Modularity of solvable Artin representations(5)

ineithercase.Similarly,(µ 1χ /χ )2is1.Theclaimedidentity(3.15)isthenaconsequence.

Thisargumentinfactshowsthatifσisnon-dihedral,thenν=1.SowemaychooseacharacterγofGFsuchthat

χχ =ResFK(γ).

Ontheotherhand,sinceσ χandσ χ areθ-invariant,thereareirreducible2-dimensionalrepesentationsτ,τ ofGFsuchthat

σ χ ResFK(τ)and

Puttingallthesetogetherweget

F ResFK(ρ) ResK(τ τ γ). σ χ ResFK(τ).θ

Also,τcannotbeaone-dimensionaltwistofτ asitwouldmakeσ σ reducible.Thenitfollowsthat

ρ τ (τ γ ),

whereγ iseitherγorγδ.Soweget(i).

Sowemayassume(forthisLemma)thatσisdihedralandthatνisanon-trivialquadraticcharacterofGK.Theθ-invarianceofσ σ χχ thenimpliesthatσ σ isisomorphictoσ σ ν.Consequently,

ν (σ∨ σ) (σ σ ) Ad(σ) Ad(σ )⊕Ad(σ)⊕Ad(σ )⊕1.

Recallthatforanyirreducible2-dimensionalrepresentationτ,Ad(τ)isreduciblei τisdihedral.Sinceσ (resp.σ)isnon-dhedral(resp.dihedral),weseethatAd(σ) Ad(σ )andAd(σ )havenoone-dimensionalsummands.Sowemusthave

ν Ad(σ).∨

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

14DinakarRamakrishnan

Thisimpliesthatσ ν σ,i.e.,thatσisinducedbyacharacterofGM,ifMdenotesthequadraticextensionofKcutoutbyν.Ontheotherhand,sinceθ2=1andν2=1, (χχ )θθ=ν 1=ν.ν= χχ

SoνistherestrictiontoKofacharacterν0,say,ofGF.DenotebyM0thequadraticextensionofFcutoutbyν0.ThenMisabiquadraticextensionofFcontainingKandM0.Notethat

KKK ResFK(ρ) IndM(χ) σ IndM(χ ResM(σ)).

Thus

K IndFM(χ ResM(σ)) ρ⊕ρ δ.

M0K ThelefthandsideisisomorphictoIndFM0(IndM(χ ResM(σ))),whichisinvariant

undertwistingbyν0.Thus

ρ⊕ρ δ ρ ν0⊕ρ δν0.

Sinceρisirreducible,wemusthave

ρ ρ β,whereβ∈{ν0,ν0δ}.

DenotebyLthequadraticextensionofFcutoutbyβ.(ItmustbeeitherofthequadraticextensionswhicharecontainedinManddi erentfromK.)Thenthereexistsanirreducible2-dimensionalrpesentationηofGLsuchthat

ρ IndFL(η),

giving(ii).Lemma3.14isnowproved.

ProofofTheoremA moduloTheoremD(contd.):

Since(i),(ii)ofLemma3.14coincidewith(i),(ii)(respectively)ofTheorem A,wemayasumefromhereonthatweareincase(II)(ofLemma3.6),withσ non-dihedral.Butasσθ σ µforacharacterµ,σisdihedrali σ is.Indeed,ifσweredihedral,itwoldadmitaself-twistbyaquadraticcharacterν=1andthiswouldconsequentlyforceσ toadmitself-twistbyνθ.Soneitherσnorσ isdihedral.Wehave

σ (σθ)θ (σ )θ µθ σ (µθ/µ).

Sinceσisnotdihedral,itdoesnotadmitanynon-trivialself-twistbyacharacter,andsowemusthaveµ=µθ.SothereexistsacharacterνofGFsuchthatµistherestrictionνKofνtoGK.Thusweget(from(3.4),

(ρ ν)K σ (σ µ) σ σθ.

Itsu cestoshowthatsomecharactertwistofρismodular.Sowemay,afterreplacingρbyitstwistbyν 1,that

(3.16)ρK σ σθ.

IfδdenotesthequadraticcharacterofGFcorrespondingtoK/F,thenρandρ δaretheonlyrepresentationsforwhich(3.16)holds.

Itiseasytoseethattheinduction(toGF)oftheexteriorsquareofσ,i.e.,det(σ),isasummandoftheexteriorsquareoftheinductionofσ.Thanksto

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)-type15

semisimplicity,wemaythende netheAsairepresentationofσ,denotedAs(σ),bythedecomposition

(3.17)FΛ2(IndFK(σ)) As(σ)⊕IndK(det(σ)).

Lemma3.18ρisisomorphictoAs(σ)orAs(σ) δ.

ProofofLemma.

that

Wecanalsowrite

(3.19)

whichimpliesthat

(3.20)β β δ.

SoAs(σ) δmustalsooccurinβ.

θOntheotherhand,sincetherestrictiontoGKofIndFK(σ)isσ⊕σ,wegetfrom

(3.19),

(3.21)22θβ IndFK(sym(σ)⊕Λ(σ)⊕(σ σ)).FFβ IndFK(σ ResK(IndK(σ))),LetβdenotethetensorsquarerepresentationofIndFK(σ),soF2β=Λ2(IndFK(σ))⊕sym(IndK(σ)).

SinceρKis(by(3.16))isomorphictoσ σθ,itmustoccurintheinductionofthelattertoGF;dittoforthetwistofρbyδ.Hence,bytheadditivityofinduction,therepresentationontherightof(3.21)isforcedtobe

F22IndFK(sym(σ))⊕IndK(Λ(σ))⊕ρ⊕(ρ δ).

Thelemmanowfollowsinviewof(3.17).

Sowemay,afterpossiblyreplacingρbyρ δ,assumethat

(3.22)ρ As(σ),

foranirreducible2-dimensional,continuousC-representationσofGKwithsolvableimage.

ForanycuspidalautomorphicrepresentationπofGL(2,AK),onemayassociatethefollowingAsaiL-function: L(s,As(σw(π))),(3.23)L(s,π,r)=

v

wherevrunsoverthesetΣFofalltheplacesofF,andforeachv∈ΣF,wdenotesaplaceofKabovevandAs(σw(π))denotestheAsairepresentationassociatedtoσw(π).Notethatthede nitionofAs(σw(π))isindependentofthechoiceofwabovev.WhenvsplitsinK,Kv=Kw×Kθw,andAs(σv(η))simplymeansthetensorproductσw(η) σθw(η).

TheL-functionontheleftof(3.23)lookslikeaLanglandsL-function,andweneedtoexplainwhywearejusti edinadoptingsuchanotation.ForthisrecallthattheL-groupoftherestrictionofscalarsofGL(2)/KtoFisthesemidirectproduct

(3.24)

(3.25)L(RK/FGL(2)/K)=(GL(2,C)×GL(2,C))×Gal(K/F),r:L(RK/FGL(2)/K)→GL(C2 C2) GL(4,C)whereθactsbyinterchangingthetwofactors.Onede nesarepresentation

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

16DinakarRamakrishnan

bysetting,forallx,yinC2,

r(g,g ;1)(x y)=g(x) g(y)

an …… 此处隐藏：5685字，全部文档内容请下载后查看。喜欢就下载吧 ……