GALOIS MODULES AND p-ADIC REPRESENTATIONS

A. AGBOOLA

Abstract. In this paper we develop a theory of class invariants associated to p-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to p-adic representations in terms of resolvends associated to torsors of finite group schemes.

1. Introduction

In this paper we shall introduce and study invariants which measure the Galois structure of certain torsors that are constructed via p-adic Galois representations. We begin by describing the background to the questions that we intend to discuss.

Let Y be any scheme, and suppose that G → Y is a finite, flat, commutative group scheme. Write G^{∗} for the Cartier dual of G. Let G̃^{∗} denote the normalisation of G^{∗}, and let i : G̃^{∗} → G^{∗} be the natural map. Suppose that π : X → Y is a G-torsor, and write π_{0} : G → Y for the trivial G-torsor. Then O_{X} is an O_{G}-comodule, and so it is also an O_{G^{∗}}-module (see e.g. [12]). As an O_{G^{∗}}-module, the structure sheaf O_{X} is locally free of rank one, and so it gives a line bundle M_{π} on G^{∗}. Set

L_{π} := M_{π} ⊗ M_{π_{0}}^{−1}.

Then the maps

ψ : H^{1}(Y, G) → Pic(G^{∗}),  [π] ↦ [L_{π}];  (1.1)
φ : H^{1}(Y, G) → Pic(G̃^{∗}),  [π] ↦ [i^{∗}L_{π}]  (1.2)

are homomorphisms which are often referred to as ‘class invariant homomorphisms’.

The initial motivation for studying class invariant homomorphisms arose from Galois module theory. Let F be a number field with ring of integers O_{F}, and suppose that Y = Spec(O_{F}). Write G^{∗} = Spec(A), G = Spec(B), and X = Spec(C). Then the algebra C is a twisted form of B, and the homomorphisms ψ and φ measure the Galois module structure of this twisted form. The homomorphism ψ was first introduced by W. Waterhouse (see [31]), and was further developed in the context of Galois module theory by M. Taylor ([28]). Taylor originally considered the case in which G is a torsion subgroup scheme of an abelian variety with complex multiplication.

Date: Version of June 19, 2006.

1991 Mathematics Subject Classification. 11Gxx, 11Rxx.

2 A. AGBOOLA

The corresponding torsors are obtained by dividing points in the Mordell-Weil groups of such abelian varieties, and they are closely related to rings of integers of abelian extensions of F. In [27], it was shown that, for elliptic curves with complex multiplication, the class invariant homomorphism ψ vanishes on the classes of torsors obtained by dividing torsion points of order coprime to 6. This implies the existence of Hopf Galois generators for certain rings of integers of abelian extensions of imaginary quadratic fields, and it may be viewed as an integral version of the Kronecker Jugendtraum (see [29], [11]). This vanishing result was extended to all elliptic curves in [2] and [20].

Since their introduction, class invariants of torsors obtained by dividing points on abelian varieties have been studied in greater generality by several authors. For example, suppose that X is a projective curve over Spec(Z) which is equipped with a free action of a finite group. In [21], it is shown that the behaviour of the equivariant projective Euler characteristic of O_{X} is partly governed by class invariants of torsors arising from torsion points on the Jacobian of X. In [5], an Arakelov (i.e. arithmetic) version of class invariants of torsors coming from points on abelian varieties is considered. There it is shown that in general such torsors are completely determined by their arithmetic class invariants, and that these invariants are related to Mazur-Tate heights on the abelian variety (see [18]). Finally we mention that in [1], [3], and [6], class invariants arising from points on elliptic curves with complex multiplication are studied using Iwasawa theory, and they are shown to be closely related to the p-adic height pairing on the elliptic curve.

The main goal of this paper is to develop a theory of class invariants for arbitrary p-adic representations, and to generalise a number of results that up to now have only been known in certain cases involving elliptic curves with complex multiplication.

We now describe the main results contained in this paper. Suppose that p is an odd prime, and let V be a d-dimensional Q_{p}-vector space. Let F^{c} be an algebraic closure of F, and write Ω_{F} := Gal(F^{c}/F). Suppose that ρ : Ω_{F} → GL(V) is a continuous representation of Ω_{F} that is ramified at only finitely many primes of F. Set V^{∗} := Hom_{Q_{p}}(V, Q_{p}(1)), and let ρ^{∗} : Ω_{F} → GL(V^{∗}) be the corresponding representation of Ω_{F}. Suppose that T ⊆ V is an Ω_{F} -stable lattice, and write T^{∗} := Hom_{Z_{p}}(T, Z_{p}(1)). (Note that for each construction in this paper that depends upon T, there is also a corresponding construction that depends upon T^{∗}; this will not always be explicitly stated.)

For each positive integer n, we may define finite, commutative group schemes G_{n} and G_{n}^{∗} over Spec(F) by

G_{n}(F^{c}) = Γ_{n} := p^{−n}T/T; G_{n}^{∗}(F^{c}) = Γ_{n}^{∗} := p^{−n}T^{∗}/T^{∗}.

Then G_{n}^{∗} is the Cartier dual of G_{n}, and we may write G_{n}^{∗} = Spec(A_{n}) for some Hopf algebra A_{n} over F.

GALOIS MODULES AND p-ADIC REPRESENTATIONS 3

For any (not necessarily finitely generated) O_{F} algebra 𝔄_{n} ⊆ A_{n} satisfying certain quite mild conditions (see Section §3 below), we use a description of H^{1}(F, G_{n}) which arises via studying the Galois structure of G_{n}-torsors in terms of A_{n} to give a new way of imposing local conditions on cohomology classes in terms of the algebra 𝔄_{n}. (Roughly speaking, if π ∈ H^{1}(F, G_{n}), then we use 𝔄_{n} to impose local conditions on the line bundle L_{π} associated to π.) This yields a certain Selmer group in H^{1}(F, G_{n}) which we denote by H_{𝔄_{n}}^{1}(F, G_{n}). Suppose that π : X → Spec(F) is any G_{n}-torsor whose isomorphism class lies in H_{𝔄_{n}}^{1}(F, G_{n}). We shall explain how to use the methods of [9], [19] and [31] to construct a natural homomorphism

ϕ_{𝔄_{n}} : H_{𝔄_{n}}^{1}(F, G_{n}) → Pic(Spec(𝔄_{n})).  (1.3)

This generalises the class invariant homomorphisms (1.1) and (1.2) above. For suppose that G_{n} is the generic fibre of a finite, flat group scheme G_{n} over Spec(O_{F}). If we choose 𝔄_{n} to be the O_{F} -Hopf algebra representing the Cartier dual G_{n}^{∗} of G_{n}, then H_{𝔄_{n}}^{1}(F, G_{n}) = H^{1}(Spec(O_{F}), G_{n}), and ϕ_{𝔄_{n}} is the same as the homomorphism (1.1) in this case. If on the other hand we take 𝔄_{n} to be the maximal O_{F} -order 𝔐_{n} in A_{n}, then Spec(𝔄_{n}) is equal to the normalisation G̃_{n}^{∗} of G_{n}^{∗}. In this case, H^{1}(Spec(O_{F}), G_{n}) is contained in H_{𝔄_{n}}^{1}(F, G_{n}), and the restriction of ϕ_{𝔄_{n}} to H^{1}(Spec(O_{F}), G_{n}) is the homomorphism (1.2). (See Example 3.5 below.)

In this paper we shall mainly be concerned with the cases

𝔄_{n} = 𝔐_{n}, 𝔄_{n} = 𝔐_{n} ⊗_{O_{F}} O_{F} [1/p] := 𝔐_{n}^{{p}}.

For each finite place v of F, let F_{v}^{nr} denote the maximal unramified extension of F_{v} in a fixed algebraic closure of F_{v}. If v ∤ p, then define

H_{f}^{1}(F_{v}, T) := Ker [H^{1}(F_{v}, T) → H^{1}(F_{v}^{nr}, T)] .  (1.4)

Following [22, §3.1.4], we set

H_{f,\{p\}}^{1}(F, T) = Ker [H^{1}(F, T) → ⊕_{v∤p} \frac{H^{1}(F_{v}, T)}{H_{f}^{1}(F_{v}, T)}] .

It may be shown that (see Remark 3.6 below)

H_{f,\{p\}}^{1}(F, T) ⊆ \underleftarrow{lim} H_{𝔐_{n}^{\{p\}}}^{1}(F, G_{n})

Here the inverse limit is taken with respect to the maps induced by the ‘multiplication by p’ maps G_{n+1} → G_{n}, and we view \underleftarrow{lim} H_{𝔐_{n}^{\{p\}}}^{1}(F, G_{n}) as being a subgroup of H^{1}(F, T) via the canonical isomorphism \underleftarrow{lim} H^{1}(F, G_{n}) ≃ H^{1}(F, T). Set

H_{u}^{1}(F, T) := \underleftarrow{lim} H_{𝔐_{n}}^{1}(F, G_{n}).

The natural inclusions G_{n}^{∗} → G_{n+1}^{∗} induce pullback homomorphisms

Pic(Spec(𝔐_{n+1})) → Pic(Spec(𝔐_{n})), Pic(Spec(𝔐_{n+1}^{{p}})) → Pic(Spec(𝔐_{n}^{{p}})).

We shall show that we may take inverse limits in (1.3) to obtain homomorphisms

Φ_{F} : H_{u}^{1}(F, T) → \underleftarrow{lim} Pic(Spec(𝔐_{n})), Φ_{F}^{\{p\}} : H_{f,\{p\}}^{1}(F, T) → \underleftarrow{lim} Pic(Spec(𝔐_{n}^{\{p\}})).

Our first result shows that the homomorphism Φ_{F}^{{p}} is closely related to a p-adic height pairing associated to T. In order to describe why this is so, we have to introduce some further notation.

Let C_{n}/F denote the n-th layer of the cyclotomic Z_{p}-extension of F, and set

𝔊_{F} (T) := {x ∈ H_{f,{p}}^{1}(F, T) | Mx ∈ ∩_{n} Cores_{C_{n}/F} (H_{f,{p}}^{1}(C_{n}, T)) for some integer M > 0}.

When T is the p-adic Tate module of an elliptic curve, 𝔊_{F} (T) is the same as the canonical subgroup that was defined by R. Greenberg in [14, p.131–132], and further studied by A. Plater in [23] and [24] (see also [17]).

Let

Loc_{F,T^{∗}} : H_{f,{p}}^{1}(F, T^{∗}) → ∏_{v|p} H^{1}(F_{v}, T^{∗})

denote the natural localisation map. In [22, Section 3.1.4], Perrin-Riou constructs a p-adic height pairing

B_{F} : H_{f,{p}}^{1}(F, T) × Ker(Loc_{F,T^{∗}}) → Q_{p},

and she shows that the group 𝔊_{F} (T) lies in the left-hand kernel of B_{F}. Write

⟨⟨ , ⟩⟩ : \frac{H_{f,\{p\}}^{1}(F, T)}{𝔊_{F} (T)} × Ker(Loc_{F,T^{∗}}) → Q_{p}  (1.5)

for the pairing induced by B_{F}. We remark that it follows from the definition of 𝔊_{F} (T) that the group H_{f,{p}}^{1}(F, T)/𝔊_{F} (T) is torsion-free. It is conjectured that ⟨⟨ , ⟩⟩ is always non-degenerate modulo torsion. If this conjecture is true, then it implies that 𝔊_{F} (T) has a natural characterisation in terms of p-adic height pairings attached to T. The following result shows that this conjecture implies that 𝔊_{F} (T) also has a natural characterisation in terms of Galois module structure (see Propositions 6.3 and 6.5 below).

Theorem 1.1. (a) If x ∈ 𝔊_{F} (T), then Φ_{F}^{{p}}(x) is of finite order.

(b) If x ∈ H_{f,{p}}^{1}(F, T) and Φ_{F}^{{p}}(x) has finite order, then x lies in the left-hand kernel of the pairing B_{F}.

Hence, if the pairing ⟨⟨ , ⟩⟩ is non-degenerate modulo torsion, then x ∈ 𝔊_{F} (T) if and only if Φ_{F}^{{p}}(x) is of finite order.

Remark 1.2. It would be interesting if one could show directly (without, of course, assuming the non-degeneracy of the pairing ⟨⟨ , ⟩⟩!) that

{x ∈ H_{f,{p}}^{1}(F, T)| Φ_{F}^{{p}}(x) has finite order} = the left-hand kernel of the pairing B_{F} .

