pacific journal of mathematics Vol. 181, No. 3, 1997

REPRESENTATIONS OF ABELIAN ALGEBRAIC GROUPS

R.P. Langlands

There^{1} is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adèle group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group.

The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle.

It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which 𝔊(K/F) acts. If T corresponds to L then T_{K}, the group of K-rational points on T, may, and shall, be identified as a 𝔊(K/F)-module with Hom(L, K^{∗}). If K is a global field and 𝔸(K) is the adèle ring of K the group T_{𝔸(K)}/T_{K} may be identified with Hom(L, C_{K}) if C_{K} is the idèle class group of K. If K is a local field C_{K} will be the multiplicative group of K.

Suppose L̂ is the lattice Hom(L, ℤ). If ℂ^{∗} is the multiplicative group of nonzero complex numbers and ℂ_{u}^{∗} the group of complex numbers of absolute value 1 we set T̂ = Hom(L̂, ℂ^{∗}) and T̂_{u} = Hom(L̂, ℂ_{u}^{∗}). There are natural actions of 𝔊(K/F) on L̂, T̂, and T̂_{u}. The semi-direct product T̂ ⋊ 𝔊(K/F) is a complex Lie group with T̂_{u} ⋊ 𝔊(K/F) as a real subgroup. If F is a local or global field the Weil group W_{K/F} is an extension

1The present paper is a reproduction, with only trivial stylistic changes, of a preprint now in circulation for 29 years. The material has, in the meantime, been treated in print by several other mathematicians. So the author is flattered that the editors of this collection still think it of some interest and is of course very pleased to be able to dedicate it to the memory of Olga Taussky-Todd.

0 → C_{K} → W_{K/F} →^{σ} 𝔊(K/F) → 0

of 𝔊(K/F) by C_{K}. We want to consider continuous homorphisms φ of W_{K/F} into T̂ ⋊ 𝔊(K/F) that make

W_{K/F}  ⟶^{σ} 𝔊(K/F)
↓^{α}  ∥
T̂ ⋊ 𝔊(K/F) ⟶ 𝔊(K/F)

commutative. Two such homomorphisms φ and φ′ are said to be equivalent if there is a t in T̂ such that φ′(w) ≡ tφ(w)t^{−1}. We write φ(w) as f(x) × σ(w). Then f is a continuous 1-cocycle on W_{K/F} with values in T̂. If φ′(w) = f′(w) × σ(w), then φ and φ′ are equivalent if and only if f and f′ are cohomologous. Thus the collection of equivalence classes may be identified with the collection H_{c}^{1}(W_{K/F} , T̂) of continuous cohomology classes.

Theorem 1. If K is a global or a local field there is a canonical isomorphism of H_{c}^{1}(W_{K/F} , T̂) with the group of generalized characters of Hom_{𝔊(K/F)}(L, C_{K}). Under this isomorphism H_{c}^{1}(W_{K/F} , T̂_{u}) will correspond to the group of ordinary characters.

Suppose F and F′ are global or local fields, K is a Galois extension of F, K′ a Galois extension of F′, and φ an isomorphism of K into K′ that takes F into F′. If F and F′ are both global or both local we want F′ to be separable over the image of F; if F is local and F′ is global we want F′ to be separable over the closure of the image of F. Under these circumstances there is associated to φ a homomorphism

φ_{w} : W_{K′/F′} → W_{K/F}

and thus a map

φ_{w}^{∗} : H_{c}^{1}(W_{K/F} , T̂) → H_{c}^{1}(W_{K′/F′}, T̂).

If K is a local field Hom_{𝔊(K/F)}(L, C_{K}) is just T_{f}, the group of F rational points on T. However, if K is a global field and I_{K} is the group of idèles of K the exact sequence

1 → K^{∗} → I_{K} → C_{K} → 0

leads to the exact sequence

1 → T_{F} → T_{𝔸(F)} → Hom_{𝔊(K/F)}(L, C_{K}) → H^{1}(𝔊(K/F), T_{K}).

T_{𝔸(F)}/T_{F} may be regarded as a subgroup of Hom_{𝔊(K/F)}(L, C_{K}). Thus we have a canonical homomorphism of H_{c}^{1}(W_{K/F} , T̂) into the group of generalized characters of T_{𝔸(F)}/T_{F}. The following theorem is a slight improvement of these immediate consequences of the first theorem.

Theorem 2. (a) If F is a local field H_{c}^{1}(W_{K/F} , T̂) is canonically isomorphic to the group of generalized characters of T_{F},

(b) If F is a global field there is a canonical homomorphism of H_{c}^{1}(W_{K/F} , T̂) onto the group of generalized characters of T_{𝔸(F)}/T_{F}. The kernel is finite and consists of those classes α such that φ_{w}^{∗}(α) = 0 whenever K′ is the completion of K with respect to some valuation, F′ is the closure of F in K′, and φ : K/F → K′/F′ is the imbedding.

Of course only the second part of this theorem will need to be proved. First we prove Theorem 1.

All 𝔊(K/F)-modules may be viewed as W_{K/F} -modules and hence as C_{K}- modules. Of course C_{K} will act trivially. The cup product gives a bilinear mapping

H_{1}(C_{K}, L) × H^{0}(C_{K}, L) → H_{1}(C_{K}, ℤ)

that commutes with the action of 𝔊(K/F) on these three groups. Since H^{0}(C_{K}, L) and H_{1}(C_{K}, ℤ) are isomorphic as 𝔊(K/F)-modules to L and C_{K} respectively we have a 𝔊(K/F) invariant homomorphism

H_{1}(C_{K}, L̂) → Hom(L, C_{K}).

