Asai cube L-functions and the local Langlands conjecture

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by  G. Henniart

ASAI CUBE L-FUNCTIONS AND
THE LOCAL LANGLANDS CONJECTURE
7
1 GUYHENNIARTANDLUISLOMEL´I
0
2
Abstract. We work over non-Archimedean local fields F of characteristic

n 0 as well as characteristic p. Let E/F be a separable cubic extension and
a let H be an ambient group of type D4, which has triality corresponding to
J E. We choose a maximal Levi subgroup M isogenous to G = GL2(E). The
6 Langlands-Shahidi method applied to(H,M)attaches anAsai cube γ-factor
toanirreduciblesmoothgenericrepresentationπofG. IfσistheWeil-Deligne

] representationcorrespondingtoπvialocalLanglands,weprovethattheAsai
T cube γ-factor ofπ istheγ-factor ofthe Weil-Delignerepresentation obtained
N from σ via cubic tensor induction from E to F. A consequence is that Asai
cubeγ-factorsbecomestableundertwistsbyhighlyramifiedcharacters.
.
h
t
a
m
[
Introduction

1
v LetGbeaquasi-splitconnectedreductivegroupdefinedoveranon-Archimedean
6 localfieldF andletr beacomplexrepresentationoftheLanglandsdualgroupLG.
1 The Langlands-Shahidi method produces γ-factors γ(s,π,r,ψ) in many situations
5 for r, a generic representation π and a smooth character ψ :F →C×. We let W
1 F
denote the absolute Weil group of F, and W′ the Weil-Deligne group. The local
0 F
. Langlands correspondence –still mostly conjectural–, attaches to π a Langlands
1
parameterφ :W′ →LG(uptoLG◦-conjugacy). Wheneversuchacorrespondence
0 π F
is established, we should have the following main equality
7
1
: (1) γ(s,π,r,ψ)=γ(s,r◦φπ,ψ).
v
i On the right hand side, they are the Galois γ-factors of Deligne and Langlands.
X
In this paper, we focus on a case which has not been looked at seriously before.
r
a Our group G is Res GL , restriction of scalars, where E is a separable cubic
E/F 2
extensionof F. In the Langlands-Shahidimethod (see [26] and [23]) we choose the
ambient quasi-split group H over F to be of type D , with triality corresponding
4
to E/F. We choose H to be simply connected, so that LH is of adjoint type. We
let M be the maximal Levi subgroup of H obtained by removing the central root
α from the diagram:
2
α
3
•
α1 • •α2
•
α
4
1
2 G.HENNIARTANDL.LOMEL´I
The local Langlands correspondence for GL is known, see [17] and the more
2
detailed account of [2]. Let π be a (generic) smooth irreducible representation of
GL (E), and σ the corresponding degree 2 representation of W′ obtained via the
2 E
local Langlands correspondence. On the one hand, we have that the Langlands-
Shahidi method studies the pair (H,M) and the adjoint action of the Langlands
dual group LM on the Lie algebra of the unipotent radical of the parabolic of LH
with Levi LM. This produces an Asai cube representation r of LM; hence, by
A
composition,adegree8representation⊗IofLG. OntheothersideoftheLanglands
correspondence,welet⊗I(σ)betherepresentationofW′ obtainedfromσbytensor
F
induction from E to F. We prove the following main equality
(2) γ(s,π,⊗I,ψ)=γ(s,⊗I(σ),ψ),
betweenLanglands-Shahidiγ-factorsandtheGaloisγ-factorsofDeligneandLang-
lands. We further have the following equalities
(3) L(s,π,⊗I)=L(s,⊗I(σ))
ε(s,π,⊗I,ψ)=ε(s,⊗I(σ),ψ)
which we derive from the main equation for γ-factors.
