ON RUBIN’S VARIANT OF THE p-ADIC BIRCH AND SWINNERTON-DYER CONJECTURE II

A. AGBOOLA

Abstract. Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when E(K) is infinite, and he proved that his conjecture is true for E(K) of rank one.

When E(K) is finite, however, the statement of Rubin’s original conjecture no longer applies, and the relevant special value of the appropriate p-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin’s conjecture when E(K) is finite.

1. Introduction

The goal of this article is to extend the results of [1] to give an unconditional proof of a certain variant of the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication.

Let E/Q be an elliptic curve with complex multiplication by O_{K}, the ring of integers of an imaginary quadratic field K; this implies that K is necessarily of class number one. Let p > 3 be a prime of good, ordinary reduction for E; then we may write pO_{K} = 𝔭𝔭^{∗}, with 𝔭 = πO_{K} and 𝔭^{∗} = π^{∗}O_{K}.

Set K_{∞} := K(E_{π^{∞}}), K_{∞}^{∗} := K(E_{π^{∗∞}}), and 𝔎_{∞} := K_{∞}K_{∞}^{∗}. Let O denote the completion of the ring of integers of K_{∞,𝔭}^{∗}. For any extension L/K we set Λ(L) := Λ(Gal(L/K)) := Z_{p}[[Gal(L/K)]], and Λ(L)_{O} := O[[Gal(L/K)]].

Date: Final version. April 1, 2010. 2000 Mathematics Subject Classification. 11G05, 11R23, 11G16. Key words and phrases. Rubin, p-adic L-function, Birch and Swinnerton-Dyer conjecture, elliptic curve. Partially supported by NSA grant no. H98230-08-1-0077. I am very grateful to the anonymous referee

for a number of very helpful comments. This paper was completed while I was visiting the Humboldt University, Berlin. I would like to thank the members of the Mathematics Department there for their generous hospitality.

Let

ψ : Gal(\overline{K}/K) → Aut(E_{π^{∞}}) ⟶^{∼} O_{K,𝔭}^{×} ⟶^{∼} Z_{p}^{×},

ψ^{∗} : Gal(\overline{K}/K) → Aut(E_{π^{∗∞}}) ⟶^{∼} O_{K,𝔭^{∗}}^{×} ⟶^{∼} Z_{p}^{×}

denote the natural Z_{p}^{×}-valued characters of Gal(\overline{K}/K) arising via Galois action on E_{π^{∞}} and E_{π^{∗∞}} respectively. We may identify ψ with the Grossecharacter associated to E (and ψ^{∗} with the complex conjugate \overline{ψ} of this Grossencharacter), as described, for example, in [12, p. 325]. We write T and T^{∗} for the 𝔭-adic and 𝔭^{∗}-adic Tate modules of E respectively.

We now recall that the Katz two-variable p-adic L-function L_{𝔭} ∈ Λ(𝔎_{∞})_{O} satisfies a p-adic interpolation formula that may be described as follows (see [12, Theorem 7.1] for the version given here, and also [5, Theorem II.4.14]. Note also that, as the notation indeicates, L_{𝔭} depends upon a choice of prime 𝔭 lying above p—cf. Remark 1.2 below). For all pairs of integers j, k ∈ Z with 0 ≤ −j < k, and for all characters χ : Gal(K(E_{p})/K) → \overline{K}^{×}, we have

L_{𝔭}(ψ^{k}ψ^{∗j}χ) = A · L(ψ^{−k}\overline{ψ}^{−j}χ^{−1}, 0).  (1.1)

Here L(ψ^{−k}\overline{ψ}^{−j}χ^{−1}, s) denotes the complex Hecke L-function, and A denotes an explicit, non-zero factor whose precise description we shall not need.

Write ⟨ψ⟩ : Gal(\overline{K}/K) → 1 + pZ_{p} for the composition of ψ with the natural projection Z_{p}^{×} → 1 + pZ_{p}, and define ⟨ψ^{∗}⟩ in a similar manner. Define

L_{𝔭}(s) := L_{𝔭}(ψ⟨ψ⟩^{s−1}), L_{𝔭}^{∗}(s) := L_{𝔭}(ψ^{∗}⟨ψ^{∗}⟩^{s−1})

for s ∈ Z_{p}. The character ψ lies within the range of interpolation of L_{𝔭}, and the behaviour of L_{𝔭} at ψ is predicted by the 𝔭-adic Birch and Swinnerton-Dyer conjecture for E (see [3, pages 133–134], [10, Theorem V.8]). Conjecturally, ord_{s=1} L_{𝔭}(s) is equal to the rank r of E(Q), and the exact value of lim_{s→1} L_{𝔭}(s)/[s − 1]^{r} may be described in terms of various arithmetic invariants associated to E.