Since L is the direct sum of a certain number of copies of ℤ this map is an isomorphism. The restriction map

Res : H_{1}(W_{K/F} , L̂) → H_{1}(C_{K}, L̂)

has an image contained in the set of 𝔊(K/F)-invariant elements of H_{1}(C_{K}, L̂). Using standard techniques [1], we shall show that it determines an isomorphism of H_{1}(W_{K/F} , L̂) and H_{1}(C_{K}, L̂)^{𝔊(K/F)} and thus an isomorphism of H_{1}(W_{K/F} , L̂) and Hom_{𝔊(K/F)}(L, C_{K}). This isomorphism can be used to turn H_{1}(W_{K/F} , L̂) into a topological group. To complete the proof of the first theorem we show that the pairings

H_{c}^{1}(W_{K/F} , T̂) × H_{1}(W_{K/F} , L̂) → H_{0}(W_{K/F} , ℂ^{∗}) = ℂ^{∗}
H_{c}^{1}(W_{K/F} , T̂_{u}) × H_{1}(W_{K/F} , L̂) → H_{0}(W_{K/F} , ℂ_{u}^{∗}) = ℂ_{u}^{∗}

associated to the natural maps T̂ × L̂ → ℂ^{∗} and T̂_{u} × L̂ → ℂ_{u}^{∗} determine isomorphisms of H_{c}^{1}(W_{K/F} , T̂) and H_{c}^{1}(W_{K/F} , T̂_{u}) with the groups of generalized and ordinary characters of H_{1}(W_{K/F} , L̂).

If x : a → x(a) is a 1-cycle on C_{K} with values in L̂ we associate to it the 1cycle on W_{K/F} that is 0 outside of C_{K} and equal to x on C_{K}. If x : w → x(w) is a 1-cycle on W_{K/F} we associate to it the 1-cycle σ → ∑_{σ=σ(w)} x(w) on 𝔊(K/F). Passing to homology classes we obtain a sequence

H_{1}(C_{K}, L̂) → H_{1}(W_{K/F} , L̂) → H_{1}(𝔊(K/F), L̂) → 0

that we shall verify is exact. Let {w_{σ} | σ ∈ 𝔊(K/F)} be representatives of the cosets of C_{K} in W_{K/F}. If x : σ → x(σ) is a 1-cycle on 𝔊(K/F) and y is the 1-cycle on W_{K/F} that is equal to x(σ) on w_{σ} but is zero otherwise, then x is the image of y. Thus the sequence is exact at H_{1}(𝔊(K/F), L̂). To show it is exact at H_{1}(W_{K/F} , L̂) we observe first of all that if G is any group and A is any G-module then a 1-cycle that vanishes off the identity is a boundary. Indeed if x(1) = x while x(g) = 0 if g ≠ 1, let y(h, g) = 0 unless h = g = 1 when y(h, g) = x. Then

dy(g) = ∑ h^{−1}(h, g) − ∑ y(gh^{−1}, h) + ∑ y(g, h) = y(1, g) = x(g).

Consequently the composition of the maps in and out of H_{1}(W_{K/F} , L̂) is 0. Suppose σ → ∑_{σ(w)=σ} x(w) is the boundary of Y (σ, τ). Let z(u, v) be the 2-chain on W_{K/F} that is 0 unless u = w_{σ} for some σ and v = w_{τ} for some τ when z(u, v) = y(σ, τ). Then

dz(u) = ∑_{σ} w_{σ}^{−1}z(w_{σ}, u) − ∑_{σ} z(uw_{σ}^{−1}, w_{σ}) + ∑_{σ} z(u, w_{σ}),

and if x′ = x − dz,

∑_{σ(w)=τ}x′(w) = ∑_{σ(w)=τ} x(w) − {∑_{σ} σ^{−1}y(σ, τ) − ∑_{σ} y(τσ^{−1}, σ) + ∑_{σ} y(τ, σ)}
= 0.

Let z′(u, v) = 0 unless (u, v) = (w_{σ}, a) for some σ and some a in C_{K} when z′(u, v) = x′(w_{σ}a). Then

dz′(u) = ∑_{σ} w_{σ}^{−1}z′(w_{σ}, u) − ∑_{σ} z′(w_{σ}, w_{σ}^{−1}u) + ∑_{a} z′(u, a)

= ∑_{σ} w_{σ}^{−1}z′(w_{σ}, u) − x′(u).

Thus x′ + dz′ has support in C_{K} and the class of x is in the image of H_{1}(C_{K}, L̂).

Let N be the map

∑_{σ∈𝔊(K/F)} σ.

It takes H_{1}(C_{K}, L̂) into H_{1}(C_{K}, L̂)^{𝔊(K/F)}, the group of 𝔊(K/F)-invariant elements. Denote its image by N(H_{1}(C_{K}, L)). By definition

H_{1}(C_{K}, L̂)^{𝔊(K/F)}/N(H_{1}(C_{K}, L̂)) = Ĥ^{0}(𝔊(K/F), H_{1}(C_{K}, L̂)).

The group on the right is isomorphic to

Ĥ^{0}(𝔊(K/F), Hom(L, C_{K}))

and the sequence

0 → N(H_{1}(C_{K}, L̂)) → H_{1}(C_{k}, L̂)^{𝔊(K/F)} → Ĥ^{0}(𝔊(K/F), Hom(L, C_{K})) → 0

is exact.

There is an obvious isomorphism of L̂ ⊗ C_{K} with Hom(L, C_{K}). It sends λ̂ ⊗ a to the homomorphism λ → a^{⟨λ,_{λ}⟩}. Since

H_{1}(𝔊(K/F), L̂) = Ĥ^{−2}(𝔊(K/F), L̂),

the cup product gives a map

H_{1}(𝔊(K/F), L̂) × Ĥ^{2}(𝔊(K/F), C_{K}) → Ĥ^{0}(𝔊(K/F), Hom(L, C_{K}))

and the cup product with the fundamental class gives a homomorphism

H_{1}(𝔊(K/F), L̂) → Ĥ^{0}(𝔊(K/F), Hom(L, C_{K})),

that is well known [2] to be an isomorphism. We have to show that the following diagram (I)

H_{1}(C_{K}, L̂) ⟶ H_{1}(W_{K/F} , L̂) ⟶  H_{1}(𝔊(K/F), L̂)  ⟶ 0
↓^{N}  ↓^{Res}  ↓
0 ⟶ N(H_{1}(C_{K}, L̂)) ⟶ H_{1}(C_{K}, L̂)^{𝔊(K/F)} ⟶ Ĥ^{0}(𝔊(K/F), Hom(L, C_{K})) ⟶ 0

is commutative. Let Λ be the group ring of W_{K/F} over ℤ and let I be the kernel of the map Λ → ℤ that sends ∑ m(g)g to ∑ m(g). The exact sequence

0 → I ⊗ L̂ → Λ ⊗ L̂ → L̂ = ℤ ⊗ L̂ → 0

gives rise to a commutative diagram

H_{1}(W_{K/F} , L̂) ⟶ H_{0}(W_{K/F} , I ⊗ L̂)
Res↓  ^{Res}↓
H_{1}(C_{K}, L̂) ⟶ H_{0}(C_{K}, I ⊗ L̂)

in which the horizontal arrows are isomorphisms. If x : w → x(w) is a 1-cycle on W_{K/F} with values in L̂, the image of its homology class in H_{0}(W_{K/F} , I⊗L̂) is the class of

∑_{w}(w^{−1} − 1)(1 ⊗ x(w)).