WhenF haspositivecharacteristic,weusedalocal-to-globalargumentin[12,14]
together with a characterization of γ-factors in order to prove the corresponding
version of equation (2) for exterior square, symmetric square, and regular Asai γ-
factors. The local-to-global result of [13] was generalized over function fields in
[8], and our previous results on (2) are a consequence of the general considera-
tions of Gan-Lomel´ı in characterisit p. We observe that equalities (2) and (3) are
true for unramified principal series over any non-Archimedean F [16]; and, in the
Archimedean case [24].
When F has characteristic0, we also use a local-to-globalargumentand a char-
acterization of γ-factors γ(s,π,r ,ψ) through a list of axioms. All but one of the
A
axiomsarelocal,andthecorrespondingaxiomsforγ(s,⊗I(σ),ψ)ontheGaloisside
areeasytoprove. Theglobalfunctionalequationistheremainingpropertyexpress-
ing the local-to-global compatibility, it involves partial L-functions and γ-factors.
We take advantage that the strong Artin conjecture is known for 2-dimensional
Σ with solvable image [18, 28]. For γ(s,⊗I(σ),ψ), such a functional equation is
availablesinceσ appearsasacomponentofaglobalrepresentationΣforwhichthe
L-functionof r◦Σ has a functional equation. The Langlands-Shahidimethod sup-
plies the functional equation on the corresponding automorphic L-functions side.
The representationsπ andσ areglobalizedin a convenientway, andthey are com-
ponentsofglobalrepresentaitonsΠandΣataplacev ofaglobalfieldk. Wethen
0
prove equality (2) at all places except, perhaps, the place v . Finally, we compare
0
both global functional equations to get our result.
It is a consequence of our method that γ-factors satisfy a stability property: if
we twistπ by a sufficiently ramifiedcharacterχ then γ(s,π⊗χ,r,ψ) depends only
on the central character π and not on π itself. See [8] for a general statement in
the case of characteristic p.
Ackwnowledgments. TheauthorswouldliketothankP.KutzkoandF.Shahidi.
ThesecondauthorwouldliketothankIHE´SandUniversit´ed’Orsayfortheirhospi-
talityduringshortvisitsin2016;hewassupportedinpartbyProjectVRIEA/PUCV
039.367/2016.
ASAI CUBE L-FUNCTIONS 3
1. Asai cube L-functions
Let E be a cubic extension of F in F¯; we view W = W(F¯/E) as an open
E
subgroup of W , of index 3, and similarly for the Weil-Deligne groups W′ and
F E
W′ .
F
In this section, we want to use the Langlands-Shahidi method to obtain the
automorphic γ-factors for GL (E) corresponding to the previous tensor induction
2
process. More precisely, we see GL (E) as G(F), where G = Res GL . The
2 E/F 2
L-groupofGis the semidirectproductLG=GLJ⋊W , where J =Hom (E,F¯),
2 F F
andW acts onGL (C)J via its naturalactionon J. The groupLG has a natural
F 2
8-dimensionalrepresentationI⊗: ifj is the givenembedding ofE intoF¯, GL (C)J
2
acts on C2 via its j-component, and W acts on C2 trivially; tensor induction to
E
LG then gives I⊗.
We now construct a quasi-split group H over F, and a maximal parabolic sub-
group P of H, with a Levi subgroup M close enough to G that the action of LM
on the Lie algebra Ln of the unipotent radical of LP gives rise to the represen-
tation I⊗ of LG. It is not necessary for us to describe H explicitly, as only the
L-group picture will be useful to us. Because H is quasi-split, its indexed root
datum[27] §16.2.1specifies Hupto isomorphism,andweworkwith the dualroot
datum (X,Φ,X∨,Φ∨), where X is the group of characters of a maximal torus in
H (the dual group of H) and Φ is the set of roots of T in H; the group X∨ is the
gbroup of cocharacters of T and Φ∨ is the set of corootbs, equbipped with a bijection
α 7→ α∨ of Φ onto Φ∨; fibnally the group Γ = Gal(F¯/F) acts linearly on X (via
F
a finite quotient), and that action preserves Φ, the dual action on X∨ preserving
Φ∨, compatibly with the bijection α7→α∨.