GALOIS MODULES AND p-ADIC REPRESENTATIONS 5

This would imply that the non-degeneracy modulo torsion of the pairing ⟨⟨ , ⟩⟩ is equivalent to the statement that x ∈ 𝔊_{F} (T) if and only if Φ_{F}^{{p}}(x) is of finite order. Unfortunately we do not know how to do this at present. □

Let S be a finite set of places of F containing the places lying over p, the places at which ρ is ramified, and the set of infinite places. Let F^{S}/F denote the maximal extension of F which is unramified outside S. A conjecture of Greenberg asserts that the group H^{2}(F^{S}/C_{∞}, V^{∗}/T^{∗}) always vanishes. This may be viewed as an analogue of a weak form of Leopoldt’s conjecture for the Galois representation V^{∗}.

Corollary 1.3. Suppose that the pairing ⟨⟨ , ⟩⟩ is non-degenerate modulo torsion, and that

H^{2}(F^{S}/C_{∞}, V^{∗}/T^{∗}) = 0.  (1.6)

Then the restriction of Φ_{F}^{{p}} to Ker(Loc_{F,T}) has finite kernel.

Proof. It follows from [22, Remark at the end of §3.4.2] that if (1.6) holds, then 𝔊_{F} (T)∩Ker(Loc_{F,T}) is finite. The result now follows from Theorem 1.1. □

We now turn to the homomorphism Φ_{F}.

For each integer n, the action of Ω_{F} on Γ_{n}^{∗} yields a representation ρ_{n}^{∗} : Ω_{F} → Aut(Γ_{n}^{∗}). Write F_{n}^{∗} for the fixed field of ρ_{n}^{∗}; then F_{∞}^{∗} := ∪_{n}F_{n}^{∗} is the extension of F cut out by ρ^{∗}. Set

ℭ_{F} (T) := {x ∈ H_{u}^{1}(F, T) | Mx ∈ ∩_{n} Cores_{F_{n}^{∗}/F} (H_{u}^{1}(F_{n}^{∗}, T)) for some integer M > 0}.

Theorem 1.4. Suppose that x ∈ ℭ_{F} (T). Then Φ_{F} (x) is of finite order.

Remark 1.5. Whether or not the the converse of Theorem 1.4 holds in general is an open question, and it appears to be a very delicate problem. If T has Z_{p}-rank one, then it may be shown that ℭ_{F} (T) = 𝔊_{F} (T) ∩ H_{u}^{1}(F, T), and that the converse to Theorem 1.4 holds in this case. □

Acknowledgements. This paper owes a great deal to ideas first introduced in [5], and I am extremely grateful to G. Pappas for many interesting and helpful conversations. I would like to thank B. Conrad, R. Greenberg and S. Howson for useful discussions. I am grateful to the anonymous referees for pointing out some mistakes in earlier versions of this paper, and for a number of very helpful comments. I would also like to thank the Mathematics Department of Harvard University for their hospitality while a part of this work was carried out. This research was partially supported by NSF grants.

Notation. Throughout this paper, we assume that p is an odd prime.

For any field L, we write L^{c} for an algebraic closure of L, and we set Ω_{L} := Gal(L^{c}/L). If L is either a number field or a local field, then we write O_{L} for its ring of integers.

6 A. AGBOOLA

If L is a number field and v is a finite place of L, then we write L_{v} for the local completion of L at v. We fix an algebraic closure L_{v}^{c} of L_{v} and we identify Ω_{L_{v}} with a subgroup of Ω_{L}. If P is any O_{L}-module, then we shall usually write write P_{v} := P ⊗_{O_{L}} O_{L_{v}}.

For any Z-module Q, we set Q̌ := \underleftarrow{lim}_{n} Q/p^{n}Q.

If R and S are rings with R ⊆ S, and if A is any R-algebra, then we often write A_{S} for A ⊗_{R} S.

2. Resolvends and cohomology groups

In this section, R denotes either a field K, or a Dedekind domain with field of fractions K. In the second case, the symbol R^{c} denotes the integral closure of R in K^{c}. We assume (in both cases) that K is of characteristic zero. Let Y = Spec(R). We shall explain how the cohomology group H^{1}(Y, G) may be described in terms of the Hopf algebra A representing G^{∗}. Set Γ := G(R^{c}) and Γ^{∗} := G^{∗}(R^{c}).

Recall that there is a canonical isomorphism

H^{1}(Y, G) ≃ Ext^{1}(G^{∗}, G_{m})

(see [31], [15, éxposé VII], or [21]). This implies that given any G-torsor π : X → Y, we can associate to it a canonical commutative extension

1 → G_{m} → G(π) → G^{∗} → 1.

The scheme G(π) is a G_{m}-torsor over G^{∗}, and its associated G^{∗}-line bundle is equal to L_{π}. (This construction is explained in detail by Waterhouse in [31].)

Over Spec(R^{c}), the G-torsors π_{0} and π become isomorphic, i.e. there is an isomorphism X ⊗_{R} R^{c} ≃ G ⊗_{R} R^{c} of schemes with G-action. (This isomorphism is not unique: it is only well-defined up to the action of an element of G(R^{c}).) Hence, via the functoriality of Waterhouse’s construction in [31], we obtain an isomorphism

ξ_{π} : L_{π} ⊗_{R} R^{c} ⟶^{∼} A_{R^{c}}.

We shall refer to ξ_{π} as a splitting isomorphism for π.

Now suppose that ψ(π) = 0. Then L_{π} is a free A-module, and so we may choose a trivialisation s_{π} : A ⟶^{∼} L_{π}. Consider the composition

A_{R^{c}} ⟶^{s_{π}⊗_{R}R^{c}} L_{π} ⊗_{R} R^{c} ⟶^{ξ_{π}} A_{R^{c}}.

This is an isomorphism of A_{R^{c}}-modules, and so it is just multiplication by an element r(s_{π}) of A_{R^{c}}^{×}. We refer to r(s_{π}) as a resolvend of s_{π} or as a resolvend associated to π. (This terminology is due to L. McCulloh, [19].) Note that r(s_{π}) depends upon the choice of ξ_{π} as well as upon s_{π}. We shall sometimes not make the dependence of r(s_{π}) upon ξ_{π} explicit.

GALOIS MODULES AND p-ADIC REPRESENTATIONS 7

Definition 2.1. Let A and R be as above. Define

H(A) := \{α ∈ A_{R^{c}}^{×} | \frac{α^{ω}}{α} ∈ Γ for all ω ∈ Ω_{K}.\} ,
H(A) := \frac{H(A)}{Γ · A^{×}} .
□

If ω ∈ Ω_{K}, then ξ_{π}^{ω} = g_{ω}ξ_{π}, where g_{ω} ∈ Γ. Since s_{π}^{ω} = s_{π}, we deduce that r(s_{π})^{ω} = g_{ω}r(s_{π}), that is, r(s_{π}) ∈ H(A). It is easy to see that changing s_{π} alters r(s_{π}) via multiplication by an element of A^{×}, while changing ξ_{π} alters r(s_{π}) via multiplication by an element of Γ. Hence the image r(π) of r(s_{π}) in H(A) depends only upon the isomorphism class of the torsor π.

The following result, in the case in which G is a constant group scheme, is equivalent to certain results of L. McCulloh (see [19, Sections 1 and 2]; note, however that McCulloh formulates his results in a rather different way from that described here). McCulloh’s methods were generalised by N. Byott to the case of arbitrary G (see [9, Lemma 1.11 and Sections 2 and 3]) using techniques from the theory of Hopf algebras. The proofs of McCulloh and Byott proceed via analysing the Ω_{K}-cohomology of the exact sequence

1 → Γ → A_{R^{c}}^{×} → A_{R^{c}}^{×}/Γ → 1

of Ω_{K}-modules. We give a different approach using a method that involves combining the functoriality of Waterhouse’s cosntruction with the theory of descent.

Theorem 2.2. Let G be a finite, flat commutative group scheme over Spec(R), and let G^{∗} = Spec(A) be the Cartier dual of G. Then the map

Υ_{R} : Ker(ψ) → H(A); [π] ↦ r(π)

is an isomorphism.

Proof. We first show that Υ_{R} is a homomorphism. Suppose that π_{1} : X_{1} → Spec(R) and π_{2} : X_{2} → Spec(R) are G-torsors satisfying ψ(π_{1}) = ψ(π_{2}) = 0. For i = 1, 2, let

ξ_{π_{i}} : L_{π_{1}} ⊗_{R} R^{c} ⟶^{∼} A_{R^{c}}

be a splitting isomorphism for π_{i}, and suppose that

s_{π_{i}} : A ⟶^{∼} L_{π_{i}}

is a trivialisation of L_{π_{i}}. Set π_{3} := π_{1} · π_{2}. Then it follows via the functoriality of Waterhouse’s construction that there is a natural isomorphism L_{π_{3}} ≃ L_{π_{1}} ⊗_{A} L_{π_{2}}. Thus, if we set

s_{π_{3}} := s_{π_{1}} ⊗ s_{π_{2}} : A ≃ A ⊗_{A} A ⟶^{∼} L_{π_{1}} ⊗_{A} L_{π_{2}},

ξ_{π_{3}} := ξ_{π_{1}} ⊗ ξ_{π_{2}} : (L_{π_{1}} ⊗_{A} L_{π_{2}}) ⊗_{R} R^{c} ⟶^{∼} A_{R^{c}} ⊗_{A_{R^{c}}} A_{R^{c}} ≃ A_{R^{c}},

8 A. AGBOOLA

then we have that r(s_{π_{3}}) = r(s_{π_{1}})r(s_{π_{2}}), where, for each i, the resolvend r(s_{π_{i}}) is defined using the splitting isomorphism ξ_{π_{i}}. This implies that r(π_{3}) = r(π_{1})r(π_{2}), as required.

We now show that Υ_{R} is surjective. For any scheme S → Spec(R), write Map_{S}(G^{∗}, G_{m}) for the set of scheme morphisms G^{∗} ×_{Spec(R)} S → G_{m} ×_{Spec(R)} S. Since G^{∗} is affine, the functor S ↦ Map_{S}(G^{∗}, G_{m}) is representable by an affine group scheme over R, which we denote by ℳ(G^{∗}, G_{m}). The group scheme G (which represents the functor S ↦ Hom_{S}(G^{∗}, G_{m})) is a closed subgroup scheme of ℳ(G^{∗}, G_{m}).

Suppose that α ∈ H(A). Then we may view α as being a Spec(R^{c})-valued point of ℳ(G^{∗}, G_{m}). Let

G_{α} := α · [G ⊗_{R} R^{c}]  (2.1)

denote the ‘translation-by-α’ in ℳ(G^{∗}, G_{m}) ⊗_{R} R^{c} of G ⊗_{R} R^{c}. Then G ⊗_{R} R^{c} acts on G_{α} via translation. Furthermore, translation by α induces an isomorphism

Ξ_{α} : G ⊗_{R} R^{c} ⟶^{∼} G_{α}  (2.2)

of schemes with G ⊗_{R} R^{c} action.

We now claim that G_{α} descends to Spec(R), i.e. that there is a scheme π_{α} : Z_{α} → Spec(R) defined over R which is such that G_{α} = Z_{α} ⊗_{R} R^{c}. (We refer the reader to [8, Chapter 6] for a good account of the theory of descent.) Since R is a Dedekind domain, and G_{α} is flat over Spec(R^{c}), it suffices to check that the generic fibre G_{α/K^{c}} of G_{α} descends to a scheme over Spec(K). This in turn follows via Galois descent, and may be seen as follows. We first note that the isomorphism Ξ_{α} induces a bijection Γ → G_{α}(K^{c}) of sets. Define z_{α} : Ω_{K} → Γ by z(ω) = α^{ω}α^{−1}; thus z_{α} is the Γ-valued cocycle of Ω_{K} associated to α. Then it is easy to check that the action of Ω_{K} on G_{α}(K^{c}) is given by

Ξ(g)^{ω} = z_{α}(ω)Ξ(g^{ω})

for all g ∈ Γ and ω ∈ Ω_{K}. This implies that G_{α/K^{c}} descends to Z_{α/K} over Spec(K). A similar argument also shows that π_{α} : Z_{α} → Spec(R) is a G-torsor over Spec(R).