Restricting to C_{K} we obtain the class of

∑_{σ} ∑_{w}{w_{σ}w^{−1}(1 ⊗ x(w)) − w_{σ}(1 ⊗ x(w))}.

If τ belongs to 𝔊(K/F) and w belongs to W_{K/F} there is a unique element δ(w_{τ}, w) of C_{K} and a unique σ in 𝔊(K/F) such that w_{τ}w = δ(w_{τ}, w)w_{σ}. The sum is equal to

∑_{τ} ∑_{w}(δ^{−1}(w_{τ}, w) − 1)w_{τ}(1 ⊗ x(w)),

which equals

∑_{a∈C_{K}} {(a^{−1} − 1) ∑_{δ(w_{τ} ,w)=a} w_{τ}(1 ⊗ x(w))} .

Since (w_{τ} − 1) ⊗ w_{τ}x(w) is in I ⊗ L̂ this is homologous to

∑_{a∈C_{K}} {(a^{−1} − 1) ∑_{δ(w_{τ} ,w)=a} 1 ⊗ w_{τ}x(w)} .

This is the image of the class in H_{1}(C_{K}, L̂) containing the 1-cycle

y : a → ∑_{δ(w_{τ} ,w)=a} w_{τ}x(w).

If x has support in C_{K},

y(a) = ∑_{w_{τ} bw_{τ}^{−1}=a} w_{τ}x(b) = ∑_{τ} τx(τ^{−1}(a))

and the first square at least of the diagram (I) is commutative.

In general the image of the class of y in Ĥ^{0}(𝔊(K/F), Hom(L, C_{K})) is the class of the homomorphism

λ → ∏_{a} a^{⟨λ,y(a)⟩} = ∏_{τ,w} δ(w_{τ}, w)^{⟨λ,w_{τ} x(w)⟩}.

If w = aw_{σ} with a in C_{K} then δ(w_{τ}, w) = w_{τ}aw_{τ}^{−1}δ(w_{τ}, w_{σ}). Thus this product equals

{ ∏_{σ,τ,a} w_{τ}aw_{τ}^{−1^{⟨λ,wτ x(awσ)⟩}}} { ∏_{σ,τ,a} δ(w_{τ}, w_{σ})^{⟨λ,w_{τ} x(aw_{σ})⟩}} .

The first product is a norm. Thus the homomorphism is cohomologous to

λ → ∏_{σ,τ} δ(w_{τ}, w_{σ})^{⟨λ,τz(σ)⟩}

if

z(σ) = ∑_{a} x(aw_{σ}).

The class of z in H_{1}(𝔊(K/F), L̂) is the image of the class of x.

To complete the verification that the diagram is commutative we have to examine the effect of taking the cup product with the fundamental class on the elements of Ĥ^{−2}(𝔊(K/F), L̂). We proceed as at the end of Chapter XI of Serre’s book [3]. We take the cup product as defined by axioms, for example, those of Serre, and by “décalage” find how it is to be expressed in terms of cycles and cocyles. Let Λ be the group algebra of 𝔊(K/F) over ℤ and let I be the kernel of the map Λ → ℤ that sends ∑ m(σ)σ to ∑ m(σ). If B is a 𝔊(K/F)-module δ will be the isomorphism

Ĥ^{p}(𝔊(K/F), B) → Ĥ^{p+1}(𝔊(K/F), I ⊗ B)

associated to the exact sequence

0 → I ⊗ B → Λ ⊗ B → B → 0.

If A is another 𝔊(K/F)-module the following diagram (II)

Ĥ^{p}(𝔊(K/F), B) ⊗ H^{q}(𝔊(K/F), A) ⟶^{δ⊗id} Ĥ^{p+1}(𝔊(K/F), I ⊗ B) ⊗ Ĥ^{q}(𝔊(K/F), A)
↓  ↓
Ĥ^{p+q}(𝔊(K/F), B ⊗ A)  ⟶^{δ}  Ĥ^{p+q+1}(𝔊(K/F), I ⊗ B ⊗ A)

in which the vertical arrows are given by the cup product is commutative. There is also a 𝔊(K/F)-invariant map from ℤ to Λ. It sends m to ∑ mg. Let J be the cokernel of this map. If B is any 𝔊(K/F)-module the sequence

0 → B → Λ ⊗ B → J ⊗ B → 0

is exact. The associated isomorphism

Ĥ^{p}(𝔊(K/F), J ⊗ B) → Ĥ^{p+1}(𝔊(K/F), B)

will be called d. The following diagram (III)

Ĥ^{p}(𝔊(K/F), J ⊗ B) ⊗ Ĥ^{q}(𝔊(K/F), A) ⟶^{d⊗id} Ĥ^{p+1}(𝔊(K/F), B) ⊗ Ĥ^{q}(𝔊(K/F), A)
↓  ↓
Ĥ^{p+q}(𝔊(K/F), J ⊗ B ⊗ A)  ⟶^{d}  Ĥ^{p+q+1}(𝔊(K/F), B ⊗ A)

is also commutative.