Our group LH has type D and to describe its root datum we use Bourbaki
4
notaion[1]. However,weprefertoseparatetherolesofX andX∨ (andnotidentify
themvia some Killingform), sothat we letX be the setofvectorsinV =R4 with
integercoordinatesaddingtoanevennumber;wewrite(e ,...,e )forthecanonical
1 4
basis of V, and choose α = e −e , α = e −e , α = e −e and α = e +e
1 1 2 2 2 3 3 3 4 4 3 4
as a basis of Φ. The vector space V∨ dual to V has the basis (e∨,...,e∨) dual to
1 4
(e ,...,e ); the simple corootsare α∨ =e∨−e∨, α∨ =e∨−e∨, α∨ =e∨−e∨ and
1 4 1 1 2 2 2 3 3 3 4
α∨ =e∨+e∨; the lattice X∨ in V∨ is generated by 1(e∨+e∨+e∨+e∨) and e∨,
4 3 4 2 1 2 3 4 2
e∨, e∨.
3 4
Writing S for the groupofpermutationsof{1,3,4}, we havea naturalactionof
SonV preservingX,fixingα andpermuting α , α , α accordingto the indices.
2 1 3 4
α
3
•
α1 • •α2
•
α
4
Any group homomorphism Γ →S then gives a group root datum for LH, where
F
H is a quasi-split group over F. For example, with E a cubic extension of F
in F¯ as at the beginning, an identification of Hom (E,F¯) with {1,3,4} gives a
F
homomorphism Γ →S producing the group LH we seek.
F
4 G.HENNIARTANDL.LOMEL´I
Remark 1.1. The construction just described is valid with any field in place of
F, and we shall use it for a global field k and a cubic separable extension, thus
producing groups H/k and LH . If v is a place of k, the root datum for the L-
k
groupLH ofH⊗ k is obtainedfromthe rootdatumfof LH by composingthe
kv k v k
action of Γ with the homomorphism Γ →Γ coming from the completion (such
k kv k
a homomorphism depends on an isomorphism of k¯ with the algebraic closure of k
ink¯ ,butchangingitchangesLH toanisomorphicgroup). Notethatevenifthe
v kv
homomorphismΓ →Sissurjective,the localhomomorphismΓ →Smightnot
k kv
be. Forexample,atasplitplacev aglobalcubic extensionl/k is l ≃k ×k ×k ,
v v v v
the local homomorphism Γ →S is trivial.
kv
WeusethemaximalparabolicsubgroupLP ofLH whichcorrespondstoomitting
the simple root α ; if M is the Levi subgroup of Φ containing T, the roots of T
2
c b b b
in M are the roots in Φ which are linear combinations of α , α , α , whereas the
1 3 4
roocts of T in Ln are the (positive) roots in Φ where α appears with a positive
2
b
coefficient. The adjoint representationof M on Ln has two irreducible components
c
r , i=1, 2, and the corresponding roots of T are the roots in Φ where α appears
i 2
with coefficient i – we use r =r . b
1
Now we relate LG and LM, and r with I⊗. With the chosen identification of J
with {1,3,4}, LG becomes GL (C){1,3,4}⋊W ; let us describe the corresponding
2 F
rootdatumandrelateittotherootdatumforLM. Forarootdatum(Y,Ψ,Y∨,Ψ∨)
for LG, we can take
Y =Y ⊕Y ⊕Y with Y =Ze ⊕Zf ,
1 2 3 i i i
where α =e −f for i=1,3,4 as simple roots; similarly
i i i
Y∨ =Y∨⊕Y∨⊕Y∨ with Y∨ =Ze∨⊕Zf∨,
1 2 3 i i i
whereα∨ =e∨−f∨ for i=1,3,4– the duality is the obviousone,andΓ acts via
i i i F
its action on {1,3,4}.