We shall now show that ψ(π_{α}) = 0. Let

ξ_{π_{α}} : L_{π_{α}} ⊗_{R} R^{c} ⟶^{∼} A_{R^{c}}

denote the splitting isomorphism of π_{α} induced by Ξ_{α}. Define an isomorphism

σ_{α} : A_{R^{c}} ⟶^{∼} L_{π_{α}} ⊗_{R} R^{c}

by σ_{α}(a) = ξ_{π_{α}}^{−1}(αa) = αξ_{π_{α}}^{−1}(a) for all a ∈ A_{R^{c}}. In order to show that ψ(π_{α}) = 0, it suffices to show that σ_{α} descends to an isomorphism σ′_{α} : A ⟶^{∼} L_{π_{α}} over R. This will in turn follow if we show that

σ_{α}^{ω}(a) = σ_{α}(a)

GALOIS MODULES AND p-ADIC REPRESENTATIONS 9

for all ω ∈ Ω_{K} and all α ∈ A_{R^{c}}. To check this last equality, we simply observe that

σ_{α}^{ω}(a) = ω[σ_{α}(a^{ω^{−1}})] = ω[αξ_{π_{α}}^{−1}(a^{ω^{−1}})]
= ω[αz_{α}(ω^{−1})ξ_{π_{α}}^{−1}(a)^{ω^{−1}}]
= ω[α^{ω^{−1}}ξ_{π_{α}}^{−1}(a)^{ω^{−1}}]
= αξ_{π_{α}}^{−1}(a)
= σ_{α}(a).

Hence ψ(π_{α}) = 0 as asserted.

To complete the proof of the surjectivity of Υ_{R}, we note that it follows from the definition of σ_{α} that we have r(σ′_{α}) = α. Hence r(π_{α}) = [α] ∈ H(A), and so Υ_{R} is surjective as claimed.

We now show that Υ_{R} is injective. Suppose that α, β ∈ H(A) with [α] = [β] ∈ H(A). Then it is easy to check that the isomorphism

Ξ_{β} ∘ Ξ_{α}^{−1} : G_{α} ⟶^{∼} G_{β}

induces an isomorphism G_{α}(K^{c}) ⟶^{∼} G_{β}(K^{c}) of Ω_{K}-modules. This implies that the G-torsors π_{α} : Z_{α} → Spec(R) and π_{β} : Z_{β} → Spec(R) are isomorphic. This completes the proof of the theorem. □

Remark 2.3. Suppose that R = K. Then it is not hard to check that (using the notation established in the proof of Theorem 2.2) the map Ω_{K} → Γ defined by ω ↦ r(s_{π})^{ω}r(s_{π})^{−1} is an Ω_{K}-cocycle representing [π] ∈ H^{1}(K, G). □

If R is a local ring, then Pic(G^{∗}) = 0, and so Ker(ψ) = H^{1}(R, G). The following result is a direct corollary of Theorem 2.2. It gives a description of the flat cohomology of G over Spec(R) in terms of resolvends. We remark that a rather different (but related) idelic description of torsors of G over a Dedekind domain has been given by M. Taylor (see [30],[10, Chapter 3, §4]) and by N. Byott (see [9, §3]).

Recall that N denotes the exponent of G.

Corollary 2.4. Suppose that R is a local ring.

(a) There is an isomorphism

Υ_{R} : H^{1}(R, G) ⟶^{∼} H(A).

(b) The map [π] ↦ r(s_{π})^{N} induces a homomorphism

η_{R} : H^{1}(R, G) → \frac{A^{×}}{(A^{×})^{N}} .

□

10 A. AGBOOLA

Remark 2.5. Suppose that R = K, and for each γ^{∗} ∈ Γ^{∗}, write K[γ^{∗}] for the smallest extension of K whose absolute Galois group fixes γ^{∗}. Let Γ^{∗}\Ω_{K} denote a set of representatives of Ω_{K}-orbits of Γ^{∗}. Then, via an argument virtually identical to that given in [1, Lemma 3.3], it may be shown that the Wedderburn decomposition of the K-algebra A is given by

A ≃ (K^{c}Γ)^{Ω_{K}} ≃ ∏_{γ^{∗}∈Γ^{∗}\Ω_{K}} K[γ^{∗}].  (2.3)
□

Proposition 2.6. Suppose that R = K, and that G^{∗} is a constant group scheme over Spec(K). Then A ≃ Map(Γ^{∗}, K), and the map η_{K} of Corollary 2.4(b) induces an isomorphism

η_{K} : H^{1}(K, G) ⟶^{∼} Hom(Γ^{∗}, K^{×}/(K^{×})^{N}) ⊂ Map(Γ^{∗}, K^{×}/(K^{×})^{N}) ≃ A^{×}/(A^{×})^{N}.

Proof. See [5, Corollary 3.4]. □

Proposition 2.7. Suppose that R = K, and let L be any algebraic extension of K. Then the following diagram is commutative:

\begin{matrix}H^{1}(K, G) & ⟶^{Υ_{K}} & H(A) \\ ^{Res}↓ & & ↓ \\ H^{1}(L, G) & ⟶^{Υ_{L}} & H(A_{L}). \end{matrix}  (2.4)

Here the left-hand vertical arrow is the restriction map on cohomology, and the right-hand vertical arrow is the homomorphism induced by the inclusion map i : H(A) → H(A_{L}).

Proof. Let π : X → Spec(K) be any G-torsor, and let s : A ⟶^{∼} L_{π} be any trivialisation of L_{π}. Then it follows via a straightforward computation that the Ω_{L}-cocycle associated to i(r(s)) is equal to the restriction of the Ω_{K}-cocycle associated to r(s) (cf. Remark 2.3). □

Remark 2.8. Suppose that R = K, and that π ∈ Ker(η_{K}). Let r(s_{π}) ∈ H(A) be any resolvend associated to π. Then r(s_{π})^{N} = α^{N} ∈ A^{×N}, and so r(α^{−1}s_{π})^{N} = 1. Hence r(α^{−1}s_{π}) ∈ A_{K(μ_{N})}^{×}, and so Proposition 2.7 implies that π lies in the kernel of the restriction map

Res_{K/K(μ_{N})} : H^{1}(K, G) → H^{1}(K(μ_{N}), G).

Conversely, if π ∈ Res_{K/K(μ_{N})}, then, since π is trivialised over K(μ_{N}), it follows that r(s_{π}) ∈ A_{K(μ_{N})}^{×} for any choice of s_{π}. We therefore deduce from Corollary 2.4(b) that r(s_{π})^{N} ∈ A^{×}∩A_{K(μ_{N})}^{×N}. Hence, if A^{×N} = A^{×} ∩ A_{K(μ_{N})}^{×N}, then r(s_{π})^{N} ∈ A^{×N}, and so π ∈ Ker(η_{K}). □

Suppose now that L is a finite Galois extension of K with [L : K] = n, say. Let ω_{1}, …, ω_{n} be a transversal of Ω_{L} in Ω_{K}. Then we have a norm homomorphism

N_{L/K} : A_{L^{c}} → A_{K^{c}};  a ↦ ∏_{i=1}^{n} a^{ω_{i}}.  (2.5)

GALOIS MODULES AND p-ADIC REPRESENTATIONS 11

This induces homomorphisms (which we denote by the same symbol)

N_{L/K} : H(A_{L}) → H(A_{K}), N_{L/K} : H(A_{L}) → H(A_{K}).

Proposition 2.9. The following diagram is commutative:

\begin{matrix}H^{1}(L, G) & ⟶^{Υ_{L}} & H(A_{L}) \\ ^{Cores}_{L/K} ↓ & & ↓^{N_{L/K}} \\ H^{1}(K, G) & ⟶^{Υ_{K}} & H(A_{K}), \end{matrix}  (2.6)

where the left-hand vertical arrow is the corestriction map on cohomology.

Proof. Let π_{L} : X_{L} → Spec(L) be any G-torsor and let s_{π_{L}} : A_{L} ⟶^{∼} L_{π_{L}} be any trivialisation of L_{π_{L}}. Then it follows via a straightforward computation that the Ω_{K}-cocycle associated to N_{L/K}(r(s_{π_{L}})) is equal to the corestriction of the Ω_{L}-cocycle associated to r(s_{π_{L}}) (cf. Remark 2.3). □

Let (G_{n})_{n≥1} be a p-divisible group over Spec(R). For each n, set G_{n}^{∗} = Spec(A_{n}), and set Γ_{n} := G_{n}(R^{c}), Γ_{n}^{∗} := G_{n}^{∗}(R^{c}). We write p_{n} := [p] : G_{n} → G_{n−1} for the multiplication-by-p map, and we use the same symbol to denote the induced map H^{1}(Spec(R), G_{n}) → H^{1}(Spec(R), G_{n−1}).

The map p_{n} induces a dual inclusion map p_{n}^{D} : G_{n−1}^{∗} → G_{n}^{∗}, and we may identify A_{n−1} with the pullback (p_{n}^{D})^{∗}A_{n} of A_{n} via p_{n}^{D}. Thus (via pullback) p_{n}^{D} induces a homomorphism q_{n} : A_{n} → A_{n−1} which extends to a homomorphism (which we denote by the same symbol) A_{n,K^{c}} → A_{n−1,K^{c}}. It is easy to check that q_{n}(H(A_{n})) ⊆ H(A_{n−1}), and that q_{n}(Γ_{n} · A_{n}^{×}) ⊆ Γ_{n−1} · A_{n−1}^{×}.

Proposition 2.10. Suppose that R is a local ring. Then the following diagram is commutative:

H^{1}(Spec(R), G_{n}) ⟶^{Υ_{R,n}} H(A_{n})
p_{n}↓  ↓^{q_{n}}
H^{1}(Spec(R), G_{n−1}) ⟶^{Υ_{R,n−1}} H(A_{n−1}).

(2.7)

Proof. Suppose that π_{n} : X_{n} → Spec(R) is any G_{n}-torsor, and let s_{π_{n}} : A_{n} ⟶^{∼} L_{π_{n}} be any trivialisation of L_{π_{n}}. Set π_{n−1} := p_{n}(π_{n}). Then it follows via the functoriality of Waterhouse’s construction in [31] that there is a natural identification L_{π_{n−1}} ≃ (p_{n}^{D})^{∗}L_{π_{n}}. Consider the trivialisation s_{π_{n−1}} := (p_{n}^{D})^{∗}s_{π_{n}} : A_{n−1} ⟶^{∼} L_{π_{n−1}} of L_{π_{n−1}} obtained by pulling back s_{π_{n}} along p_{n}^{D}. We have

Υ_{R,n−1}(π_{n−1}) = [r(s_{π_{n−1}})] = [r((p_{n}^{D})^{∗}s_{π_{n}})] = [q_{n}(r(s_{π_{n}})].

This establishes the result. □

Suppose now that R = K. Fix a positive integer n, and assume that G_{n}^{∗} is a constant group scheme. Then we have

A_{n}^{×}/(A_{n}^{×})^{p^{n}} ≃ Map(Γ_{n}^{∗}, K^{×}/(K^{×})^{p^{n}}).

For each element P : Spec(K) → G_{n}^{∗} in Γ_{n}^{∗}, write χ_{P} : G_{n} → μ_{p^{n}} for the corresponding character of G_{n}. Then χ_{P} induces a homomorphism (which we denote by the same symbol):

χ_{P} : H^{1}(K, G_{n}) → H^{1}(K, μ_{p^{n}}); [π] ↦ [π(χ_{P})].

Write

ev_{P} : A_{n}^{×}/(A_{n}^{×})^{p^{n}} ≃ Map(Γ_{n}^{∗}, K^{×}/(K^{×})^{p^{n}}) → K^{×}/(K^{×})^{p^{n}}

for the map a ↦ a(P) given by ‘evaluation at P ’. The following result shows how to describe the map η_{K} of Corollary 2.4 in terms of Kummer theory.

Proposition 2.11. Let the hypotheses and notation be as above. Then the following diagram is commutative:

H^{1}(K, G_{n}) ⟶^{χ_{P}} H^{1}(K, μ_{p^{n}})
η_{K} ↓ ↑^{Kummer}
A_{n}^{×}/(A_{n}^{×})^{p^{n}} ⟶^{ev_{P}} K^{×}/(K^{×})^{p^{n}}.

(2.8)

(Here the right-hand vertical arrow is the natural isomorphism afforded by Kummer theory.)

Proof. See [5, Proposition 3.2]. □

Corollary 2.12. Let the hypotheses and notation be as above. For each integer n, let

r_{n} : Hom(Γ_{n}^{∗}, K^{×}/(K^{×})^{p^{n}}) → Hom(Γ_{n−1}^{∗}, K^{×}/(K^{×})^{p^{n−1}})

be the homomorphism given by f ↦ f|_{Γ_{n−1}^{∗}}. Then the following diagram commutes:

H^{1}(K, G_{n}) ⟶^{η_{K}} Hom(Γ_{n}^{∗}, K^{×}/(K^{×})^{p^{n}})
p_{n}↓ ↓^{r_{n}}
H^{1}(K, G_{n−1}) ⟶^{η_{K}} Hom(Γ_{n−1}^{∗}, K^{×}/(K^{×})^{p^{n−1}}).