If a is an invariant element of A the map b → b ⊗ a is a 𝔊(K/F)- homomorphism f_{a} of B into B ⊗ A. If \overline{a} is the class of a in Ĥ^{0}(𝔊(K/F), A) and \overline{x} belongs to Ĥ^{p}(𝔊(K/F), B) the equality

\overline{x} · \overline{a} = f_{a}^{∗}(\overline{x})

is true by definition for all B when p = 0. The diagrams (I) and (II) together with the commutativity of

Ĥ^{p}(𝔊(K/F), B) ⟶^{δ}  Ĥ^{p+1}(𝔊(K/F), I ⊗ B)
f_{a}^{∗}↓  ^{f_{a}^{∗}}↓
Ĥ^{p}(𝔊(K/F), B ⊗ A) ⟶^{δ} Ĥ^{p+1}(𝔊(K/F), I ⊗ B ⊗ A)

and

Ĥ^{p}(𝔊(K/F), J ⊗ B) ⟶^{d}  Ĥ^{p+1}(𝔊(K/F), B)
f_{a}^{∗}↓  ↓
Ĥ^{p}(𝔊(K/F), J ⊗ B ⊗ A) ⟶^{d} Ĥ^{p+1}(𝔊(K/F), B ⊗ A)

allow it to be verified, by induction up and down from 0, for all p. We are going to apply the equality with the roles of A and B interchanged.

Suppose a belongs to A and Na = 0 and f is a 1-cocycle with values in B. We now verify that \overline{f} · \overline{a} is the class of ∑_{σ} f(σ) ⊗ σa in Ĥ^{0}(𝔊(K/F), B ⊗ A). The image of f(σ) in Λ ⊗ B is

∑_{τ} τ ⊗ f(σ) = ∑_{τ} στ ⊗ f(σ) = ∑ στ ⊗ f(στ) − ∑ στ ⊗ σf(τ)

which, in turn, equals

∑_{τ} τ ⊗ f(τ) − σ (∑_{τ} τ ⊗ f(τ)) .

Thus \overline{f} is the image of the class in Ĥ^{0}(𝔊(K/F), J ⊗ B) of the image b in J ⊗ B of − ∑_{τ} τ ⊗ f(τ). The class of b ⊗ a is \overline{b} ∘ \overline{a}. Since diagram (III) is commutative \overline{f}·\overline{a} = d(\overline{b}·\overline{a}). By the definition of d on Ĥ^{−1}(𝔊(K/F), J⊗B⊗A) this is the class of the element of B ⊗ A corresponding to

− ∑_{σ,τ} στ ⊗ σf(τ) ⊗ σa.

This expression is equal to

− ∑_{σ,τ} στ ⊗ f(στ) ⊗ σa + ∑_{σ,τ} στ ⊗ f(σ) ⊗ σa.

The first sum is

(∑_{τ} τ ⊗ f(τ)) ⊗ (∑_{σ} σa) = 0.

The second is

∑_{τ} τ ⊗ (∑_{σ} f(σ) ⊗ σa)

and corresponds to ∑_{σ} f(σ) ⊗ σa as required.

Next we verify that, if f is a 1-cocycle with values in A and x is a 1-cycle with values in B, \overline{x}·\overline{f} is the class in Ĥ^{−1}(𝔊(K/F), B⊗A) of − ∑ x(σ)⊗f(σ). The image of this class in H̃^{0}(𝔊(K/F), I ⊗ B ⊗ A) is the class of

− ∑_{τ,σ} τ ⊗ τx(σ) ⊗ τf(σ).

The image of \overline{x} in Ĥ^{−1}(𝔊(K/F), I ⊗ B) is the class of

b = ∑_{σ}(σ^{−1} − 1)(1 ⊗ x(σ)).

Now apply the previous remark with A and B replaced respectively by I ⊗B and A. Then \overline{b} ∘ \overline{f} is the class of

− ∑_{τ,σ} τσ^{−1} ⊗ τσ^{−1}x(σ) ⊗ f(τ) + ∑_{τ,σ} τ ⊗ τx(σ) ⊗ f(τ)

which equals

− ∑_{τ,σ} τ ⊗ τx(σ) ⊗ f(τσ) + ∑_{τ,σ} τ ⊗ τx(σ) ⊗ f(τ)

which in turn equals

− ∑_{τ,σ} τ ⊗ τx(σ) ⊗ τf(σ).

Finally we have to show that if f is a 2-cocycle with values in B and x is 1-cycle with values in A then \overline{f} · \overline{x} is the class of

∑_{σ,τ} f(τ, σ) ⊗ τx(σ).

The image of f(τ, σ) in Λ ⊗ B is

∑_{ρ} ρ ⊗ f(τ, σ) = ∑_{ρ} τσρ ⊗ f(τ, σ)
= ∑ τσρ ⊗ {τf(σ, ρ) − f(τσ, ρ) + f(τ, σρ)}

= τ (∑ σρ ⊗ f(σ, ρ))−∑ τσρ ⊗ f(τσ, ρ)+∑ τρ ⊗ f(τ, ρ).

If g : σ → ∑_{ρ} σρ ⊗ f(σ, ρ) this is just dg(τ, σ). Thus \overline{f} is the image of the class of the 1-cocycle h with values in J ⊗ B that takes σ to the image in J ⊗ B of g(σ). The class of − ∑ h(σ) ⊗ x(σ) in Ĥ^{−1}(𝔊(K/F), J ⊗ B ⊗ A) is \overline{h} · \overline{x}. Applying d we obtain the class of the element in B ⊗ A corresponding to

− ∑_{τ,σ,ρ} τσρ ⊗ τf(σ, ρ) ⊗ τx(σ).

This expression equals

∑_{τ,σ,ρ}τσρ⊗f(τσ, ρ)⊗τx(σ)−∑_{τ,σ,ρ}τσρ⊗f(τ, σρ)⊗τx(σ)+∑_{τ,σ,ρ}τσρ⊗f(τ, σ)⊗τx(σ).

The first sum is equal to

∑_{τ,σ,ρ} τρ ⊗ f(τ, ρ) ⊗ τσ^{−1}x(σ),

and the second is equal to

∑_{τ,σρ} τρ ⊗ f(τ, ρ) ⊗ τx(σ).

Since ∑ σ^{−1}x(σ) = ∑ x(σ) their difference is 0. The third sum is equal to

∑_{τ,σ,ρ} ρ ⊗ f(τ, σ) ⊗ τx(σ),

which corresponds to

∑_{τ,σ} f(τ, σ) ⊗ τx(σ)

as required. In this case \overline{f} · \overline{x} = \overline{x} · \overline{f} modulo the identification of B ⊗ A and A ⊗ B.