Consider the quotient LG of LG by the central subgroup made out of central
elements(x ,x ,x )inGL (C){1,3,4} suchthatx x x =1;thecorrespondingroot
1 3 4 2 1 3 4
datum is (Z,Ψ,Z¯∨,Ψ∨), where Z is the sublattice of Y of elements a e +b f +
1 1 1 1
a e +b d +a e +b f such that a +b = a +b = a +b ; then Z¯∨ is the
3 3 3 3 4 4 4 4 1 1 3 3 4 4
corresponding quotient of Y∨ and Ψ∨ is the image of Ψ∨ in Z¯∨. The action of Γ
F
is, again, obtained via the action on {1,3,4}. One verifies immediately that the
root datum (Z,Ψ,Z¯∨,Ψ) is isomorphic to the one for LM by sending α to α , for
i i
i=1,3,4,andf +f +f toα ,sothatduallyα¯∨ inZ¯∨ issenttoα∨,fori=1,3,
1 2 3 2 i i
4,wherasthe imageofe∨+f∨ inZ¯∨ (whichisthe sameasthe imageofe∨+f∨ or
1 1 2 2
e∨+f∨),issenttoα∨+α∨+α∨−2α∨ (sothate∨ issenttoα∨−α∨+1(α∨+α∨)).
3 3 1 3 4 2 1 1 2 2 3 4
ItfollowsthatwecanchooseanisomorphismϕofLG¯ ontoLM,compatiblewith
theisomorphismofrootdatajustdescribed. Sincetherepresentationr ofM onLn
c
has roots α , α +α , α +α , α +α , α +α +α , α +α +α , α +α +α ,
2 1 2 2 3 2 4 1 2 3 1 2 4 2 3 4
α +α +α +α ,weseethatthroughG→G→M rgivesrisetothetensorproduct
1 2 3 4
representationofG=GL (C){1,3,4},{bg ,gb,g }c7→g ⊗g ⊗g . Thatidentification
2 1 3 4 1 3 4
evenextendstoLGb anditsactionviaI⊗: indeedintherepresentationofLM onLn
we can choose bases for the root subspaces so that the action of S (hence Γ ) on
F
these vectors is given by its action on the roots via the representation of {1,3,4}.
It is then clear that via LG→LM, r does indeed give I⊗.
ASAI CUBE L-FUNCTIONS 5
2. LS method and Asai cube
Let A denote the class of triples (E/F,π,ψ) consisting of: a degree 3 ´etale
algebra E over a non-Archimedean local field F; a smooth irreducible (complex)
representationπ of GL (E); a non-trivial character ψ :F →C×.
2
The Langlands-Shahidi method produces γ-factors
γ(s,π,I⊗,ψ) for (E/F,π,ψ)∈A.
More precisely, the LS method attaches a factor γ(s,π′,r ,ψ) to a representation
A
π′ of M; which we identify as a subgroup of G
M = g ∈G|N (detg)=1 .
(cid:8) E/F (cid:9)
Thefactorγ(s,π,I⊗,ψ)isbydefinitiontheLanglands-Shahidifactorγ(s,π′,r ,ψ),
A
where π′ is the (unique) ψ-generic component of the restricition of π to M.
We also have the global class Aglob of quadruples (K/k,Π,Ψ,S) consisting of:
a separable cubic extension of global fields K/k; a (unitary) globally generic rep-
resentation Π = ⊗′Π of GL (A ); a non-trivial character Ψ : k\A → C×; and
v n K k
S a finite set of places of K containing all Archimedean v and such that Π is
v
unramified for v ∈/ S.
2.1. Characterizationofγ-factors. LetusrecallthemaintheoremoftheLanglands-
Shahidimethodforγ-factors. Namely,Theorem3.5of[26]andTheorem4.1of[23].
