthecontextofabelianvari-eties,letusrstconsiderthemorefamiliarsettingofArtinmotives(varietiesofdimensionzero).
1.
1.
Arstexample.
Letf∈Z[x]beasquarefreepolynomialofdegreed.
Foreachprimep,letfp∈(Z/pZ)[x]Fp[x]denotethereductionoffmodulop,anddeneNf(p):=#{x∈Fp:fp(x)=0},whichwenoteisanintegerbetween0andd.
WewouldliketounderstandhowNf(p)varieswithp.
ThetablebelowshowsthevaluesofNf(p)whenf(x)=x3x+1forprimesp≤60:p:235711131719232931374143475359Nf(p)00111011200101013Theredoesnotappeartobeanyobviouspattern(andweshouldknownottoexpectone,becausetheGaloisgroupoffisnonabelian).
Theprimep=23isexceptionalbecauseitdividesdisc(f)=23,whichmeansthatf23(x)hasadoubleroot.
AsweareinterestedinthedistributionofNf(p)asptendstoinnity,wearehappytoignoresuchprimes,whicharenecessarilyniteinnumber.
Thistinydatasetdoesnottellusmuch.
Letusnowconsiderprimesp≤BforincreasingboundsB,andcomputetheproportionsci(B)ofprimesp≤BwithNf(p)=i.
Weobtainthefollowingstatistics:Bc0(B)c1(B)c2(B)c3(B)1030.
3233530.
5209580.
0059880.
1556891040.
3314330.
5105860.
0008140.
1579801050.
3336460.
5028670.
0001040.
1634871060.
3331850.
5007830.
0000130.
1660321070.
3333600.
5002660.
0000020.
1663731080.
3333370.
5000580.
0000000.
1666051090.
3333280.
5000160.
0000000.
16665610120.
3333330.
5000000.
0000000.
166666TheauthorwassupportedbyNSFgrantsDMS-1115455andDMS-1522526.
1arXiv:1604.
01256v5[math.
NT]22Jan2018Thisleadsustoconjecturethatthefollowinglimitingvaluesciofci(B)asB→∞arec0=1/3,c1=1/2,c2=0,c3=1/6.
Thereisofcourseanaturalmotivationforthisconjecture(whichis,infact,atheorem),onethatwouldallowustocorrectlypredicttheasymptoticratiosciwithoutneedingtocomputeanystatistics.
LetusxanalgebraicclosureQofQ.
TheabsoluteGaloisgroupGal(Q/Q)actsontherootsoff(x)bypermutingthem.
ThisallowsustodenetheGaloisrepresentation(acontinuoushomomorphism)ρf:Gal(Q/Q)→GLd(C),whoseimageisasubgroupofthepermutationmatricesinOd(C)GLd(C);hereOddenotestheorthog-onalgroup(wecouldreplaceCwithanyeldofcharacteristiczero).
NotethatGal(Q/Q)andGLd(C)aretopologicalgroups(theformerhastheKrulltopology),andhomomorphismsoftopologicalgroupsareunderstoodtobecontinuous.
Inordertoassociateapermutationoftherootsoff(x)toamatrixinGLd(C)weneedtoxanorderingoftheroots;thisamountstochoosingabasisforthevectorspaceCd,whichmeansthatourrepresentationρfisreallydenedonlyuptoconjugacy.
Thevalueρftakesonσ∈Gal(Q/Q)dependsonlyontherestrictionofσtothesplittingeldLoff,sowecouldrestrictourattentiontoGal(L/Q).
ThismakesρfanArtinrepresentation:acontinuousrepresentationGal(Q/Q)→GLd(C)thatfactorsthroughanitequotient(byanopensubgroup).
Butinthemoregeneralsettingswewishtoconsiderthismaynotalwaysbetrue,andevenwhenitis,wetypicallywillnotbegivenL;itisthusmoreconvenienttoworkwithGal(Q/Q).
Tofacilitatethisapproach,weassociatetoeachprimepanabsoluteFrobeniuselementFrobp∈Gal(Q/Q)thatmaybedenedasfollows.
FixanembeddingQinQpandusethevaluationidealPofQp(themaximalidealofitsringofintegers)todeneacompatiblesystemofprimesqL:=P∩L,whereLrangesoverallniteextensionsofQ.
ForeachprimeqL,letDqLGal(L/Q),denoteitsdecompositiongroup,IqLDqLitsinertiagroup,andFqL:=ZL/qLitsresidueeld,whereZLdenotestheringofintegersofL.
Takingtheinverselimitoftheexactsequences1→IqL→DqL→Gal(FqL/Fp)→1overniteextensionsL/Qorderedbyinclusiongivesanexactsequenceofpronitegroups1→Ip→Dp→Gal(Fp/Fp)→1.
WenowdeneFrobp∈DpGal(Q/Q)byarbitrarilychoosingapreimageoftheFrobeniusautomor-phismx→xpinGal(Fp/Fp)underthemapintheexactsequenceabove.
WeactuallymadetwoarbitrarychoicesinourdenitionofFrobp,sincewealsochoseanembeddingofQintoQp.
OurabsoluteFrobe-niuselementFrobpisthusfarfromcanonical,butitexists.
ItskeypropertyisthatifL/QisaniteGaloisextensioninwhichpisunramied,thentheconjugacyclassconjL(Frobp)inGal(L/Q)oftherestrictionofFrobp:Q→QtoLisuniquelydetermined,independentofourchoices;notethatwhenpisunramied,IpistrivialandDpGal(Fp/Fp).
EverythingwehavesaidappliesmutatismutandiifwereplaceQbyanumbereldK:putK:=Q,replacepbyaprimepofK(anonzeroprimeidealofZK),andreplaceFpbytheresidueeldFp:=ZK/p.
Wenowmakethefollowingobservation:foranyprimepthatdoesnotdividedisc(f)wehave(1)Nf(p)=trρf(Frobp).
2Thisfollowsfromthefactthatthetraceofapermutationmatrixcountsitsxedpoints.
Sincepisunramiedinthesplittingeldoff,theinertiagroupIpGal(Q/Q)actstriviallyontherootsoff(x),andtheactionofFrobpontherootsoff(x)coincides(uptoconjugation)withtheactionoftheFrobeniusautomorphismx→xpontherootsoffp(x),bothofwhicharedescribedbythepermutationmatrixρf(Frobp).
TheChebotarevdensitytheoremimpliesthatwecancomputecivia(1)bycountingmatricesinρf(Gal(Q/Q))withtracei,anditisenoughtodeterminethetraceandcardinalityofeachconjugacyclass.
Theorem1.
1.
CHEBOTAREVDENSITYTHEOREMLetL/KbeaniteGaloisextensionofnumbereldswithGaloisgroupG:=Gal(L/K).
ForeverysubsetCofGstableunderconjugationwehavelimB→∞#{N(p)≤B:conjL(Frobp)C}#{N(p)≤B}=#C#G,whereprangesoverprimesofKandN(p):=#FpisthecardinalityoftheresidueeldFp:=ZK/p.
Proof.
SeeCorollary2.
13inSection2.
Remark1.
2.
InTheorem1.
1theasymptoticratioontheleftdependsonlyonprimesofinertiadegree1(thosewithprimeresidueeld),sincethesemakeupallbutanegligibleproportionoftheprimespforwhichN(p)≤B.
TakingC={1G}showsthataconstantproportionoftheprimesofKsplitcompletelyinLandinparticularhaveprimeresidueelds;thisspecialcaseisalreadyimpliedbytheFrobeniusdensitytheorem,whichwasprovedmuchearlier(intermsofDirichletdensity).
InourstatementofTheorem1.
1wedonotbothertoexcludeprimesofKthatareramiedinLbecausenomatterwhatvalueconjL(Frobp)takesontheseprimesitwillnotchangethelimitingratio.
Inourexamplewithf(x)=x3x+1,onendsthatGf:=ρf(Q/Q)isisomorphictoS3,theGaloisgroupofthesplittingeldoff(x).
Itsthreeconjugacyclassesarerepresentedbythematrices010001100,100001010,100010001,withtraces0,1,3.
Thecorrespondingconjugacyclasseshavecardinalities2,3,1,respectively,thusc0=1/3,c1=1/2,c2=0,c3=1/6,asweconjectured.
IfweendowthegroupGfwiththediscretetopologyitbecomesacompactgroup,andthereforehasaHaarmeasurethatisuniquelydeterminedoncewenormalizeitsothat(Gf)=1(whichwealwaysdo).
RecallthattheHaarmeasureofacompactgroupGisatranslation-invariantRadonmeasure(so(gS)=(Sg)=(S)foranymeasurablesetSandg∈G),andisuniqueuptoscaling.
1FornitegroupstheHaarmeasureisjustthenormalizedcountingmeasure.
Wecancomputetheexpectedvalueoftrace(andmanyotherstatisticalquantitiesofinterest)byintegratingagainsttheHaarmeasure,whichinthiscaseamountstosummingoverthenitegroupGf:E[tr]=Gftr=1#Gfg∈Gftr(g)=di=0cii.
1ForlocallycompactgroupsGonedistinguishesleftandrightHaarmeasures,butthetwocoincidewhenGiscompact;see[22]formorebackgroundonHaarmeasures.
3TheChebotarevdensitytheoremimpliesthatthisisalsotheaveragevalueofNf(p),thatis,limB→∞p≤BNf(p)p≤B1=E[tr].
Thisaverageis1inourexample,becausef(x)isirreducible;seeExercise1.
1.
Thequantitiescideneaprobabilitydistributionontheset{tr(g):g∈Gf}oftracesthatwecanalsoviewasaprobabilitydistributionontheset{Nf(p):pprime}.
Pickingarandomprimepinsomelargeinterval[1,B]andcomputingNf(p)isthesamethingaspickingarandommatrixginGfandcomputingtr(g).
Moreprecisely,thesequence(Nf(p))pindexedbyprimespisequidistributedwithrespecttothepushforwardoftheHaarmeasureunderthetracemap.
WediscussthenotionofequidistributionmoregenerallyintheSection2.
1.
2.
Momentsequences.
Thereisanotherwaytocharacterizetheprobabilitydistributionontr(g)givenbytheci;wecancomputeitsmomentsequence:M[tr]:=(E[trn])n≥0,whereE[trn]=Gftrn.
ItmightseemsillytoincludethezerothmomentE[tr0]=E[1]=1,butinSection4wewillseewhythisconventionisuseful.
InourexamplewehavethemomentsequenceM[tr]=(1,1,2,5,14,41,.
.
.
,12(3n1+1),.
.
.
).
ThesequenceM[tr]uniquelydetermines2thedistributionsoftracesandthuscapturesalltheinforma-tionencodedintheci.
Itmaynotseemveryusefultoreplaceanitesetofrationalnumberswithaninnitesequenceofintegers,butwhendealingwithcontinuousprobabilitydistributions,asweareforcedtodoassoonasweleaveourweightzerosetting,momentsequencesareapowerfultool.
Ifwepickanothercubicpolynomialf∈Z[x],wewilltypicallyobtainthesameresultaswedidinourexample;whenorderedbyheightalmostallcubicpolynomialsfhaveGaloisgroupGfS3.
Butthereareexceptions:iffisnotirreducibleoverQthenGfwillbeisomorphictoapropersubgroupofS3,andthisalsooccurswhenthesplittingeldoffisacycliccubicextension(thishappenspreciselywhendisc(f)isasquareinQ*;thepolynomialf(x)=x33x1isanexample).
UptoconjugacytherearefoursubgroupsofS3,eachcorrespondingtoadifferentdistributionofNf(p):f(x)Gfc0c1c2c3M[tr]x3x10001(1,3,9,27,81,.
.
.
)x3+xC201/201/2(1,2,5,14,41,.
.
.
)x33x1C32/3001/3(1,1,3,19,27,.
.
.
)x3x+1S31/31/201/6(1,1,2,5,14,.
.
.
)Onecandothesamethingwithpolynomialsofdegreed>3.
Ford≤19theresultsareexhaustive:foreverytransitivesubgroupGofSdthedatabaseofKlünersandMalle[51]containsatleastonepolynomialf∈Z[x]withGfG(includingall1954transitivesubgroupsofS16).
Thenon-transitive2Notallmomentsequencesuniquelydetermineanunderlyingprobabilitydistribution,butallthemomentsequenceweshallconsiderdo(becausetheysatisfyCarleman'scondition[52,p.
126],forexample).
4casescanbeconstructedasproducts(ofgroupsandofpolynomials)oftransitivecasesoflowerdegree.
Itisanopenquestionwhetherthiscanbedoneforalld(eveninprinciple).
ThisamountstoastrongformoftheinverseGaloisproblemoverQ;weareaskingnotonlywhethereverynitegroupcanberealizedasaGaloisgroupoverQ,butwhethereverytransitivepermutationgroupofdegreedcanberealizedastheGaloisgroupofthesplittingeldofanirreduciblepolynomialofdegreed.
1.
3.
Zetafunctions.
Forpolynomialsfofdegreed=3thereisaone-to-onecorrespondencebetweensubgroupsofSdanddistributionsofNf(p).
Thisisnottrueford≥4.
Forexample,thepolynomialsf(x)=x4x3+x2x+1withGfC4andg(x)=x4x2+1withGgC2*C2bothhavec0=3/4,c1=c2=c3=0,andc4=1/4,correspondingtothemomentsequenceM[tr]=(1,1,4,16,64,.
.
.
).
Wecandistinguishthesecasesif,inadditiontoconsideringthedistributionofNf(p),wealsoconsiderthedistributionofNf(pr):=#{x∈Fpr:fp(x)=0}forintegersr≥1.
InourquarticexamplewehaveNg(p2)=4foralmostallp,whereasNf(p2)is4or2dependingonwhetherpisasquaremodulo5ornot.
IntermsofthematrixgroupGfwehave(2)Nf(pr)=trρf(Frobp)rforallprimespthatdonotdividedisc(f).
Toseethis,notethatthepermutationmatrixρf(Frobp)rcorrespondstothepermutationoftherootsoffp(x)givenbytherthpoweroftheFrobeniusautomor-phismx→xp.
Itsxedpointsarepreciselytherootsoffp(x)thatlieinFpr;takingthetracecountstheseroots,andthisyieldsNf(pr).
Thisnaturallyleadstothedenitionofthelocalzetafunctionoffatp:(3)Zfp(T):=exp∞r=1Nf(pr)Trr,whichcanbeviewedasageneratingfunctionforthesequence(Nf(p),Nf(p2),Nf(p3),.
.
.
).
Thispar-ticularformofgeneratingfunctionmayseemstrangewhenrstencountered,butithassomeveryniceproperties.
Forexample,iff,g∈Z[x]aresquarefreepolynomialswithnocommonfactor,thentheirproductfgisalsosquarefree,andforallpdisc(fg)wehaveZ(fg)p=Zfpgp=ZfpZgp.
Remark1.
3.
Theidentity(2)canbeviewedasaspecialcaseoftheGrothendieck-Lefschetztracefor-mula.