Proof. Suppose that P ∈ Γ_{n−1}^{∗}. Then, by definition, the following diagram commutes:

\begin{matrix}H^{1}(K, G_{n}) & ⟶^{χ_{P}} & H^{1}(K, μ_{p^{n}}) ≃ K^{×}/(K^{×})^{p^{n}} \\ ^{p_{n}}↓ & & ↓_{red} \\ H^{1}(K, G_{n−1}) & ⟶^{χ_{P}} & H^{1}(K, μ_{p^{n−1}}) ≃ K^{×}/(K^{×})^{p^{n−1}}. \end{matrix}  (2.9)

(Here the right-hand vertical arrow denotes the natural reduction map.) The result now follows from Propositions 2.11 and 2.6. □

GALOIS MODULES AND p-ADIC REPRESENTATIONS 13

3. Selmer conditions and Galois structure

In this section we shall apply the results of §2 to explain how resolvends may be used to impose local conditions on G-torsors. This enables us to define certain Selmer groups. We then show that there are natural homomorphisms from these Selmer groups into suitable locally free classgroups. These generalise the class invariant homomorphisms described at the begining of the introduction to this paper.

In what follows, F will denote either a number field or a local field (depending upon the context), with ring of integers O_{F}. We suppose given a finite, flat, commutative group scheme G over Spec(F), and we let G^{∗} = Spec(A). As usual, we set Γ = G(F^{c}), and we write 𝔐 for the unique O_{F} -maximal order contained in A.

Let 𝔄 denote any O_{F} -algebra in A satisfying the following conditions:

(i) F · 𝔄 = A;

(ii) Γ ⊆ 𝔄_{O_{F^{c}}};

(iii) If F is a number field, then 𝔄_{v} = 𝔐_{v} for all but finitely many places v of F.

Note that we do not assume that 𝔄 is finitely generated over O_{F}.

Set

H(𝔄) := \{α ∈ 𝔄_{O_{F^{c}}}^{×} | \frac{α^{ω}}{α} ∈ Γ for all ω ∈ Ω_{K}.\} .

We shall be interested in using the groups H(𝔄) and H(A) to impose Selmer-type conditions on elements of H^{1}(F, G). The following definition is motivated by Corollary 2.4(a). (Recall that the isomorphism Υ_{F} below was defined in Corollary 2.4.)

Definition 3.1. Suppose that F is a local field. Then we define the subgroup H_{𝔄}^{1}(F, G) of H^{1}(F, G) by:

H_{𝔄}^{1}(F, G) = \{x ∈ H^{1}(F, G)|Υ_{F} (x) ∈ \frac{H(𝔄) · A^{×}}{Γ · A^{×}} ⊆ \frac{H(A)}{Γ · A^{×}} = H(A)\} .

Hence a G-torsor π : X → Spec(F) lies in H_{𝔄}^{1}(F, G) if and only if there exists a trivialisation s_{π} : A ⟶^{∼} L_{π} with r(s_{π}) ∈ H(𝔄) ⊆ H(A). The resolvend r(s_{π}) of such a trivialisation is welldefined up to multiplication by an element of Γ · 𝔄^{×}. □

Definition 3.2. If F is a number field, then we define H_{𝔄}^{1}(F, G) by

H_{𝔄}^{1}(F, G) = Ker [H^{1}(F, G) → ∏_{v∤∞} \frac{H^{1}(F_{v}, G)}{H_{𝔄_{v}}^{1}(F_{v}, G)}] .

□

Remark 3.3. Suppose that F is either a number field or a local field, and assume that π ∈ H^{1}(F, G) lies in Ker(η_{F}). Then it follows from the discussion in Remark 2.8 that there exists a

14 A. AGBOOLA

resolvend r(s_{π}) ∈ H(A) associated to π such that r(s_{π})^{N} = 1. Hence r(s_{π}) ∈ H(𝔐), and so π ∈ H_{𝔐}^{1}(F, G). □

Now suppose that F is a number field. Let J_{f}(A) denote the group of finite ideles of A, i.e. J_{f}(A) is the restricted direct product of the groups A_{v}^{×} with respect to the subgroups 𝔐_{v}^{×} for v ∤ ∞. We view A^{×} as being a subgroup of J_{f}(A) via the obvious diagonal embedding. Write Cl(𝔄) for the locally free classgroup of 𝔄. Thus, Cl(𝔄) is the Grothendieck group of locally free 𝔄-modules of finite rank, and it may be identified with the group Pic(Spec(𝔄)). Then it is a standard result from the theory of classgroups (see e.g. [13, §52]) that there is a natural isomorphism

Cl(𝔄) ≃ \frac{J_{f}(A)}{(∏_{v∤∞} 𝔄_{v}^{×}) · A^{×}} .  (3.1)

Theorem 3.4. Let F be a number field.

(a)There is a natural homomorphism

ϕ_{𝔄} : H_{𝔄}^{1}(F, G) → Cl(𝔄).

(b) The isomorphism

Υ_{F} : H^{1}(F, G) ⟶^{∼} H(A)

of Corollary 2.4 induces an isomorphism

Υ_{F,𝔄} : Ker(ϕ_{𝔄}) ⟶^{∼} H(𝔄) ⊆ H(A).

(c) We have Ker(η_{F}) ⊆ Ker(ϕ_{𝔐}).

Proof. (a) Suppose that π : X → Spec(F) is a G-torsor with [π] ∈ H_{𝔄}^{1}(F, G), and let ξ_{π} : L_{π} ⊗_{F} F^{c} ⟶^{∼} A_{F^{c}} be a splitting isomorphism for π. Fix a trivialisation s_{π} : A ⟶^{∼} L_{π}. Then the resolvend r(s_{π}) ∈ H(A) (defined using ξ_{π}) is well-defined up to multiplication by an element of A^{×}.

For each finite place v of F, write π_{v} for the torsor X⊗_{F} F_{v} → Spec(F_{v}). Since [π_{v}] ∈ H_{𝔄_{v}}^{1}(F_{v}, G), we may choose a trivialisation t_{π_{v}} : A_{v} ⟶^{∼} L_{π_{v}} whose resolvend r(t_{π_{v}}) (computed using the local completion ξ_{π_{v}} of ξ_{π} at v) satisfies r(t_{π_{v}}) ∈ H(𝔄_{v}). Then r(t_{π_{v}}) is well-defined up to multiplication by an element of 𝔄_{v}^{×}, and r(t_{π_{v}})r(s_{π})^{−1} ∈ A_{v}^{×}. We also note that r(t_{π_{v}})r(s_{π})^{−1} ∈ A_{v}^{×} is in fact independent of the choice of the splitting isomorphism ξ_{π}, because changing ξ_{π} alters both r(s_{π}) and r(t_{π_{v}}) by multiplication by the same element of Γ. Furthermore, for all but finitely many places v, both r(s_{π}) and r(t_{π_{v}}) lie in H(𝔐_{v}), and so r(t_{π_{v}})r(s_{π})^{−1} ∈ 𝔐_{v}^{×} for all such v.

It therefore follows that the element (r(t_{π_{v}})r(s_{π})^{−1})_{v} lies in J_{f}(A), and that its image in

\frac{J_{f}(A)}{(∏_{v∤∞} 𝔄_{v}^{×}) · A^{×}} ≃ Cl(𝔄)

is well-defined. We define

ϕ_{𝔄}(π) = [(r(t_{π_{v}})r(s_{π})^{−1})_{v}] ∈ Cl(𝔄).

GALOIS MODULES AND p-ADIC REPRESENTATIONS 15

We now show that ϕ_{𝔄} is a homomorphism. Suppose that π_{i} : X_{i} → Spec(F) (i = 1, 2) are G-torsors. For each i, fix a splitting isomorphism ξ_{π_{i}} of π_{i}, and let s_{π_{i}} and t_{πi,v} (v ∤ ∞) be defined analogously to s_{π} and t_{π} above. Then it follows from the functoriality of Waterhouse’s construction that there is a natural isomorphism L_{π_{3}} ≃ L_{π_{1}} ⊗_{A} L_{π_{2}}. Thus, if we set

ξ_{π_{3}} := ξ_{π_{1}} ⊗ ξ_{π_{2}} : (L_{π_{1}} ⊗_{A} L_{π_{2}}) ⊗_{F} F^{c} ⟶^{∼} A_{F^{c}},

s_{π_{3}} := s_{π_{1}} ⊗ s_{π_{2}} : L_{π_{1}} ⊗_{A} L_{π_{2}} ⟶^{∼} A,

t_{π_{3,v}} := t_{π_{1,v}} ⊗ t_{π_{2,v}} : L_{π_{1,v}} ⊗_{A_{v}} L_{π_{2,v}} ⟶^{∼} A_{v},

then r(s_{π_{3}}) = r(s_{π_{1}})r(s_{π_{2}}), and r(t_{π_{3,v}}) = r(t_{π_{1,v}})r(t_{π_{2,v}}) (where, for each i the resolvends r(s_{π_{i}}) and r(t_{π_{i,v}}) are defined using the splitting isomorphisms ξ_{π_{i}} and ξ_{π_{i,v}} respectively).

Hence it follows that

ϕ_{𝔄}(π_{3}) = [(r(t_{π_{3,v}})r(s_{π_{3}})^{−1})_{v}]
= [(r(t_{π_{1,v}})r(s_{π_{1}})^{−1})_{v}][(r(t_{π_{2,v}})r(s_{π_{2}})^{−1})_{v}]
= ϕ_{𝔄}(π_{1})ϕ_{𝔄}(π_{2}),

as asserted.

(b) Suppose that π : X → Spec(F) is a G-torsor satisfying r(s_{π}) ∈ H(𝔄) for some choice of trivialisation s_{π} : A ⟶^{∼} L_{π} of L_{π}. Write s_{π,v} : A_{v} ⟶^{∼} L_{π_{v}} for the trivialisation of L_{π_{v}} induced by s_{π}. Then r(s_{π,v}) ∈ H(𝔄_{v}) for all finite places v of F. Hence π ∈ H_{𝔄}^{1}(F, G), and we may take t_{π_{v}} = s_{π,v} in the definition of ϕ_{𝔄}(π) given in part (a). This in turn gives ϕ_{𝔄}(π) = 0.

Now suppose conversely that π ∈ H_{𝔄}^{1}(F, G) with ϕ_{𝔄}(π) = 0. Then for any trivialisations s_{π} and r(t_{π_{v}}) chosen as in part (a), we have

(r(t_{π_{v}})r(s_{π})^{−1})_{v} = α · (β_{v})_{v} ∈ A^{×} · ∏_{v∤∞}𝔄_{v}^{×}.

Hence if we replace s_{π} by s′_{π} := αs_{π}, then

(r(t_{π_{v}})r(s′_{π})^{−1})_{v} = (β_{v})_{v} ∈ ∏_{v∤∞}𝔄_{v}^{×} ⊆ ∏_{v∤∞}H(𝔄_{v}).

This implies that r(s′_{π}) ∈ H(𝔄_{v}) for each place v ∤ ∞, and so it follows that r(s′_{π}) ∈ H(𝔄). Hence Υ_{F} (π) ∈ H(𝔄) ⊆ H(A), as claimed.

(c) This follows directly from Remark 3.3 and part (b) above. □

Example 3.5. Suppose that F is a number field. Let G be a finite, flat, commutative group scheme over Spec(O_{F}), with generic fibre G. Let G^{∗} = Spec(𝔄) denote the Cartier dual of G^{∗}; then

16 A. AGBOOLA

G^{∗} = Spec(A) is the generic fibre of G^{∗}. Corollary 2.4(a) implies that H_{𝔄_{v}}^{1}(F_{v}, G) = H^{1}(O_{F_{v}}, G) for each finite place v of F, and so it follows that

H_{𝔄}^{1}(F, G) = H^{1}(O_{F} , G).

Hence we obtain a description of the flat Selmer group of G in terms of resolvends. In this case, the map

ϕ_{𝔄} : H^{1}(O_{F} , G) → Cl(𝔄) ≃ Pic(G^{∗})

is the same as the class invariant homomorphism (1.1) for the group G.

Also, we have

H^{1}(O_{F} , G) = H_{𝔄}^{1}(F, G) ⊆ H_{𝔐}^{1}(F, G),

and Spec(𝔐) is the normalisation of G^{∗}. The restriction of the homomorphism

ϕ_{𝔐} : H_{𝔐}^{1}(F, G) → Cl(𝔐) ≃ Pic(Spec(𝔐))

to H^{1}(O_{F} , G) is the same as the class invariant homomorphism (1.2). □

Remark 3.6. Suppose that F is a number field, and let N denote the exponent of G. If v is a place of F with v ∤ N, set

H_{f}^{1}(F_{v}, G) := Ker [H^{1}(F_{v}, G) → H^{1}(F_{v}^{nr}, G)] ,

where F_{v}^{nr} is the maximal unramified extension of F_{v} in a fixed algebraic closure of F_{v}.

