Modularity of solvable Artin representations

Let F be a number field, and (ρ, V) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F) on a finite-dimensional C-vector space V. Denote by L(s, ρ) the associated L-function, which is known to be meromorphic

MODULARITYOFSOLVABLEARTINREPRESENTATIONSOF

GO(4)-TYPE

DINAKARRAMAKRISHNAN

Tomyteacher,Herv´eJacquet

Introduction

LetFbeanumber eld,and(ρ,V)acontinuous,n-dimensionalrepresentationoftheabsoluteGaloisgroupGal(F/F)ona nite-dimensionalC-vectorspaceV.DenotebyL(s,ρ)theassociatedL-function,whichisknowntobemeromorphicwithafunctionalequation.Artin’sconjecturepredictsthatL(s,ρ)isholomorphiceverywhereexceptpossiblyats=1,whereitsorderofpoleisthemultiplicityofthetrivialrepresentationinV.ThemodularityconjectureofLanglandsforsuchrepresentations([La3]),oftencalledthestrongArtinconjecture,assertsthatthereshouldbeanassociated(isobaric)automorphicformπ=π∞ πfonGL(n)/FsuchthatL(s,ρ)=L(s,πf).SinceL(s,πf)possessestherequisiteproperties([JS]),themodularityconjectureimpliestheArtinconjecture.

Forany eldk,letGO(n,k)denotethesubgroupofGL(n,k)consistingoforthogonalsimilitudes,i.e.,matricesgsuchthattgg=λgI,withλg∈k .

Wewillsaythatak-representation(ρ,V)isofGO(n)-typei dim(V)=nanditfactorsasσρ=[Gal(F/F) →GO(n,k) GL(V)].

Inthisarticleweprove

TheoremALetFbeanumber eldandlet(ρ,V)beacontinuous,4-dimensionalC-representationofGal(F/F)whoseimageissolvableandliesinGO(4,C).Thenρismodular,i.e.,L(s,ρ)=L(s,πf)forsomeisobaricautomorphicrepresentationπ=π∞ πfofGL(4,AF).Moreover,πiscuspidali ρisirreducible.

Amongthe nitegroupsGshowingupastheimagesofsuchρarethose ttingintoanexactsequence

1→C→H×H→G→{±1}→1,

whereHisany nitesolvablegroupinGL(2,C)withscalarsubgroupC,theembe-dingofCinH×Hisgivenbyx→(x,x 1),andtheactionof{±1}on(H×H)/CisinducedbythepermutationofthetwofactorsofH×H.Thisisduetothewellknownfact(seesection1)thatGO(4,C)containsasubgroupofindex2whichisaquotientofGL(2,C)×GL(2,C)byC .OfparticularinterestaretheexampleswhereHisacentralextensionofS4orA4(cf.section8).

PARTIALLYSUPPORTEDBYTHENSFGRANTDMS-9801328

1

Let F be a number field, and (ρ, V) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F) on a finite-dimensional C-vector space V. Denote by L(s, ρ) the associated L-function, which is known to be meromorphic

2DinakarRamakrishnan

OnecanaskifthishelpsfurnishnewexamplesofArtin’sconjecture,andtheanswerisyes.

CorollaryBLetFbeanumber eld,andletρ,ρ becontinuousC-representationsofGal(F/F)ofsolvableGO(4)-type.ThenArtin’sconjectureholdsforρ ρ .

Wewillshowinsection8thatinfactthereis,foreachF,adoublyin nitefamilyofsuchexampleswheretherepresentationsρ ρ areirreducibleandprimitive(i.e.,notinduced)ofdimension16.PrimitivityisimportantbecauseArtinL-functionsareinductive,andonewantstomakesurethattheseexamplesdonotcomebyinductionfromknown(solvable)casesinlowdimensions.Wewillthenshow(insecton9)that,givenanyρasinTheoremAwithcorrespondingextensionK/F,thestrongDedekindconjectureholdsforcertainnon-normalextensionsN/FcontainedinK/F,forinstancewhen[N:F]=3a,a≥1(seeTheorem9.3);theassertionisthattheratioζN(s)/ζF(s)isthestandardL-functionofanisobaricautomorphicformηonGL(3a 1)/F.ItimpliesthatforanycuspformπonGL(n)/F,theformallyde nedEulerproductL(s,πN)(seethediscussionbeforeCorollary9.4)admitsameromorphiccontinuationandfunctionalequation,andmoreimportantly,itisdivisiblebyL(s,π).ItshouldbenotedthatwedonotknowifπNisautomorphic.ForacuriousconsequenceofthisresultseeRemark9.6.

Ithasbeenknownforalongtime,thankstotheresultsofArtinandHecke,thatmonomialrepresentationsofGal(F/F),i.e.,thoseinducedbyone-dimensionalrepresentationsofGal(F/K)withK/F nite,satisfyArtin’sconjecture.(Infactthisholdsforanymultipleofamonomialrepresentation.)ButthestrongArtinconjectureisstillopenfortheseexceptwhenK/Fisnormalandsolvable([AC])andwhen[K:Q]=3([JPSS1].Thework[AC]ofArthurandClozelimpliesthatthestrongconjectureholdsforρwithnilpotentimage,infactwheneverρisaccessible([C]),i.e.,apositiveintegrallinearcombinationofrepresentationsinducedfromlinearcharactersofopen,subnormalsubgroups.ItshouldhoweverbenotedthatDadehasshown([Da])thatallaccessiblerepresentationsofsolvablegroupsaremonomial.

Theodddimensionalorthogonalrepresentationswithsolvableimagearesimplerthantheevendimensionalones.Indeedwehave

PropositionCLetρbeacontinuous,irreducible,solvableC-representationofGal(F/F)ofGO(n)-typewithnodd.Thenρismonomialandhencesatis estheArtinconjecture.Ifρisinadditionself-dual,itmustbeinducedbyaquadraticcharacter.

BythegroundbreakingworkofLanglandsintheseventies([La1]),andtheensu-ingtheoremofTunnellin1980([Tu]),oneknowsthatthestrongArtinconjectureholdsforalltwo-dimensionalrepresentationsσwithsolvableimage.Theraisond’etreforthisarticleisthedesireto ndirreducible,solvable,butnon-monomial,evenprimitive,examplesinhigherdimensionssatisfyingtheArtincojecture.WewereledtothisproblemafewyearsagobyaremarkofJ.-P.Serre.Onecouldlookatthesymmetricpowersofatwo-dimensionalsolvableσ,buttheyareallinthelinearspanofmonomialrepresentationsand1-dimensionaltwistsofσ.Thefactsthatthesymmetricsquareofσismodular(byGelbart-Jacquet[GeJ])andmonomial(forσsolvable)wereusedcruciallyin[La1]and[Tu].

Let F be a number field, and (ρ, V) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F) on a finite-dimensional C-vector space V. Denote by L(s, ρ) the associated L-function, which is known to be meromorphic

SolvableArtinrepresentationsofGO(4)-type3

Inthenon-solvabledirection,whichisorthogonal(nopunintended)totheonepursuedinthispaper,therehasbeensomespectacularprogressrecently.Forodd2-dimensionalsρofGal(Q/Q)withprojectivizationρofA5-type,atheoremofBuz-zard,Dickinson,Shepherd-BarronandTaylor([BDST])establishesthemodularityconjectureassumingthefollowing:(i)ρisunrami e …… 此处隐藏：6595字，全部文档内容请下载后查看。喜欢就下载吧 ……