Modularity of solvable Artin representations(2)

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

4DinakarRamakrishnan

wedenotebyIndKM(χ).)Anyhowwemanagetosolvethisproblemandestablishthefollowingresult,whereResKMdenotes,foranyextensionM/Kofnumber elds,therestrictionofrepresentationsofWKtoWM:

TheoremDLetK/Fbeaquadraticextensionofnumber eldswithnon-trivialautomorphismθ,andletπbeacuspidalautomorphicrepresentationofGL(2,AK).Then

(a)As(π)isautomorphic;

(b)Ifπisnon-dihedral,thenAs(π)isacuspformi π θisnotisomorphicto

π µ,foranyideleclasscharacterµofK;

(c)Ifπisdihedral,i.e.,associatedtoarepresentationσ=IndKM(χ)ofWKfora

characterχofaquadraticextensionMofK,thenAs(π)iscuspidali M/F

θisnon-GaloisandtherepresentationResKM(σ) χofWMdoesnotextend

toarepresentationofWF.

ItmaybehelpfulforthereadertonotethefollowingconcretedescriptionoftheAsaiL-functioninaspecialcase.SupposeF=Q,Karealquadratic eldofclassnumber1withringofintegersOK,andπtherepresentationde nedbyaholomorphicHilbertmodularnewformfofweight2with L(s,π)=c(a)N(a) s.

a

Thenwehave

L(s,As(π))=ζ(2s 2)

m≥1c(mOK)m s.

Theinterestinthiscomes,ontheanalyticside,fromthefactthatonesumsoverasparsesubsetofthenon-zeroidealsaofOKtogetthisL-function,andonthegeometricside,fromthefactthatL(s,As(π))isafactorofthedegree2L-functionoftheassociatedHilbertmodularsurface(cf.[Ra2],sec.4,forexample).ItwillbenaturalifoneisremindedofthesymmetricsquareL-functionofacuspformonGL(2)/Q,wheree ectivelyonesumsoverthesquaresofpositiveintegers.Indeed,foranyquadraticextensionK/Fofnumber eldsandforanycuspformπonGL(2)/K,ifπisthebasechangeofaformπ0onGL(2)/F,L(s,As(π))factorsasL(s,sym2(π0) δ)L(s,ω0δ),whereω0isthecentralcharacterofπ0.

Wewillnowmakesomeremarksabouttheproofsoftheseresults.

Insection3weshowhowtoreducetheproofofTheoremAtothefollowingTheoremA FixaquadraticextensionK/Fofnumber eldswithassociatedquadraticcharacterδofGal(F/F).Letρbeanirreducible4-dimensional,solv-ableC-representationofGal(F/F)whoserestrictionρKtoGal(F/K)isatensorproductoftwo2-dimensionalrepresentations.Then

(a)ρismodular,i.e.,thereisacuspidalautomorphicrepresentationΠofGL(4,AF)

suchthatL(s,ρ)=L(s,Πf);

(b)IfρKisirreducible,oneofthefollowinghappens:

(i)ρ τ τ overF,withdim(τ)=dim(τ )=2;

(ii)ρ IndFL(η),withL/Fquadratic,L=K,andηa2-dimensionalrepre-

sentationofGal(F/L);

(iii)ρ As(σ) β,withσa2-dimensionalrepresentationofGal(F/K)and

βacharacterofGal(F/F).

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)-type5

Thenweshow,stillinsection3,howtodeduceTheoremA moduloTheoremD;infactweneedTheoremDonlyinthe(crucial)casewhen(iii)occurs.When(i)or(ii)occurs,themodularityalreadyfollowsfromTheoremMof[Ra1]andbasechange([AC]),withthedesiredΠbeingincase(i)(resp.(ii))theRankin-SelbergFproductπ(τ) π(τ )(resp.theautomorphicinductionIL(π(η)));hereτ→π(τ)theLanglands-Tunnellmaponsolvable2-dimensionalGaloisrepresentations.

TheproofofTheoremDisaccomplishedinsections4through7.Theapproachissimilarto,butsomewhatmoresubtlethan,theproofoftheexistenceof :A(GL(2)/F)×A(GL(2)/F)→A(GL(4)/F)in[Ra1].(Foranyn≥1,A(GL(n)/F)denotesthesetofisomorphismclassesofisobaricautomorphicrepresentationsofGL(n,AF).)Insection4weshowwhyitsu cestohavetherequisitepropertiesatalmostallplaces.Theninsection5thedistinguishedcase,i.e.,whenπ θisanabeliantwistofπ,istreatedseparately.Inthegeneralsituation,i.e.,whenπ (π θ)iscuspidal,wecruciallyusetheconversetheoremforGL(4)duetoCogdellandPiatetski-Shapiro([CoPS1],whichrequiresknowledgeofthenicenessofthetwistedL-functionsL(s,As(π)×π )forautomorphicformsπ ofGL(2)/Fforasuitableclassofπ .Manypropertiesofcertaincloselyrelatedfunctions,tobedenotedL1(s,As(π)×π ),wereestablishedbyPiatetski-ShapiroandRallis([PS-R]),andbyIkeda([Ik1,2]),viaanintegralrepresentation,whichweuse.ThereisanotherpossibleapproachtostudyingL(s,As(π)×π )viatheLanglands-Shahidimethod[Sh1],whichyieldsanotherfamilyofcloselyrelatedL-functions,denotedL2(s,As(π)×π ),withboundednesspropertiesestablishedrecentlyin[GeSh],butwewillnotuseitandourargumenthereishewedtomakeuseoftheintegralrepresentation. Foralmostall niteplacesv,thelocalfactorsL(s,As(πv)×πv)andL1(s,As(πv)× πv)agree.Butathornyproblemariseshowever,duetoourinabilitytoidentifythebadandarchimedeanfactors.Infact,whenFv=R,onedoesnotevenhaveacomputationofthecorrespondingL1-factorwhenπ,π areunrami ed.Luckily,thingssimplifyquiteabitundersuitable,solvablebasechangesK/FwithKto-tallycomplex,andafterconstructingthebase-changedcandidatesAs(π)Kforanin nitudeofsuchK,wedescendtoFasinsections3.6,3.7of[Ra1].Wealsohavetocontroltheintersectionoftherami cationlociofAs(π),π andK/F.

OneofthereasonswhywehavetoworkwithL(s,As(π)×π )isthatweknowhowitslocalε-factorsbehaveundertwistingbyahighlyrami edcharacter,andthisisnotapriorithecasewithLj(s,As(π)×π )forj=1or2.Indeedthiswillpresentadi cultyfortheliftingof(generic)automorphicformsfromGO(2n)toGL(2n),andforlargenonecannot,asofyet,makeuseofbasechangeanddescentaswedohere.FortheliftingfromGO(2n+1)toGL(2n),see[CoKPSSh].

nglandsandtheInstituteforAdvancedStudy,Princeton,fortheirhospitalityduringtheyear1999-2000,andtheAmericanIn-stituteofMathematics,PaloAlto,forsupportduringthemonthofAugustin1999.ThisprojectwaspartiallyfundedbytheNSFandAMIAS.WealsowanttoacknowledgewiththankssomehelpfulconversationswithMichaelAschbacherandDavidWalesconcerningtherepresentationsof nite …… 此处隐藏：5958字，全部文档内容请下载后查看。喜欢就下载吧 ……