If π ∈ H_{f}^{1}(F_{v}, G), and r(s_{π}) is any resolvend asociated to π, then Proposition 2.7 implies that r(s_{π}) ∈ A_{v,F_{v}^{nr}}^{×}. Since F_{v}^{nr}/F_{v} is unramified, it follows (via considering the Wedderburn decomposition (2.3) of A_{v}) that there exists α ∈ A_{v}^{×} such that α^{−1}r(s_{π}) = r(α^{−1}s_{π}) ∈ 𝔐_{v,O_{F{nr \atop v}}}^{×}. This implies that π ∈ H_{𝔐}^{1}(F, G), and so

H_{f}^{1}(F_{v}, G) ⊆ H_{𝔐}^{1}(F_{v}, G).

Suppose further that G is unramified at v. Then G := Spec(𝔐_{v}) is a finite, flat, commutative O_{F_{v}}-group scheme, and it is a standard result that H_{f}^{1}(F_{v}, G) = H^{1}(O_{F_{v}}, G). We therefore deduce that in this case, we have H_{f}^{1}(F_{v}, G) = H_{𝔐}^{1}(F_{v}, G). □

Remark 3.7. It is not difficult to define refinements of the homomorphism ϕ_{𝔄} taking values in relative algebraic K-groups as in [4], or in Arakelov Picard groups as in [5]. However, for the sake of brevity, we shall not go into this here. □

Now suppose that F is a number field, and let L/F be a finite extension. It is not hard to check that the homomorphism N_{L/F} of (2.5) induces a homomorphism

N_{L/K} : Cl(𝔄_{O_{L}}) → Cl(𝔄).

GALOIS MODULES AND p-ADIC REPRESENTATIONS 17

Proposition 3.8. If F is a number field, and L/F is a finite extension, then the following diagram is commutative:

H_{𝔄_{O_{L}}}^{1}(L, G) ⟶^{ϕ_{𝔄_{O_{L}}}} Cl(𝔄_{O_{L}})
Cores_{L/F} ↓  ^{N_{L/F}} ↓
H_{𝔄}^{1}(F, G) ⟶^{ϕ_{𝔄}} Cl(𝔄).

(3.2)

Proof. Let π : X → Spec(L) be a G-torsor with [π] ∈ H_{𝔄_{O_{L}}}^{1}(L, G), and let s_{π} : A_{L} ⟶^{∼} L_{π} be any trivialisation of L_{π}. For each finite place v of L, let t_{π_{v}} : A_{L_{v}} ⟶^{∼} L_{π_{v}} be a trivialisation of L_{π_{v}} satisfying r(t_{π_{v}}) ∈ H(𝔄_{O_{Lv}}), Then

ϕ_{𝔄_{O_{L}}}(π) = [(r(t_{π_{v}})r(s_{π})^{−1})_{v}] ∈ Cl(𝔄_{O_{L}}).

The result now follows via a similar argument to that used in the proof of Proposition 2.9. □

For the rest of this paper, we shall mainly be concerned with the special cases in which 𝔄 = 𝔐 or 𝔄 = 𝔐 ⊗_{O_{F}} O_{F} [1/p] := 𝔐^{{p}}. We identify Pic(Spec(𝔐)) and Pic(Spec(𝔐^{{p}})) with the locally free classgroups Cl(𝔐) and Cl(𝔐^{{p}}) of 𝔐 and 𝔐^{{p}} respectively. We set

H_{u}^{1}(F, G) := H_{𝔐}^{1}(F, G), H_{u,{p}}^{1}(F, G) := H_{𝔐^{{p}}}^{1}(F, G),

and we write

ϕ : H_{u}^{1}(F, G) → Cl(𝔐), ϕ^{{p}} : H_{u,{p}}^{1}(F, G) → Cl(𝔐^{{p}})

for the homomorphisms given by Theorem 3.4.

Proposition 3.9. Let F be a number field, and suppose that G^{∗} is constant over Spec(F).

(i) If v is any finite place of F, then the isomorphism η_{F_{v}} of Proposition 2.6 induces an isomorphism

H_{u}^{1}(F_{v}, G) ⟶^{∼} Hom(Γ^{∗}, O_{F_{v}}^{×}/(O_{F_{v}}^{×})^{N}).

(ii) The isomorphism η_{F} induces isomorphisms

Ker(ϕ) ⟶^{∼} Hom(Γ^{∗}, O_{F}^{×}/(O_{F}^{×})^{N}),

Ker(ϕ^{{p}}) ⟶^{∼} Hom(Γ^{∗}, O_{F} [1/p]^{×}/(O_{F} [1/p]^{×})^{N})

Proof. Since G^{∗} is constant over Spec(F), we have

A ≃ Map(Γ^{∗}, F),  𝔐 ≃ Map(Γ^{∗}, O_{F}),  𝔐^{{p}} ≃ Map(Γ^{∗}, O_{F} [1/p]).

Proposition 2.6 implies that we have isomorphisms

H^{1}(F, G) ≃ Hom(Γ^{∗}, F^{×}/(F^{×})^{N}),  (3.3)
H^{1}(F_{v}, G) ≃ Hom(Γ^{∗}, F_{v}^{×}/(F_{v}^{×})^{N}).  (3.4)

Hence (i) follows from (3.4) and the definition of H^{1}(F_{v}, G), while (ii) follows from (3.3) together with Theorem 3.4(b). □

4. p-adic representations

In this section, we shall apply our previous work to the situation described in the introduction. We first recall the relevant notation.

Let F be a number field and V be a d-dimensional Q_{p}-vector space. Suppose that ρ : Ω_{F} → GL(V) is a continuous representation which is ramified at only finitely many primes of F. We set V^{∗} := Hom_{Q_{p}}(V, Q_{p}(1)), and we write ρ^{∗} : Ω_{F} → GL(V^{∗}) for the corresponding representation of Ω_{F}. Let T ⊆ V be any Ω_{F} -stable lattice, and write T^{∗} := Hom_{Z_{p}}(T, Z_{p}(1)). For each positive integer n, we define finite group schemes G_{n} and G_{n}^{∗} over Spec(F) by

G_{n}(F^{c}) := Γ_{n} = p^{−n}T/T; G_{n}^{∗}(F^{c}) := Γ_{n}^{∗} = p^{−n}T^{∗}/T^{∗}.

Then G_{n}^{∗} is the Cartier dual of G_{n} with G_{n}^{∗} = Spec(A_{n}) for the Hopf algebra A_{n} = (F^{c}Γ_{n})^{Ω_{F}} over F.

Recall that q_{n} : A_{n} → A_{n−1} is the homomorphism induced by the dual p_{n}^{D} of the multiplicationby-p map p_{n} : G_{n} → G_{n−1}. Suppose that, for each n, we are given an O_{F} -algebra 𝔄_{n} ⊆ A_{n} satisfying the conditions stated at the begining of Section 3. Suppose also that q_{n}(𝔄_{n}) = 𝔄_{n−1} for each n. Then it is easy to check that q_{n} induces homomorphisms

H(A_{n}) → H(A_{n−1}), and H(𝔄_{n,v}) → H(𝔄_{n−1,v})

for each finite place v of F. This implies that the natural maps

H^{1}(F_{v}, G_{n}) → H^{1}(F_{v}, G_{n−1}), H^{1}(F, G_{n}) → H^{1}(F, G_{n−1})

induce homomorphisms

H_{𝔄_{n,v}}^{1}(F_{v}, G_{n}) → H_{𝔄_{n−1,v}}^{1}(F_{v}, G_{n−1}), H_{𝔄_{n}}^{1}(F, G_{n}) → H_{𝔄_{n−1}}^{1}(F, G_{n−1})

via restriction.

Set 𝔄(T) := \underleftarrow{lim} 𝔄_{n} and 𝔄_{v}(T) := \underleftarrow{lim} 𝔄_{n,v} (where the inverse limits are taken with respect to the maps q_{n}), and let

H(𝔄_{v}(T)) := \{α ∈ 𝔄_{v}(T)_{O_{F^{c}_{v}}}^{×} | \frac{α^{ω}}{α} ∈ T for all ω ∈ Ω_{F_{v}^{c}}.\} ,
H(𝔄_{v}(T)) := \frac{H(𝔄_{v}(T))}{T · 𝔄_{v}(T)^{×}} .

Define H(𝔄(T)) and H(𝔄(T)) in a similar way. Write

H_{𝔄_{v}(T)}^{1}(F_{v}, T) := \underleftarrow{lim} H_{𝔄_{n,v}}^{1}(F_{v}, G_{n}), H_{𝔄(T)}^{1}(F, T) := \underleftarrow{lim} H_{𝔄_{n}}^{1}(F, G_{n}).

GALOIS MODULES AND p-ADIC REPRESENTATIONS 19

Proposition 4.1. For each finite place v of F, we have

H_{𝔄_{v}(T)}^{1}(F_{v}, T) ≃ \frac{H(𝔄_{v}(T))}{T · 𝔄_{v}^{×}(T)}.

Proof. It follows from the definition of H_{𝔄_{n,v}}^{1}(F_{v}, G_{n}) (see Definition 3.1) that, for each n, there is an exact sequence

1 → G_{n} · 𝔄_{n,v}^{×} → H(𝔄_{n,v}) → H_{𝔄_{n,v}}^{1}(F_{v}, G_{n}) → 0.

Passing to inverse limits, and using the fact that the inverse system {G_{n} · 𝔄_{n,v}^{×}}_{n} satisfies the Mittag-Leffler condition yields

H_{𝔄_{v}(T)}^{1}(F_{v}, T) ≃ \frac{\underleftarrow{lim} H(𝔄_{n,v})}{T · 𝔄_{v}(T)^{×}}.

It follows easily from the definitions that

H(𝔄_{v}(T)) = \underleftarrow{lim} H(𝔄_{n,v}),

and this implies the result. □

It is easy to check that p_{n}^{D} induces pullback homomorphisms

(p_{n}^{D})^{∗} : Cl(𝔄_{n}) → Cl(𝔄_{n−1}).

Let

ϕ_{𝔄_{n}} : H_{𝔄_{n}}^{1}(F, G_{n}) → Cl(𝔄_{n})

denote the natural homomorphism afforded by Theorem 3.4.

Theorem 4.2. The following diagram is commutative:

\begin{matrix}H_{𝔄_{n}}^{1}(F, G_{n}) & ⟶^{ϕ_{𝔄n}} & Cl(𝔄_{n}) \\ ^{p_{n}}↓ & & ↓^{(p_{n}^{D})^{∗}} \\ H_{𝔄_{n−1}}^{1}(F, G_{n−1}) & ⟶^{ϕ_{𝔄_{n−1}}} & Cl(𝔄_{n−1}), \end{matrix}  (4.1)

Proof. The proof of this is similar to that of Proposition 2.10. Let π_{n} : X_{n} → Spec(F) be a G_{n}-torsor with [π_{n}] ∈ H_{𝔄_{n}}^{1}(F, G_{n}), and write π_{n−1} := p_{n}(π_{n}). Fix a splitting isomorphism ξ_{π_{n}} of π_{n}. Let s_{π_{n}} : A_{n} ⟶^{∼} L_{π_{n}} be any trivialisation of L_{π_{n}}, and for each finite place v of F, let t_{π_{n},v} : A_{n,v} ⟶^{∼} L_{π_{n},v} be a trivialisation of L_{π_{n},v} satisfying r(t_{π_{n},v}) ∈ H(𝔄_{n,v}) (where r_{π_{n},v} is defined using the splitting isomorphism ξ_{π_{n},v} of π_{n,v} induced by ξ_{π_{n},v} .

Then, via functoriality, we have that (p_{n}^{D})^{∗}L_{π_{n}} ≃ L_{π_{n−1}}. The pullbacks (p_{n}^{D})^{∗}s_{π_{n}} and (p_{n}^{D})^{∗}t_{π_{n},v} of s_{π_{n}} and t_{π_{n},v} along p_{n}^{D} give trivialisations of L_{π_{n−1}} and L_{π_{n−1},v} respectively, while (p_{n}^{D})^{∗}ξ_{π_{n}} and (p_{n}^{D})^{∗}ξ_{π_{n},v} are splitting isomorphisms of π_{n−1} and π_{n−1,v} respectively. We have that

r((p_{n}^{D})^{∗}t_{π_{n},v}) = q_{n,v}(t_{π_{n},v}) ∈ H(𝔄_{n−1,v}),

20 A. AGBOOLA

where r((p_{n}^{D})^{∗}t_{π_{n},v}) is defined using (p_{n}^{D})^{∗}ξ_{π_{n},v}. The result now follows from the definitions of ϕ_{𝔄_{n}} and ϕ_{𝔄_{n−1}}. □

Set

Cl(𝔄(T)) := \underleftarrow{lim} Cl(𝔄_{n}).