Thereexistsasystemofγ-factorsonA,satisfyinganduniquelydeterminedbythe
following properties:
(i) (Naturality). Let (E/F,π,ψ) ∈ A. Let η : E′/F′ → E/F be an isomor-
phism of degree 3 extensions over the non-Archimedean local fields F and
F′. Let (E′/F′,π′,ψ′)∈A be the triple obtained via η. Then
γ(s,π,⊗I,ψ)=γ(s,π′,⊗I,ψ′).
(ii) (Isomorphism). Let (E/F,π ,ψ)∈A, j =1, 2. If π ∼=π , then
j 1 2
γ(s,π ,⊗I,ψ)=γ(s,π ,⊗I,ψ).
1 2
(iii) (Compatibility atunramifiedrepresentations). Let(E/F,π,ψ)∈A besuch
that π has an Iwahori fixed vector. Let φ : W′ →LM be the Langlands
π F
parameter corresponding to π. Then
γ(s,π,⊗I,ψ)=γ(s,⊗I(φ ),ψ).
π
(iv) (Compatibility at real places). Let(E/F,π,ψ)∈A be genericwithF =R.
Letφ :W′ →LM betheLanglandsparametercorrespondingtoπ. Then
π F
γ(s,π,⊗I,ψ)=γ(s,⊗I(φ ),ψ).
π
(v) (Multiplicativity). Let (E/F,π,ψ)∈A be such that
π ֒→Ind(χ ⊗χ ),
1 2
the genericconstituentobtainedfromparabolicinductionwithχ , i=1,2,
i
charactersofE×. Identify χ with a characterofW viaclass fieldtheory.
i E
Then
γ(s,π,⊗I,ψ)=γ(s,χ1|F×,ψ)γ(s,χ2|F×,ψ)
γ(s,IE/F(χ1χ−21)⊗χ2|F×,ψ)γ(s,IE/F(χ2χ−11)⊗χ1|F×,ψ),
where I denotes induction from W to W .
E/F E F
6 G.HENNIARTANDL.LOMEL´I
(vi) (Dependence on ψ). Let(E/F,π,ψ)∈A. Givena∈F×, letψa :F →C×
be the character given by ψa(x)=ψ(ax). Then
γ(s,π,⊗I,ψa)=ω (a)2|a|4(s−21)·γ(s,π,⊗I,ψ).
π F
(vii) (Functional equation). Let (k,Π,Ψ,S)∈Aglob. Then
LS(s,Π,⊗I)= Yγ(s,Πv,⊗Iv,Ψv)LS(1−s,Π,⊗I).
e
v∈S
2.2. Additional properties of γ-factors. From the axioms,one can deduce the
following important property.
(viii) (Local functional equation). Let (E/F,π,ψ)∈A. Then
γ(s,π,⊗I,ψ)γ(1−s,π˜,⊗I,ψ)=1.
We let ν denote the character of GL (E) given by |det(·)|. We also have
2
(ix) (Twists by unramified characters). Let (E/F,π,ψ)∈A. Then
γ(s,π·νs0,⊗I,ψ)=γ(s+s ,π,⊗I,ψ).
0
3. Main equality
LetG be the classoftriples(E/F,σ,ψ) consistingof: adegree3´etalealgebraE
overanon-ArchimedeanlocalfieldF;anirreducibleddimensionalFrob-semisimple
representationσ of W′ ; a non-trivial character ψ :F →C×.
E
If σ is a representationofW′ , ofdimensiond, we mayformits tensorinduction
E
I⊗(σ), which is a representationof W′ , of dimensiond3 [5]. Let Ver:Wab →Wab
F F E
be the transfer [5]. If σ has dimension one, given by a character η of Wab, then
E
I⊗(σ) is givenby the characterη◦Ver of Wab. In a similar vein, if η is a character
F
ofWab,twistingσ byη givesI⊗(σ)twistedbyη◦Ver. Recallthat,ifη corresponds
E
via classfield theoryto a characterχ ofE×, then η◦Ver correspondsto χ|F×. We
fix d=2.