ItallowsustoexpressthezetafunctionZfp(T)asasumoverpowersofthetracesoftheimageofFrobpundertheGaloisrepresentationρf.
IngeneraloneconsidersthetraceoftheFrobeniusendo-morphismactingonétalecohomology,butindimensionzerotheonlyrelevantcohomologyisH0.
Whiledenedasapowerseries,infactZfp(T)isarationalfunctionoftheformZfp(T)=1Lp(T),whereLp(T)isanintegerpolynomialwhoserootslieontheunitcircle.
ThiscanbeviewedasaconsequenceoftheWeilconjecturesindimensionzero,3butinfactitfollowsdirectlyfrom(2).
Indeed,3Providedoneaccountsforthefactthatf(x)=0doesnotdeneanirreduciblevarietyunlessdeg(f)=1;inthiscaseNf(pr)=1andLp(T)=1T,whichisconsistentwiththeusualformulationoftheWeilconjectures(seeTheorem1.
8).
5foranymatrixA∈GLd(C)wehavetheidentity(4)exp∞r=1tr(Ar)Trr=det(1AT)1,whichcanbeprovedbyexpressingthecoefcientsonbothsidesassymmetricfunctionsintheeigen-valuesofA;seeExercise1.
2.
Applying(2)and(4)tothedenitionofZfp(T)in(3)yieldsZfp(T)=1det(1ρf(Frobp)T),thusLp(T)=det(1ρf(Frobp)T).
ThepolynomialLp(T)ispreciselythepolynomialthatappearsintheEulerfactoratpofthe(partial)ArtinL-functionL(ρf,s)fortherepresentationρf:L(ρf,s):=pLp(ps)1,atleastforprimespthatdonotdividedisc(f);forthedenitionoftheEulerfactorsatramiedprimes(andtheGammafactorsatarchimedeanplaces),see[60,Ch.
2].
4TheEulerproductforL(ρf,s)denesafunctionthatisholomorphicandnonvanishingonRe(s)>1.
WeshallnotbeconcernedwiththeEulerfactorsatramiedprimes,otherthantonotethattheyareholomorphicandnonvanishingonRe(s)>1.
Remark1.
4.
Everyrepresentationρ:Gal(Q/Q)→GLd(C)withniteimagegivesrisetoanArtinL-functionL(ρ,s),andArtinprovedthateverydecompositionofρintosub-representationsgivesrisetoacorrespondingfactorizationofL(ρ,s)intoArtinL-functionsoflowerdegree.
TherepresentationρfwehavedenedisdeterminedbythepermutationactionofGal(Q/Q)ontheformalC-vectorspacewithbasiselementscorrespondingtorootsoff.
ThelinearsubspacespannedbythesumofthebasisvectorsisxedbyGal(Q/Q),soford>1wecanalwaysdecomposeρfasthesumofthetrivialrepresentationandarepresentationofdimensiond1,inwhichcaseL(ρf,s)istheproductoftheRiemannzetafunction(theArtinL-functionofthetrivialrepresentation),andanArtinL-functionofdegreed1.
TheArtinL-functionsL(ρf,s)wehavedenedarethusimprimitivefordegf>1.
Returningtoourinterestinequidistribution,theHaarmeasureonGf=ρf(Gal(Q/Q))allowsustodeterminethedistributionofL-polynomialsLp(T)thatweseeaspvaries.
EachpolynomialLp(T)isthereciprocalpolynomial(obtainedbyreversingthecoefcients)ofthecharacteristicpolynomialofρf(Frobp).
IfwexapolynomialP(T)ofdegreed=degf,andpickaprimepatrandomfromsomelargeinterval,theprobabilitythatLp(T)=P(T)isequaltotheprobabilitythatthereciprocalpolynomialTdP(1/T)isthecharacteristicpolynomialofarandomelementofGf(thisprobabilitywillbezerounlessP(T)hasaparticularform;seeExercise1.
3).
Remark1.
5.
Ford≤5thedistributionofcharacteristicpolynomialsuniquelydetermineseachsub-groupofSd(uptoconjugacy).
Thisisnottrueford≥6,andford≥8onecanndnon-isomorphicsubgroupsofSdwiththesamedistributionofcharacteristicpolynomials;thetransitivepermutationgroups8T10and8T11whichariseforx813x6+44x417x2+1andx8x52x4+4x2+x+1(respectively)areanexample.
4Thealertreaderwillnotethatprimesdividingthediscriminantoffneednotramifyinitssplittingeld;wearehappytoignoretheseprimesaswell,justaswemayignoreprimesofbadreductionforacurvethataregoodprimesforitsJacobian.
61.
4.
Computingzetafunctionsindimensionzero.
LetusnowbrieyaddressthepracticalquestionofefcientlycomputingthezetafunctionZfp(T),whichamountstocomputingthepolynomialLp(T).
ItsufcestocomputetheintegersNf(pr)forr≤d,whichisequivalenttodeterminingthedegreesoftheirreduciblepolynomialsappearinginthefactorizationoffp(x)inFp[x].
Thesedeterminethecycletype,andthereforetheconjugacyclass,ofthepermutationoftherootsoffp(x)inducedbytheactionoftheFrobeniusautomorphismx→xp,whichinturndeterminesthecharacteristicpolynomialofρf(Frobp)andtheL-polynomialLp(T)=det(1ρf(Frobp)T);seeExercise1.
3.
Todeterminethefactorizationpatternoffp(x),onecanapplythefollowingalgorithm.
Algorithm1.
6.
Givenasquarefreepolynomialf∈Fp[x]ofdegreed>1,computethenumberniofirreduciblefactorsoffinFp[x]ofdegreei,for1≤i≤dasfollows:1.
Letg1(x)bef(x)mademonicandputr0(x):=x.
2.
Forifrom1tod:a.
Ifi>deg(gi)/2thenfori≤j≤dputnj:=1ifj=deg(gi)andnj:=0otherwise,andthenproceedtostep3.
b.
UsingbinaryexponentiationintheringFp[x]/(gi),computeri:=rpi1modgi.
c.
Computehi(x):=gcd(gi,ri(x)x)=gcd(gi(x),xpix)usingtheEuclideanalgorithm.
d.
Computeni:=deg(hi)/iandgi+1:=gi/hiusingexactdivision.
e.
Ifdeg(gi+1)=0thenputnj:=0fori1,whereasthechord-and-tangentlawdoesnot.
TheAbel–JacobimapP→PP∞givesabijectionbetweenpointsonEandpointsonJac(E)thatcommuteswiththegroupoperation,sothetwoapproachesareequivalent.
Foreachprimepthatdoesnotdividethediscriminant:=16(4A3+27B2)wecanreduceourequationforEmoduloptoobtainanellipticcurveEp/Fp;inthiscasewesaythatpisaprimeofgoodreductionforE(orsimplyagoodprime).
Weshouldnotethatthediscriminantisnotnecessarilyminimal;thecurveEmayhaveanothermodelwithgoodreductionatprimesthatdivide(possiblyincluding2),butwearehappytoignoreanynitesetofprimes,includingthosethatdivide.
77AllellipticcurvesoverQhaveaglobalminimalmodelforwhichtheprimesofbadreductionarepreciselythosethatdividethediscriminant,butthismodelisnotnecessarilyoftheformy2=x3+Ax+B.
OvergeneralnumbereldsKglobalminimalmodelsdonotalwaysexist(theydowhenKhasclassnumberone).
8ForeveryprimepofgoodreductionforEwehaveNE(p):=#Ep(Fp)=p+1tp,wheretheintegertpsatisestheHasse-bound|tp|≤2p.
Incontrasttoourrstexample,theintegersNE(p)nowtendtoinnitywithp:wehaveNE(p)=p+1+O(p).
Inordertostudyhowtheerrortermvarieswithpwewanttoconsiderthenormalizedtracesxp:=tp/p∈[2,2].
Wearenowinapositiontoconductthefollowingexperiment:givenanellipticcurveE/Fp,computexpforallgoodprimesp≤Bandseehowthexparedistributedovertherealinterval[2,2].
OnecanseeanexamplefortheellipticcurveE:y2=x3+x+1inFigure1,whichshowsahistogramwhosex-axisspanstheinterval[2,2].
Thisintervalissubdividedintoapproximatelyπ(B)subintervals,eachofwhichcontainsabarrepresentingthenumberofxp(forp≤B)thatlieinthesubinterval.
Thegraylineshowstheheightoftheuniformdistributionforscale(notethattheverticalandhorizontalscalesarenotthesame).
For0≤n≤10,themomentstatisticsMn:=p≤Bxnpp≤B1,areshownbelowthehistogram.
Theyappeartoconvergetotheintegers1,0,1,0,2,0,5,0,14,0,42,whichisthestartofsequenceA126120intheOnlineEncyclopediaofIntegerSequences(OEIS)[64]).
FIGURE1.
Clickimagetoanimate(requiresAdobeReader),orvisitthiswebpage.
9TheSato–TateconjectureforellipticcurvesoverQ(nowatheorem)impliesthatforalmostallE/Q,wheneverwerunthisexperimentwewillseetheasymptoticdistributionofFrobeniustracesvisibleinFigure1,withmomentstatisticsthatconvergetothesameintegersequence.
Inordertomakethisconjectureprecise,letusrstexplainwheretheconjectureddistributioncomesfrom.
InourrstexamplewehadacompactmatrixgroupGfassociatedtotheschemeX=SpecZ[x]/(f)whoseHaarmeasuregovernedthedistributionofNf(p).
Infactweshowedthatmoreistrue:thereisadirectrelationshipbetweencharacteristicpolynomialsofelementsofGfandtheL-polynomialsLp(T)thatappearinthelocalzetafunctionsZfp(T).
Thesameistruewithourellipticcurveexample.
InordertoidentifyacandidategroupGEwhoseHaarmeasurecontrolsthedistributionofnormalizedFrobeniustracesxpweneedtolookatthelocalzetafunctionsZEp(T).
LetusrecallwhattheWeilconjectures[96](provedbyDeligne[18,19])tellusaboutthezetafunctionofavarietyoveraniteeld.
Thecaseofone-dimensionalvarieties(curves)wasprovedbyWeil[94],whoalsoprovedananalogousresultforabelianvarieties[95].
Thiscoversallthecasesweshallconsider,butletusstatethegeneralresult.
RecallthatforacompactmanifoldXoverC,theBettinumberbiistherankofthesingularhomologygroupHi(X,Z),andtheEulercharacteristicχofXisdenedbyχ:=(1)ibi.
Theorem1.
8(WEILCONJECTURES).
LetXbeageometricallyirreduciblenon-singularprojectivevarietyofdimensionndenedoveraniteeldFqanddenethezetafunctionZX(T):=exp∞r=1NX(qr)Trr,whereNX(qr):=#X(Fqr).
Thefollowinghold:(i)Rationality:ZX(T)isarationalfunctionoftheformZX(T)=P1(T)···P2n1(T)P0(T)···P2n(T),withPi∈1+TZ[T].
(ii)FunctionalEquation:therootsofPi(T)arethesameastherootsofTdegP2niP2ni(1/(qnT)).
8(iii)RiemannHypothesis:thecomplexrootsofPi(T)allhaveabsolutevalueqi/2.
(iv)BettiNumbers:ifXisthereductionofanon-singularvarietyYdenedoveranumbereldKC,thenthedegreeofPiisequaltotheBettinumberbiofY(C).
ThecurveEpisacurveofgenusg=1,sowemayapplytheWeilconjecturesindimensionn=1,withBettinumbersb0=b2=1andb1=2g=2.
Thisimpliesthatitszetafunctioncanbewrittenas(5)ZEp(T)=Lp(T)(1T)(1pT),whereLp∈Z[T]isapolynomialoftheformLp(T)=pT2+c1T+1,with|c1|≤2p(bytheRiemannHypothesis).
Ifweexpandbothsidesof(5)aspowerseriesinZ[[T]]weobtain1+NE(p)T+···=1+(p+1+c1)T+···,8Moreover,onehasZX(T)=±qnχ/2TχZX(1/(qnT)),whereχistheEulercharacteristicofX,whichisdenedastheintersectionnumberofthediagonalwithitselfinX*X.
10sowemusthaveNE(p)=p+1+c1,andthereforec1=NE(p)p1=tp.
ItfollowsthatthesingleintegerNE(p)completelydeterminesthezetafunctionZEp(T).
Correspondingtoournormalizationxp=tp/p,wedenethenormalizedL-polynomialLp(T):=Lp(T/p)=T2+a1T+1,wherea1=c1/p=xpisarealnumberintheinterval[2,2]andtherootsofLp(T)lieontheunitcircle.
InourrstexampleweobtainedthegroupGfasasubgroupofpermutationmatricesinGLd(C).
HerewewantasubgroupofGL2(C)whoseelementshaveeigenvaluesthat(a)areinverses(bythefunctionalequation);(b)lieontheunitcircle(bytheRiemannhypothesis).
Constraint(a)makesitclearthateveryelementofGEshouldhavedeterminant1,soGESL2(C).
Constraints(a)and(b)togetherimplythatinfactGESU(2).
Asintheweightzerocase,weexpectthatGEshouldingeneralbeaslargeaspossible,thatis,GE=SU(2).
Wenowconsiderwhatitmeansforanellipticcurvetobegeneric.
9RecallthattheendomorphismringofanellipticcurveEnecessarilycontainsasubringisomorphictoZ,correspondingtothemultiplication-by-nmapsP→nP.
HerenP=P+···+Pdenotesrepeatedadditionunderthegrouplaw,andwetaketheadditiveinverseifnisnegative.
Forellipticcurvesovereldsofcharacteristiczero,thistypicallyaccountsforalltheendomorphisms,butinspecialcasestheendomorphismringmaybelarger,inwhichcaseitcontainselementsthatarenotmultiplication-by-nmapsbutcanbeviewedas"multiplication-by-α"mapsforsomeα∈C.
Onecanshowthattheminimalpolynomialsoftheseextraendomorphismsarenecessarilyquadratic,withnegativediscriminants,sosuchanαnecessarilyliesinanimaginaryquadraticeldK,andinfactEnd(E)ZQK.
WhenthishappenswesaythatEhascomplexmultiplication(CM)byK(ormoreprecisely,bytheorderinZKisomorphictoEnd(E)).
WecannowstatetheSato-Tateconjecture,asindependentlyformulatedinthemid1960'sbyMikioSato(basedonnumericaldata)andJohnTate(asanapplicationofwhatisnowknownastheTateconjecture[88]),andnallyprovedinthelate2000'sbyRichardTayloretal.
[6,7,32].
Theorem1.
9(SATO–TATECONJECTURE).
LetE/QbeanellipticcurvewithoutCM.
ThesequenceofnormalizedFrobeniustracesxpassociatedtoEisequidistributedwithrespecttothepushforwardoftheHaarmeasureonSU(2)underthetracemap.
Inparticular,foreverysubinterval[a,b]of[2,2]wehavelimB→∞#{p≤B:xp∈[a,b]}#{p≤B}=12πba4t2dt.