Then passing to inverse limits over the diagrams (4.1) yields a homomorphism

Φ_{𝔄(T)} : H_{𝔄(T)}^{1}(F, T) → Cl(𝔄(T)).  (4.2)

Proposition 4.3. Suppose that L/F is a finite extension. Then the map N_{L/K} (see (2.5)) induces a homomorphism

N_{L/K} : Cl(𝔄(T)_{O_{L}}) → Cl(𝔄(T)),

and the following diagram is commutative:

H_{𝔄(T)_{O_{L}}}^{1}(F, G) ⟶^{ϕ_{𝔄(T)_{O_{L}}}} Cl(𝔄(T)_{O_{L}})
Cores_{L/F} ↓  ^{N_{L/F}} ↓
H_{𝔄(T)}^{1}(F, G) ⟶^{ϕ_{𝔄(T)}} Cl(𝔄(T)).

(4.3)

Proof. This follows from Proposition 3.8. □

Proposition 4.4. There is an isomorphism

Υ_{F,𝔄(T)} : Ker(Φ_{𝔄(T)}) ⟶^{∼} H(𝔄(T)) ⊆ H(A(T)).

Proof. This follows easily from Theorem 3.4(b). □

Write

H_{u}^{1}(F, T) := \underleftarrow{lim} H_{u}^{1}(F, G_{n}), H_{u,\{p\}}^{1}(F, T) := \underleftarrow{lim} H_{u,\{p\}}^{1}(F, G_{n}).

Then (4.2) yields a homomorphism

Φ_{F} := Φ_{𝔐(T)} : H_{u}^{1}(F, T) → Cl(𝔐(T)).  (4.4)

From Remark 3.6, we see that

H_{f,{p}}^{1}(F, T) ⊆ H_{u,{p}}^{1}(F, T).

Hence, restricting Φ_{𝔐^{{p}}(T)} to H_{f,{p}}^{1}(F, T) yields a homomorphism

Φ_{F}^{{p}} : H_{f,{p}}^{1}(F, T) → Cl(𝔐^{{p}}(T)).

Remark 4.5. Let S be any finite set of places of F containing all places lying above p, as well as all places at which T is ramified, and let F^{S}/F denote the maximal extension of F which is unramified outside S. Then it follows from the definitions that H_{u}^{1}(F, T) ⊆ H^{1}(F^{S}/F, T), and so we deduce that H_{u}^{1}(F, T) is always a finitely generated Z_{p}-module. □

GALOIS MODULES AND p-ADIC REPRESENTATIONS 21

Remark 4.6. Suppose that A is an abelian scheme over Spec(O_{F}), and let T denote its p-adic Tate module. For each positive integer n, let G_{n} denote the O_{F} -group scheme of p^{n}-torsion on A, and write G_{n}^{∗} for its Cartier dual. Then taking inverse limits of the homomorphisms

ψ_{n} : H^{1}(Spec(O_{F}), G_{n}) → Pic(G_{n}^{∗})

yield a homomorphism

Ψ_{F} : H_{f}^{1}(F, T) → \underleftarrow{lim} Pic(G_{n}^{∗})

(see [1], [6], [5]). It seems reasonable to conjecture that Ψ_{F} is injective modulo torsion. In [6], this conjecture is shown to be true (subject to certain technical hypotheses) when A_{/F} is an elliptic curve and p is a prime of ordinary reduction. □

Example 4.7. Let v be a place of F lying above p. In general, the group H_{u}^{1}(F_{v}, T) is not equal to the group H_{f}^{1}(F_{v}, T) introduced by Bloch and Kato in [7]. In order to illustrate this, we apply the theory developed above to the example of the Tate twist T = Z_{p}(i) (i ∈ Z) for an odd prime p.

Assume for simplicity of exposition that F_{v} contains no non-trivial roots of unity of p-power order. Fix a generator (ζ_{p^{n}})_{n≥0} of Z_{p}(1); such a choice also determines a generator (which we shall denote by (ζ_{p^{n}}^{⊗i})_{n}) of Z_{p}(i) for each i ∈ Z. Write G_{n}^{(i)} for the group scheme over Spec(F_{v}) defined by

G_{n}^{(i)}(F_{v}^{c}) = Γ_{n}^{(i)} := p^{−n}Z_{p}(i)/Z_{p}(i).

The Cartier dual of G_{n}^{(i)} is G_{n}^{(1−i)}, and we have G_{n}^{(i)} = Spec(A_{n}^{(i)}), where

A_{n}^{(i)} = (F_{v}^{c}[Γ_{n}^{(1−i)}])^{Ω_{Fv}} .

For each i ∈ Z, and each non-negative integer j, let F_{v}[ζ_{p^{j}}^{⊗i}] denote the smallest extension of F_{v} whose absolute Galois group fixes ζ_{p^{j}}^{⊗i} (cf. Remark 2.5), i.e.

F_{v}[ζ_{p^{j}}^{⊗i}] = \begin{cases}F_{v}(ζ_{p^{j}}), & if i ≠ 0; \\ F_{v} & if i = 0. \end{cases}

Then, if i ≠ 0, Remark 2.5 implies that the Wedderburn decomposition of A_{n}^{(i)} is given by

A_{n}^{(i)} ≃ ⊕_{j=0}^{n}F_{v}[ζ_{p^{j}}^{⊗i}] = ⊕_{j=0}^{n}F_{v}(ζ_{p^{j}}),

and if i = 0 (so G_{n}^{(i)} is a constant group scheme), then

A_{n}^{(0)} ≃ ⊕_{j=0}^{p^{n}−1}F_{v}.

Let k_{n}^{(i)} denote the following sequence of maps:

H^{1}(F_{v}, Z_{p}(1)) → H^{1}(F_{v}, μ_{p^{n}}^{⊗i}) → H^{1}(F_{v}[ζ_{p^{n}}^{⊗(1−i)}], μ_{p^{n}}^{⊗i}) ⟶^{∼}

H^{1} (Γ_{n}^{(1−i)}, \frac{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×}}{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×p^{n}}} ) → \frac{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×}}{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×p^{n}}} .

Here:

• the first arrow is induced by the natural quotient map Z_{p}(i) → μ_{p^{n}}^{⊗i};

• the second arrow is given by corestriction;

• the third arrow is defined via the isomorphism afforded by Proposition 2.6;

• the fourth arrow is induced by “evaluation at ζ_{p^{n}}^{⊗(1−i)}”.

Suppose now that c ∈ H^{1}(F_{v}, Z_{p}(i)). It follows from the definitions that c ∈ H_{u}^{1}(F_{v}, Z_{p}(i)) if and only if, for each n ≥ 0, we have

k_{n}^{(i)}(c) ∈ \frac{O_{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]}^{×} · F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×p^{n}}}{F_{v}[ζ_{p^{n}}^{⊗(1−i)}]^{×p^{n}}} .

The cohomology groups H^{1}(F_{v}, Z_{p}(i)) may be described using ‘twisted Kummer theory’ in the following way (see [26]). Set

X := \underleftarrow{lim} F_{v}(ζ_{p^{n}})^{×}, Y := \underleftarrow{lim} O_{F_{v}(ζ_{p^{n}})}^{×},

where the inverse limits are taken with respect to the norm maps N_{n} : F_{v}(ζ_{p^{n}}) → F_{v}(ζ_{p^{n−1}}). Let H_{∞} := Gal(F_{v}(ζ_{p^{∞}})/F_{v}). A theorem of Iwasawa (see [16, Theorem 25]) implies that X and Y are Z_{p}[[H_{∞}]]-modules of rank [F_{v} : Q_{p}].

For each integer i, we write

X(i) := X ⊗_{Z_{p}} Z_{p}(i), Y (i) := Y ⊗_{Z_{p}} Z_{p}(i).

The group H_{∞} acts on X(i) and Y (i) diagonally. We write X(i)_{H_{∞}} and Y (i)_{H_{∞}} for the group of coinvariants of the H_{∞}-modules X(i) and Y (i) respectively.

Define a homomorphism

φ̃_{i} : X(i − 1)_{H_{∞}} → H^{1}(F_{v}, Z_{p}(i))

by

φ̃_{i}((u_{n} ⊗ ζ_{p^{n}}^{⊗(i−1)})_{n}) = (N_{n}(u_{n} ∪ z_{n}^{i−1}))_{n},

where z_{n} ∈ H^{0}(F_{v}(ζ_{p^{n}}), μ_{p^{n}}) is the element corresponding to ζ_{p^{n}}, and u_{n} ∈ F_{v}(ζ_{p^{n}})^{×}/F_{v}(ζ_{p^{n}})^{×p^{n}}. It is shown in [26, §2] that φ̃_{i} is well-defined, and is an isomorphism for i ≥ 2. It may be checked that the same proof shows that φ̃_{i} is also an isomorphism if i ≤ −1. The theorem of Iwasawa

GALOIS MODULES AND p-ADIC REPRESENTATIONS 23

mentioned above (together with standard Kummer theory and local Tate duality for the cases i = 0 and i = 1) then leads to the following result (cf. [26, p. 390, Remark 1]):

rk_{Z_{p}} (H^{1}(F_{v}, Z_{p}(i))) = \begin{cases}[F_{v} : Q_{p}], & if i ≤ −1; \\ [F_{v} : Q_{p}] + 1, & if i = 0 or i = 1; \\ [F_{v} : Q_{p}], & if i ≥ 2. \end{cases}

If i ≠ 1 and c ∈ φ̃_{i}(Y (i − 1)_{H_{∞}}), then it may be checked that

k_{n}^{(i)}(c) ∈ \frac{O_{F_{v}(ζ_{p^{n}})}^{×} · F_{v}(ζ_{p^{n}})^{×p^{n}}}{F_{v}(ζ_{p^{n}})^{×p^{n}}}

for all n ≥ 0. This implies that φ̃_{i}(Y (i − 1)_{H_{∞}}) ⊆ H_{u}^{1}(F_{v}, Z_{p}(i)) if i ≠ 1. It may be shown using Iwasawa’s theorem, together with (4.5) and a separate analysis of the case i = 1, that

rk (H_{u}^{1}(F_{v}, Z_{p}(i))) = [F_{v} : Q_{p}]

for all i ∈ Z.

On the other hand, it follows from the theory of Bloch and Kato (see [7, Example 3.9]) that

rk_{Z_{p}} (H_{f}^{1}(F_{v}, Z_{p}(i))) = \begin{cases}0, & if i ≤ −1; \\ 1, & if i = 0; \\ [F_{v} : Q_{p}], & if i ≥ 1. \end{cases}

Hence, if i ≤ −1, for example, then H_{u}^{1}(F_{v}, Z_{p}(i)) is never equal to H_{f}^{1}(F_{v}, Z_{p}(i)). □

5. Proof of Theorem 1.4

In this section we give the proof of Theorem 1.4.

For each integer n the action of Ω_{F} on Γ_{n}^{∗} yields a representation

ρ_{n}^{∗} : Ω_{F} → Aut(Γ_{n}^{∗}).

Write F_{n}^{∗} for the fixed field of ρ_{n}^{∗}; then F_{∞}^{∗} = ∪_{n}F_{n}^{∗}, where F_{∞}^{∗} is the extension of F cut out by ρ^{∗}. The group scheme G_{n}^{∗} is constant over Spec(F_{n}^{∗}), and we write

η_{n,F_{n}^{∗}} : Ker(ϕ_{n,F_{n}^{∗}}) ⟶^{∼} Hom(Γ_{n}^{∗}, O_{F_{n}^{∗}}^{×}/(O_{F_{n}^{∗}}^{×})^{p^{n}})

for the isomorphism afforded by Proposition 3.9(ii).

Consider the map

d_{n} : H^{1}(F_{n}^{∗}, Γ_{n}) ⟶^{p_{n}} H^{1}(F_{n}^{∗}, Γ_{n−1}) ⟶^{Cores_{F^{∗}_{n}/F{∗ \atop n−1}}} H^{1}(F_{n−1}^{∗}, Γ_{n−1}).

