On Artin’s L-Functions*

by

R.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 of the zeta function but little is known about their poles. It seems to be expected that the L-functions associated to irreducible representations of the Galois group or another closely related group, the Weil group, will be entire. One way that has been suggested to show this is to show that the Artin L-functions are equal to L-functions associated to automorphic forms. This is a more difficult problem than we can consider at present.

In certain cases the converse question is much easier. It is possible to show that if the Lfunctions associated to irreducible two-dimensional representations of the Weil group are all entire then 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.

A global field will be just an algebraic number field of finite degree over the rationals or a function field in one variable over a finite field. A local field is the completion of a global field at some place. If F is a local field let C_{F} be the multiplicative group of F and if F is a global field let C_{F} be the idèle class group of F. If K is a finite Galois extension of F the Weil group W_{K/F} is an extension of 𝔊(K/F), the Galois group of K/F, by C_{K}. Thus there is an exact sequence

1 → C_{K} → W_{K/F} → 𝔊(K/F) → 1.

If L/F is also Galois and L contains K there is a continuous homomorphism τ_{L/F,K/F} of W_{L/F} onto W_{K/F}. It is determined up to an inner automorphism of W_{K/F} by an element of C_{K}. In particular W_{F/F} = C_{F} and the kernel of τ_{K/F,F/F} is the commutator subgroup of W_{K/F}. Also if F ⊆ E ⊆ K we may regard W_{K/E} as a subgroup of W_{K/F}. If F is global and v is a place of F we also denote by v any extension of v to K. There is a homomorphism α_{v} of W_{K_{v}/F_{v}} into W_{K/F} which is determined up to an inner automorphism by an element of C_{K}.

A representation σ of W_{K/F} is a continuous homomorphism of W_{K/F} into the group of invertible linear transformations of a finite-dimensional complex vector space such that σ(w) is

On Artin’s L-Functions 2

* First appeared in Rice University Studies, vol. 56 (1970) diagonalizable for all w in W_{K/F}. If F ⊆ K ⊆ L then σ ∘ τ_{L/F,K/F} is a representation of W_{L/F} which we need not distinguish from σ. In particular, one-dimensional representations of W_{K/F} may be identified with quasi-characters of C_{F}, that is, continuous homomorphisms of C_{F} into the multiplicative group of complex numbers. If ω is such a quasi-character and σ a representation of W_{K/F} then ω ⊗ σ has the same dimension as σ. If F ⊆ E ⊆ K and ρ is a representation of W_{K/E} on X let Y be the space of functions ϕ on W_{K/F} with values in X such that

ϕ(uw) = ρ(u)ϕ(w)

for all u in W_{K/E}. If ν is in W_{K/F} we set

σ(v)ϕ(w) = ϕ(wv);

σ is a representation of W_{K/F} on Y and we write

σ = Ind(W_{K/F} , W_{K/E}, ρ).

If F is a global field and σ is a representation of W_{K/F} then, for all places v, σ_{v} = σ ∘ α_{v} is a representation of W_{K_{v}/F_{v}} whose class depends only on that of σ.

Now take F to be a local field. Suppose ω is a quasi-character of C_{F}. If F is nonarchimedean and ω is ramified, we set L(s, ω) = 1 but if ω is unramified we set

L(s, ω) = \frac{1}{1 − ω(ϖ)|ϖ|^{s}};

ϖ is a uniformizing parameter for F. If F is real and ω(x) = (sgn x)^{m}|x|^{r} with m equal to 0 or 1 we set

L(s, ω) = π^{− \frac{1}{2} (s+r+m)}Γ (\frac{s + r + m}{2});

if F is complex and

ω(z) = (zz̄)^{r−(m+n)/2}z^{m}z̄^{n}

with m + n ≥ 0, mn = 0 then

L(s, ω) = 2(2π)^{−(s+r+(m+n)/2)}Γ (s + r + \frac{m + n}{2}) .

The local L-functions L(s, σ) associated to representations of the Weil groups W_{K/F} are characterized by the following four properties.

(i) L(s, σ) depends only on the class of σ.

(ii) If σ is of dimension 1 and thus associated to a quasi-character ω of C_{F} then

L(s, σ) = L(s, ω).

(iii)

L(s, σ_{1} ⊕ σ_{2}) = L(s, σ_{1})L(s, σ_{2}).

(iv) If F ⊆ E ⊆ K, ρ is a representation of W_{K/E}, and

σ = Ind(W_{K/F} , W_{K/E}, ρ),

then

L(s, σ) = L(s, ρ).

Now suppose ψ = ψ_{F} is a nontrivial additive character of F. If E is a finite separable extension of F let

ψ_{E/F} (x) = ψ_{F} (T r_{E/F} x).

Suppose E is any local field and ψ_{E} is a nontrivial additive character of E. If E is nonarchimedean and ω_{E} is a quasi-character of C_{E} let n = n(ψ_{E}) be the largest integer such that ψ_{E} is trivial on 𝔓_{E}^{−n} and let m = m(ω_{E}) be the order of the conductor of ω_{E}. If U_{E} is the group of units in the ring of integers of E and γ is any element of F with order m + n let

Δ(ω_{E}, ψ_{E}) = ω_{E}(γ)\frac{∫_{U_{E}} ψ_{E} (\frac{α}{γ}) ω_{E}^{−1}(α)dα}{|∫_{U_{E}} ψ_{E} (\frac{α}{γ}) ω_{E}^{−1}(α)dα|}.