Wehavenotdenedxpforprimesofbadreduction,butthereisnoneedtodoso;thistheoremispurelyanasymptoticstatement.
Toseewheretheexpressionintheintegralcomesfrom,weneedtoun-derstandtheHaarmeasureonSU(2)anditspushforwardontothesetofconjugacyclassesconj(SU(2))(infactweonlycareaboutthelatter).
AconjugacyclassinSU(2)canbedescribedbyaneigenangle9Thecriteriongivenhereintermsofendomorphismringssufcesforellipticcurves(andcurvesofgenusg≤3orabelianvarietiesofdimensiong≤3),butingeneralonewantstheGaloisimagetobeaslargeaspossible,whichisastrictlystrongerconditionforg>3.
ThisissueisdiscussedfurtherinSection3.
11θ∈[0,π];itseigenvaluesarethene±iθ(aconjugatepairontheunitcircle,asrequired).
Intermsofeigenangles,thepushforwardoftheHaarmeasuretoconj(SU(2))isgivenby=2πsin2θdθ(seeExercise2.
4),andthetraceist=2cosθ;fromthisonecandeducethetracemeasure12π4t2dton[2,2]thatappearsinTheorem1.
9.
WecanalsousetheHaarmeasuretocomputethenthmomentofthetrace(6)E[tn]=2ππ0(2cosθ)nsin2θdθ=0ifnisodd,1m+12mmifn=2miseven,andndthatthe2mthmomentisthemthCatalannumber.
101.
7.
Exercises.
Exercise1.
1.
Letf∈Z[x]beanonconstantsquarefreepolynomial.
ProvethattheaveragevalueofNf(p)overp≤BconvergestothenumberofirreduciblefactorsoffinZ[x]asB→∞.
Exercise1.
2.
Provethattheidentityin(4)holdsforallmatricesA∈GLd(C).
Exercise1.
3.
Letfp∈Fp[x]denoteasquarefreepolynomialofdegreed>0andletLp(T)denotethedenominatorofthezetafunctionZfp(T).
WeknowthattherootsofLp(T)lieontheunitcircleinthecomplexplane;showthatinfacteachisannthrootofunityforsomen≤d.
Thengiveaone-to-onecorrespondencebetween(i)cycle-typesofdegree-dpermutations,(ii)possiblefactorizationpatternsoffpinFp[x],and(iii)thepossiblepolynomialsLp(T).
Exercise1.
4.
Constructamonicsquarefreequinticpolynomialf∈Z[x]withnorootsinQsuchthatfp(x)hasarootinFpforeveryprimep.
Computec0,.
.
.
,c5andGf.
Exercise1.
5.
LetXbethearithmeticschemeSpecZ[x,y]/(f,g),wheref(x,y):=y22x3+2x22x2,g(x,y):=4x22xy+y22.
BycomputingZXp(T)=Lp(T)1forsufcientlymanysmallprimesp,constructalistofthepolynomialsLp∈Z[T]thatyoubelieveoccurinnitelyoften,andestimatetheirrelativefrequencies.
UsethisdatatoderiveacandidateforthematrixgroupGX:=ρX(Gal(Q/Q),whereρXistheGaloisrepresentationdenedbytheactionofGal(Q/Q)onX(Q).
YoumaywishtouseofcomputeralgebrasystemsuchasSage[67]orMagmaor[11]tofacilitatethesecalculations.
2.
EQUIDISTRIBUTION,L-FUNCTIONS,ANDTHESATO-TATECONJECTUREFORELLIPTICCURVESInthissectionweintroducethenotionofequidistributionincompactgroupsGandrelateittoanalyticpropertiesofL-functionsofrepresentationsofG.
Wethenexplain(followingTate)whytheSato-Tateconjectureforellipticcurvesfollowsfromtheholomorphicityandnon-vanishingofacertainsequenceofL-functionsthatonecanassociatetoanellipticcurveoverQ(oranynumbereld).
10ThisgivesyetanotherwaytodenetheCatalannumbers,onethatdoesnotappeartobeamongthe214listedin[84].
122.
1.
Equidistribution.
Wenowformallydenethenotionofequidistribution,following[71,§1A].
ForacompactHausdorffspaceX,weuseC(X)todenotetheBanachspaceofcomplex-valuedcontinuousfunctionsf:X→Cequippedwiththesup-normf:=supx∈X|f(x)|.
ThespaceC(X)isclosedunderpointwiseadditionandmultiplicationandcontainsallconstantfunctions;itisthusacommutativeC-algebrawithunit1X(thefunctionx→1).
11ForanyC-valuedfunctionsfandg(continuousornot),wewritef≤gwheneverfandgarebothR-valuedandf(x)≤g(x)forallx∈X;inparticular,f≥0meansim(f)R≥0.
ThesubsetofR-valuedfunctionsinC(X)formadistributivelatticeunderthisorderrelation.
Denition2.
1.
A(positivenormalizedRadon)measureonacompactHausdorffspaceXisacontinuousC-linearmap:C(X)→Cthatsatises(f)≥0forallf≥0and(1X)=1.
Example2.
2.
Foreachpointx∈Xthemapf→f(x)denestheDiracmeasureδx.
Thevalueofonf∈C(X)isoftendenotedusingintegralnotationXf:=(f),andweshallusethetwointerchangeably.
12HavingdenedthemeasureasafunctiononC(X),wewouldliketouseittoassignvaluesto(atleastsome)subsetsofX.
ItistemptingtodenethemeasureofasetSXasthemeasureofitsindicatorfunction1S,butingeneralthefunction1SwillnotlieinC(X);thisoccursifandonlyifSisbothopenandclosed(whichwenoteappliestoS=X).
Instead,foreachopensetSXwedene(S)=sup(f):0≤f≤1S,f∈C(X)∈[0,1],andforeachclosedsetSXwedene(S)=1(XS)∈[0,1].
IfSXhasthepropertythatforeveryε>0thereexistsanopensetUSofmeasure(U)≤ε,thenwedene(S)=0andsaythatShasmeasurezero.
IftheboundaryS:=SS0ofasetShasmeasurezero,thenwenecessarilyhave(S0)=(S)anddene(S)tobethiscommonvalue;suchsetsaresaidtobe-quarrable.
Forthepurposeofstudyingequidistribution,weshallrestrictourattentionto-quarrablesetsS.
Thistypicallydoesnotincludeallmeasurablesetsintheusualsense,bywhichwemeanelementsoftheBorelσ-algebraΣofXgeneratedbytheopensetsundercomplementsandcountableunionsandintersections(seeExercise2.
1).
However,ifwearegivenaregularBorelmeasureonXoftotalmass1,bywhichwemeanacountablyadditivefunction:Σ→R≥0forwhich(S)=inf{(U):SU,Uopen}and(X)=1,itiseasytocheckthatdening(f):=Xfforeachf∈C(X)yieldsameasureunderDenition2.
1;see[41,§1]fordetails.
Thismeasureisdeterminedbythevaluestakeson-quarrablesets[99].
Inparticular,ifXisacompactgroupthenitsHaarmeasureinducesameasureonXinthesenseofDenition2.
1.
11Infact,itisacommutativeC-algebrawithcomplexconjugationasitsinvolution,butwewillnotmakeuseofthis.
12Notethatthisisadenition;withameasure-theoreticapproachoneavoidstheneedtodevelopanintegrationtheory.
13Denition2.
3.
Asequence(x1,x2,x3,.
.
.
)inXissaidtobeequidistributedwithrespectto,orsimply-equidistributed,ifforeveryf∈C(X)wehave(f)=limn→∞1nni=1f(xi).
Remark2.
4.
Whenwespeakofequidistribution,notethatwearetalkingaboutasequence(xi)ofele-mentsofXinaparticularorder;itdoesnotmakesensetosaythatasetisequidistributed.
Forexample,supposewetookthesetofoddprimesandarrangedtheminthesequence(5,13,3,17,29,7,.
.
.
)wherewelisttwoprimescongruentto1modulo4followedbyoneprimecongruentto3modulo4.
These-quenceobtainedbyreducingthissequencemodulo4isnotequidistributedwithrespecttotheuniformmeasureon(Z/4Z)*,eventhoughthesequenceofoddprimesintheirusualorderis(byDirichlet'sthe-oremonprimesinarithmeticprogressions).
However,localrearrangementsthatchangetheindexofanelementbynomorethanaboundedamountdonotchangeitsequidistributionproperties.
Thisapplies,inparticular,tosequencesindexedbyprimesofanumbereldorderedbynorm;theequidistributionpropertiesofsuchasequencedonotdependonhowweorderprimesofthesamenorm.
If(xi)isasequenceinX,foreachreal-valuedfunctionf∈C(X)wedenethekth-momentofthesequence(f(xi))byMk[(f(xi)]:=limn→∞1nni=1f(xi)k.
Iftheselimitsexistforallk≥0,wethendenethemomentsequenceM[f(xi)]:=(M0[(f(xi)],M1[(f(xi)],M2[(f(xi)],.
.
.
).
If(xi)is-equidistributed,thenMk[f(xi)]=(fk)andthemomentsequence(7)M[f(xi)]=((f0),(f1),(f2),.
.
.
)isindependentofthesequence(xi);itdependsonlyonthefunctionfandthemeasure.
Remark2.
5.
Thereisapartialconversethatisrelevanttosomeofourapplications.
Tosimplifymatters,letusmomentarilyrestrictourattentiontoreal-valuedfunctions;forthepurposesofthisremark,letC(X)denotetheBanachalgebraofreal-valuedfunctionsonXandreplaceCwithRinDenition2.
1.
Let(xi)beasequenceinXandletf∈C(X).
Thenf(X)isacompactsubsetofR,andwemayview(f(xi))asasequenceinf(X).
IfthemomentsMk[f(xi)]existforallk≥0,thenthereisauniquemeasureonf(X)withrespecttowhichthesequence(f(xi))isequidistributed;thisfollowsfromtheStone-Weierstrasstheorem.
IfisameasureonC(X),wedenethepushforwardmeasuref(g):=(gf)onC(f(X)andseethatthesequence(f(xi))isf-equidistributedifandonlyif(7)holds.
Thisgivesanecessary(butingeneralnotsufcient)conditionfor(xi)tobe-equidistributedthatcanbecheckedbycomparingmomentsequences.
Ifwehaveacollectionoffunctionsfj∈C(X)suchthatthepushforwardmeasuresfjuniquelydetermine,weobtainanecessaryandsufcientconditioninvolvingthemomentsequencesofthefjwithrespectto.
Onecangeneralizethisremarktocomplex-valuedfunctionsusingthetheoryofC-algebras.
Moregenerally,wehavethefollowinglemma.
Lemma2.
6.
Let(fj)beafamilyoffunctionswhoselinearcombinationsaredenseinC(X).
If(xi)isasequenceinXforwhichthelimitlimn→∞1nni=1fj(xi)convergesforeveryfj,thenthereisauniquemeasureonXforwhich(xi)is-equidistributed.
14Proof.
See[71,LemmaA.
1,p.
I-19].
Proposition2.
7.
If(xi)isa-equidistributedsequenceinXandSisa-quarrablesetinXthen(S)=limn→∞#{xi∈S:i≤n}n.
Proof.
SeeExercise2.
2.
Example2.
8.
IfX=[0,1]andistheLebesguemeasurethenasequence(xi)is-equidistributedifandonlyifforevery0≤a1.
Theorem2.
12.
LetGand(xp)beasabove,andsupposeL(ρ,s)ismeromorphiconRe(s)≥1withnozerosorpolesexceptpossiblyats=1,foreveryirreduciblerepresentationρofG.
Thesequence(xp)isequidistributedifandonlyifforeachρ=1,theL-functionL(ρ,s)extendsanalyticallytoafunctionthatisholomorphicandnonvanishingonRe(s)≥1.
Proof.
Seethecorollaryto[71,Thm.
A.
2],orsee[24,Thm.
2.
3].
AnotablecaseinwhichthehypothesisofTheorem2.
12isknowntoholdiswhenL(ρ,s)correspondstoanArtinL-function.
AsinSection1.
1,toeachprimepinKweassociateanabsoluteFrobeniuselementFrobp∈Gal(K/K),andforeachniteGaloisextensionL/KweuseconjL(Frobp)todenotetheconjugacyclassinGal(L/K)oftherestrictionofFrobptoL.
Corollary2.
13.
LetL/KbeaniteGaloisextensionwithG:=Gal(L/K)andletPbethesequenceofunramiedprimesofKorderedbynorm(breaktiesarbitrarily).
Thesequence(conjL(Frobp))p∈Pisequidistributedinconj(G);inparticular,theChebotarevdensitytheorem(Theorem1.
1)holds.
Proof.
Forthetrivialrepresentation,theL-functionL(1,s)agreeswiththeDedekindzetafunctionζK(s)uptoanitenumberofholomorphicnonvanishingfactors,and,asoriginallyprovedbyHecke,ζK(s)isholomorphicandnonvanishingonRe(s)≥1exceptforasimplepoleats=1;see[62,Cor.
VII.
5.
11],forexample.
Foreverynontrivialirreduciblerepresentationρ:G→GLd(C),theL-functionL(ρ,s)agreeswiththecorrespondingArtinL-functionforρ,uptoanitenumberofholomorphicnonvanishingfactors,and,asoriginallyprovedbyArtin,L(ρ,s)isholomorphicandnonvanishingonRe(s)≥1;see[14,p.
225],forexample.
ThecorollarythenfollowsfromTheorem2.
12.
2.
4.
Sato–TateforCMellipticcurves.
AsasecondapplicationofTheorem2.
12,letusproveanequidistributionresultforCMellipticcurves.
TodosoweneedtointroduceHeckecharacters,whichwewillviewas(quasi-)charactersoftheidèleclassgroupofanumbereld.
17Denition2.
14.
LetKbeanumbereldandletIKdenoteitsidèlegroup.
AHeckecharacterisacontinuoushomomorphismψ:IK→C*whosekernelcontainsK*.
TheconductorofψistheZK-idealf:=ppepinwhicheachepistheminimalnonnegativeintegerforwhich1+pepZ*Kp→IKliesinthekernelofψ(allbutnitelymanyeparezerobecauseψiscontinuous);herepdenotesthemaximalidealofthevaluationringZKpofKp,thecompletionofKwithrespecttoitsp-adicabsolutevalue.
EachHeckecharacterψhasanassociatedHeckeL-functionL(ψ,s):=pf(1ψ(p)N(p)s)1,whereψ(p):=ψ(πp)foranyuniformizerπpofp(wehaveomittedthegammafactorsatarchimedeanplaces).
Wenowwanttoconsiderthesequenceofunitarizedvaluesxp:=ψ(p)|ψ(p)|∈U(1)indexedbyprimespforderedbynorm.
Lemma2.
15.
Thesequence(xp)isequidistributedinU(1).
Proof.
AsintheproofofProposition2.
11,thenontrivialirreduciblecharactersofU(1)arethoseoftheformφa(z)=zawitha∈Znonzero,andineachcasethecorrespondingL-functionisaHeckeL-function(ifψisaHeckecharacter,soisψaanditsunitarizedversion).