24 A. AGBOOLA

Lemma 5.1. Passing to the inverse limit of the maps

d_{n} : H^{1}(F_{n}^{∗}, Γ_{n}) → H^{1}(F_{n−1}^{∗}, Γ_{n−1})

induces isomorphisms

\underleftarrow{lim} H^{1}(F_{n}^{∗}, Γ_{n}) ⟶^{∼} \underleftarrow{lim} H^{1}(F_{n}^{∗}, T),  (5.1)
\underleftarrow{lim} H^{1}(F_{n,v}^{∗}, Γ_{n}) ⟶^{∼} \underleftarrow{lim} H^{1}(F_{n,v}^{∗}, T)  (5.2)

Proof. See e.g. [25, Lemma B.3.1]. □

Let h_{n} denote the composition

h_{n} : Hom(Γ_{n}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}}) ⟶^{r_{n}} Hom(Γ_{n−1}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n−1}})

⟶^{N_{F^{∗}_{n}/F{∗ \atop n−1}}} Hom(Γ_{n−1}^{∗}, F_{n−1}^{∗×}/(F_{n−1}^{∗×})^{p^{n−1}}),

where r_{n} is defined in Corollary 2.12, and N_{F_{n}^{∗}/F_{n−1}^{∗}} is induced by the norm map from F_{n}^{∗} to F_{n−1}^{∗}. We remind the reader that, for any Z-module Q, we set

Q̌ := \underleftarrow{lim}_{n} Q/p^{n}Q.

Lemma 5.2. Passing to the inverse limit of the maps

h_{n} : Hom(Γ_{n}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}}) → Hom(Γ_{n−1}^{∗}, F_{n−1}^{∗×}/(F_{n−1}^{∗×})^{p^{n−1}})

induces isomorphisms

\underleftarrow{lim} Hom(Γ_{n}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}}) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} F̌_{n}^{∗×}),

\underleftarrow{lim} Hom(Γ_{n}^{∗}, O_{F_{n}^{∗}}^{×}/(O_{F_{n}^{∗}}^{×})^{p^{n}}) ⟶^{∼} Hom(T^{∗}, Ǒ_{F_{n}^{∗}}^{×}).

Proof. This is proved by applying Lemma 5.1 to the p-divisible group schemes (Z/p^{n}Z)_{n≥1} and (μ_{p^{n}})_{n≥1}. We have

\underleftarrow{lim}[Hom(Γ_{n}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}})] = \underleftarrow{lim}[Hom(Γ_{n}^{∗}, Z/p^{n}Z) ⊗_{Z_{p}} (F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}})]
= [\underleftarrow{lim} Hom(Γ_{n}^{∗}, Z/p^{n}Z)] ⊗_{Z_{p}} [\underleftarrow{lim}(F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}})]
= Hom(T^{∗}, Z_{p}) ⊗_{Z_{p}} \underleftarrow{lim} H^{1}(F_{n}^{∗}, μ_{p^{n}})
= Hom(T^{∗}, Z_{p}) ⊗_{Z_{p}} \underleftarrow{lim} H^{1}(F_{n}^{∗}, Z_{p}(1))
= Hom(T^{∗}, Z_{p}) ⊗_{Z_{p}} \underleftarrow{lim} F̌_{n}^{∗×}
= Hom(T^{∗}, \underleftarrow{lim} F̌_{n}^{∗×}).

The second isomorphism may be established in a similar manner. □

GALOIS MODULES AND p-ADIC REPRESENTATIONS 25

Lemma 5.3. The following diagram is commutative:

H^{1}(F_{n}^{∗}, Γ_{n})  ⟶^{η_{n,F^{∗}_{n}}}  Hom(Γ_{n}^{∗}, F_{n}^{∗×}/(F_{n}^{∗×})^{p^{n}})
d_{n}↓  ↓^{h_{n}}
H^{1}(F_{n−1}^{∗}, Γ_{n−1}) ⟶^{η_{n−1,F{∗ \atop n−1}}} Hom(Γ_{n−1}^{∗}, F_{n−1}^{∗×}/(F_{n−1}^{∗×})^{p^{n−1}}).

(5.3)

Proof. Note that G_{n}^{∗} is constant over Spec(F_{n}^{∗}). The result now follows from Proposition 2.9 and Corollary 2.12. □

Passing to inverse limits over the diagrams (5.3), and applying Lemmas 5.1 and 5.2 yields a natural isomorphism

β : \underleftarrow{lim} H^{1}(F_{n}^{∗}, T) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} F̌_{n}^{∗×}).  (5.4)

For each finite place v of F, similar arguments to those given above show that there is also a local isomorphism

β_{v} : \underleftarrow{lim} H^{1}(F_{n,v}^{∗}, T) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} F̌_{n,v}^{∗×}).  (5.5)

Proposition 5.4. (i) For each finite place v of F, the map β_{v} induces an isomorphism (which we denote by the same symbol)

β_{v} : \underleftarrow{lim} H_{u}^{1}(F_{n,v}^{∗}, T) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} Ǒ_{F_{n,v}^{∗}}^{×}).  (5.6)

(ii) The map β induces an isomorphism

β : \underleftarrow{lim} H_{u}^{1}(F_{n}^{∗}, T) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} Ǒ_{F_{n}^{∗}}^{×}).  (5.7)

Proof. (i) It is easy to check that (5.2) induces an isomorphism

\underleftarrow{lim} H_{u}^{1}(F_{n,v}^{∗}, Γ_{n}) ⟶^{∼} \underleftarrow{lim} H_{u}^{1}(F_{n,v}^{∗}, T).

The result now follows from the isomorphism

H_{u}^{1}(F_{n,v}^{∗}, G_{n}) ⟶^{∼} Hom(Γ_{n}^{∗}, O_{F_{n,v}^{∗}}^{×}/(O_{F_{n,v}^{∗}}^{×})^{p^{n}})

afforded by Proposition 3.9(i).

(ii) Suppose that f ∈ Hom(T^{∗}, \underleftarrow{lim} F̌_{n}^{∗×}). For each finite place v of F, write f_{v} for the image of f in Hom(T^{∗}, \underleftarrow{lim} F̌_{n,v}^{∗×}). Then f_{v} ∈ Hom(T^{∗}, \underleftarrow{lim} O_{F_{n}^{∗},v}^{×}) for all v if and only if f ∈ Hom(T^{∗}, \underleftarrow{lim} Ǒ_{F_{n}^{∗}}^{×}). The result now follows from (i) above. □

Corollary 5.5. There is a natural isomorphism

\underleftarrow{lim} H_{u}^{1}(F_{n}^{∗}, T) ≃ \underleftarrow{lim} Ker(Φ_{F_{n}^{∗}}).

Proof. This follows directly from (5.6), Lemma 5.2, and Proposition 3.9(ii). □

26 A. AGBOOLA

Corollary 5.5 implies that we have

∩_{n} Cores_{F_{n}^{∗}/F} (H_{u}^{1}(F_{n}^{∗}, T)) ⊆ Ker(Φ_{F}).

Hence, if x ∈ ℭ_{F} (T), then for some integer M > 0, we have Φ_{F} (Mx) = MΦ_{F} (x) = 0, and so Φ_{F} (x) is of finite order. This proves Theorem 1.4.

6. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1.

Recall that C_{∞} = ∪_{n}C_{n} denotes the cyclotomic Z_{p}-extension of F. We begin by showing that if x ∈ 𝔊_{F} (T), then Φ_{F}^{{p}}(x) is of finite order.

Lemma 6.1. Suppose that m ≥ 1. Then

\underleftarrow{lim} Cl(𝔐_{m,O_{Cn}}^{\{p\}}) = 0,

where the inverse limit is taken with respect to the maps N_{C_{n}/C_{n−1}}.

Proof. In order to ease notation somewhat, we write 𝔐^{{p}} for 𝔐_{m}^{{p}}. Using the notation of Remark 2.5, we have

𝔐_{C_{n}}^{{p}} ≃ ∏_{γ^{∗}∈Γ_{m}\Ω_{Cn}} O_{C_{n}[γ^{∗}]}[1/p],

and so

Cl(𝔐_{C_{n}}^{{p}}) ≃ ∏_{γ^{∗}∈Γ_{m}\Ω_{Cn}} Cl(O_{C_{n}[γ^{∗}]}[1/p]).

Choose d ≥ 1 sufficiently large that

Γ_{m}^{∗}\Ω_{C_{d}} = Γ_{m}^{∗}\Ω_{C_{i}}

for all i ≥ d. Then, for each γ^{∗} ∈ Γ_{m}^{∗}\Ω_{C_{d}}

C_{∞}[γ^{∗}] := ∪_{i≥d}C_{i}[γ^{∗}]

is the cyclotomic Z_{p}-extension of C_{d}[γ^{∗}], and the map

N_{C_{i}/C_{d}} : Cl(𝔐_{O_{C_{i}}}^{{p}}) → Cl(𝔐_{C_{d}}^{{p}})

is induced by the usual norm maps

Cl(O_{C_{i}[γ^{∗}]}[1/p]) → Cl(O_{C_{d}[γ^{∗}]}[1/p])

for each γ^{∗} ∈ Γ_{m}^{∗}\Ω_{C_{i}} = Γ_{m}^{∗}\Ω_{C_{d}}.

However, if L is any number field, and L_{∞} = ∪_{j}L_{j} is the cyclotomic Z_{p}-extension of L, then it is easy to show that \underleftarrow{lim} Cl(O_{L_{j}}[1/p]) = 0. Hence we have

\underleftarrow{lim} Cl(O_{C_{i}[γ^{∗}]}[1/p]) = 0 (i ≥ d),

GALOIS MODULES AND p-ADIC REPRESENTATIONS 27

and so it follows that

\underleftarrow{lim} Cl(𝔐_{O_{Cn}}^{\{p\}}) = 0,

as claimed. □

Corollary 6.2. We have \underleftarrow{lim} Cl(𝔐_{O_{Cn}}^{\{p\}} (T)) = 0, where the inverse limit is taken with respect to the maps N_{C_{n}/C_{n−1}}. □

Proposition 6.3. Suppose that x ∈ 𝔊_{F} (T). Then Φ_{F}^{{p}}(x) is of finite order.

Proof. Taking inverse limits over (4.3) yields a homomorphism

\underleftarrow{lim} Φ_{C_{n}}^{\{p\}} : \underleftarrow{lim} H_{𝔐(T)_{O_{Cn}}}^{1} (C_{n}, T) → \underleftarrow{lim} Cl(𝔐_{O_{Cn}}^{\{p\}} (T)),

and Corollary 6.2 implies that this is the zero map. Hence

∩_{n} Cores_{C_{n}/F} (H_{𝔐(T)_{O_{Cn}}}^{1} (C_{n}, T)) ⊆ Ker(Φ_{F}^{{p}}).

Thus, if x ∈ 𝔊_{F} (T), then MΦ_{F}^{{p}}(x) = Φ_{F}^{{p}}(Mx) = 0 for some integer M > 0. This establishes the result. □

We now recall the definition of the pairing

B_{F} : H_{f,{p}}^{1}(F, T) × Ker(Loc_{F,T^{∗}}) → Q_{p}

given in [22, Section 3.1.4].

Fix x ∈ H_{f,{p}}^{1}(F, T) and y ∈ Ker(Loc_{F,T^{∗}}). Then viewing y as an element of H^{1}(F, T^{∗}) ≃ Ext_{Ω_{F}}^{1} (Z_{p}, T^{∗}) yields an extension

0 → T^{∗} → T′_{y} → Z_{p} → 0.  (6.1)

Taking Z_{p}(1)-duals of (6.1) yields an exact sequence

1 → Z_{p}(1) → T_{y} → T → 0.  (6.2)

We may consider the global and local Galois cohomology of (6.2) for each finite place v of F:

\begin{matrix}H^{1}(F, Z_{p}(1)) & ⟶^{i} & H^{1}(F, T_{y}) & ⟶^{j} & H^{1}(F, T) & ⟶ & H^{2}(F, Z_{p}(1)) \\ ↓ & & ↓ & & ↓ & & ↓ \\ H^{1}(F_{v}, Z_{p}(1)) & ⟶^{i_{v}} & H^{1}(F_{v}, T_{y}) & ⟶^{j_{v}} & H^{1}(F_{v}, T) & ⟶ & H^{2}(F_{v}, Z_{p}(1)). \end{matrix}  (6.3)

It may be shown via Tate local duality that

H_{f}^{1}(F_{v}, T) ⊆ j_{v}(H_{f}^{1}(F_{v}, T_{y}))

for all places v ∤ p.