Thus, if B is C_{K} and f(τ, σ) = δ(w_{τ}, w_{σ}), so that \overline{f} is the fundamental class, and A is L̂, then \overline{x} · \overline{f} is the class of ∑ τx(σ) ⊗ δ(w_{τ}, w_{σ}). Applying the isomorphism of L̂ ⊗ C_{K} with Hom(L, C_{K}) we obtain the homomorphism

λ → ∏_{σ,τ} δ(w_{τ}, w_{σ})^{⟨λ,τx(σ)⟩}.

The commutativity of the second square is established.

To show that

Res : H_{1}(W_{K/F} , L̂) → H_{1}(C_{K}, L̂)^{𝔊(K/F)}

is an isomorphism all we have to do is show that the kernel of the map

H_{1}(C_{K}, L̂) → H_{1}(W_{K/F} , L̂)

consists precisely of the elements of norm 0. This is equivalent to showing that the image of the adjoint map

Hom(H_{1}(W_{K/F} , L̂), ℚ/ℤ) → Hom(H_{1}(C_{K}, L̂), ℚ/ℤ)

consists of the homomorphisms that vanish on the elements of norm 0.

If G is any group that acts on L̂ and Q any abelian group on which G acts trivially, the cup product gives a pairing

H^{1}(G, Hom(L̂, Q)) × H_{1}(G, L̂) → H_{0}(G, Q) = Q

and thus a homomorphism

φ : H^{1}(G, Hom(L̂, Q)) → Hom(H_{1}(G, L̂), Q).

We need to show that φ is an isomorphism if Q is injective as an abelian group. If f is a 1-cocycle and x a 1-cycle the pairing sends \overline{f} × \overline{x} to

∑_{g}⟨f(g), x(g)⟩.

Suppose φ(\overline{f}) = 0. Then

x → ∑⟨f(g), x(g)⟩

is a homomorphism of the group of 1-chains into Q that vanishes on the cycles. Thus it defines a mapping of the boundaries in dimension 0 into Q. This map can be extended to a homomorphism f′ of L̂ into Q. Then

∑_{g}⟨f(g) − df′(g), x(g)⟩

is 0 for all 1-chains. This means f = df′ and \overline{f} = 0. Thus φ is injective. Conversely suppose ψ is a homomorphism of H_{1}(G, L̂) into Q. Then ψ determines a homomorphism of the 1-cycles into Q that may be extended to a homomorphism ψ′ of the group of 1-chains into Q. If g belongs to G define f(g) in Hom(L̂, Q) by ⟨f(g), λ̂⟩ = ψ′(x) if x is the 1-chain satisfying x(g) = λ̂ and x(h) = 0 if h ≠ g. Then for any 1-chain x,

ψ′(x) = ∑_{g}⟨f(g), x(g)⟩.

Suppose g_{1} and g_{2} are two elements of G and x is the 2-chain defined by x(g_{1}, g_{2}) = λ̂, while x(h, g) = 0 if h ≠ g_{1} or g ≠ g_{2}. Then

dx(g) = ∑_{h} h^{−1}x(h, g) − ∑_{h} x(gh^{−1}, h) + ∑_{h} x(g, h).

The first sum is 0 unless g = g_{2} when it is g_{1}^{−1}λ̂; the second is 0 unless g = g_{1}g_{2} when it is λ̂; the third is 0 unless g = g_{1} when it too is λ̂. Thus

0 = ψ′(dx) = ⟨f(g_{2}), g_{1}^{−1}λ̂⟩ − ⟨f(g_{1}g_{2}), λ̂⟩ + ⟨f(g_{1}), λ̂⟩
= ⟨g_{1}f(g_{2}) − f(g_{1}g_{2}) + f(g_{1}), λ̂⟩.

Consequently f is a cocycle, ψ = φ(f), and, in conclusion, φ is an isomorphism.

The necessary functorial properties are easily checked so that to handle the original problem we need only verify that the image of H^{1}(W_{K/F} , Hom(L̂, ℚ/ℤ)) in H^{1}(C_{K}, Hom(L̂, ℚ/ℤ)) consists of those elements corresponding to homomorphisms vanishing on elements of norm 0. This problem may be reformulated. 𝔊(K/F) acts on Hom(L̂, ℚ/ℤ), and we can form the semi-direct product Hom(L̂, ℚ/ℤ) × 𝔊(K/F). Suppose we have a commutative diagram

0 ⟶  C_{K}  ⟶  W_{K/F}  ⟶^{σ} 𝔊(K/F) ⟶ 0
↓^{ψ}  ↓^{ψ′}  ↓^{id}
0 ⟶ Hom(L̂, ℚ/ℤ) ⟶ Hom(L̂, ℚ/ℤ) × 𝔊(K/F) ⟶ 𝔊(K/F) ⟶ 0.

Define the 1-cochain f by ψ′(w) = f(w) × σ(w). Then f is a cocycle. Conversely given a cocycle this equation determines a ψ′ that, together with its restriction ψ to C_{K}, will make the diagram commutative. What we have to show is that given ψ we can find a ψ′ making the diagram commutative if and only if the associated homomorphism φ : H_{1}(C_{K}, L̂) → ℚ/ℤ vanishes on the elements of norm 0. It is well known [3] that ψ can be so extended if and only if it is 𝔊(K/F) invariant and sends the fundamental class H^{2}(𝔊(K/F), C_{K}) to 0. The first condition is satisfied if φ vanishes on the elements of norm 0 because the map from ψ to φ is 𝔊(K/F) invariant. We need to verify that the diagram

L̂ ⊗ C_{K} ⟶^{id⊗ψ} L̂ ⊗ Hom(L̂, ℚ/ℤ)
↓  ↓
H_{1}(C_{K}, L̂) ⟶^{φ}  ℚ/ℤ

is commutative. The arrow on the left is the composition of the isomorphism of L̂ ⊗ C_{K} with Hom(L, C_{K}) and Hom(L, C_{K}) with H_{1}(C_{K}, L̂) previously introduced. It sends λ̂ ⊗ a to the class of the 1-cycle that is 0 except at a where it is λ̂. Applying φ we obtain ⟨ψ(a), λ̂⟩ which is, of course, the result obtained by moving around the diagram in the other direction. If ψ, and hence φ, is 𝔊(K/F)-invariant we are led to the commutative diagram

Ĥ^{−3}(𝔊(K/F),L̂) ⊗Ĥ^{2}(𝔊(K/F), C_{K}) ⟶ Ĥ^{−3}(𝔊(K/F),L̂) ⊗Ĥ^{2}(𝔊(K/F), Hom(L̂, ℚ/ℤ))
↓^{μ}  ↓^{ν}
Ĥ^{−1}(𝔊(K/F), H_{1}(C_{K},L̂))  ⟶  Ĥ^{−1}(𝔊(K/F), ℚ/ℤ).