Ifψistrivialthen,asintheproofofCorollary2.
13,L(1,s)isholomorphicandnonvanishingonRe(s)≥1exceptforasimplepoleats=1,sincethesameistrueofζK(s).
Heckeproved[40]thatwhenψisnontrivialL(ψ,s)isholomorphicandnonvanishingonRe(s)≥1,andthelemmathenfollowsfromTheorem2.
12.
AsanapplicationofLemma2.
15,wecannowprovetheSato-TateconjectureforCMellipticcurves.
LesusrstconsiderthecasewhereKisanimaginaryquadraticeldandE/KisanellipticcurvewithCMbyK(soKEnd(E)ZQ).
Asexplainedbelow,thegeneralcase(includingK=Q)followseasily.
LetfbetheconductorofE;thisisaZK-idealdivisibleonlybytheprimesofbadreductionforE;see[81,§IV.
10]foradenition.
AclassicalresultofDeuring[81,Thm.
II.
10.
5]impliestheexistenceofaHeckecharacterψEofKofconductorfsuchthatforeachprimepfwehave|ψE(p)|=N(p)1/2andψE(p)+ψE(p)=tp,wheretp:=trπE=N(p)+1#Ep(Fp)∈ZisthetraceofFrobeniusofthereductionofEmodulop.
Proposition2.
16.
LetKbeanimaginaryquadraticeldandletE/KbeanellipticcurveofconductorfwithCMbyK.
LetPbethesequenceofprimesofKthatdonotdivideforderedbynorm(breaktiesarbitrarily),andforp∈Pletxp:=tp/N(p)1/2∈[2,2]bethenormalizedFrobeniustraceofEp.
Thesequence(xp)isequidistributedon[2,2]withrespecttothemeasurecm:=1πdz4z2.
Proof.
Bythepreviouslemma,thesequence(ψE(p)/N(p)1/2)p∈PisequidistributedinU(1).
AsshownintheproofofProposition2.
11,themeasurecmisthepushforwardoftheHaarmeasureonU(1)to18[2,2]underthemapu→u+u.
Foreachp∈PtheimageofψE(p)/N(p)1/2underthismapisψE(p)N(p)1/2+ψE(p)N(p)1/2=tpN(p)1/2=xp.
Figure2showsatracehistogramfortheCMellipticcurvey2=x3+1overitsCMeldQ(3).
FIGURE2.
Clickimagetoanimate(requiresAdobeReader),orvisitthiswebpage.
LetusnowconsiderthecaseofanellipticcurveE/QwithCMbyF.
ForprimespofgoodreductionthatareinertinF,theendomorphismalgebraEnd(Ep)Q:=End(Ep)ZQofthereducedcurveEpcontainstwodistinctimaginaryquadraticelds,onecorrespondingtotheCMeldFEnd(E)QandtheothergeneratedbytheFrobeniusendomorphism(thetwocannotcoincidebecausepisinertinFbuttheFrobeniusendomorphismhasnormpinEnd(Ep)Q).
ItfollowsthatEnd(Ep)Qmustbeaquaternionalgebra,Epissupersingular,andforp>3wemusthavetp=0,sincetp≡0modpand|tp|≤2p;see[82,III,9,V.
3]fortheseandotherfactsaboutendomorphismringsofellipticcurves.
Atsplitprimesp=ppthereducedcurveEpwillbeisomorphictothereductionmodulopofitsbasechangetoF(bothofwhichareellipticcurvesoverFp=Fp),andwillhavethesametraceofFrobeniustp=tp.
BytheChebotarevdensitytheorem,thesplitandinertprimesbothhavedensity1/2,anditfollowsthatthesequenceofnormalizedFrobeniustracesxp:=tp/p∈[2,2]isequidistributedwithrespecttothemeasure12δ0+12cm,whereweusetheDiracmeasureδ0toputhalfthemassat0toaccountfortheinertprimes.
ThiscanbeseeninFigure3,whichshowsatracehistogramfortheCMellipticcurvey2=x3+1overQ;thethinspikeinthemiddleofthehistogramatzerohasarea1/2(onecanalsoseethatthenontrivialmomentsarehalfwhattheywereinFigure2).
19FIGURE3.
Clickimagetoanimate(requiresAdobeReader),orvisitthiswebpage.
AsimilarargumentapplieswhenEisdenedoveranumbereldKthatdoesnotcontaintheCMeldF.
Forthesakeofprovinganequidistributionresultwecanrestrictourattentiontothedegree-1primespofK,thoseforwhichN(p)=pisprime.
HalfoftheseprimespwillsplitinthecompositumKF,andthesubsequenceofnormalizedtracesxpattheseprimeswillbeequidistributedwithrespecttothemeasurecm,andhalfwillbeinertinKF,inwhichcasexp=tp=0.
2.
5.
Sato–Tatefornon-CMellipticcurves.
WecannowstatetheSato-TateconjectureintheformoriginallygivenbyTate,following[71,§1A].
Tate'sseminalpaper[88]describeswhatisnowknownastheTateconjecture,whichcomesintwoconjecturallyequivalentformsT1andT2,thelatterofwhichisstatedintermsofL-functions.
TheSato-TateconjectureisobtainedbyapplyingT2toallpowersofaxedellipticcurveE/Q(asproductsofabelianvarieties);see[66]foranintroductiontotheTateconjectureandanexplanationofhowtheSato-Tateconjecturetswithinit.
LetGbethecompactgroupSU(2)of2*2unitarymatriceswithdeterminant1.
TheirreduciblerepresentationsofGarethemthsymmetricpowersρmofthenaturalrepresentationρ1ofdegree2givenbytheinclusionSU(2)GL2(C).
EachelementofX:=conj(G)canbeuniquelyrepresentedbyamatrixoftheformeiθ00eiθ,whereθ∈[0,π]istheeigenangleoftheconjugacyclass.
Itfollowsthateachf∈C(X)canbeviewedasacontinuousfunctionf(θ)onthecompactset[0,π].
20ThepushforwardoftheHaarmeasureofGtoXisgivenby(8)=2πsin2θdθ(seeExercise2.
4),whichmeansthatforeachf∈C(X)wehave(f)=2ππ0f(θ)sin2θdθ.
LetE/QbeanellipticcurvewithoutCM,letP:=(p)bethesequenceofprimesthatdonotdividetheconductorNofE,inorder,andforeachp∈Pletxp∈XtobetheelementofXcorrespondingtotheuniqueθp∈[0,π]forwhich2cosθpp=tp:=p+1#Ep(Fp)isthetraceofFrobeniusofthereducedcurveEp.
Wearenowinthesettingof§2.
3.
WehaveacompactgroupG:=SU(2),itsspaceofconjugacyclassesX:=conj(G),anumbereldK=Q,asequencePcontainingallbutnitelymanyprimesofKorderedbynorm,asequence(xp)inXindexedbyP,andforeachintegerm≥1,anirreduciblerepresentationρm:G→GLm+1(C).
TheL-functioncorrespondingtoρmisgivenbyL(ρm,s):=pNdet(1ρm(xp)ps)1=pNmk=0(1ei(m2k)θpps)1.
ForeachpN,letαpandαpbetherootsofT2tpT+p,sothatαp=eiθpp1/2.
IfwenowdeneL1m(s):=pNmr=0(1αmrpαrpps)1,thenform≥1wehaveL(ρm,s)=L1m(sm/2).
Tateconjecturedin[88]thatL1m(s)isholomorphicandnonvanishingonRe(s)≥1+m/2,whichimpliesthateachL(ρm,s)isholomorphicandnonvanishingonRe(s)≥1.
Assumingthisistrue,Theorem2.
12impliesthatthesequence(xp)is-equidistributed,whichisequivalenttotheSato-Tateconjecture.
WenowrecallthemodularitytheoremforellipticcurvesoverQ,whichstatesthatthereisaone-to-onecorrespondencebetweenisogenyclassesofellipticcurvesE/QofconductorNandmodularformsf(z)=n≥1ane2πinz∈S2(Γ0(N))new(an∈Zwitha1=1)thatareeigenformsfortheactionoftheHeckealgebraonthespaceS2(Γ0(N))ofcuspformsofweight2andlevelNandnewatlevelN,meaningnotcontainedinS2(Γ0(M))foranypositiveintegerMproperlydividingN.
Suchmodularformsfarecalled(normalized)newforms,ofweight2andlevelN,withrationalcoefcients.
ThemodularitytheoremwasprovedforsquarefreeNbyTaylorandWiles[91,98],andextendedtoallconductorsNbyBreuil,Conrad,Diamond,andTaylor[12].
ThemodularformfisasimultaneouseigenformforalltheHeckeoperatorsTn,andthenormalizationa1=1ensuresthatforeachprimepN,thecoefcientapistheeigenvalueoffforTp.
Underthecorrespondencegivenbythemodularitytheorem,theeigenvalueapisequaltothetraceofFrobeniustpofthereducedcurveEp,whereEisanyrepresentativeofthecorrespondingisogenyclass.
HereweareusingthefactthatifEandEareisogenousellipticcurvesoverQtheynecessarilyhavethesameconductorNandthesametraceofFrobeniustpateverpN.
ThereisanL-functionL(f,s)associatedtothemodularformf,andthemodularitytheoremguar-anteesthatitcoincideswiththeL-functionL(E,s)ofE.
Sonotonlydoesap=tpforallpN,theEuler21factorsatthebadprimesp|Nalsoagree.
WeneednotconcernourselveswithEulerfactorsattheseprimes,otherthantonotethattheyareholomorphicandnonvanishingonRe(s)≥3/2.
AfterremovingtheEulerfactorsatbadprimes,theL-functionsL(E,s)andL(f,s)bothhavetheformpN(1apps+p12s)1=pN1r=0(1α1rpαrpps)1=L11(s),whereαpandαparetherootsofT2apT+p=T2tpT+p.
TheL-functionL(f,s)isholomorphicandnonvanishingonRe(s)≥3/2;see[21,Prop.
5.
9.
1].
ThemodularitytheoremtellsusthatthesameistrueofL(E,s),andthereforeofL11(s).
ThusthemodularitytheoremprovesthatTate'sconjectureregardingL1m(s)holdswhenm=1.
ToprovetheSato-Tateconjectureoneneedstoshowthatthisholdsforallm≥1.
Theorem2.
17.
Letf(z):=n≥1ane2πizn∈S2(Γ0(N)newbeanormalizednewformwithoutCM.
ForeachprimepNletαp,αpbetherootsofT2apT+p.
ThenpNmr=0(1αmrpαrpps)1=L1m(s)isholomorphicandnonvanishingonRe(s)≥1+m/2.
Proof.
Apply[7,TheoremB.
2]withweightk=2,trivialnebentypusψ=1,andtrivialcharacterχ=1(asnotedin[7],thisspecialcasewasalreadyaddressedin[32]).
Corollary2.
18.
TheSato-Tateconjecture(Theorem1.
9)holds.
Remark2.
19.
TheSato-Tateconjectureisalsoknowntoholdforellipticcurvesovertotallyrealelds,andoverCMelds(imaginaryquadraticextensionsoftotallyrealelds).
Thetotallyrealcasewasinitiallyprovedforellipticcurveswithpotentiallymultiplicativereductionatsomeprimein[32,90];itwaslatershownthistechnicalassumptioncanberemoved(seetheintroductionof[6]).
Thegeneral-izationtoCMeldswasobtainedatarecentIASworkshop[3]andstillintheprocessofbeingwrittenupindetail.
AsaconsequenceofthisresulttheSato-Tateconjectureforellipticcurvesisnowknownforallnumbereldsofdegree1or2(butnotforanyhigherdegrees).
2.
6.
Exercises.
Exercise2.
1.
LetXbeacompactHausdorffspace.
ShowthatasetSXis-quarrableforeverymeasureonXifandonlyifthesetSisbothopenandclosed.
Exercise2.
2.
ProveProposition2.
7.
Exercise2.
3.
LetEanellipticcurveoverFqandletαbearootofthecharacteristicpolynomialoftheFrobeniusendomorphismπE.
Provethatα/qisarootofunityifandonlyifEissupersingular.
Exercise2.
4.
ShowthatthesetofconjugacyclassesofSU(2)isinbijectionwiththesetofeigenanglesθ∈[0,π].
ThenprovethatthepushforwardoftheHaarmeasureofSU(2)onto[0,π]isgivenby:=2πsin2θdθ(hint:showthatSU(2)isisomorphictothe3-sphereS3andusethisisomorphismtogetherwiththetranslationinvarianceoftheHaarmeasuretodetermine)Exercise2.
5.
ComputethetracemomentsequenceforSU(2)(thatis,prove(6)).
EmbedU(1)inSU(2)viathemapu→u00uandcomputeitstracemomentsequence(comparetoFigure2).
NowdeterminethenormalizerN(U(1))ofU(1)inSU(2)andcomputeitstracemomentsequence(comparetoFigure3).
223.
SATO-TATEGROUPSIntheprevioussectionweshowedthattherearethreedistinctSato-TatedistributionsthatariseforellipticcurvesEovernumbereldsK(onlytwoofwhichoccurwhenK=Q).
AllthreedistributionscanbeassociatedtotheHaarmeasureofacompactsubgroupGSU(2),inwhichweembedU(1)viathemapu→u00u.
WeareinterestedinthepushforwardoftheHaarmeasureontoconj(G),whichcanbeexpressedintermsoftheeigenangleθ∈[0,π].
ThethreepossibilitiesforGarelistedbelow.
U(1):wehave(θ)=1πdθandtracemoments:(1,0,2,0,6,0,20,0,70,0,252,.
.
.
).
ThiscasearisesforCMellipticcurvesdenedoveraeldthatcontainstheCMeld.
N(U(1)):wehave(θ)=12πdθ+12δπ/2andtracemoments:(1,0,1,0,3,0,10,0,35,0,126,.
.
.
).
ThiscasearisesforCMellipticcurvesdenedoveraeldthatdoesnotcontaintheCMeld.
SU(2):wehave(θ)=2πsin2θdθandtracemoments:(1,0,1,0,2,0,5,0,14,0,42,.
.
.
).
Thiscasearisesforallnon-CMellipticcurves(conjecturallysowhenKnottotallyrealorCM).
Wehavewrittenintermsofθ,butwemayviewitasalinearfunctionontheBanachspaceC(X),whereweidentifyX:=conj(G)with[0,π],bydening(f):=π0f(θ)(θ),asin§2.
1.
Inparticular,assignsavaluetothetracefunctiontr:X→[2,2],wheretr(θ)=2cosθ,andtoitspowerstrn,whichallowsustocomputethetracemomentsequence((trn))n≥0.
OurgoalinthissectionistodenethecompactgroupGasaninvariantoftheellipticcurveE,theSato-TategroupofE,andtothengeneralizethisdenitiontoabelianvarietiesofarbitrarydimension.
ThiswillallowustostatetheSato-TateconjectureforabelianvarietiesasanequidistributionstatementwithrespecttotheHaarmeasureoftheSato-Tategroup.
