On Artin s L-Functions. Robert P. Langlands

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1 On Artin s L-Functions Robert P. Langlands The nonabelian Artin L functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that ofthezetafunctionbutlittleisknownabouttheirpoles. Itseemstobeexpectedthatthe L functions associated to irreducible representations of the Galois group or another closely relatedgroup,theweilgroup,willbeentire.onewaythathasbeensuggestedtoshowthisis to show that the Artin L functions are equal to L functions associated to automorphic forms. Thisisamoredifficultproblemthanwecanconsideratpresent. Incertaincasestheconversequestionismucheasier. Itispossibletoshowthatifthe L functions associated to irreducible two dimensional representations of the Weil group are all entire these L functions are equal to certain L functions associated to automorphic forms on GL(2). Before formulating this result precisely we review Weil s generalization of Artin s L functions. Aglobalfieldwillbejustanalgebraicnumberfieldoffinitedegreeovertherationalsora functionfieldinonevariableoverafinitefield.alocalfieldisthecompletionofaglobalfield atsomeplace.if Fisalocalfieldlet C F bethemultiplicativegroupof Fandif Fisaglobal fieldlet C F betheidèleclassgroupof F.If KisafiniteGaloisextensionof FtheWeilgroup W K/F isanextensionof G(K/F),theGaloisgroupof K/F,by C K. Thusthereisanexact sequence 1 C K W K/F G(K/F) 1. If L/F isalsogaloisand Lcontains Kthereisacontinuoushomomorphism τ L/F,K/F of W L/F onto W K/F.Itisdetermineduptoaninnerautomorphismof W K/F byanelementof C K. Inparticular W F/F = C F andthekernelof τ K/F,F/F isthecommutatorsubgroupof W K/F.Alsoif F E Kwemayregard W K/E asasubgroupof W K/F.If Fisglobaland visaplaceof Fwealsodenoteby vanyextensionof vto K.Thereisahomomorphism α v of W Kv /F v into W K/F whichisdetermineduptoaninnerautomorphismbyanelementof C K. Arepresentation σof W K/F isacontinuoushomomorphismof W K/F intothegroupof invertible linear transformations of a finite dimensional complex vector space such that σ(w) isdiagonalizableforall win W K/F. If F K L σ τ L/F,K/F isarepresentationof W L/F whichweneednotdistinguishfrom σ.inparticular,one dimensionalrepresentations of W K/F maybeidentifiedwithquasi charactersof C F,thatis,continuoushomomorphisms FirstappearedinRiceUniversityStudies,vol.56(1970)

2 On Artin s L Functions 2 of C F intothemultiplicativegroupofcomplexnumbers.if ωissuchaquasi characterand σ arepresentationof W K/F ω σhasthesamedimensionas σ. If F E Kand ρis arepresentationof W K/E on Xlet Y bethespaceoffunctions φon W K/F withvaluesin X such that φ(uw) = ρ(u)φ(w) forall uin W K/E.If νisin W K/F weset σ(v)φ(w) = φ(wv); σisarepresentationof W K/F on Yandwewrite σ =Ind(W K/F, W K/E, ρ). If Fisaglobalfieldand σisarepresentationof W K/F,forallplaces v, σ v = σ α v isa representationof W Kv /F v whoseclassdependsonlyonthatof σ. Nowtake F tobealocalfield. Suppose ωisaquasi characterof C F. If F isnonarchimedeanand ωisramified,weset L(s, ω) = 1butif ωisunramifiedweset L(s, ω) = 1 1 ω( ) s; isauniformizingparameterfor F.If Fisrealand ω(x) = (sgn x) m x r with mequalto 0 or 1weset ( ) s + r + m L(s, ω) = π 1 2 (s+r+m) Γ ; 2 if Fiscomplexand with m + n 0, mn = 0 ω(z) = (z z) r (m+n)/2 z m z n ( L(s, ω) = 2(2π) (s+r+(m+n)/2) Γ s + r + m + n ). 2 Thelocal L functions L(s, σ)associatedtorepresentationsoftheweilgroups W K/F are characterized by the following four properties. (i) L(s, σ)dependsonlyontheclassof σ. (ii)if σisofdimension1andthusassociatedtoaquasi character ωof C F L(s, σ) = L(s, ω).

3 On Artin s L Functions 3 (iii) L(s, σ 1 σ 2 ) = L(s, σ 1 )L(s, σ 2 ). (iv)if F E K, ρisarepresentationof W K/E,and σ =Ind(W K/F, W K/E, ρ), L(s, σ) = L(s, ρ). Nowsuppose ψ = ψ F isanontrivialadditivecharacterof F. If Eisafiniteseparable extensionof Flet ψ E/F (x) = ψ F (Tr E/F x). Suppose Eisanylocalfieldand ψ E isanontrivialadditivecharacterof E. If Eisnonarchimedeanand ω E isaquasi characterof C E let n = n(ψ E )bethelargestintegersuchthat ψ E istrivialon P n E andlet m = m(ω E)betheorderoftheconductorof ω E. If U E isthe groupofunitsintheringofintegersof Eand γisanyelementof Fwithorder m + nlet U E ψ E ( (ω E, ψ E ) = ω E (γ) ( U E ψ E α γ α γ ) ω 1 E (α)dα ) ω 1 E (α)dα. If Eisrealand ψ E (z) = e 2πiux while ω E (x) = (sgn x) m x r If Eiscomplexand ψ E (z) = e 4πiRe(wz) while (ω E, ψ E ) = (isgn u) m u r. ω E (z) = (z z) r (m+n)/2 z m z n (ω E, ψ E ) = i m+n ω E (w). Theorem 1. Suppose F is a local field and ψ F is a nontrivial additive character of F. It is possible in exactly one way to associate to every finite separable extension E of F a complex number λ(e/f, ψ F ) and to every representation σ of any Weil group W K/E a complex number ǫ(σ, ψ E/F ) so that the following conditions are satisfied. (i) ǫ(σ, ψ E/F ) depends only on the class of σ.

