Modularity of solvable Artin representations(3)

(1.3)1→k →GL(2,k)→SO(3,k)→1,

wherethemapsinthemiddlearec→cI2andg→Ad(g).

Theevendimensionalcasen=2mismoreinteresting.SinceforanyginGO(2m,k),thesquareofitsdeterminantisλ(g)2m,wecande neahomomorphism,calledthesimilitudenorm

(1.4)ν:GO(2m,k)→{±1},

bysendinggtoλ(g) mdet(g).

Thekernelofν,denotedSGO(2m,k),iscalledthespecialorthogonalsimili-tudegroup.(SomepeoplewriteGSO(2m,k)instead.)Themapνdoesnotsplit.

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)-type7

SinceνisjustthedeterminantmaponO(2m,k),theintersectionofSGO(2m,k)withO(2m,k)isSO(2m,k).Whenk=C,SGO(2m,k)(resp.SO(2m,k))istheconnectedcomponentofGO(2m,k)(resp.O(2m,k)).SGO(2,k)isatorus.

Notethatν(cI2m)is1,andthatSGO(2m,k)isgeneratedbySO(2m,k)andZ2m(k);buttheirintersectionis{±I2m}.

Wewillconcludethissectionbyrecallingalowdimesionalisomorphismfork=k,whichwewillneed,betweenSGO(4,k)andaquotientofGL(2,k)×GL(2,k).

LetWbek2withthestandardsymplecticformgivenbythedeterminant.ThentheinducedbilinearformBonthetensorproductW Wisnon-degenerateandsymmetric.Thereisanisometrybetween(W W,B)and(k4,B0).SinceGL(2,k)isthesymplecticsimilitudegroupof(W,det),wegetanexactsequence

(1.5)1→k →GL(2,k)×GL(2,k)→GO(4,k),

wherethemaponk isjustgivenbyc→(cI2,c 1I2).Themapβ,say,ontherightcanbedescribedexplicitlyasfollows.Thequadraticspace(k4,B0)isalsoisometricto(M2(k),B1),whereB1isthesymmetricbilinearmap(X,Y)→tr(tXY).Underthisidenti cation,β(g,g )is,forallg,g inGL(2,k),theautomorphismofk4givenbyX→tgXg .Clearlythekernelofβconsistsofpairs(cI2,c 1I2)withc∈k ,provingtherequisiteexactness.

Notethatλ(β(g,g ))isdet(g)det(g ),whilethedeterminantofβ(g,g )isitssquare.Henceνistrivialontheimageofβ.ItiseasytoseethatZ4(k)liesintheimageofβ,andthatβ(SL(2,k)×SL(2,k))isasubgroupofSO(4,k)properlycontaining{±I4}.TheabelianizationofSO(4,k)isk /k 2([D],p.57),andsincewehaveassumedthatk=k,SO(4,k)isperfect,i.e.,itequalsitsowncommutatorsubgroup.Thenbythediscussiononpage59ofloc.cit,SO(4,k)/{±I4}isiso-morphictoPSL(2,k)×PSL(2,k).ItfollowsthatβmapsSL(2,k)×SL(2,k)ontoSO(4,k)withkernel{±(I2,I2)}.Puttingeverythingtogether,weobtain

(1.6)β(GL(2,k)×GL(2,k))=SGO(4,k).

2.Thereduciblecase

SupposewearegivenarepresentationρasinthestatementofTheoremA,whichisreducible.ThankstoMaschke’stheoremwemaywriteρ ⊕jρj,witheachρj irreducibleofdimensionnj,andjnj=4.Supposewehavefound,foreachj,acuspidalautomorphicrepresentationπj=πj,∞ πj,fofGL(nj,AF)suchthatL(s,ρj)=L(s,πj,f).ThenwecanconsidertheisobaricsumofLanglands([La2],

[JS])

(2.1)

whichisautomorphicandsatis es

L(s,π)=π= jπj,

jL(s,πj).

SincetheL-functionsofArtinarealsoadditive,wegetL(s,ρ)=L(s,πf)asdesired.Soitremainsto ndtheπj.

NotethatcuspidalautomorphicrepresentationsofGL(1,AF)arejustideleclasscharactersofF.Sowhennj=1,theexistenceofπjfollowsfromclass eldtheory.

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

8DinakarRamakrishnan

Sincetheimageofρisbyhypothesissolvable,thesamewillbetrueforeachρj.Soifnj=2,wemayapplythecelebratedtheoremofLanglands([La1])andTunnell([Tu])toconcludetheexistenceofπj.

Itremainstoconsiderthecasewhennjis3forsomej,sayforj=1.Thenwemusthaveadecomposition

ρ ρ1⊕ρ2,

withρ1(resp.ρ2)irreducibleofdimension3(resp.1).Sincebyhypothesis,theimageofρlandsisGO(4,C),andsincetherecanbenointertwiningbetweenρ1andρj,wemusthave

im(ρ1) GO(3,C).

ThankstoLemma1.2,GO(3,C)isSO(3,C)×C .Sowemaywrite

(2.2)ρ1 ρ χ,

whereχisacharacterGF→C ,andρ isa3-dimensionalrepresentationofGFwithimageinSO(3,C).

Moreover,theexactsequence(1.3),whichcanbeviewedasanexactsequenceoftrivialmodulesunderGF=Gal(F/F),furnishesthecohomologyexactsequence(2.3)Hom(GF,GL(2,C))→Hom(GF,SO(3,C))→H2(GF,C ),

withρ belongingtothemiddlegroup.Ontheotherhand,atheoremofTate(see

[Se],foraproof)assertsthatthegroupontherighthandsideof(2.3)istrivialasFisanumber eld.Thuswemayliftρ toanelementofthelefthandsidegroupof(2.3).Inotherwords,wecan nda(non-unique)2-dimensionalrepresentationτ1ofGFsuchthat

(2.4)ρ1 Ad(τ1) χ.

Sinceρ1hassolvableimage,τ1isalsoforcedtohavethesameproperty.ApplyingLanglands-Tunnellonceagain,wegetanisobaricautomorphicrepresentationη1,whichmustinfactbecuspidalasρ1andhenceτ1areirreducible,suchthatL(s,τ1)equalsL(s,η1,f).

By[GeJ]oneknowsthat,givenanycuspidalautomorphicrepresentationηofGL(2,AF),thereexistsafunctoriallyassociated(isobaric)automorphicrepresen-tationAd(η)suchthat

L(s,Ad(σv(η))),(2.5)L(s,Ad(η))=

v

wheretheproductisoveralltheplacesvofF,andσv(η)(resp.σ(ηv))isthe3-dimensionalrepresentationoftheWeilgroup(resp.Weil-Delignegroup)WFv (resp.WF)whenvisarchimedean(resp.non-archimedean),associatedtoηvbyvthelocalLanglandscorrespondenceforGL(n)([HaT],[He]).

Thenitfollowsthat

L(s,ρ1)=L(s,(Ad(η1) χ)f).

Sowearedonebysettingπ1=Ad(η1) 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

SolvableArtinrepresentationsofGO(4)-type9

3.ModularitymoduloTheoremD

InthissectionwewillshowhowtoproveTheoremAifweadmitthetruthofTheoremD.Wewillalsoneedtomakeuseofthemaintheoremof[Ra1].Thankstothediscussionintheprevioussection,wemayassumethatρisirr …… 此处隐藏：5804字，全部文档内容请下载后查看。喜欢就下载吧 ……