3.
1.
TheSato-Tategroupofanellipticcurve.
ThusfarthelinkbetweentheellipticcurveEandthecompactgroupGwhoseHaarmeasureisclaimed(andinmanycasesproved)togovernthedistributionofFrobeniustraceshasbeenmadeviathemeasure.
Thatis,wehaveanequidistributionclaimforthesequence(xp)ofnormalizedFrobeniustracesassociatedtoEthatisphrasedintermsofameasurethathappenstobeinducedbytheHaarmeasureofacompactgroupG.
WenowwanttoestablishadirectrelationshipbetweenEandGthatdenesGasanarithmeticinvariantofE,withoutassumingtheSato-Tateconjecture.
InSection1.
1weconsideredtheGaloisrepresentationρf:Gal(Q/Q)→GLd(C)denedbytheactionofGal(Q/Q)ontherootsofasquarefreepolynomialf∈Z[x].
WetherebyobtainedacompactgroupGfandamapthatsendseachprimepofgoodreductionforftoanelementofconj(Gf)(namely,themapp→ρf(Frobp)).
WewerethenabletorelatetheimageofpunderthismaptothequantityNf(p)ofinterest,via(1).
Thisconstructiondidnotinvolveanydiscussionofequidistribution,butwecouldthenprove,viatheChebotarevdensitytheorem,thattheconjugacyclassesρf(p)areequidistributedwithrespecttothepushforwardoftheHaarmeasuretoconj(Gf).
Wetakeasimilarapproachhere.
ToeachellipticcurveEoveranumbereldKwewillassociateacompactgroupGthatisconstructedviaaGaloisrepresentationattachedtoE,equippedwithamapthatsendseachprimepofgoodreductionforEtoanelementxpofconj(G)thatwecandirectlyrelatetothequantityNE(p):=p+1tpwhosedistributionwewishtostudy.
Wemaythenconjecture(andprove,whenEhasCMorKisatotallyrealorCMeld),thatthesequence(xp)isequidistributedinX:=conj(G)(withrespecttothepushforwardoftheHaarmeasureofG).
ThegroupGistheSato–TategroupofE,andwillbedenotedST(E).
ItisacompactsubgroupofSU(2),andourconstructionwillmakeiteasytoshowthatST(E)isalwaysoneofthethreegroupsU(1),N(U(1)),SU(2)listedabove,dependingonwhetherEhasCMornot,andifso,whetherthe23CMeldiscontainedinthegroundeldornot.
Noneofthisdependsonanyequidistributionresults.
ThisconstructionwillbeourprototypeforthedenitionoftheSato-Tategroupofanabelianvarietyofarbitrarydimensiong,sowewillworkouttheg=1caseinsomedetail.
InordertoassociateaGaloisrepresentationtoE/K,weneedasetonwhichGal(K/K)canact.
Foreachintegern≥1,letE[n]:=E(K)[n]denotethen-torsionsubgroupofE(K),afreeZ/nZ-moduleofrank2(see[82,Cor.
III.
6.
4]).
ThegroupGal(K/K)actsonpointsinE(K)coordinate-wise,andE[n]isinvariantunderthisactionbecauseitisthekernelofthemultiplication-by-nmap[n],anendomorphismofEthatisdenedoverK;onecanconcretelydeneE[n]asthezerolocusofthen-divisionpolynomials,whichhavecoefcientsinK.
TheactionofGal(K/K)onE[n]inducesthemod-nGaloisrepresentationGal(K/K)→Aut(E[n])GL2(Z/nZ).
ThisGaloisrepresentationisinsufcientforourpurposes,becausetheimageMpofFrobpinGL2(Z/nZ)doesnotdeterminetp,weonlyhavetp≡trMpmodn;weneedtoletGal(K/K)actonabiggerset.
Soletusxaprime(anyprimewilldo),andconsidertheinversesystem···[]→E[3][]→E[2][]→E[].
TheinverselimitT:=lim←nE[n]isthe-adicTate-moduleofE;itisafreeZ-moduleofrank2.
ThegroupGal(K/K)actsonTviaitsactiononthegroupsE[n],andthisactioniscompatiblewiththemultiplication-by-map[]becausethismapisdenedoverK(itcanbewrittenasarationalmapwithcoefcientsinK).
Thisyieldsthe-adicGaloisrepresentationρE,:Gal(K/K)→Aut(T)GL2(Z).
TherepresentationρE,enjoysthefollowingproperty:foreveryprimepofgoodreductionforEtheimageofFrobpisamatrixMp∈GL2(Z)thathasthesamecharacteristicpolynomialastheFrobeniusendomorphismofEp,namely,T2tpT+N(p),wheretp:=trπEp.
NotethatthematrixMpisdeterminedonlyuptoconjugacy;thereisambiguitybothinourchoiceofFrobp(see§1.
1)andinourchoiceofabasisforT,whichxestheisomorphismAut(T)GL2(Z).
WeshouldthusthinkofρE,(Frobp)asrepresentingaconjugacyclassinGL2(Z).
WeprefertoworkovertheeldQ,ratherthanitsringofintegersZ,soletusdenetherationalTatemoduleV:=TZQ,whichisa2-dimensionalQ-vectorspaceequippedwithanactionofGal(K/K).
ThisallowsustoviewtheGaloisrepresentationρE,ashavingimageGGL2(Q).
Wealsoprefertoworkwithanalgebraicgroup,soletusdeneGzartobetheQ-algebraicgroupobtainedbytakingtheZariskiclosureofGinGL2(Q).
ThismeansthatGzaristheafnevarietydenedbytheidealofQ-polynomialsthatvanishonthesetG;itisasubvarietyofGL2/QthatisclosedunderthegroupoperationandthusanalgebraicgroupoverQ.
ThealgebraicgroupGzaristhe-adicmonodromygroupofE(itisalsodenotedGalg).
Background3.
1(Algebraicgroups).
Anafne(orlinear)algebraicgroupoveraeldkisagroupobjectinthecategoryof(notnecessarilyirreducible)afnevarietiesoverk.
Theonlyprojectivealgebraicgroupsweshallconsideraresmoothandconnected,henceabelianvarieties,sowhenweusetheterm24algebraicgroupwithoutqualication,wemeananafnealgebraicgroup.
14ThecanonicalexampleisGLn,whichcanbedenedasanafnevarietyinAn2+1(overanyeld)bytheequationtdetM=1(heredetMdenotesthedeterminantpolynomialinn2variablesMij),withmorphismsm:GLn*GLn→GLnandi:GLn→GLndenedbypolynomialmapscorrespondingtomatrixmultiplicationandinversion(oneusestastheinverseofdetAwhendeningi).
TheclassicalgroupsSLn,Sp2n,Un,SUn,On,SOnareallafnealgebraicgroups(assumechar(k)=2forOnandSOn),asarethegroupsUSp2n:=Sp2n∩U2nandGSp2nthatareofparticularinteresttous;theRandCpointsofthesegroupsareLiegroups(differentiablemanifoldswithagroupstructure).
IfGisanafnealgebraicgroupoverkandL/kisaeldextension,theZariskiclosureofanysubgroupHG(L)oftheL-pointsofGisequaltothesetofrationalpointsofanafnevarietydenedoverLthatisalsoanalgebraicgroupviathemorphismsmandideningG.
ThuseverysubgroupHG(L)uniquelydeterminesanalgebraicgroupoverLwhoserationalpointscoincidewiththeZariskiclosureofH;asanabuseofterminologywemayrefertothisalgebraicgroupastheZariskiclosureofHinG(L)(orinGL,thebasechangeofGtoL).
TheconnectedandirreduciblecomponentsofanalgebraicgroupGcoincide,andarenecessarilyniteinnumber.
TheconnectedcomponentG0oftheidentityisitselfanalgebraicgroup,anormalsubgroupofGcompatiblewithbasechange.
Formoreonalgebraicgroupsseeanyoftheclassictexts[10,42,83],orsee[55]foramoremoderntreatment.
HavingdenedtheQ-algebraicgroupGzar,wenowrestrictourattentiontothesubgroupG1,zarobtainedbyimposingthesymplecticconstraintMtM0110,whichcorrespondstoputtingasymplecticform(anondegeneratebilinearalternatingpairing)onthevectorspaceV(wecouldofcoursechooseanythatdenessuchaform).
Thisconditioncanclearlybeexpressedbyapolynomial(aquadraticforminfact),thusG1,zarisanalgebraicgroupoverQcontainedinSp2.
WeremarkthatSp2=SL2,sowecouldhavejustrequireddetM=1,butthisisanaccidentoflowdimension:theinclusionSp2nSL2nisstrictforalln>1.
Finally,letuschooseanembeddingι:Q→C,andletG1,zar,ιbetheC-algebraicgroupobtainedfromG1,zarbybasechangetoC(viaι).
ThegroupG1,zar,ι(C)isasubgroupofSp2(C)thatwemayviewasaLiegroupwithnitelymanyconnectedcomponents.
Itthereforecontainsamaximalcompactsubgroupthatisuniqueuptoconjugacy[63,Thm.
IV.
3.
5],andwetakethisastheSato–TategroupST(E)ofE(whichisthusdenedonlyuptoconjugacy).
ItisacompactsubgroupofUSp(2)=SU(2)(thisequalityisanotheraccidentoflowdimension).
ForeachprimepofgoodreductionforE,letMpdenotetheimageofFrobpunderthemapsGal(K/K)ρE,→G→Gzar(Q)→Gzar,ι(C),wherethemapinthemiddleisinclusionandweusetheembeddingι:Q→Ctoobtainthelastmap.
WenowconsiderthenormalizedFrobeniusimageMp:=N(p)1/2Mp;14Thereareinterestingalgebraicgroups(groupschemesofnitetypeoveraeld)thatareneitherafnenorprojective(evenifwerestrictourattentiontothosethataresmoothandconnected),butweshallnotconsiderthemhere.
25itisamatrixwithtracetp/N(p)1/2∈[2,2]anddeterminant1,anditseigenvaluese±iθplieontheunitcircle.
15TheeigenangleθpdeterminesauniqueconjugacyclassinST(E),whichwetakeasxp.
ThecharacteristicpolynomialofxpisthenormalizedL-polynomialLp(T):=Lp(N(p)1/2T),whereLp(T)isthenumeratorofthezetafunctionofEp,andLp(N(p)s)istheEulerfactoratpintheL-seriesL(E,s).
TheSato–Tateconjecturethenamountstothestatementthatthesequence(xp)inX:=conj(ST(E))isequidistributed.
NoticethatthestatementisthesameinboththeCMandnon-CMcases,butthemeasureonXisdifferent,becauseST(E)isdifferent.
Indeed,therearethreepossibilitiesforST(E),correspondingtothethreedistributionsthatwenotedatthebeginningofthissection.
Theorem3.
2.
LetEbeanellipticcurveoveranumbereldK.
UptoconjugacyinSU(2)wehaveST(E)=U(1)ifEhasCMdenedoverK,N(U(1))ifEhasCMnotdenedoverK,SU(2)ifEdoesnothaveCM,whereU(1)isembeddedinSU(2)viau→u00u.
Proof.
IfEhasCMdenedoverKthenGisabelian,becausetheactionofGal(K/K)onVfactorsthroughtheabeliangroupGal(L/K),whereL:=K(E[∞])isobtainedbyadjoiningthecoordinatesofthe-powertorsionpointsofE;thisfollowsfrom[81,Thm.
II.
2.
3].
ThereforeGliesinaCartansubgroupofGL2(Q)(amaximalabeliansubgroup),whichnecessarilysplitswhenwepasstoGzar,ι(C),whereitisconjugatetothegroupofdiagonalmatrices.
ThisimpliesthatST(E)isconjugatetoU(1),thesubgroupofdiagonalmatricesinSU(2).
IfEhasCMnotdenedoverK,thenGliesinthenormalizerofaCartansubgroupofGL2(Q),butnotintheCartanitself,andST(E)isconjugatetothenormalizerN(U(1))ofU(1)inSU(2);theargumentisasabove,butnowtheactionofGal(K/K)factorsthroughGal(FL/K),whereFistheCMeldandGal(FL/K)containstheabeliansubgroupGal(FL/FK)withindex2.
IfEdoesnothaveCMthenSerre'sopenimagetheorem(see[71,§IV.
3]and[72])impliesthatGisaniteindexsubgroupofGL2(Z);wethereforehaveG1,zar=SL2,whichimpliesST(E)=SU(2).
ItfollowsfromTheorem3.
2that(uptoconjugacy),theSato–TategroupST(E)doesnotdependonourchoiceoftheprimeortheembeddingι:Q→Cthatweused.
WeshouldalsonotethatST(E)dependsonlyontheisogenyclassofE;thisfollowsfromthefactthatweusedtherationalTatemoduleVtodeneit(indeed,twoabelianvarietiesoveranumbereldareisogenousifandonlyiftheirrationalTatemodulesareisomorphicasGaloismodules,byFaltings'isogenytheorem[23],butweareonlyusingtheeasydirectionofthisequivalencehere).
3.
2.
TheSato–Tategroupofanabelianvariety.
WenowwishtogeneralizeourdenitionoftheSato–Tategroupofanellipticcurvetoabelianvarieties.
Recallthatanabelianvarietyisasmoothconnectedprojectivevarietythatisalsoanalgebraicgroup,wherethegroupoperationsarenowgivenbymorphismsofprojectivevarieties;onanyafnepatchtheycanbedenedbyapolynomialmap.
Remarkably,thefactthatabelianvarietiesarecommutativealgebraicgroupsisnotpartofthedenition,itisaconsequence;see[54,Cor.
1.
4].
Wealsorecallthatanisogenyofabelianvarietiesissimplyanisogenyofalgebraicgroups,asurjectivemorphismwithnitekernel.
15NotethatweembedGzar(Q)inGzar,ι(C)beforenormalizingbyN(p)1/2;aspointedoutbySerre[77,p.
131],wewanttotakethesquarerootinCwhereitisunambiguouslydened.
26AbelianvarietiesofdimensiongmayariseastheJacobianJac(C)ofasmoothprojectivecurveC/kofgenusg.
IfChasak-rationalpoint(aswhenCisanellipticcurve),onecanfunctoriallyidentifyJac(C)withthedivisorclassgroupPic0(C),thegroupofdegree-zerodivisorsmoduloprincipaldivisors,butonecanunambiguouslydenetheabelianvarietyJac(C)inanycase;see[54,Ch.
III]fordetails.
IfCisasmoothprojectivecurveoveranumbereldKandA:=Jac(C)isitsJacobian,thenforeveryprimepofgoodreductionforC,theabelianvarietyAalsohasgoodreductionatp,16andtheL-polynomialLp(T)appearinginthenumeratorofthezetafunctionZCp(T)isreciprocaltothecharac-teristicpolynomialχp(T)oftheFrobeniusendomorphismπApofAp,whichactsonpointsofAviatheN(p)-powerFrobeniusautomorphism(coordinate-wise).