4 On Artin s L Functions 4 (ii) If σ is one-dimensional and corresponds to the quasi-character ω E ǫ(σ, ψ E/F ) = (ω E, ψ E/F ). (iii) (iv) If ǫ(σ 1 σ 2, ψ E/F ) = ǫ(σ 1, ψ E/F )ǫ(σ 2, ψ E/F ). σ = Ind(W K/F, W K/E, ρ), ǫ(σ, ψ F ) = ǫ(ρ, ψ E/F )λ(e/f, ψ E ) dim ρ. If sisacomplexnumberand a F isormalizedabsolutevalueon Fset α s F (a) = a s F and set ǫ(s, σ, ψ F ) = ǫ(α s 1 2 σ, ψf ). If Fisaglobalfieldand σisarepresentationof W K/F theglobal L function L(s, σ)isdefined by L(s, σ) = L(s, σ v ). v The product is taken over all places including the archimedean ones. It converges in a right half plane. L(s, σ) can be analytically continued to the whole plane as a meromorphic function. If ψisanontrivialcharacterof F \A,thequotientoftheringofadèlesby F,,foreach place v,therestriction ψ v of ψto F v isnontrivial. Foralmostall vthefactor ǫ(s, σ v, ψ v )is1 and ǫ(s, σ) = ǫ(s, σ v, ψ v ) v doesnotdependonthechoiceof ψ. Theorem 2. If σ is the contragradient of σ the functional equation is satisfied. L(s, σ) = ǫ(s, σ)l(1 s, σ) Theproofsofthesetwotheoremsmaybefoundin[2]. Nowwehavetosaysomething about the L functions associated to automorphic forms on GL(2). If Fisalocalfieldand πisanirreduciblerepresentationofthelocallycompactgroup GL(2, F)wecanintroducealocal L function L(s, π).givenanontrivialadditivecharacter ψ

5 On Artin s L Functions 5 of Fwecanalsointroduceafactor ǫ(s, π, ψ).if ωisaquasi characterof C F ω πwill be the representation g ω(detg)π(g). If Fisaglobalfieldand ηisaquasi characterof C F A 0 (η)willbethespaceofall measurable functions φ on GL(2, F)\GL(2, A) satisfying: (i)foreveryidèlea φ (( ) ) a 0 g = η(a)φ(g); 0 a (ii)if Z A isthegroupofscalarmatricesin GL(2, A) is finite; (iii)foralmostall gin GL(2, A) Z A GL(2,F)\GL(2,A) F \A φ(g) 2 η(detg) 1 dg (( ) ) 1 x φ 0 1 g g dx = 0. GL(2, A)actson A 0 (η)byrighttranslations. Underthisaction A 0 (η)breaksupinto the direct sum of mutually orthogonal, invariant, irreducible subspaces. Let π be the representationof GL(2, A)ononeofthesesubspaces.Onecandecompose πintoatensor product v π v where π v isanirreduciblerepresentationof GL(2, F v ).Theproduct L(s, π) = v L(s, π v ) convergesinahalfplaneand L(s, π)canbeanalyticallycontinuedtoanentirefunction. If ψisanontrivialcharacterof F \Aand ǫ(s, π) = v ǫ(s, π v, ψ v ) L(s, π) = ǫ(s, π)l(1 s, π). Nowsuppose F isalocalfieldand σisatwo dimensionalrepresentationof W K/F. Then detσisaone dimensionalrepresentationof W K/F andmaythereforeberegardedasa quasi characterof C F.Thereisatmostoneirreduciblerepresentation π = π(σ)of GL(2, F), satisfying the following conditions.

6 On Artin s L Functions 6 (i)forall ain C F π (( )) a 0 = detσ(a)i 0 a if Iistheidentity. (ii)foreveryquasi character ωof C F L(s, ω π) = L(s, ω σ), and L(s, ω π) = L(s, ω σ). (iii)forone,andhenceforevery,nontrivialcharacter ψof F forall ω. ǫ(s, ω π, ψ) = ǫ(s, ω σ, ψ) Theorem 3. Suppose F is a global field and σ is a two-dimensional representation of W K/F. Let η = detσ. Suppose that for every quasi-character ω of C F both L(s, ω σ) and L(s, ω 1 σ) are entire functions which are bounded in vertical strips of finite width. Then π v = π(σ v ) exists for each v and the representation v π v of GL(2, A) occurs in the representation of GL(2, A) on A 0 (η). This theorem is proved in[1], written in collaboration with H. Jacquet. References 1. Jacquet, H. and R.P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, No. 114, Springer Verlag(1970). 2. Langlands, R.P., On the functional equation of the Artin L-functions,(in preparation). Yale University