28 A. AGBOOLA

At places v | p the extension (6.2) splits locally at v because y ∈ Ker(Loc_{F,T^{∗}}), and so we have a corresponding splitting

H^{1}(F_{v}, T_{y}) = H^{1}(F_{v}, Z_{p}(1)) ⊕ H^{1}(F_{v}, T)  (6.4)

on the level of cohomology groups. Hence we have

H_{f}^{1}(F_{v}, T) = j_{v}(H_{f}^{1}(F_{v}, T_{y}))

in this case, and in fact every element z ∈ H^{1}(F_{v}, T) has a canonical lifting to an element of H^{1}(F_{v}, T_{y}) given by z ↦ (0, z).

Global classfield theory implies that the natural map

H^{2}(F, Z_{p}(1)) → ⨁_{v} H^{2}(F_{v}, Z_{p}(1))

is injective, and so we deduce from (6.3) that

H_{f,{p}}^{1}(F, T) ⊆ j(H_{f,{p}}^{1}(F, T_{y})).

Choose a global lifting x̃ ∈ H_{f,{p}}^{1}(F, T_{y}) of x ∈ H_{f,{p}}^{1}(F, T). For each place v with v ∤ p, choose any local lifting λ_{v} ∈ H_{f}^{1}(F_{v}, T_{y}) of x_{v} ∈ H_{f}^{1}(F_{v}, T). For places v with v | p, define λ_{v} ∈ H^{1}(F_{v}, T_{y}) to be the canonical lifting of x_{v} afforded by the splitting (6.4). Then for each place v of F, we have x̃_{v} − λ_{v} ∈ i_{v}(H^{1}(F_{v}, Z_{p}(1))). If v | p, then i_{v} is injective, and so we may in fact identify i_{v}(H^{1}(F_{v}, Z_{p}(1))) with H^{1}(F_{v}, Z_{p}(1)).

Let

l_{χ} : ⨁_{v} H^{1}(F_{v}, Z_{p}(1)) → Q_{p}

denote the composition

⨁_{v} H^{1}(F_{v}, Z_{p}(1)) ≃ ⊕_{v}F̌_{v}^{×} ⟶^{L_{χ}} Q_{p},

where L_{χ} is defined by

L_{χ}((u_{v})_{v}) = ∑_{v|p} log_{p} N_{F_{v}/Q_{p}}(u_{v}) − ∑_{v∤p}(log_{p} q_{v}) ord_{v}(u_{v}).

(Here q_{v} denotes the cardinality of the residue field of F_{v}, and we choose Iwasawa’s branch of the p-adic logarithm, so that log_{p}(p) = 0.)

It may be shown that l_{χ} induces a well-defined map on ⊕_{v}i_{v}(H^{1}(F_{v}, Z_{p}(1))). We define

B_{F} (x, y) = l_{χ} (x̃_{v} − ∑_{v} λ_{v}) ∈ Q_{p}.

It is shown in [22, Section 3.1.4] that B_{F} induces a pairing

⟨⟨ , ⟩⟩ : \frac{H_{f,\{p\}}^{1}(F, T)}{𝔊_{F} (T)} × Ker(Loc_{F,T^{∗}}) → Q_{p},

GALOIS MODULES AND p-ADIC REPRESENTATIONS 29

and it is conjectured that this pairing is always non-degenerate. We shall relate this pairing to the homomorphism Φ_{𝔐^{{p}}(T)} by interpreting the pairing B_{F} in terms of resolvends.

In order to ease notation in what follows, we shall write Φ_{Z_{p}(1)}^{{p}}, Φ_{T_{y}}^{{p}} and Φ_{T}^{{p}} for Φ_{𝔐^{{p}}(Z_{p}(1))}, Φ_{𝔐^{{p}}(T_{y})}, and Φ_{𝔐^{{p}}(T)} respectively.

Proposition 6.4. Set h_{F} := | Cl(O_{F} [1/p])|, and suppose that Φ_{T}^{{p}}(x) = 0. Then there exists x̃ ∈ H_{f,{p}}^{1}(F, T_{y}) such that j(x̃) = h_{F} x and Φ_{T_{y}}^{{p}}(x̃) = 0.

Proof. It is not hard to check that (6.2) yields sequences (which are exact in the middle, and where we denote maps on resolvends by the same symbols as the corresponding maps on cohomology groups):

H(A(Z_{p}(1))) ⟶^{i} H(A(T_{y})) → H(A(T)),

H(A_{v}(Z_{p}(1))) ⟶^{i_{v}} H(A_{v}(T_{y})) → H(A_{v}(T)),

H(𝔐(Z_{p}(1))) ⟶^{i} H(𝔐(T_{y})) → H(𝔐(T)),

H(𝔐_{v}(Z_{p}(1))) ⟶^{i_{v}} H(𝔐_{v}(T_{y})) → H(𝔐_{v}(T)).

It therefore follows from (3.1) that (6.2) induces a sequence

Cl(𝔐^{{p}}(Z_{p}(1))) → Cl(𝔐^{{p}}(T_{y})) → Cl(𝔐^{{p}}(T))

of classgroups which is exact in the middle. We therefore deduce via functoriality that the following diagram (whose rows are exact in the middle) commutes:

Cl(𝔐^{{p}}(Z_{p}(1))) ⟶ Cl(𝔐^{{p}}(T_{y})) ⟶ Cl(𝔐^{{p}}(T))
Φ_{Zp(1)}^{{p}}↑  ^{Φ_{Ty}^{{p}}}↑  _{Φ_{T}^{{p}}}↑
H_{f,{p}}^{1}(F, Z_{p}(1)) ⟶^{i} H_{f,{p}}^{1}(F, T_{y}) ⟶^{j} H_{f,{p}}^{1}(F, T).

(6.5)

For each integer n > 0, the Wedderburn decomposition of 𝔐^{{p}}(μ_{p^{n}}) (cf. (2.3)) is given by

𝔐^{{p}}(μ_{p^{n}}) ≃ ⨁_{i=0}^{p^{n}−1} O_{F} [1/p].

Hence h_{F} · Cl(𝔐^{{p}}(μ_{p^{n}})) = 0 for each n, and so it follows that h_{F} · Cl(𝔐^{{p}}(Z_{p}(1))) = 0 also. The result now follows from the commutativity of (6.5). □

Proposition 6.5. Assume that the pairing ⟨⟨ , ⟩⟩ is non-degenerate, and suppose that Φ_{T}^{{p}}(x) is of finite order. Then x ∈ 𝔊_{F} (T).

Proof. We first observe from the definition of 𝔊_{F} (T) that if M_{1}x ∈ 𝔊_{F} (T) for any integer M_{1} > 0, then x ∈ 𝔊_{F} (T) also. Hence, we see from Proposition 6.4 that, without loss of generality, we may

30 A. AGBOOLA

assume that there exists a lift x̃ ∈ H_{f,{p}}^{1}(F, T_{y}) of x ∈ H_{f,{p}}^{1}(F, T) such that Φ_{T_{y}}^{{p}}(x̃) = 0. We shall make this assumption from now on.

It follows from the standard identification of H^{1}(F, T^{∗}) with Ext_{Ω_{F}}^{1} (Z_{p}, T^{∗}) that we may write

T_{y} = Z_{p}(1) × T  (6.6)

with Ω_{F} -action given by

(ζ, t)^{σ} = (ζ^{σ} · {f(σ^{−1})(t)}^{σ}, t^{σ})

for any fixed choice of Ω_{F} -cocycle f representing y ∈ H^{1}(F, T^{∗}). This implies that there is an isomorphism of F^{c}-algebras (but not of Ω_{F} -modules)

A_{F^{c}}(T_{y}) ⟶^{∼} A_{F^{c}}(Z_{p}(1)) ⊗_{F^{c}} A_{F^{c}}(T);  α ↦ k_{1}(α) ⊗ k_{2}(α),  (6.7)

where k_{1} and k_{2} are induced by the projection maps Z_{p}(1) × T → Z_{p}(1) and Z_{p}(1) × T → T respectively.

We now make a choice of resolvends associated to x̃ and λ_{v} for each v. Since x̃ ∈ Ker(Φ_{T_{y}}^{{p}}), we may choose a resolvend α ∈ H(𝔐^{{p}}(T_{y})) associated to x̃. For each place v ∤ p, we choose an arbitrary resolvend ν_{v} ∈ H(𝔐_{v}^{{p}}(T_{y})) associated to λ_{v}. For each place v | p, we set

ν_{v} = (1 ⊗ k_{2}(α))_{v}

(cf. (6.7)); this is a resolvend associated to λ_{v} since y is locally trivial at v. Then, for each place v of F, we have

τ_{v} := α_{v}ν_{v}^{−1} ∈ i_{v}(H(𝔐_{v}^{{p}}(Z_{p}(1)))),

and τ_{v} is a resolvend associated to x̃_{v} − λ_{v} ∈ i_{v}(H^{1}(F_{v}, Z_{p}(1))).

If v ∤ p, then Z_{p}(1) is unramified at v, and so Remark 3.6 implies that we have

H(𝔐_{v}^{{p}}(Z_{p}(1))) = H(𝔐_{v}(Z_{p}(1))) ≃ H_{f}^{1}(F_{v}, Z_{p}(1)) ≃ Ǒ_{F_{v}}^{×}.

Hence it follows that (log_{p}(q_{v})) ord_{v}(x̃_{v} − λ_{v}) = 0.

If v | p, then τ_{v} = (k_{1}(α) ⊗ 1)_{v}, and so ⊕_{v|p}τ_{v} is the image of k_{1}(α) ∈ 𝔐^{{p}}(Z_{p}(1))_{O_{F^{c}}}^{×} under the localisation map

δ : 𝔐^{{p}}(Z_{p}(1))_{O_{F^{c}}}^{×} → ⨁_{v|p} 𝔐_{v}^{{p}}(Z_{p}(1))_{O_{F^{c}_{v}}}^{×} .

For each integer n ≥ 1, write

δ_{n} : 𝔐^{{p}}(μ_{p^{n}})_{O_{F^{c}}}^{×} → ⨁_{v|p} 𝔐_{v}^{{p}}(μ_{p^{n}})_{O_{F^{c}_{v}}}^{×}

for the localisation map, and let τ_{v,n} denote the image of τ_{v} in 𝔐_{v}^{{p}}(μ_{p^{n}})_{O_{F^{c}_{v}}}^{×} under the natural map 𝔐_{v}^{{p}}(Z_{p}(1))_{O_{F^{c}_{v}}}^{×} → 𝔐_{v}^{{p}}(μ_{p^{n}})_{O_{F^{c}_{v}}}^{×}. Let (x̃_{v} −λ_{v})_{n} denote the image of (x̃_{v} −λ_{v}) in H^{1}(F_{v}, μ_{p^{n}}) under the natural map H^{1}(F_{v}, Z_{p}(1)) → H^{1}(F_{v}, μ_{p^{n}}).

Since ⨁_{v|p} τ_{v} lies in the image of δ, we have that ⨁_{v|p} τ_{v,n} lies in the image of δ_{n}. It therefore follows that

⊕_{v|p}(x̃_{v} − λ_{v})_{n} ∈ ⨁_{v|p} H^{1}(F_{v}, μ_{p^{n}}) ≃ ⨁_{v|p} F_{v}^{×}/F_{v}^{×p^{n}}

lies in the image of O_{F} [1/p]^{×}F^{×p^{n}}/F^{×p^{n}} under the localisation map

F^{×}/F^{×p^{n}} → ⨁_{v|p} F_{v}^{×}/F_{v}^{×p^{n}}.

This in turn implies that

⨁_{v|p} N_{F_{v}/Q_{p}}(x̃_{v} − λ_{v})_{n} ∈ H^{1}(Q_{p}, μ_{p^{n}}) ≃ Q_{p}^{×}/Q_{p}^{×p^{n}}

lies in the image of Z[1/p]^{×}Q^{×p^{n}}/Q^{×p^{n}} under the localisation map

H^{1}(Q, μ_{p^{n}}) ≃ Q^{×}/Q^{×p^{n}} → H^{1}(Q_{p}, μ_{p^{n}}) ≃ Q_{p}^{×}/Q_{p}^{×p^{n}},

for all n ≥ 1. As the function log_{p}(z) vanishes on the image of Z[1/p]^{×} in Q_{p}^{×}, we conclude that

∑_{v|p} log_{p} (N_{F_{v}/Q_{p}}(x̃_{v} − λ_{v})) = 0.

We therefore deduce that B_{F} (x, y) = 0. Hence x ∈ 𝔊_{F} (T), since by hypothesis ⟨⟨ , ⟩⟩ is nondegenerate. □

Theorem 1.1 now follows immediately from Propositions 6.3 and 6.5.

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Department of Mathematics, University of California, Santa Barbara, CA 93106.

   E-mail address: agboola@math.ucsb.edu