Remark 3.1. As in the Asai representation in § 1, the construction is valid over
a globalfield k and a separable cubic extension K/k. At a split place v of a global
cubic extension K/k is K ≃ k × k × k , the cubic induction process simply
v v v v
consists in taking the tensor product σ ⊗σ ⊗σ of 2 dimensionalrepresentations
1 2 3
of W′ . At a partially split place v of K/k, we have K is the separable algebra
F v
K ×k ,whereK /k isaseparablequadraticextensionofnon-Archimedeanlocal
w v w v
fields. Then cubic tensor induction gives a tensor product ⊗I(σ )⊗σ , where σ
1 2 1
andσ are2dimensionalrepresentationsofW andW ,respectively,and⊗I(σ )
2 Kw kv 1
denotes quadratic tensor induction from W′ to W′ .
Kw kv
3.1. Main equality. We consider triples (E/F,π,ψ) ∈ A and (E/F,σ,ψ) ∈ G
such that π ↔ σ correspond to each other via the local Langlands conjecture for
GL [17]. Additionally,wesaythatatriple(E/F,σ,ψ)∈G foracubicextensionof
2
non-Archimedeanfields is dihedral, tetrahedraloroctahedral,depending onσ. We
notethaticosahedralGaloisrepresentationsdonotariseassmoothrepresentations
of GL .
2
Theorem 3.2. Let (E/F,π,ψ) ∈ A and (E/F,σ,ψ) ∈ G be such that π ↔σ are
related via the local Langlands correspondence. Then
γ(s,π,I⊗,ψ)=γ(s,I⊗(σ),ψ).
ASAI CUBE L-FUNCTIONS 7
Proof. As in [12, 13, 14, 8], we use a local-global approach which goes back to the
early days of local class field theory. It is at the core of the Langlands-Shahidi
method, and also in Deligne’s construction of partial L-functions and γ-factors for
representations of W′ .
F
Let (E/F,π,ψ)∈A and (E/F,σ,ψ)∈G be as in the statemen of the theorem.
From § 2.1, Properties (v) and (ix), we can assume π is supercuspidal unitary.
Cuspidal representations of GL are generic. Suppose we are given a global field
2
K with a place v such that K ∼= F. We will concoct a globally generic cusp-
0 v0
idal automorphic representation Π = ⊗Π of GL (A ), with Π ∼= π, where we
v 2 K v0
excercise control over its ramification and its compatibility with the local Lang-
landsconjecture. IfK isanumberfield,weuseShahidi’sγ-factorsatArchimedean
places, where we the main equality is known [24]. Let Ψ=⊗Ψ :K\A →C× be
v K
a non-trivialcharacter,which we canassume satisfies Ψ =ψ, by Property(vi) of
v0
§ 2.1. We have the global functional equation
(4) LS(s,Π,⊗I)= Yγ(s,Πv,⊗Iv,Ψv)LS(s,Π,⊗I).
e
v∈S
Let us go through the argument in the case of char(F) = p. The globalization
is done via Theorem 3.3 of [13], being GL , the number of additional places where
2
the representation is not necessarily unramified is one, which we call v . We can
∞
also use the more general Theorem 1.1 of [8] for the globalization. At v , we
∞
have that Π is a tamely ramified principal series representation. We can apply
v∞
multiplicativity, Property (v), to get compatibility with the Galois side at v . At
∞
every v ∈/ {v ,v }, we have an unramified principal series representation, where
0 ∞
the main equality is known. We then proceed as in the proof of Theorem 5.1 of
[8] to obtain the main equality at the remaining place v . However,being GL , we
0 2
have Drinfeld’s proof of the global Langlands conjecture [7]. Thus, let Σ be the
absolutely irreducible 2-dmensional λ-adic representation of the Weil group W
K
corresponding to Π under Drinfeld’s correspondence. Then we have the functional
equation, proved by Deligne on the Galois side
(5) LS(s,⊗I(Σ))= Yγ(s,⊗Iv(Σv),Ψv)LS(s,⊗I(Σ˜)).
v∈S
Now, if we compare equations (4) and (5), we obtain
γ(s,Π ,⊗I ,Ψ )=γ(s,⊗I (Σ ),Ψ ).
v v0 v0 v0 v0 v0
Which completes the proof of the main equality in characteristic p.