The vertical arrows are given by the cup products. If α is the fundamental class ψ(α) = 0 if and only if ν(β ⊗ ψ(α)) = 0 for all β. On the other hand γ → μ(γ ⊗ α) is an isomorphism of Ĥ^{−3}(𝔊(K/F), L̂) and Ĥ^{−1}(𝔊(K/F), H_{1}(C_{K}, L̂)). Thus ψ(α) = 0 if and only if the lower horizontal arrow is 0. By its definition this is so if and only if φ vanishes on the elements of norm 0.

We now know that H_{1}(W_{K/F} , L̂) is isomorphic to Hom_{𝔊(K/F)}(L, C_{K}) and that, since both ℂ^{∗} and ℂ_{u}^{∗} are injective, there are isomorphisms

φ : H^{1}(W_{K/F} , T̂) → Hom(H_{1}(W_{K/F} , L̂), ℂ^{∗})
φ : H^{1}(W_{K/F} , T̂_{u}) → Hom(H_{1}(W_{K/F} , L̂), ℂ_{u}^{∗}).

We need only verify that φ(\overline{f}) is continuous if and only if f is a continuous cocycle.

If U_{K} consists of the elements of norm 1 in C_{K} we have an exact sequence

1 → U_{K} → C_{K} → M_{K} → 1.

M_{K} is ℤ or ℝ and 𝔊(K/F) acts trivially on it. The sequence splits as a sequence of abelian groups and

0 → Hom(L, U_{K}) →^{λ} Hom(L, C_{K}) →^{μ} Hom(L, M_{K}) → 0

is exact. There is an obvious map

N(Hom(L, C_{K})) ∩ Hom(L, U_{K})/N(Hom(L, U_{K}))

→ Ĥ^{−1}(𝔊(K/F), Hom(L, M_{K}))/μĤ^{−1}(𝔊(K/F), Hom(L, C_{K})).

If z = Nx belongs to Hom(L, U_{K}) and y is the image of x in Hom(L, M_{K}), then Ny = 0. We send z to the image of y in the group on the right. The image is independent of x and if x can be chosen in Hom(L, U_{K}) it is 0. If the image is 0 we can so choose x that y = ∑_{σ} σ^{−1}v_{σ} − v_{σ}. If u_{σ} in Hom(L, C_{K}) maps to v_{σ} then x′ = x − ∑ σ^{−1}u_{σ} − u_{σ} lies in Hom(L, U_{K}) and Nx′ = Nx. Thus the map is an injection. Since the group on the right is finite so is the one on the left. Since Hom(L, U_{K}) is compact, N(Hom(L, U_{K})), and hence N(Hom(L, C_{K})) ∩ Hom(L, U_{K}), is closed in Hom(L, U_{K}). Since Hom_{𝔊(K/F)}(L, U_{K}) is an open subgroup of Hom_{𝔊(K/F)}(L, C_{K}), except when K is archimedean, and

N(Hom(L, C_{K})) ∩ Hom_{𝔊(K/F)}(L, U_{K}) = N(Hom(L, C_{K})) ∩ Hom(L, U_{K})

is closed, N(Hom(L, C_{K})) is closed in Hom_{𝔊(K/F)}(L, C_{K}). Since it is of finite index, it is also open. In the archimedean case the sequence

0 → U_{K} → C_{K} → M_{K} → 0

splits as a sequence of 𝔊(K/F)-modules and

Hom_{𝔊(K/F)}(L, C_{K}) = Hom_{𝔊(K/F)}(L, U_{K}) × Hom_{𝔊(K/F)}(L, M_{K}).

Moreover

N(Hom(L, M_{K})) = Hom_{𝔊(K/F)}(L, M_{K}),

so that, in this case too, N(Hom(L, C)K) is closed in Hom_{𝔊(K/F)}(L, C_{K}).

The upshot is that a homomorphism φ of Hom_{𝔊(K/F)}(L, C_{K}) or, what is the same, H_{1}(C_{K}, L̂)^{𝔊(K/F)} into ℂ_{u}^{∗} or ℂ^{∗} is continuous if and only if φ ∘ N is continuous. Of course an element of H^{1}(W_{K/F} , T̂_{u}) or H^{1}(W_{K/F} , T̂) is continuous if and only if its restriction to C_{K} is. Since the diagrams

H^{1}(W_{K/F} , T̂) ↔ Hom(H_{1}(W_{K/F} , L̂), ℂ^{∗})
↓  ↓
H^{1}(C_{K}, T̂) ↔ Hom(H_{1}(C_{K}, L̂), ℂ^{∗})

H^{1}(W_{K/F} , T̂_{u}) ↔ Hom(H_{1}(W_{K/F} , L̂), ℂ_{u}^{∗})
↓  ↓
H^{1}(C_{K}, T̂_{u}) ↔ Hom(H_{1}(C_{K}, L̂), ℂ_{u}^{∗})

in which the right hand vertical arrows are adjoint to the corestriction are commutative, we need only check that \overline{f} in H^{1}(C_{K} T̂) is continuous if and only if the corresponding element of Hom(H_{1}(C_{K}, L̂), ℂ^{∗}) or Hom(L̂ ⊗ C_{K}, ℂ^{∗}) is continuous. Since that homomorphism sends λ̂ ⊗ a to ⟨λ̂, f(a)⟩ this is clear.

We have still to prove the second part of Theorem 2. The diagram

T_{𝔸(K)} ⟶ Hom(L, C_{K}) ⟶ 0
N↓  ^{N}↓
T_{𝔸(F)} ⟶ Hom_{𝔊(K/F)}(L, C_{K})

is commutative and the upper row is exact. Thus T_{𝔸(F)}/T_{F} contains N(Hom(L, C_{K})) and is a closed subgroup of finite index in Hom_{𝔊(K/F)}(L, C_{K}).