Inparticular,wehavetheidentity(9)Lp(T)=T2gχp(T1),inwhichbothsidesareintegerpolynomialsofdegree2gwhosecomplexrootshaveabsolutevalueN(p)1/2.
Aswithellipticcurves,onecanconsidertheL-functionL(A,s)attachedtoA,whichisdenedasanEulerproductwithfactorsLp(N(p)s)ateachprimepwhereAhasgoodreduction.
17StudyingthedistributionofthenormalizedL-polynomialsLp(T)associatedtoCisthusequivalenttostudyingthedistributionofthenormalizedcharacteristicpolynomialsofπAp,andalsoequivalenttostudyingthedistributionofthenormalizedEulerfactorsofL(A,s).
Remark3.
3.
Eachofthesethreeperspectivesissuccessivelymoregeneralthantheprevious,thelastvastlyso.
ThereareabelianvarietiesoverKthatarenottheJacobianofanycurvedenedoverK,andL-functionsthatcanbewrittenasEulerproductsoverprimesofKthatarenottheL-functionofanyabelianvariety.
OnecanmoregenerallyconsiderthedistributionofnormalizedEulerfactorsofmotivicL-functions,whichwealsoexpecttobegovernedbytheHaarmeasureofaSato-Tategroupassociatedtotheunderlyingmotive,asdenedin[76,77];see[26]forsomeconcreteexamplesinweight3.
TherecipefordeningtheSato-TategroupST(A)ofanabelianvarietyA/Kofgenusgisadirectgeneralizationoftheg=1case.
Weproceedasfollows:1.
Pickaprime,denetheTatemoduleT:=lim←nA[n],afreeZ-moduleofrank2g,andtherationalTatemoduleV:=TZQ,aQ-vectorspaceofdimension2g.
2.
UsetheGaloisrepresentationρA,:Gal(K/K)→Aut(V)GL2g(Q)todeneG:=imρA,.
3.
LetGzarbetheZariskiclosureofGinGL2g(Q)(asanalgebraicgroup),anddeneG1,zarbyaddingthesymplecticconstraintMtM=,sothatG1,zarisaQ-algebraicsubgroupofSp2g.
4.
Pickanembeddingι:Q→CanduseittodeneG1,zar,ιasthebase-changeofG1,zartoC.
5.
DeneST(A)USp(2g)asamaximalcompactsubgroupofG1,zar,ι(C),uniqueuptoconjugacy.
6.
Foreachgoodprimep,letMpbetheimageofFrobpinGzar,ι(C)anddenexp∈conj(ST(A))tobetheconjugacyclassofMp:=N(p)1/2Mp,inST(A).
Step6requiressomejustication;itisnotobviouswhyMpshouldnecessarilybeconjugatetoanelementofST(A).
Herewearerelyingontwokeyfacts.
16Forg>1theconversedoesnothold(ingeneral);thisimpactsonlynitelymanyprimespandwillnotconcernus.
17ExplicitlydeterminingtheEulerfactorsatbadprimesisdifcultwhendimA>1.
Practicalmethodsareknownonlyinspecialcases,suchaswhenAistheJacobianofahyperellipticcurve(eveninthiscasethereisstillroomforimprovement).
27First,theimageGofρA,inGL2g(Q)actuallyliesinGSp2g(Q),thegroupofsymplecticsimilitudes.
ThealgebraicgroupGSp2gisdenedbyimposingtheconstraintMtM=λ,:=0IgIg0,whereλisnecessarilyanelementofthemultiplicativegroupGm:=GL1,sinceMisinvertible.
ThemorphismGSp2g→Gmdenedbyλisthesimilitudecharacter,andwehaveanexactsequenceofalgebraicgroups1→Sp2g→GSp2gλ→Gm→1.
TheactionofGal(K/K)ontheTatemoduleiscompatiblewiththeWeilpairing,andthisforcestheimageGofρE,tolieinGSp2g(Q);seeExercise3.
1.
ByxingasymplecticbasisforVinstep1wecanviewρA,asacontinuoushomomorphismρA,:Gal(K/K)→GSp2g(Q)GL2g(Q)Forg=1wehaveGL2=GSp2,butforg>1thealgebraicgroupGSp2gisproperlycontainedinGL2g.
Second,wearerelyingonthefactthatMp,andthereforeMp,issemisimple(diagonalizable,sinceweareworkingoverC).
ThisfollowsfromTate'sproofoftheTateconjectureforabelianvarietiesoverniteelds(combinethemaintheoremandpart(a)ofTheorem2in[89]).
ThematrixMpisthusdiagonalizableandhaseigenvaluesofabsolutevalue1;itthereforeliesinacompactsubgroupofG1,zar,ι(C)(taketheclosureofthegroupitgenerates).
ThiscompactgroupisnecessarilyconjugatetoasubgroupofthemaximalcompactsubgroupST(A),whichmustcontainanelementconjugatetoMp.
Remark3.
4.
WhendeningtheSato-Tategroupinmoregeneralsettingsoneinsteadusesthesemisim-plecomponentofthe(multiplicative)Jordandecomposition(see[10,Thm.
I.
4.
4])ofMptodenexp,asin[77,§8.
3.
3].
ThisavoidstheneedtoassumetheconjecturedsemisimplicityofFrobenius,whichisknownforabelianvarietiesbutnotingeneral.
Background3.
5(Weilpairing).
IfAisanabelianvarietyoveraeldkandA∨isitsdualabelianvariety(see[54,§I.
8]),thenforeachn≥1primetothecharacteristicofk,theWeilpairingisanondegeneratebilinearmapA[n]*A∨[n]→n(k)thatcommuteswiththeactionofGal(k/k);herendenotesthegroupofnthrootsofunity(thealgebraicgroupdenedbyxn=1).
Lettingnvaryoverpowersofaprime=char(k)andtakinginverselimitsyieldsabilinearmaponthecorrespondingTatemodules:e:T*T∨→∞(k):=lim←nn(k).
Givenapolarization,anisogenyφ:A→A∨,wecanuseittodeneabilinearpairingeφ:T*T→∞(k)(x,y)→e(x,φ(y))thatisalsocompatiblewiththeactionofGal(k/k).
Onecanalwayschooseapolarizationφsothatthepairingeφisnondegenerateandskewsymmetric,meaningthateφ(a,b)=eφ(b,a)1foralla,b∈T;see[54,Prop.
I.
13.
2].
WhenAistheJacobianofacurveitisnaturallyequippedwithaprincipalpolarizationφ,anisomorphismA→A∨,forwhichthisautomaticallyholds;inthissituationitiscommontosimplyidentifyewitheφwithoutmentioningφexplicitly.
28WeshouldnotethatourdenitionoftheSato-TategroupST(A)requiredustochooseaprimeandanembeddingι:Q→C.
UptoconjugacyinUSp(2g)oneexpectstheSato-Tategrouptobeindependentofthesechoices;thisisknownforg≤3(see[4]),butopeningeneral.
WeshallneverthelessrefertoST(A)as"the"Sato-TategroupofA,withtheunderstandingthatwearexingonceandforallaprimeandanembeddingι:Q→C(notethatthesechoicesdonotdependonAorevenitsdimensiong).
3.
3.
TheSato-Tateconjectureforabelianvarieties.
HavingdenedtheSato-TategroupofanabelianvarietyoveranumbereldwecannowstatetheSato-Tateconjectureforabelianvarieties.
Conjecture3.
6.
LetAbeanabelianvarietyoveranumbereldK,letST(A)denoteitsSato-Tategroup,andlet(xp)bethesequenceofconjugacyclassesofnormalizedimagesofFrobeniuselementsinST(A)atprimespofgoodreductionforA,orderedbynorm(breaktiesarbitrarily).
Thenthesequence(xp)isequidistributed(withrespecttothepushforwardoftheHaarmeasureofST(A)toitsspaceofconjugacyclasses).
3.
4.
TheidentitycomponentoftheSato-Tategroup.
TherearetwoalgebraicgroupsthatonecanassociatetoanabelianvarietyAoveranumbereldKthatarecloselyrelatedtoitsSato–Tategroup,theMumford–TategroupandtheHodgegroup,bothofwhichconjecturallydeterminetheidentitycomponentoftheSato–Tategroup(provablysowhenevertheMumford–Tateconjectureisknown,whichincludesallabelianvarietiesofdimensiong≤3,asshownin[4]).
InordertodenethesegroupsweneedtorecallsomefactsaboutcomplexabelianvarietiesandtheirassociatedHodgestructures.
Background3.
7(complexabelianvarieties).
LetAbeanabelianvarietyofdimensiongoverC.
ThenA(C)isaconnectedcompactLiegroupandthereforeisomorphictoatorusV/Λ,whereVCgisacomplexvectorspaceofdimensiongandΛZ2gisafulllatticeinVthatweviewasafreeZ-module;onecanobtainΛasthekerneloftheexponentialmapexp:T0(A(C))→A(C),whereT0(A(C))denotesthetangentspaceattheidentity.
Whileeverycomplexabelianvarietycorrespondstoacomplextorus,theconverseistrueonlywheng=1.
ThecomplextoriX:=V/Λthatcorrespondtoabelianvarietiesarethosethatadmitapolarization(orRiemannform),apositivedeniteHermitianformH:V*V→CwithImH(Λ,Λ)=Z(hereImmeansimaginarypart).
GivenapolarizationHonX,themapv→H(v,·)denesanisogenytothedualtorusX∨:=V/Λ,whereV:={f:V→C:f(αv)=αf(v)andf(v1+v2)=f(v1)+f(v2)},andΛ:={f∈V:Imf(Λ)Z}.
ThisisogenyisapolarizationofXasanabelianvariety;conversely,anypolarizationonA(onealwaysexists)canbeusedtodeneapolarizationonthecomplextorusA(C).
OnecanthenshowthatthemapA→A(C)denesanequivalenceofcategoriesbetweencomplexabelianvarietiesandpolarizablecomplextori.
Formorebackgroundoncomplexabelianvarieties,seetheoverviewsin[54,§1]or[59,§1],orsee[8]foracomprehensivetreatment.
NowletAbeanabelianvarietyoveranumbereldK,xanembeddingK→C,andletCg/ΛbethecomplextoruscorrespondingtoA(C).
WemayidentifyΛwiththesingularhomologygroupH1(A(C),Z),andwesimilarlyhaveΛR:=ΛZRH1(A(C),R)foranyringR.
TheisomorphismsA(C)Cg/ΛandA(C)R2g/ΛofcomplexandrealLiegroupsallowustoviewΛRH1(A(C),R)asarealvectorspaceofdimension2gequippedwithacomplexstructure,bywhichwemeananR-algebrahomomorphismh:C→End(ΛR).
InthelanguageofHodgetheory,thisamountstothestatementthat(Λ,h)isanintegralHodgestructure(pureofweight1).
29WecanalsoviewhasmorphismofR-algebraicgroupsh:S→GLΛR.
HereSdenotestheDelignetorus(alsoknownastheSerretorus),obtainedbyviewingC*asanR-algebraicgroup(thisamountstotakingtherestrictionofscalarsofGm:=GL1fromCtoR;seeExercise3.
2).
ThemorphismhcanbedenedoverRbecauseCg/Λisapolarizabletorus,sinceitcomesfromanabelianvariety(ingeneralthisneednothold).
TherealLiegroupS(R)C*isgeneratedbyR*andU(1)={z∈C*:zz=1},whichintersectin{±1};takingZariskiclosuresyieldsR-algebraicsubgroupsGmandU1ofSthatintersectin2.
RestrictinghtoU1SyieldsamorphismU1→GLΛRwiththefollowingproperty:theimageofeachu∈U1(R)=U(1)haseigenvaluesu,u1withmultiplicityg;see[8,Prop.
17.
1.
1].
TheimageofsuchamapisknownasaHodgecircle.
TherationalHodgestructure(ΛQ,h)isobtainedbyreplacingΛwithΛQ:=ΛZQandcanbeusedtodenetheMumford-Tategroup.
Denition3.
8.
TheMumford–TategroupMT(A)isthesmallestQ-algebraicgroupGinGLΛQforwhichh(S)G(R);equivalently,itistheQ-Zariskiclosureofh(S(R))inGLΛR.
TheHodgegroupHg(A)issimilarlydenedastheQ-Zariskiclosureofh(U(1))inGLΛR.
Asdenedabove,theMumford–TategroupMT(A)isaQ-algebraicsubgroupofGL2g.
ButthecomplextorusCg/Λispolarizable,whichmeansthatwecanputasymplecticformonΛRthatiscompatiblewithh,andthisimpliesthatinfactMT(A)isaQ-algebraicsubgroupofGSp2g.
Similarly,theHodgegroupHg(A)isaQ-algebraicsubgroupofSp2g,andinfactHg(A)=MT(A)∩Sp2g;thisissometimesusedasanalternativedenitionofHg(A).
MuchoftheinterestintheHodgegrouparisesfromthefactthatitgivesusanisomorphismofQ-algebrasEnd(AC)QEnd(ΛQ)Hg(A),whereEnd(AC)Q:=End(AC)ZQandHg(A)actsonEnd(ΛQ)byconjugation;see[8,Prop.
17.
3.
4].
Toseewhythisisomorphismisuseful,letusnoteoneapplication.
Theorem3.
9.
ForanabelianvarietyAofdimensiongoveranumbereldK,theHodgegroupHg(A)iscommutativeifandonlyiftheendomorphismalgebraEnd(AK)QcontainsacommutativesemisimpleQ-algebraofdimension2g.
Proof.
See[8,Prop.
17.
3.
5].
Forg=1theabelianvarietiesAthatsatisfythetwoequivalentpropertiesofTheorem3.
9areCMellipticcurves.
Moregenerally,suchabelianvarietiesaresaidtobeofCM-type.
Forabelianvarietiesofgeneraltypeonehastheoppositeextreme:End(AK)Q=QandHg(A)=Sp2g;see[8,Prop.
17.
4.
2].
IntheprevioussectionwedenedtwoQ-algebraicgroupsGzarGSp2gandG1,zarSp2gassociatedtoA.
ItisreasonabletoaskhowtheyarerelatedtotheQ-algebraicgroupsMT(A)andHg(A).
UnlikethegroupsGzarandG1,zar,thealgebraicgroupsMT(A)andHg(A)arenecessarilyconnected(bycon-struction).
18DeligneprovedthattheidentitycomponentofGzarisalwaysasubgroupofMT(A)QQ,equivalently,thattheidentitycomponentofG1,zarisasubgroupofHg(A)QQ);see[20].
Itisconjec-turedthattheseinclusionsareinfactequalities.
Conjecture3.
10(MUMFORD–TATECONJECTURE).
TheidentitycomponentofGzarisequaltoMT(A)QQ;equivalently,theidentitycomponentofG1,zarisequaltoHg(A)QQ.
18Thisistruemoregenerallyforallmotivesofoddweight.
Formotivesofevenweightthesituationismoredelicate;complicationsarisefromthefactthatwearethenworkingwithorthogonalgroupsratherthansymplecticgroups;see[4,5].
30Thisconjectureisknowntoholdforabelianvarietiesofdimensiong≤3;see[4,Th.