Wenowgivemoredetailinthehardercharacteristiczerocase. AssumethusF is
anextensionofQ . IfE istheseparablealgebraF×F×F,thenπ isgivenbyπ ⊗
p 1
π ⊗π , where eachπ , j =1,...,3,is a smoothgeneric representationof GL (F).
2 3 j 2
For each j, let σ be the corresponding degree 2 Weil-Deligne representation σ
j j
under local Langlands. Then
γ(s,π,⊗I,ψ)=γ(s,π ×π ×π ,ψ)
1 2 3
=γ(s,σ ⊗σ ⊗σ ,ψ).
1 2 3
IfE istheseparablealgebraE′×F,whereE′/F isaseparablequadraticextension
ofnon-Archimedeanlocalfields. Thenπ isasmoothgenericrepresentationπ ⊗π
1 2
ofGL (E′)×GL (F). Foreachj,letσ bethecorrespondingdegree2Weil-Deligne
2 2 j
8 G.HENNIARTANDL.LOMEL´I
representation. Then
γ(s,π,⊗I,ψ)=γ(s,⊗I(σ )⊗σ ,ψ),
1 2
where ⊗I(σ ) on the right hand side is obtained via the usual (quadratic) Asai
1
representation.
We then assume E/F is a cubic extension of p-adic fields. First, consider the
case σ is dihedral (that is, the image of σ in PGL (C) is a dihedral group); note
2
that σ is always dihedral when p is odd. Then there is a quadratic extension E′ of
E and a character χ:E′ →C×, such that π =π(χ,ψ) is the Weil representation.
We lift F to anumberfieldk, whichis totallyrealandwithonlyone placelying
above 2 if p = 2; see for example [11]. We can also lift E/F to a cubic extension
K/k, which is again totally real. We further lift E′/E to a quadratic extension
K′/K. Let χ be the characterofE′ fromwhichσ is induced. Then, we canalways
extend χ to a character X of WK′ which is unramified at all finite places except
the place v corresponding to E′. We thus have the representation
0
Σ=Ind(X)
oftheWeilgroupofK. TheGaloisL-functionL(s,⊗I(Σ))hasafunctionalequation
due to Artin, andthe decompositionofthe globalconstants into a product oflocal
constants was shown by Deligne.
LetΠ=Π(Σ)be the correspondingWeilrepresentation,sothatΠ corresponds
v
toΣ atallplaces. ThenΠsatisfiesthefunctionalequation,Property(vii)of§2.1.
v
We know the equality at all unramified places, in addition to real places, since we
areinthetotallyrealsetting. Bycomparingbothfunctionalequations,weconclude
the main equality at the remaining place, i.e.,
γ(s,π,I⊗,ψ)=γ(s,⊗I(σ),ψ).
We now assume σ is tetrahedral or octahedral, so that p = 2. Twisting by an
unramified character, if necessary, we may assume that σ is a Galois local repre-
sentation. We can find an extension of global fields K/k (with only one place v
0
over p) and a Galois representation Σ which gives σ at v . And, indeed has the
0
sameimageasσ. Moreprecisely,σ factorizesthroughaGaloisgroupE′/E,andwe
find a global K′/K of the same degree, yielding E′/E at v . Hence, seeing σ as a
0
representation of Gal(K′/K) we get Σ. Now the strong Artin conjecture is known
for Σ, so let Π be the corresponding cuspidal representation, for which Π is π.
v0
Wehaveagainfunctionalequationsonbothsides,withthe realplacesgivingtriple
products. At unramified places, there being no problem. And, notice that at finite
placesotherthanv ,haveoddresiduecharacteristis,henceγ-factorsagreeatthese
0
places by the dihedral case discussed above. And, we only have the remaining one
place v above 2, where the representation can be tetrahedral or octahedral. By
0
comparing both functional equations, we conclude the main equality at v .