The character ψ^{∗}, however, lies outside the range of interpolation of L_{𝔭}, and the function L_{𝔭}^{∗}(s) has not been studied nearly as much as L_{𝔭}(s). The first results concerning the behaviour of L_{𝔭}^{∗}(s) were obtained by Karl Rubin (see [12], [13]). When r ≥ 1, Rubin formulated a variant of the 𝔭-adic Birch and Swinnerton-Dyer conjecture for L_{𝔭}^{∗}(s) which predicts that that ord_{s=1} L_{𝔭}^{∗}(s) is equal to r − 1, and which gives a formula for lim_{s→1}[L_{𝔭}^{∗}(s)/(s − 1)^{r−1}]. Under suitable hypotheses, Rubin showed that his conjecture is equivalent to the usual 𝔭-adic Birch and Swinnerton-Dyer conjecture, and he proved both conjectures when r = 1. In the case r = 1, he then used these results to give the first examples of a p-adic construction of a global point of infinite order in E(Q) directly from the special value of a p-adic L-function.

BIRCH AND SWINNERTON-DYER CONJECTURE II 3

On the other hand, when r = 0, matters become quite different, and the results of [12],

[13] do not apply. It is not hard to show that the functional equation satisfied by L_{𝔭} (see [5, §6]) implies that ord_{s=1} L_{𝔭}^{∗}(s) ≥ 1. Rubin speculated that perhaps ord_{s=1} L_{𝔭}^{∗}(s) = 1, and assuming that this was true, he also raised the question of determining the value of lim_{s→1}[L_{𝔭}^{∗}(s)/(s − 1)] (see [13, Remark on p.74]).

A natural approach to studying the behaviour of L_{𝔭} at ψ^{∗} is to try to find a suitable Selmer group that governs the behaviour of L_{𝔭}^{∗}(s) at s = 1. This may be done by using the two-variable main conjecture to study certain Iwasawa modules associated naturally to L_{𝔭}^{∗}(s), as described in detail in [1, §1–§3]. In [1], we defined a restricted Selmer group Σ̌_{𝔭^{∗}}(T^{∗}) ⊆ H^{1}(K, T^{∗}). This restricted Selmer group is defined by reversing the Selmer conditions above 𝔭 and 𝔭^{∗} that are used to define the usual Selmer group Sel(K, T^{∗}). The O_{K,𝔭^{∗}}-module Σ̌_{𝔭^{∗}}(K, T^{∗}) is free of rank |r − 1|, and if r ≥ 1, then in fact Σ̌_{𝔭^{∗}}(K, T^{∗}) ⊆ Sel(K, T^{∗}) (see [1, Lemma 3.6, Proposition 6.5, and Proposition 6.7]). We also defined a similar group Σ̌_{𝔭}(K, T) ⊆ H^{1}(K, T), and we constructed a p-adic height pairing

[ , ]_{K,𝔭^{∗}} : Σ̌_{𝔭}(K, T) × Σ̌_{𝔭^{∗}}(K, T^{∗}) → O_{K,𝔭^{∗}}.

If r ≥ 1, and if the 𝔭^{∗}-adic Birch and Swinnerton-Dyer conjecture is true, then [ , ]_{K,𝔭^{∗}} is non-degenerate. We conjectured that [ , ]_{K,𝔭^{∗}} is also non-degenerate when r = 0.

It was shown in [1] that if [ , ]_{K,𝔭^{∗}} is non-degenerate and the p-primary part of � (E/K) is finite, then

ord_{s=1} L_{𝔭}^{∗}(s) = rank_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) = |r − 1|.

Under these assumptions, we also determined the value of lim_{s→1} L_{𝔭}^{∗}(s)/(s − 1)^{|r−1|} up to multiplication by a p-adic unit (thereby recovering a weak form of [12, Corollary 11.3] in the case r ≥ 1). In particular, our results implied that if r = 0 (in which case � (E/K) is known to be finite; see [11]) and [ , ]_{K,𝔭^{∗}} is non-degenerate, then ord_{s=1} L_{𝔭}^{∗}(s) = 1, as was guessed by Rubin in [13].

Suppose now that r = 0. In this paper, we strengthen the results of [1] by giving an unconditional proof of the fact that ord_{s=1}(L_{𝔭}^{∗}(s)) = 1, and we also determine the exact value of the first derivative of L_{𝔭}^{∗}(s) at s = 1. We do this via an approach involving elliptic units and explicit reciprocity laws (cf. [13]), rather than the two-variable main conjecture and Galois cohomology, as in [1]. In order to state our main result, we must introduce some further notation.