If E is real and ψ_{E}(z) = e^{2πiux} while ω_{E}(x) = (sgn x)^{m}|x|^{r} then

Δ(ω_{E}, ψ_{E}) = (i sgn u)^{m}|u|^{r}.

If E is complex and ψ_{E}(z) = e^{4πiRe(wz)} while

ω_{E}(z) = (zz̄)^{r−(m+n)/2}z^{m}z̄^{n}

then

Δ(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 ϵ(σ, ψ_{E/F}) so that the following conditions are satisfied.

(i) ϵ(σ, ψ_{E/F}) depends only on the class of σ. (ii) If σ is one-dimensional and corresponds to the quasi-character ω_{E} then

ϵ(σ, ψ_{E/F}) = Δ(ω_{E}, ψ_{E/F}).

(iii)

ϵ(σ_{1} ⊕ σ_{2}, ψ_{E/F}) = ϵ(σ_{1}, ψ_{E/F})ϵ(σ_{2}, ψ_{E/F}).

(iv) If

σ = Ind(W_{K/F} , W_{K/E}, ρ),

then

ϵ(σ, ψ_{F}) = ϵ(ρ, ψ_{E/F})λ(E/F, ψ_{E})^{dim ρ}.

If s is a complex number and |a|_{F} is the normalized absolute value on F set α_{F}^{s} (a) = |a|_{F}^{s} and set

ϵ(s, σ, ψ_{F}) = ϵ(α^{s− \frac{1}{2}} ⊗ σ, ψ_{F}).

If F is a global field and σ is a representation of W_{K/F} the global L-function L(s, σ) is defined by

L(s, σ) = ∏_{v} L(s, σ_{v}).

The product is taken over all places including the archimedean ones. It converges in a right halfplane. L(s, σ) can be analytically continued to the whole plane as a meromorphic function. If ψ is a nontrivial character of F \A, the quotient of the ring of adèles by F, then, for each place v, the restriction ψ_{v} of ψ to F_{v} is nontrivial. For almost all v the factor ϵ(s, σ_{v}, ψ_{v}) is 1 and

ϵ(s, σ) = ∏_{v} ϵ(s, σ_{v}, ψ_{v})

does not depend on the choice of ψ.

Theorem 2. If σ̃ is the contragradient of σ the functional equation

L(s, σ) = ϵ(s, σ)L(1 − s, σ̃)

is satisfied.

The proofs of these two theorems may be found in [2]. Now we have to say something about the L-functions associated to automorphic forms on GL(2).

If F is a local field and π is an irreducible representation of the locally compact group GL(2, F) we can introduce a local L-function L(s, π). Given a nontrivial additive character ψ of F we can also introduce a factor ϵ(s, π, ψ). If ω is a quasi-character of C_{F} then ω⊗π will be the representation g → ω(detg)π(g).

If F is a global field and η is a quasi-character of C_{F} then A_{0}(η) will be the space of all measurable functions ϕ on GL(2, F)\GL(2, A) satisfying:

(i) For every idèle a

ϕ (( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) g) = η(a)ϕ(g);

(ii) If Z_{A} is the group of scalar matrices in GL(2, A)

∫_{ZAGL(2,F)\GL(2,A)} |ϕ(g)|^{2}|η(detg)|^{−1}dg

is finite; (iii) For almost all g in GL(2, A)

∫_{F \A} ϕ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} g) g) dx = 0.

GL(2, A) acts on A_{0}(η) by right translations. Under this action A_{0}(η) breaks up into the direct sum of mutually orthogonal, invariant, irreducible subspaces. Let π be the representation of GL(2, A) on one of these subspaces. One can decompose π into a tensor product ⊗_{v}π_{v} where π_{v} is an irreducible representation of GL(2, F_{v}). The product

L(s, π) = ∏_{v} L(s, π_{v})

converges in a half plane and L(s, π) can be analytically continued to an entire function. If ψ is a nontrivial character of F \A and

ϵ(s, π) = ∏_{v} ϵ(s, π_{v}, ψ_{v})

then

L(s, π) = ϵ(s, π)L(1 − s, π̃).

Now suppose F is a local field and σ is a two-dimensional representation of W_{K/F}. Then detσ is a one-dimensional representation of W_{K/F} and may therefore be regarded as a quasi-character of C_{F}. There is at most one irreducible representation π = π(σ) of GL(2, F), satisfying the following conditions.

(i) For all a in C_{F}

π (( \begin{matrix}a & 0 \\ 0 & a \end{matrix} )) = detσ(a)I

if I is the identity. (ii) For every quasi-character ω of C_{F}

L(s, ω ⊗ π) = L(s, ω ⊗ σ),

and

L(s, ω ⊗ π̃) = L(s, ω ⊗ σ̃).

(iii) For one, and hence for every, nontrivial character ψ of F

ϵ(s, ω ⊗ π, ψ) = ϵ(s, ω ⊗ σ, ψ)

for all ω.

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} ⊗ σ̃) 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