Modularity of solvable Artin representations(6)

As(σ(πw χw)) As(σ(πw)) χ0,v.

Consequently,

(4.0)L(s,π χ;r)=L(s,π;r χ0).

Oneknows,cf.[HLR],thatL(s,π;r)admitsameromorphiccontinuationtothewholes-planeandsatis esastandardfunctionalequation.

Proposition4.1LetπbeacuspidalautomorphicrepresentationofGL(2,AK).SupposewehaveconstructedaweakAsailifting,i.e.,anisobaricautomorphicrepresentationΠofGL(4,AF)satisfyingthefollowingidentityatalmostallplacesvofF:

(Lv)L(s,Πv)=L(s,πw;r),

wherewisanyplaceofKabovev.Thenwehave(Lv)and(εv)atallthe niteplacesv,and(L∞)aswell.Inaddition,thecuspidalitycriterionofTheoremDholds.

Proof.LetSbethe( nite)setofplacesofFoutsidewhich(Lv)holds.Note rstthatthecentralcharacter ofΠissimplytherestrictionω0toFofthecentralcharacterωofπ.Indeedatanyv,ω0correspondstothetransferoftheGaloischaracterattachedtoω,whichalsogivesthedeterminantofAs(σw(π)),withwbeingaplaceofKabovev.Sincebyde nition,L(s,πw;r)equalsL(s,As(σw(π)),itfollowsthattheideleclasscharacters andω0agreeoutsideS,andhenceagreeeverywherebyaclassicalresultofHecke.

Nowsomenotations.Iff(s),g(s)aretwomeromorphicfunctionsofssuchthattheirquotientisinvertible,wewillwritef(s)≡g(s).Atanyplacev,givena characterνofFv,wecanwriteitasν0|.|z,foraunitarycharacterν0andacomplexnumberz.Therealpartofzisuniquelyde ned;wewillcallittheexponentofν,anddenoteite(ν).

Choosea niteordercharacterµofCFwithµ∞=1(whichmeansthatµistotallyeven)suchthatµvissu cientlyrami edatevery niteplaceuinSsoastomaketheL-factorsatuofΠµu,(π,r µu),andtheircontragredients,allequal1.ThisisevidentforL(s,π;r µ)asitslocalfactorsarede nedheretocoincidewiththecorrespondingGaloisfactors,anditispossibleforL(s,Π µ)bytheresults

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

18DinakarRamakrishnan

of[JPSS1].ComparingtheglobalfunctionalequationsofbothL-functions,andnotingthattwistingbyµdoesnotchangeanythingatin nity,weget ∨(4.2)L(s,Πv)L(1 s,πw;r))≡L(1 s,Π∨v)L(s,πw;r).

v|∞v|∞

Foranyplacev,archimedeanorotherwise,foranyn≥1,andforanycuspidalautomorphicrepresentationηofGL(n,AF),itisknownthatL(s,ηv)isholomor-phicin (s)>12 t,forsomet=t(η,v)>0(see[BaR],Prop.2.1,partB).Consequently,L(s,Πv)hasnopoleincommonwithL(1 s,Π∨v).

ThusthepolesofL(s,Π∞)arecontainedinthoseofL(s,π∞;r).Fortheconversedirection,weappealtothefactorizationformula

(4.3)L(s,π×(π θ))=L(s,π;r)L(s,π;r δ),

whereδisthequadraticcharacterofFassociatedtoK/F.Thisformulaisevidentfromthede nition(3.23).Sincethelocalfactorsareneverzeroatanyplacev,thev-factorofL(s,π;r)canhaveapolesomewhereonlyifthev-factorofL(s,π×(π θ))alsodoes.Ontheotherhand,byTheoremMof[Ra1],thereisauniqueisobaricautomorphicrepresentationπ (π θ)ofGL(4,AF)whosestandardL-functioncoincideswithL(s,π×π ).So,applying[BaR],Proposition2.1,again,weseethatL(s,πv;r)isholomorphicin (s)>1 t,forsomet>0.Hencethepolesof2∨L(s,π∞;r)aredistinctfromthoseofL(1 s;π∞;r),andsomustcoincidewiththose

)andΓC(s)=ofL(s,Π∞).LetususethecustomarynotationΓR(s)=π s/2Γ(s22(2π) sΓ(s).UsingtheduplicationformulaexpressingΓC(s)asΓR(s)ΓR(s+1),

andappealingtothefactthatL(s,Π∞)isastandardL-factorofGL(4,F∞)andthatL(s,π∞;r)is(byde nition)aGaloisL-factor,wemaywrite

L(s,Π∞)=n

j=1s+ajΓR()and2L(s,π∞;r)=n

j=1ΓR(s+bj),2

forsomecomplexnumbersaj,bj,1≤j≤n=4[F:Q].(Thefactthatnis4times

[F:Q]willplaynorole.)Wemayrenumbertheajandbjandassumethatthereexistsanintegerm,with0≤m≤n,suchthataj=bjforallj>mandthesets{aj|j≤m}and{bj|j≤m}aretotallydisjoint.Wehavenothingtoproveifm=0,soassumethatmispositive.Nowweappealtothefollowing

BabyLemmaLetm>0beanintegerandlet{aj|j≤m},{bj|j≤m}besubsets s+ajofCwithemptyintersection.ThenthepolardivisorsofL1(s):=nj=1ΓR(2) ns+bandL2(s):=j=1ΓR(2j)cannotbethesame.

cProofofBabyLemma.ClearlythesupportofthepolardivisorofΓR(s+)is,2foranyc∈C,theset{ c 2k|k∈Z,k≥0}.SupposethepolesofL1(s)ands+bL2(s)coincide.Then a1mustbeapoleofsomeΓR(2j).Afterrenumbering

thebj,wemaythenassumethata1=b1+2k1forsomepositiveintegerk1.(Sinces+aa1=b1,k1cannotbe0.)Now b1willneedtobeapoleofsomeΓR(2j),andjmustbe≥2.Afterrenumberingtheajwemayassumethatb1=a2+2 1forsome 1>0.Wemaycontinuethusandwrite,aftersuitablerenumberingsoftheajandthebj,a2=b2+2k2,b2=a3+2 2,andsoon.Thisleadstothestringofinequalitiesa1<b1<a2<b2<a3···<am<bm.Then bmisapoleofL2(s)anditisnotapoleofL1(s).

Theidentity(L∞)nowfolows.

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)-type19

Next xa niteplacevinS.Nowchooseacharacterµwhichisunrami edatv,butishighlyrami edateveryuinS {v}.Comparingthefunctionalequationsandarguingexcatlyasinthearchimedeancase,wegetthedesiredidentity(Lv).

Nextweprovetheidentityofepsilonfactors.FixanyvinSandnotethatby

[JPSS2],forµusu cientlyrami edatu∈S {v},theepsilonfactorofΠu µudependsonlyonµuandω0,u,andthedependenceissimple.Similarly,by[DeH],theepsilonfactoratuof(π,r µ)hasthesamedependenceonµuandω0,u;thereasonwecanapply[DeH]isthatwehavede nedthelocalfactorsof(π,r µ)asthoseassociatedtothecorrespondingrepresentationsofthelocalWeilgroups.Theanalogousstatementsholdforthecontragredients,andthisresultsintheidentity(εv)aswealreadyknowthattheL-factorsagree.This nishestheproofofthe rstpartofProposition4.1.

Itislefttoprovethecuspidalitycriterion.

Notethat(4.3)impliestheidentity

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