6.
11]whereitisshownthatthisfollowsfrom[57].
Whenitholds,theMumford–Tategroup(andtheHodgegroup)uniquelydeterminestheidentitycomponentoftheSato–Tategroup,uptoconjugationinUSp(2g);see[25,Lemma2.
8].
NeithertheMumford–TategroupnortheHodgegrouptellusanythingaboutthecomponentgroupsofGzar,G1,zar,ST(A)(thethreeareisomorphic;see[77,§8.
3.
4]),butthereisacloselyrelatedQ-algebraicgroupthatconjecturallydoes.
Conjecture3.
11(ALGEBRAICSATO–TATECONJECTURE).
ThereexistsaQ-algebraicsubgroupAST(A)ofSp2gsuchthatG1,zar=AST(A)QQ.
BanaszakandKedlaya[4]haveshownthatthisconjectureholdsforg≤3viaanexplicitdescriptionofAST(A)usingtwistedLefschetzgroups.
3.
5.
ThecomponentgroupoftheSato-Tategroup.
WehaveseenthattheMumford–Tategroupcon-jecturallydeterminestheidentitycomponentST(A)0oftheSato–TategroupST(A)ofanabelianvarietyAoveranumbereldK(provablysoindimensiong≤3).
TheidentitycomponentST(A)0isanormalniteindexsubgroupofST(A),andwenowwanttoconsiderthecomponentgroupST(A)/ST(A)0.
Asabove,foranyeldextensionL/K,weuseALtodenotethebasechangeofAtoL.
Theorem3.
12.
LetAbeanabelianvarietyoveranumbereldK.
ThereisauniqueniteGaloisextensionL/KwiththepropertythatST(AL)isconnectedandGal(L/K)ST(A)/ST(A)0.
TheextensionL/KisunramiedoutsidetheprimesofbadreductionforA,andforeverysubextensionF/KofL/KwehaveGal(L/F)ST(AF)/ST(AF)0.
Proof.
Asexplainedin[77,§8.
3.
4],thecomponentgroupsofGzarandST(A)areisomorphic.
LetΓbetheGaloisgroupofthemaximalsubextensionKSofGal(K/K)thatisunramiedawayfromthesetSconsistingoftheprimesofbadreductionforAandtheprimesofKlyingabove.
The-adicGaloisrepresentationρA,:Gal(K/K)→Aut(V)inducesacontinuoussurjectivehomomorphismΓ→Gzar/(Gzar)0,whosekernelisanormalopensubgroupΓ0ofΓ.
ThecorrespondingxedeldLisaniteGaloisextensionofK,anditistheminimalGaloisextensionofKforwhichST(AL)isconnected.
ItisclearlyuniquelydeterminedandunramiedoutsideS,andwehaveisomorphismsGal(L/K)Γ/Γ0Gzar/(Gzar)0ST(A)/ST(A)0.
AsshownbySerre[75],thecomponentgroupofGzar,andthereforeofST(A),isindependentof,andtheaboveargumentappliestoanychoiceof.
ThusL/KcanberamiedonlyatprimesofbadreductionforA.
ForanysubextensionF/KofL/K,replacingAbyAFintheargumentaboveyieldsthesameeldL,withGal(L/F)ST(AF)/ST(AF)0.
3.
6.
Exercises.
Exercise3.
1.
LetAbeanabelianvarietyofdimensiongoveranumbereldK.
ShowthatonecanchooseabasisforV=TZQsothatthematrixMdescribingtheactionofanyσ∈Gal(K/K)onVsatisesMtM=λforsomeλ∈Q*,where:=0II0.
ConcludethattheimageofthecorrespondingGaloisrepresentationliesinGSp2g(Q)anddescribethemapGal(K/K)→Q*inducedbythesimilitudecharacterλ.
31Exercise3.
2.
DenetheDelignetorusSasanR-algebraicgroupinA4(giveequationsthatdeneitasanafnevarietyandpolynomialmapsforthegroupoperations),andthenexpresstheR-algebraicgroupsGmandU1assubgroupsofSthatintersectin2.
ProvethatS(R)andC*areisomorphicasrealLiegroups(giveexplicitmapsinbothdirections).
Exercise3.
3.
LetL/Kbeaniteseparableextensionofdegreed,withL=K(α).
GivenanafneL-varietyYdenedbypolynomialsPk∈L[y1,.
.
.
,yn],wecanconstructanafneK-varietyResL/K(Y)bywritingeachyi=d1j=0xijαjintermsoftheK-basis{1,α,.
.
.
,αd1}forLandusingtheminimalpolynomialofαtoreplaceeachPk(y1,.
.
.
,yn)byapolynomialinK[x11,.
.
.
,x1d,.
.
.
,xn1.
.
.
,xnd].
TheK-varietyResL/K(Y)istheWeilrestriction(orrestrictionofscalars)ofY.
ProvethattheR-algebraicgroupS(theDelignetorus)istheWeilrestrictionoftheC-algebraicgroupGm,thatis,S=ResC/R(Gm).
4.
SATO–TATEAXIOMSANDGALOISENDOMORPHISMTYPESInthissectionwepresenttheSato-TateaxiomsandconsidertheproblemofclassifyingSato-Tategroupsofabelianvarietiesofagivendimensiong.
WethencomputetracemomentsequencesofallconnectedSato-Tategroupsofabelianvarietiesofdimensiong≤3andpresentformulasforthetracemomentsequenceofUSp(2g)(thegenericcase)thatapplytoallg,4.
1.
Sato–Tateaxioms.
In[77,§8.
2]SerregivesasetofaxiomsthatanySato–Tategroupisexpectedtosatisfy.
SerreconsidersSato–Tategroupsinamoregeneralcontextthanwedohere,sowewillstatetheaxiomsastheyapplytoSato–Tategroupsofabelianvarieties.
Asin§3.
4,foraLiegroupGwedeneaHodgecircletobeasubgroupHofGthatistheimageofacontinuoushomomorphismθ:U(1)→G0whoseelementsθ(u)haveeigenvaluesuandu1withmultiplicityg(notethatHnecessarilyliesintheidentitycomponentG0ofG).
Denition4.
1.
AgroupGsatisestheSato–Tateaxioms(forabelianvarietiesofdimensiong≥1)ifandonlyifthefollowinghold:(ST1)(Liecondition)GisaclosedsubgroupofUSp(2g).
(ST2)(Hodgecondition)TheHodgecirclesinGgenerateadensenon-trivialsubgroupofG0.
19(ST3)(rationalitycondition)ForeachcomponentHofGandirreduciblecharacterχofGL2g(C),wehaveHχ∈Z,whereistheHaarmeasureonGnormalizedsothat(1H)=1.
Remark4.
2.
Denition4.
1generalizeseasilytoself-dualmotiveswithrationalcoefcients.
Givenanintegerweightw≥0andHodgenumbershp,q∈Z≥0indexedbyp,q∈Z≥0withp+q=wsuchthathp,q=hq,pwhenwisodd,letd:=hp,q.
Forabelianvarietieswehavew=1andh1,0=h0,1=g.
Inaxiom(ST1)werequireGtobeaclosedsubgroupofUSp(d)(resp.
O(d))whenwisodd(resp.
even),andinaxiom(ST2)werequireelementsθ(u)ofaHodgecircletohaveeigenvaluesupqwithmultiplicityhp,q;axiom(ST3)isunchanged.
Axiom(ST1)impliesthatGisacompactLiegroup,and(ST2)rulesoutnitegroups,sinceGmustcontainatleastoneHodgecircleandthereforecontainsasubgroupisomorphictoU(1).
WhenGisconnected,(ST3)holdsautomaticallyandonly(ST1)and(ST2)needtobechecked;thisisaneasyapplicationofrepresentationtheory,see[49,Prop.
2].
Axiom(ST3)playsnorolewheng=1(seetheproofofProposition4.
4below),butforg>1itiscrucial.
Wheng=2,forexample,foreveryintegern≥1wecandiagonallyembedU(1)*U(1)[n]inUSp(4)togetinnitelymanynon-conjugateclosed19Thestatementof(ST2)in[25]inadvertentlyomitstherequirementthattheHodgecirclesgenerateadensesubgroup.
32groupsGUSp(4)whoseidentitycomponentisaHodgecircle.
Allofthesegroupssatisfy(ST1)and(ST2),butonlynitelymanysatisfy(ST3).
Indeed,ifwetakeχandletCbeacomponentonwhichtheprojectiontoU(1)[n]hasordern,wehaveCχ=ζn+ζn∈Zonlyforn∈{2,3,4,6}.
Moregenerally,wehavethefollowingtheorem.
Theorem4.
3.
Uptoconjugacy,foranyxeddimensiong≥1thenumberofsubgroupsofUSp(2g)thatsatisfytheSato–Tateaxiomsisnite.
Proof.
See[25,Rem.
3.
3]Theorem4.
3motivatesthefollowingclassicationproblem:givenanintegerg≥1,determinethesubgroupsofUSp(2g)thatsatisfytheSato–Tateaxioms.
Thecaseg=1iseasy.
Proposition4.
4.
Forg=1thethreegroupsU(1),N(U(1)andSU(2)listedinTheorem3.
2aretheonlygroupsthatsatisfytheSato–Tateaxioms(uptoconjugacy).
Proof.
SupposeGsatisestheSato–Tateaxioms.
ThenG0containsaconjugateofU(1)embeddedinUSp(2)viau→u00u,asinTheorem3.
2,anditmustbeacompactconnectedLiegroup.
TheonlynontrivialcompactconnectedLiegroupsinUSp(2)=SU(2)areU(1)andSU(2)itself(thisfollowsfromtheclassicationofcompactconnectedLiegroupsbutiseasytoseedirectly).
ThuseitherG0=SU(2),inwhichcaseG=SU(2),orG0isconjugatetoU(1)andmustbeanormalsubgroupofG(theidentitycomponentofacompactLiegroupisalwaysanormalsubgroupofniteindex).
ThegroupU(1)hasindex2initsnormalizer,soU(1)andN(U(1))aretheonlypossibilitiesforGwhenG0=U(1).
Corollary4.
5.
Forg=1agroupGsatisestheSato–TateaxiomsifandonlyifitistheSato–Tategroupofanellipticcurveoveranumbereld.
Theclassicationproblemforg=2ismoredifcult,butithasbeensolved.
Theorem4.
6.
UptoconjugacyinUSp(4)thereare55groupsthatsatisfytheSato–Tateaxiomsforg=2.
Ofthese55,thefollowing6areconnected:U(1)2,SU(2)2,U(1)*U(1),U(1)*SU(2),SU(2)*SU(2),USp(4),wereU(1)2denotesU(1)=u00u:u∈C*diagonallyembeddedinUSp(4),andsimilarlyforSU(2)2.
Proof.
See[25,Thm.
3.
4],whichgivesanexplicitdescriptionofthe55groups.
Remark4.
7.
ThosefamiliarwiththeclassicationofconnectedcompactLiegroupsmaynoticethatthegroupU(2),whichcanbeembeddedinUSp(4),ismissingfromTheorem4.
6.
ThisisbecauseitfailstosatisfytheHodgecondition(ST2);itcontainssubgroupsisomorphictoU(1),butthereisnowaytoembedU(1)→U(2)→USp(4)andgeteigenvaluesuandu1withmultiplicity2;see[26,Rem.
2.
3].
However,formotivesofweight3andHodgenumbersh3,0=h2,1=h1,2=h0,3=1themodiedHodgeconditionnotedinRemark4.
2issatisedbyasubgroupofUSp(4)isomorphictoU(2);see[26]fordetails,includingtwoexamplesofweight3motiveswithSato-TategroupU(2).
Corollary4.
5doesnotholdforg=2.
Theorem4.
8.
Ofthe55groupsappearinginTheorem4.
6,only52ariseastheSato–Tategroupofanabeliansurfaceoveranumbereld.
Ofthese,34ariseforabeliansurfacesdenedoverQ.
33Proof.
See[25,Thm.
1.
5].
ThethreesubgroupsofUSp(4)thatsatisfytheSato–TateaxiomsbutarenottheSato–TategroupofanyabeliansurfaceoveranumbereldarethenormalizerofU(1)*U(1)inUSp(4),whosecomponentgroupisthedihedralgroupoforder8,andtwoofitssubgroups,oneofindex2andoneofindex4.
TheproofthatthesethreegroupsdonotoccurisobtainedbyrstestablishingabijectionbetweenGaloisendomorphismtypes(seeDenition4.
10below)andSato–Tategroups,andthenshowingthatthereareonly52Galoisendomorphismtypesofabeliansurfaces.
Explicitexamplesofgenus2curveswhoseJacobiansrealizethese52possibilitiescanbefoundin[25,Table11],andanimatedhistogramsoftheirSato–Tatedistributionsareavailableathttp://math.
mit.
edu/~drew/g2SatoTateDistributions.
htmlTheclassicationproblemforg=3remainsopen,buttheconnectedcaseshavebeendetermined(seeTable2inthenextsection).
BeforeleavingourdiscussionoftheSato–Tateaxioms,itisreasonabletoaskwhetherSato–Tategroupsnecessarilysatisfythem.
Ofcourseweexpectthistobethecase,butitisdifculttoproveingeneral.
However,itcanbeprovedtoholdinallcaseswheretheMumford–Tateconjectureisknown,includingallcaseswithg≤3.
Proposition4.
9.
LetAbeanabelianvarietyofdimensiongoveranumbereldKforwhichtheMumford–Tateconjectureholds.
ThenST(A)satisestheSato–Tateaxioms.
Proof.
See[25,Prop.
3.
2].
4.
2.
Galoisendomorphismtypes.
Wewillworkintheabstractcategorywhoseobjectsarepairs(G,E)ofanitegroupGandanR-algebraEequippedwithanR-linearactionofG,andwhosemor-phismsΦ:(G,E)→(G,E)arepairs(φG,φE),whereφG:G→Gisamorphismofgroups,andφE:E→EisanequivariantmorphismofR-algebras,meaningthat(10)φE(eg)=φE(e)φG(g)forallg∈Gande∈E.
ToeachabelianvarietyA/Kwenowassociateanisomorphismclass[G,E]inasfollows.
TheminimalextensionL/KforwhichEnd(AL)=End(AK)isaniteGaloisextensionofK;weshalltakeGtobeGal(L/K)andEtobetherealendomorphismalgebraEnd(AL)R:=End(AL)ZR.
TheGaloisgroupGal(L/K)actsonEnd(AL)viaitsactiononthecoefcientsoftherationalmapsdeningeachelementofEnd(AK);thisinducesanR-linearactionofGal(L/K)onEnd(AL)RviacompositionwiththenaturalmapEnd(AL)→End(AL)R.
Thepair(Gal(L/K),End(AL)R)isthusanobjectof.
Denition4.
10.
TheGaloisendomorphismtypeGT(A)ofanabelianvarietyA/Kistheisomorphismclassofthepair(Gal(L/K),End(AL)R)inthecategory,whereListheminimalextensionofKforwhichEnd(AL)=End(AK).