0
(cid:3)
4. Consequences
An important consequence of the main equality is that local factors become
stable under highly ramified twists. Indeed, this follows by using Theorem 3.2 and
applying the results of [6] to the Galois side of the equality.
ASAI CUBE L-FUNCTIONS 9
Corollary 4.1 (Stability). Let (E/F,π ,ψ)∈A, for i=1 or 2, be such that their
i
central characters satisfy ω = ω . Then, for a sufficiently ramified character η
π1 π2
of F×:
γ(s,π ⊗η,r,ψ)=γ(s,π ⊗η,r,ψ).
1 2
4.1. Equality for L-functions and ε-factors. Inordertoobtaintheequalityfor
the remaining local factors, we recall how to obtain L-funtions and ε-factors from
γ-factors. We begin with the tempered L-function conjecture, now proved for all
Langlands-Shahidi L-functions in [10].
(x) (Tempered L-functions). For (E/F,π,ψ) ∈ A tempered, let P (t) be the
π
unique polynomialwith P (0)=1 andsuchthat P (q−s) is the numerator
π π F
of γ(s,π,⊗I,ψ). Then
1
L(s,π,⊗I)= .
P (q−s)
π F
is holomorphic and non-zero for ℜ(s)>0.
Property (x) when combined with the Property (viii) of γ-factors, gives us the
following well defined notion of ε-factors.
(xi) (Tempered ε-factors). Let (E/F,π,ψ)∈A be tempered, then
L(s,π,⊗I)
ε(s,π,⊗I,ψ)=γ(s,π,⊗I,ψ)
L(1−s,π˜,⊗I)
is a monomial in q−s.
F
We also have
(xii) (Twists by unramified characters). Let (E/F,π,ψ)∈A, then
L(s+s ,π,⊗I)=L(s,π·νs0,⊗I),
0
ε(s+s ,π,⊗I,ψ)=ε(s,π·νs0,⊗I,ψ).
0
Finally, we have multiplicativity of local factors
(xiii) (Multiplicativity). Let (E/F,π,ψ)∈A be such that
π ֒→Ind(χ ⊗χ ),
1 2
the genericconstituentobtainedfromparabolicinductionwithχ , i=1,2,
i
charactersofE×. Identify χ with a characterofW viaclass fieldtheory.
i E
Then
L(s,π,⊗I)=L(s,χ1|F×)L(s,χ2|F×)
L(s,IE/F(χ1χ−21)⊗χ2|F×)L(s,IE/F(χ2χ−11)⊗χ1|F×),
ε(s,π,⊗I,ψ)=ε(s,χ1|F×,ψ)γ(s,χ2|F×,ψ)
ε(s,IE/F(χ1χ−21)⊗χ2|F×,ψ)ε(s,IE/F(χ2χ−11)⊗χ1|F×,ψ),
where I denotes induction from W to W .
E/F E F
We deduce the main equality for L-functions and rood numbers from Theo-
rem3.2, with the aid of Langlands classificationfor GL(2) [15]. First for tempered
representations,byProperties(x)and(xi),whereweobservethatallsupercuspidal
representations π of GL (E) are generic. Then in general, relying on Properties
2
(xii) and (xiii).
10 G.HENNIARTANDL.LOMEL´I
Corollary 4.2. Let (E/F,π,ψ)∈A and (E/F,σ,ψ) ∈G be such that π ↔σ are
related via the local Langlands correspondence. Then
L(s,π,⊗I)=L(s,⊗I(σ)),
ε(s,π,⊗I,ψ)=ε(s,⊗I(σ),ψ).
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