Modularity of solvable Artin representations(4)

10DinaarRamarishnan

Wemayhenceforthassumethat[K:F]=2,whichisthesubtlercase.De-notebyθthenon-trivialautomorphismofKoverF.Wecanagain ndcuspidalautomorphicrepresentationsπ,π ofGL(2,AK)suchthat

(3.5) L(s,ρK)=L(s,πf πf),

whichprovesthattherestrictionρKofρtoGKismodular.

Wewillnowexplainwhythiscaseisdi cult.Theidentity(3.5)impliesthatπ π isθ-invariant,sobythebasechangetheoremofArthurandClozel([AC]),wecan ndanisobaricautomorphicrepresentationΠofGL(4,AF)suchthatitsbasechangeΠKisisomorphictoπ π .OnecanalsoseeeasilythatthelocalfactorsofΠandρagreeatalltheplacesofFwhichsplitinK.Butoneisstuckatthispointandcannoteasilydeducetherequisiteidentityattheinertplaces,exceptwhenρKisnolongerirreducible.

ItisnowclearthatTheoremAisaconsequenceofTheoremA .Sowewilladdressthefollowing:

ProofofTheoremA moduloTheoremD:

SupposeρKisreducible.Thenitmustcontainanirreduciblesummandτofdimension≤2.ByFrobeniusreciprocity,ρshouldintertwinewiththeinductionFIndFK(τ)ofτtoGF.Asρisirreducibleofdimension4andIndK(τ)isatmostofdimension4,weareforcedtohave

ρ IndFK(τ),

withdim(τ)=2.Thesolvabilityoftheimageofρimpliesthesameaboutthatofτ,andsowemayapplyLanglands-Tunnelltogetacuspidalautomorphicrep-resentationηofGL(2,AK)associatedtoτ,andwearedonebytakingΠtobeFtheautomorphicallyinducedrepresentationIK(τ)(see[AC],andalso[Ra1],sec.

2).Sinceρisirreducible,τ,andhenceη,cannotbeθ-invariant,whereθisthekkFnon-trivialautomorphismofK/F.Consequently,IK(η)mustbecuspidal.

Sowemay,andwewill,assumethatρKisirreducible.Sinceitistherestrictionofρ,wehave

θ(σ σ )θ ρθK ρK σ σ,

whereθisthenontrivialautomorphismofKoverF.

Wewill rstprovepart(b)ofTheoremA .

Lemma3.6TheirreducibilityofρK=σ σ impliesthatatleastoneoftherepresentationsσ,σ isnon-dihedral,andσ cannotbeaone-dimensionaltwistofσ.Furthermore,sinceρKisθ-invariant,oneofthefollowingmusthappen:(I)ThereexistsacharacterνofGal(F/K)suchthat

σθ σ νand(σ )θ σ ν 1;

(σ )θ σ µ 1.(II)ThereexistsacharacterµofGal(F/K)suchthatσθ σ µand

ProofofLemma3.6.Letω,resp.ω ,denotethedeterminantofσ,resp.σ .Clearly,ifσ σ λforsomecharacterλ,thenρKisreducible,asσ σ willthen

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)-type11

be(sym2(σ) λ)⊕ωλ.Wewillnowshowthatnotbothσ,σ canbedihedral.Wehave

1) (σ ω End(ρK) ρK ρ∨K ρK (σ ω 12 1) ρ .K (ωω)

SinceρKisirreducible,anylinearcharacteroccurringinEnd(ρK)musthavemul-tiplicityone.Weclaimthatifσandσ arebothdihedral,thenEnd(ρK)containsthetrivialrepresentationwithmultiplicity>1.Indeedsupposeσ=IndKM(χ)andK σ=IndN(χ),whereM,NarequadraticextensionsofK,andχ,χcharactersof222GM,GNrespectively.Sinceρ Kissym(ρK)⊕Λ(ρK),itsu cestoprovethatωω occurstwiceinsym2(ρK).Notethat

2sym2(σ) IndKM(χ)⊕ω andsym2(σ ) IndKM(χ)⊕ω ,2

where (resp. )isthequadraticcharacterofGKcorrespondingtoM/K(resp.N/K).Since

(3.7)

weget

KKK2 2 ⊕2,sym2(ρK) IndKM(χ) IndM(χ)⊕IndM(χ) ω ⊕IndM(χ) ω ⊕(ωω)22sym2(σ σ ) sym2(σ) sym2(σ )⊕Λ2(σ) Λ2(σ ),

asasserted.Soatleastoneofσ,σ mustbenon-dihedral.Finallybyθ-invariance,

σθ (σ )θ σ σ ,

whichevidentlyimplies,byusingirreducibility,thatwemustbeincase(I)or(II).Lemma3.8Supposeweareincase(I)ofLemma3.6.Thenthereexistcharactersχ,χ ofGKsuchthatσ χandσ χ arebothθ-invariant.

ProofofLemma3.8.Wemayassume,afterinterchangingσ,σ ifnecessary,thatσ isnon-dihedral.Weclaimthat

(3.9)(ωω )θ=ωω .

Tobegin,notethattheθ-invarianceofρKimpliesthesameforitscontragredientanditssymmetricandexteriorpowers.ThedeterminantofρKiseasilyseentobe(ωω )2,andthisshowsthat

β:=(ωω )θ/ωω

hasorder≤2.Supposetheorderis2.Thentheidentity(3.7)showsthat

β Ad(σ) Ad(σ ),

whereforany2-dimensionalτ,

1,Ad(τ):=sym2(τ) ωτ

theadjointrepresentation.Theself-dualityofAd(σ),plustheirreducibilityofAd(σ ),thenimpliesthatAd(σ )isisomorphictoAd(σ) β,whichisthesame,asβ2=1,asAd(σ β).Butwehavethefollowing

Theorem3.10Letτ,τ beirreducible,2-dimensionalrepresentationsofGKwithisomorphicadjointrepresentations.ThenthereexistsacharacterχofGKsuchthat

τ τ χ.

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

12DinakarRamakrishnan

Foraproofsee[Ra2];itisimportanttonotethatitholdswhetherornotτisdihedral.ThisistheGaloisversionofthesocalledmultiplicityoneforSL(2).Itsautomorphicversionwasprovedin[Ra1](seeTheorem4.1.2).

Applyingthiswithτ=σ βandτ =σ ,weseethatσ isaonedimensionaltwistofσ,whichcontradictstheirreducibilityofρK.Henceβmustbe1andtheclaimisproved.

Nextweclaimthat

(3.11)(sym2(σ) ω )θ sym2(σ) ω

2 θ2and(sym2(σ ) ω)θ sym2(σ ) ω. θ2 2Indeed,sinceσθisbyhypothesisσ µ,wehave(3.12)(sym(σ) ω) sym(σ µ) (ω) (sym(σ) ω) µ (ω )θ

ω .

Ontheotherhand,comparingthedeterminantsofσθandσ µ,weget

ωθ

.(3.13)µ=ω

The rsthalfoftheassertedidentity(3.11)nowfollowsbycombining(3.9),(3.12)and(3.13).Theproofofthesecondhalfisthesame.2

Wewillnowprovetheθ-invarianceofσ χforasuitableχ.Thecaseofσ issimilarandwillbelefttothereader.

Notethatsym2(σ ω )isofGO(3)-type.Explicitly,

sym2(sym2(σ) ω ) (sym4(σ)⊕ω2) ω ,

andso(ω …… 此处隐藏：5355字，全部文档内容请下载后查看。喜欢就下载吧 ……