Example4.
11.
LetEbeanellipticcurveoveranumbereldK.
IfEdoesnothaveCM,orifithasCMdenedoverK,thenitsendomorphismsarealldenedoverL=K;otherwise,itsendomorphismsarealldenedoveritsCMeldL,animaginaryquadraticextensionofK.
TherealendomorphismalgebraEnd(EL)RisisomorphictoRwhenEdoesnothaveCM,andisomorphictoCwhenEdoeshaveCM.
WethereforehaveGT(E)=[C1,C]ifEhasCMdenedoverK[C2,C]ifEhasCMnotdenedoverK[C1,R]ifEdoesnothaveCM34HereCndenotesthecyclicgroupofordern;inthecase[C2,C]theactionofC2onCcorrespondstocomplexconjugation.
ThethreeGaloisendomorphismtypeslistedinExample4.
11correspondtothethreeSato-TategroupslistedinTheorem3.
2.
Underthiscorrespondence,therealendomorphismalgebraEnd(EL)RdeterminestheidentitycomponentST(E)0(uptoconjugacy),andtheGaloisgroupGal(L/K)isisomorphictothecomponentgroupST(E)/ST(E)0.
Moreover,theeldLispreciselytheeldLgivenbyTheorem3.
12.
Theorem4.
12.
LetAbeanabelianvarietyAofdimensiong≤3denedoveranumbereldKandletLbetheminimaleldforwhichEnd(AL)=End(AK).
TheconjugacyclassoftheSato-TategroupST(A)determinestheGaloisendomorphismtypeGT(A);moreover,theconjugacyclassoftheidentitycomponentST(A)0determinestheisomorphismclassofEnd(AL)RandST(A)/ST(A)0Gal(L/K).
Forg≤2theconverseholds:theGaloisendomorphismtypeGT(A)determinestheSato–TategroupST(A)uptoconjugacy.
Proof.
SeeProposition2.
19andTheorem1.
4in[25].
ItisexpectedthatinfacttheSato–TategroupalwaysdeterminestheGaloisendomorphismtype,andthattheconverseholdsforg≤3.
Forg=3weatleastknowthattherealendomorphismalgebraEnd(AL)RdeterminestheidentitycomponentST(A)0andthatGal(L/K)ST(A)/ST(A)0.
AtrstglanceitmightseemthatthisshoulddetermineST(A),butitdoesnot,evenwheng=2.
OneneedstoalsounderstandhowGal(L/K)actsonEnd(AL)RandrelatethistotheSato-TategroupST(A).
In[25]thisisaccomplishedforg=2bylookingatthelatticeofR-subalgebrasofEnd(AL)RxedbysubgroupsofGal(L/K)andshowingthatthisisenoughtouniquelydetermineST(A);see[25,Thm.
4.
3].
Toapplythesameapproachwheng=3weneedamoredetailedclassicationofthepossibleGaloisendomorphismtypesandSato–Tategroupsforg=3thaniscurrentlyavailable.
Forg=4theGaloisendomorphismtypedoesnotalwaysdeterminetheSato–Tategroup.
ThisisduetoanexceptionalcounterexampleconstructedbyMumfordin[58],inwhichheprovestheexistenceofanabelianfour-foldAforwhichEnd(AK)=ZbutMT(A)=GSp8.
ThefactthatMT(A)isproperlycontainedinGSp8impliesthatST(A)mustbeproperlycontainedinUSp(8)(thisdoesnotdependontheMumford–Tateconjecture,hereweareonlyusingtheinclusionprovedbyDeligne).
Ontheotherhand,foranabelianvarietyofgeneraltypeonehasEnd(AK)=ZandST(A)=USp(2g);see[31,100]foranexplicitcriterionthatappliestoalmostallJacobiansofhyperellipticcurves.
Forg>4onecanconstructexceptionalexamplesasaproductofanabelianvarietywithoneofMumford'sexceptionalfour-folds,soingeneraltheGaloisendomorphismtypecannotdeterminetheSato–Tategroupforanyg≥4.
However,suchexampleswillnotbesimpleandwillhaveEnd(A)=Z.
In[74]Serreprovesananalogofhisopenimagetheoremforellipticcurvesthatappliestoabelianvarietiesofdimensiong=2,6andgodd.
Forthesevaluesofg,ifEnd(AK)=ZthenST(A)=USp(2g)andnodirectanalogofMumford'sconstructionexists.
Remark4.
13.
Forg≤3,theeldLinTheorem3.
12(theminimalLforwhichST(AL)isconnected)isthesameastheeldLinTheorem4.
12(theminimalLforwhichEnd(AL)=End(AK)).
Inanycase,theformeralwayscontainsthelatter:ifST(AL)isconnectedthenwenecessarilyhaveEnd(AK)=End(AL).
ThiscanbeseenasaconsequenceofBogomolov'stheorem[9],whichstatesthatGisopeninGzar(Q),andFaltings'theorem[23]thatEnd(A)QEnd(V(A))G.
IfST(A)(andthereforeGzar)isconnected,thenEnd(A)isinvariantunderbasechange(nowapplythistoA=AL).
Tables1and2belowlisttherealendomorphismalgebrasandcorrespondingidentitycomponentsofSato-Tategroupsthatariseindimensionsg=2,3.
Acompletelistofthe52GaloisendomorphismtypesandcorrespondingSato-Tategroupsforg=2canbefoundin[25,Thm.
4.
3]and[25,Table9].
35geometrictypeofabeliansurfaceEnd(AK)RST(A)0squareofCMellipticcurveM2(C)U(1)2QMabeliansurfaceM2(R)SU(2)2squareofnon-CMellipticcurveCMabeliansurfaceC*CU(1)*U(1)productofCMellipticcurvesproductofCMandnon-CMellipticcurvesC*RU(1)*SU(2)RMabeliansurfaceR*RSU(2)*SU(2)productofnon-CMellipticcurvesabeliansurfaceofgeneraltypeRUSp(4)TABLE1.
RealendomorphismalgebrasandSato–Tateidentitycomponentsforabeliansurfacesgeometrictypeofabelianthree-foldEnd(AK)RST(A)0cubeofaCMECM3(C)U(1)3cubeofanon-CMECM3(R)SU(2)3productofCMECandsquareofCMECC*M2(C)U(1)*U(1)2productofCMECandQMabeliansurfaceC*M2(R)U(1)*SU(2)2productofCMECandsquareofnon-CMECproductofnon-CMECandsquareofCMECR*M2(C)SU(2)*U(1)2productofnon-CMECandQMabeliansurfaceR*M2(R)SU(2)*SU(2)2productofnon-CMECandsquareofnon-CMECCMabelianthreefoldC*C*CU(1)*U(1)*U(1)productofCMECandCMabeliansurfaceproductofthreeCMECsproductofnon-CMECandCMabeliansurfaceC*C*RU(1)*U(1)*SU(2)productofnon-CMECandtwoCMECsproductofCMECandRMabeliansurfaceC*R*RU(1)*SU(2)*SU(2)productofCMECandtwonon-CMECsRMabelianthreefoldR*R*RSU(2)*SU(2)*SU(2)productofnon-CMECandRMabeliansurfaceproductof3non-CMECsproductofCMECandabeliansurfaceC*RU(1)*USp(4)productofnon-CMECandabeliansurfaceR*RSU(2)*USp(4)quadraticCMabelianthreefoldCU(3)genericabelianthreefoldRUSp(6)TABLE2.
RealendomorphismalgebrasandSato–Tateidentitycomponentsforabelianthreefolds36Ascanbeseeninthetwotablesabove,theSato–Tategroupisinsomerespectsarathercoarseinvari-ant;forexample,itcannotdistinguishaproductofnon-CMellipticcurvesfromageometricallysimpleabeliansurfacewithrealmultiplication(RM).
Ontheotherhand,theHaarmeasuresofthe52Sato–Tategroupsofabeliansurfacesovernumbereldsallgiverisetodistinctdistributionsofcharacteristicpoly-nomials,which,undertheSato–Tateconjecture,matchthedistributionofnormalizedL-polynomials,andtherearesomerathernedistinctionsamongthesedistributionsthattheSato–Tategroupdetects.
Forexample,thereareonly37distincttracedistributionsamongthe52groups,oneneedstolookatboththelinearandquadraticcoefcientsofthecharacteristicpolynomialsinordertodistinguishthem.
Itispossiblefortwonon-conjugateSato–Tategroupstobeisomorphicasabstractgroupsyetgiverisetodistincttracedistributions.
Forexample,theconnectedSato-TategroupsSU(2)*U(1)2andU(1)*SU(2)2thatappearinTable2arebothabstractlyisomorphictotherealLiegroupU(1)*SU(2),butthesetwoembeddingsofU(1)*SU(2)inUSp(6)havedifferenttracedistributions.
Asshownbytheexamplebelow,thisphenomenoncanalsooccurfordisconnectedSato-Tategroupswiththesameidentitycomponent.
Example4.
14.
ConsiderthehyperellipticcurvesC1:y2=x6+3x5+15x420x3+60x260x+28,C2:y2=x6+6x515x4+20x315x2+6x1,andletA1:=Jac(C1)andA2:=Jac(C2)denotetheirJacobians.
OverQbothA1andA2areisoge-noustothesquareoftheellipticcurvey2=x3+1,whichhasCMbyQ(3).
WenecessarilyhaveST(A1)0=ST(A2)0=U(1)2,andthecomponentgroupsarebothisomorphictothedihedralgroupoforder12.
However,theirSato–Tategroupsaredifferent:intermsofthelabelsusedin[25],wehaveST(A1)=D6,1,whileST(A2)=D6,2(see[25,§3.
4]forexplicitdescriptionsofthesegroupsintermsofgenerators),andtheirnormalizedtracedistributionsarequitedifferent.
ForC1theden-sityofzerotracesis3/4,whereasforC2itis7/12(theseratiosrepresenttheproportionofSato–Tategroupcomponentsonwhichthetraceisidenticallyzero),andtheirnormalizedtracemomentsequencesare(1,0,1,0,9,0,110,0,1505,0,21546,.
.
.
)and(1,0,2,0,18,0,200,0,2450,0,31752,.
.
.
),respectively.
TheSato-Tateconjectureforthesetwocurveswasprovedin[27],sothisdifferenceinSato-TategroupsprovablyimpactsthenormalizedtracedistributionsofA1andA2.
4.
3.
Sato–Tatemeasures.
OnceweknowtheSato–TategroupST(A)ofanabelianvarietyA,weareinapositiontocomputevariousstatisticrelatedtothedistributionofitsconjugacyclasses,suchasthemomentsofcharacteristicpolynomialcoefcients(oranyotherconjugacyclassinvariant).
WecanthentesttheSato–TateconjecturebycomparingthesetocorrespondingstatisticsobtainedbycomputingnormalizedL-polynomialsLp(T)forallprimespofgoodreductionforAuptosomenormboundB.
TherststepistodeterminetheHaarmeasureonST(A)0.
Forg=1thereareonlytwopossibilities:eitherST(A)0=U(1)orST(A)0=SU(2),where,asusualweembedU(1)inSU(2)viau→u00u.
Intermsoftheeigenangleθ,thepushforwardmeasureonconj(ST(A)0)isoneofU(1):=1πdθ,SU(2):=2πsin2θdθ,with0≤θ≤π.
ThisalsoaddressestwoofthepossibilitiesforST(A)0thatarisewheng=2,thegroupsU(1)2andSU(1)2listedinthersttworowsofTable1;thesedenotetwoidenticalcopiesofU(1)andSU(2)diagonallyembeddedinUSp(4).
Whenexpressedintermsoftheeigenangleθ,themeasureU(1)237isexactlythesameasU(1)(andsimilarlyforSU(2)2),butnotethatwewillgetadifferentdistributiononcharacteristicpolynomials(whichnowhavedegree4ratherthandegree2),becauseeacheigenvaluenowoccurswithmultiplicity2;inparticular,thetracebecomes4cosθratherthan2cosθ.
ForthegroupsST(A)0thatappearinthenextthreerowsofTable1,themeasureonconj(ST(A)0)isaproductofmeasuresthatwealreadyknow:U(1)*U(1):=1π2dθ1dθ2,U(1)*SU(2):=2π2sin2θ2dθ1dθ2,SU(2)*SU(2):=4π2sin2θ1sin2θ2dθ1dθ2.
ToobtainthemeasureforthegenericcaseST(A)=ST(A)0=USp(4),weusetheWeylintegrationformulaforUSp(2g)(whichincludesthecaseUSp(2)=SU(2)thatwealreadyknow):(11)USp(2g):=1g!
1≤j0,themomentsequencesMUSp(2g)[tr]andMUSp(2g)[tr]mustagreeuptothe2gthmoment;seeExer-cise4.
3.
ThusthemomentssequencesMUSp(2g)[tr]convergetoalimitingsequenceasg→∞:40MUSp(2)[tr]=(1,0,1,0,2,0,5,0,14,0,42,.
.
.
),MUSp(4)[tr]=(1,0,1,0,3,0,14,0,84,0,594,.
.
.
),MUSp(6)[tr]=(1,0,1,0,3,0,15,0,104,0,909,.
.
.
),MUSp(8)[tr]=(1,0,1,0,3,0,15,0,105,0,944.
.
.
).
.
.
.
MUSp(∞)[tr]=(1,0,1,0,3,0,15,0,105,0,945,.
.
.
).
ThelimitingsequenceMUSp(∞)[tr]ispreciselythemomentsequenceofthestandardnormaldistri-bution(mean0andvariance1);thenthmomentiszeroifnisodd,andforevennitisgivenby(n1)!
!
:=n(n2)(n4)···3·1.
Figure4showsthea1-distributionsforg=1,2,3,4,normalizedtothesamescale,whichillustratesconvergencetothestandardnormaldistribution.
4.
5.
Exercises.
Exercise4.
1.
Givecombinatorialproofsoftheidentitiesusedin(13),(14),(15).
Exercise4.
2.
UsingthecombinatorialinterpretationofthetracemomentsequenceMUSp(2g)[tr],provethatforg>gthemomentsequencesMUSp(2g)[tr]andMUSp(2g)[tr]agreeuptothe2gthmomentbutdisagreeatthe(2g+2)thmoment.
ThenshowthatthelimitingtracemomentsequenceMUSp(∞)[tr]isequaltothemomentsequenceofthestandardnormaldistribution.
Exercise4.
3.
Characterizeeachofthe6tracemomentsequencesthatariseforconnectedSato–Tategroupsindimensiong=2byshowingthateachsequencecountsreturningwalksonan2-dimensionalintegerlatticethatareconstrainedtoacertainregionoftheplane.
Exercise4.
4.
Similarlycharacterizethe14tracemomentsequencesthatariseforconnectedSato–Tategroupsindimensiong=3intermsofreturningwalksona3-dimensionalintegerlattice.
Exercise4.
5.
Foreachofthe5non-genericconnectedSato–Tategroupsthatariseindimensiong=2computethemomentsequencefora2,thequadraticcoefcientofthecharacteristicpolynomial.
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