If K′ is the completion of K with respect to some valuation and F′ is the closure of F in K′ there is a mapping T_{F′} → T_{𝔸(F)} and thus a mapping T_{F′} → Hom_{𝔊(K/F)}(L, C_{K}). A generalized character of Hom_{𝔊(K/F)}(L, C_{K}) is trivial on T_{𝔸(F)}/T_{F} if and only if it is trivial on the image of T_{F′}, for all choices of K′.

T_{F′} = Hom_{𝔊(K′/F′)}(L, C_{K′})

and, if E = K ∩ F′, the natural map C_{K′} → C_{K} gives a map

Hom_{𝔊(K′/F′)}(L, C_{K′}) → Hom_{𝔊(K/E)}(L, C_{K}).

If 𝔊(K/F) is the disjoint union ⋃_{i=1}^{r} σ_{i}𝔊(K/E) we compose this with the map ∑_{i=1}^{r} σ_{1} of Hom_{𝔊(K/E)}(L, C_{K}) into Hom_{𝔊(K/F)}(L, C_{K}) to obtain the map of T_{F′} into Hom_{𝔊(K/F)}(L, C_{K}).

On the other hand if φ is the imbedding K/F → K′/F′, the map φ_{w} : W_{K′/F′} → W_{K/F} determines maps

\begin{matrix}H_{c}^{1}(W_{K/F} , T̂) & ⟶ & H_{c}^{1}(W_{K′/F′}, T̂) \\ H_{1}(W_{K′/F′}, L̂) & ⟶ & H_{1}(W_{K/F} , L̂) \end{matrix}

that are adjoint to each other. All we need do is verify that

H_{1}(W_{K′/F′}, L̂) ⟶ H_{1}(W_{K/F} , L̂)
↓  ↓
Hom_{𝔊(K′/F′)}(L, C_{K′}) ⟶ Hom_{𝔊(K/F)}(L, C_{K})

is commutative.

Let x′ be a 1-cycle on W_{K′/F′} with values in L̂ and let x be the 1-cycle on W_{K/F} that sends w to ∑_{w′→w} x′(w′). Restriction to C_{K′} sends the class of x′ to the class of y′ if

y′(a′) = ∑_{δ(w_{τ} ,w′)=a′} w_{τ′}x′(w′).

Restriction to C_{K} sends the class of x to the class of y if

y(a) = ∑_{δ(w_{τ} ,w)=a} w_{τ}x(w).

The class of y′ corresponds to the homomorphism

λ → ∏_{τ′} ∏_{w′} δ(w_{τ′}, w′)^{⟨λ,w_{τ′} x′(w′)⟩}

and that of y to the homomorphism

λ → ∏_{τ} ∏_{w} δ(w_{τ}, w)^{⟨λ,w_{τ} x(w)⟩}.

The inner product need only be taken over W_{K/E}.

If σ = σ_{i}τ with τ in 𝔊(K/E) we take w_{σ} = w_{σ_{i}}w_{τ}. Then if w belongs to W_{K/E}

w_{σ_{i}}w_{τ}w = w_{σ_{i}}δ(w_{τ}, w)w_{σ_{i}}^{−1}w_{σ_{i}}w_{ρ}

for some ρ in 𝔊(K/E). Thus the homomorphism corresponding to y sends λ to

∑_{i=1}^{r} σ_{i}(φ(σ_{i}^{−1}λ)),

if φ is the homomorphism

λ → ∏_{τ∈𝔊(K/E)} ∏_{w∈W_{K/E}} δ(w_{τ}, w)^{⟨λ,w_{τ} x(w)⟩}

of L into C_{K}. The homomorphism φ_{w} is an injection and we may use it to identify W_{K′/F′} with a subgroup of W_{K/E}. We may also identify 𝔊(K/E) and 𝔊(K′/F′). If τ′ ↔ τ we take w_{τ′} ↔ w_{τ}. If w′ ↔ w

δ(w_{τ′}, w′) ↔ δ(w_{τ}, w)

and the above product may be written

∏_{τ′} ∏_{w′} δ(w_{τ′}, w′)^{⟨λ,w_{τ′} x′(w′)⟩}

so that φ is the homomorphism corresponding to y′. The diagram is commutative and Theorem 2 is proved.

It is convenient to add now one or two observations that will be used elsewhere to compare two kinds of L-series. Suppose F is a non-archimedean local field and K/F is unramified. I claim that

N : Hom(L, U_{K}) → Hom_{𝔊(K/F)}(L, U_{K})

is surjective. To see this let U_{k}^{n} = {x ∈ U_{K} | x ≡ 1 (mod P_{K}^{n})} if n ≥ 1. These subgroups are 𝔊(K/F)-invariant and we need only verify that

N : Hom(L, U_{K}/U_{K}^{1}) → Hom_{𝔊(K/F)}(L, U_{K}/U_{K}^{1})
N : Hom(L, U_{K}^{n}/U_{K}^{n+1}) → Hom_{𝔊(K/F)}(L, U_{K}^{n}/U_{K}^{n+1})

are surjective. Let κ = O_{K}/P_{K} be the residue field of O_{K}. The group U_{K}^{n}/U_{K}^{n+1} is isomorphic as a 𝔊(K/F)-module to κ; so we consider

N : Hom(L, κ) → Hom_{𝔊(K/F)}(L, κ).

If k = O_{F} /P_{F} and Λ is the group ring of 𝔊(K/F) then κ is isomorphic as a 𝔊(K/F)-module to Λ⊗k, and Hom(L, Λ⊗k) is isomorphic to Λ⊗Hom(L, k), so that

Ĥ^{0}(𝔊(K/F), Λ ⊗ Hom(L, k)) = 0

and N is surjective. The group U_{K}/U_{K}^{1} is isomorphic as a 𝔊(K/F) module to κ^{∗} the multiplicative group of κ; so we consider

N : Hom(L, κ^{∗}) → Hom_{𝔊(K/F)}(L, κ^{∗}).

To show this is surjective is the same as showing that Ĥ^{0}(𝔊(K/F), Hom(L, κ^{∗})) is 0. Since 𝔊(K/F) is cyclic and Hom(L, κ^{∗}) is finite all the groups Ĥ^{p}(𝔊(K/F), Hom(L, κ^{∗})) have the same order. We show that

Ĥ^{1}(𝔊(K/F), Hom(L, κ^{∗})) = H^{1}(𝔊(K/F), Hom(L, κ^{∗}))

is 0. If \overline{κ} is the algebraic closure of κ and 𝔊 is the subgroup of 𝔊(\overline{κ}/κ) generated by the Frobenius substitution, the sequence

0 → H^{1}(𝔊(K/F), Hom(L, κ^{∗})) → H^{1}(𝔊, Hom(L, \overline{κ}^{∗}))

is exact. We show that the group on the right is 0. If σ_{0} is the Frobenius substitution any 1-cycle f on 𝔊 is determined by f(σ_{0}). To prove that f is a boundary we have to show that f(σ_{0}) = σ_{0}φ − φ for some φ in Hom(L, \overline{κ}^{∗}). Let λ_{1}, ⋯, λ_{n} be a basis for L and let

σ_{0}^{−1} = ∑_{j} a_{ji}λ_{j}.

The matrix (a_{ij}) is integral and all its eigenvalues are algebraic integers. Thus, if q is the number of elements in κ, (qa_{ij} − δ_{ij}), where δ_{ij} is Kronecker’s delta, is non-singular. We may write it as a product (b_{ij})(m_{i}δ_{ij})(c_{ij}) where (b_{ij}) and (c_{ij}) are integral with integral inverses (b′_{ij}) and (c′_{ij}), and m_{1}, ⋯, m_{n} are integers. Let f(σ_{0}) = ψ and let ψ(λ_{i}) = α_{i}. If φ(λ_{i}) = β_{i},

σ_{0}φ(λ_{i}) = σ_{0}(φ(σ^{−1}λ_{i})) = ∏_{j=1}^{n} β_{j}^{qa_{ji}},

and the equation ψ = σ_{0}φ − φ is equivalent to the n equations:

α_{i} = ∏_{j=1}^{n} β_{j}^{qa_{ji}−δ_{ji}}.

If γ_{i} = ∏_{k=1}^{n} α_{k}^{c′_{ki}} and δ_{j} = ∏_{k} β_{k}^{b_{kj}} these equations are equivalent to

γ_{i} = δ_{i}^{m_{i}}.

This equation can be solved for δ_{1}, ⋯, δ_{n}. Setting

β_{j} = ∏_{k} δ_{k}^{b′_{kj}}

we solve the original equations.

There is an exact sequence

0 → U_{K} → W_{K/F} →^{μ} ℤ → 0

such that μ(w) = 1 implies that σ(w) is the Frobenius substitution in 𝔊(K/F). Then μ(w) is 1 if and only if the transfer of w in C_{K} generates the prime ideal of O_{K}. The preceding discussion shows that under the isomorphism

H_{1}(W_{K/F} , L̂) ↔ Hom_{𝔊(K/F)}(L, C_{K})

the image of H_{1}(U_{K}, L̂) in H_{1}(W_{K/F} , L̂) corresponds to Hom_{𝔊(K/F)}(L, U_{K}). Thus the generalized character associated to an element of H_{c}^{1}(W_{K/F} , T̂) is trivial on Hom_{𝔊(K/F)}(L, U_{K}) if and only if that element is in the image of the inflation

H^{1}(ℤ, L̂) → H^{1}(W_{K/F} , L̂).

The action of ℤ on L̂ is determined by that of W_{K/F}.

The sequence

0 → U_{K} → C_{K} →^{ν} ℤ → 0

in which ν(a) = 1 if and only if a generates the prime ideal of O_{K} splits as a sequence of 𝔊(K/F)-modules and leads to the exact sequence

0 → Hom_{𝔊(K/F)}(L, U_{K}) → Hom_{𝔊(K/F)}(L, C_{K}) → Hom_{𝔊(K/F)}(L, ℤ) → 0,

and the generalized characters of Hom_{𝔊(K/F)}(L, C_{K}) that are trivial on Hom_{𝔊(K/F)}(L, U_{K}) are just the generalized characters of Hom_{𝔊(K/F)}(L, ℤ), which is contained in Hom(L, ℤ) = L̂.

If λ̂ is an invariant element of L̂ and w belongs to W_{K/F}, consider the 1-cycle x, where x(w) = λ̂ while x(u) = 0 if u ≠ w. The image of the class of x in Hom(L, C_{K}) is the homomorphism

λ → {∏_{τ} δ(w_{τ}, w)}^{⟨λ,_{λ}⟩} .

Now ∏_{τ} δ(w_{τ}, w) is just the transfer of w. Applying ν we obtain the homomorphism μ(w)λ̂. Thus if f is a 1-cocycle on W_{K/F} with values in T̂ associated to a 1-cocycle on ℤ, μ(w) = 1, λ̂ is an invariant element of L̂, and χ is the generalized character of L̂^{𝔊(K/F)} associated to f, then

f(w)(λ̂) = χ(λ).

Given χ we can take any extension of χ to L̂, take f(w) to be that extension when μ(w) = 1, and extend f to all of W_{K/F} by the cocycle condition.

Suppose ρ is a rational representation of the complex algebraic group T̂ × 𝔊(K/F). Given χ we define a local L-function L(s, χ, ρ) by extending χ to L̂ and thus defining t in T̂, choosing the Frobenius substitution σ in 𝔊(K/F), and finally setting

L(s, χ, ρ) = \frac{1}{det(I − ρ(t × σ)| ∏ |^{s})}

if ∏ is a generator of the prime ideal of O_{K}.

But f determines a homomorphism φ of W_{K/F} into T̂ × 𝔊(K/F):

φ(w) = f(w) × σ(w).

Thus ρ ∘ φ is a representation of W_{K/F}. The associated local L-function is

L(s, ρ ∘ φ) = \frac{1}{det(I − ρ ∘ φ(w)| ∏ |^{s})}

if μ(w) = 1. Since f(w) may be taken equal to t, the two functions are equal.

References

[1] E. Artin and J. Tate, Class Field Theory, Harvard, 1961.

[2] S. Lang, Rapport sur la Cohomologie des Groupes, New York, 1966.

[3] J.P. Serre, Corps Locaux, Hermann, Paris, 1962.

Institute for Advanced Study Princeton, NJ 08540 E-mail address: rpl@ias.edu