Suppose that y ∈ Σ̌_{𝔭}(K, T) and y^{∗} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}) are of infinite order, and let

exp_{𝔭}^{∗} : H^{1}(K_{𝔭}, T^{∗}) → Q_{p}, exp_{𝔭^{∗}}^{∗} : H^{1}(K_{𝔭^{∗}}, T) → Q_{p}

4 A. AGBOOLA

denote the Bloch-Kato dual exponential maps. Via localisation, these induce maps (which we denote by the same symbols):

exp_{𝔭}^{∗} : Σ̌_{𝔭^{∗}}(K, T^{∗}) → Q_{p}, exp_{𝔭^{∗}}^{∗} : Σ̌_{𝔭}(K, T) → Q_{p}.

Write 𝔣 ⊆ O_{K} for the conductor of the Grossencharacter associated to E, and let N(𝔣) denote the norm of this ideal. Set

L′_{𝔭}(ψ^{∗}) := lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{s − 1}.

The following result may perhaps be viewed as being an analogue of a similar exceptional zero phenomenon observed in the work of Mazur, Tate and Teitelbaum concerning p-adic Birch and Swinnerton-Dyer conjectures for elliptic curves without complex multiplication (see [6, especially Conjecture 2], [7, especially page 38]). It relates the value of L_{𝔭} at a point within the range of p-adic interpolation to the derivative of L_{𝔭} at a point lying outside the range of p-adic interpolation.

Theorem A. Suppose that L_{𝔭}(ψ) ≠ 0.

(a) The p-adic height pairing [ , ]_{K,𝔭^{∗}} is non-degenerate and ord_{s=1} L_{𝔭}^{∗}(s) = 1.

(b) We have that

(p − 1)· (1 − \frac{1}{ψ(𝔭^{∗})}) · (1 − \frac{1}{ψ^{∗}(𝔭)}) · \frac{Ω_{𝔭} · L′_{𝔭}(ψ^{∗})}{[y, y^{∗}]_{K,𝔭^{∗}}} =

p · N(𝔣) · (1 − \frac{ψ^{∗}(𝔭)}{p}) · (1 − \frac{ψ(𝔭^{∗})}{p}) · \frac{L_{𝔭}(ψ)}{Ω_{𝔭^{∗}} · exp_{𝔭}^{∗}(y^{∗}) · exp_{𝔭^{∗}}^{∗}(y)},

where Ω_{𝔭} and Ω_{𝔭^{∗}} are certain p-adic periods defined using the formal group associated to E (see Section 4 below).

It is interesting to compare Theorem A with [12, Theorem 10.1]. Both of these results are examples of the following more general phenomenon concerning certain special values of the Katz two-variable p-adic L-function L_{𝔭}. Let k ≥ 0 be an integer, and set

ϕ_{k} := ψ^{k+1}ψ^{∗−k}, ϕ_{k}^{∗} := ψ^{−k}ψ^{∗k+1}.

Then we see from (1.1) that ϕ_{k} lies within the range of interpolation of L_{𝔭}; for k ≥ 1, the behaviour of L_{𝔭} at ϕ_{k} is predicted by various conjectures due to Bloch, Beilinson, Kato and Perrin-Riou. On the other hand, the character ϕ_{k}^{∗} lies outside the range of interpolation of L_{𝔭}, and as far the present author is aware, the behaviour of L_{𝔭} at ϕ_{k}^{∗} for k ≥ 1 does not appear to have previously been studied. Write ⟨ϕ_{k}⟩ (respectively ⟨ϕ_{k}^{∗}⟩) for the composition of ϕ_{k} (respectively ϕ_{k}^{∗}) with the natural projection Z_{p}^{×} → 1 + pZ_{p}, and define

L_{𝔭}(ϕ_{k}, s) := L_{𝔭}(ϕ_{k}⟨ϕ_{k}⟩^{s−1}), L_{𝔭}(ϕ_{k}^{∗}, s) := L_{𝔭}(ϕ_{k}^{∗}⟨ϕ_{k}^{∗}⟩^{s−1})

BIRCH AND SWINNERTON-DYER CONJECTURE II 5

for s ∈ Z_{p}.

Using the techniques of [12], [1] and the present paper, it may be shown that the orders of vanishing of L_{𝔭}(ϕ_{k}, s) and L_{𝔭}(ϕ_{k}^{∗}, s) at s = 1 are of opposite parity, and that the values of the first non-vanishing derivatives of L_{𝔭}(ϕ_{k}, s) and L_{𝔭}(ϕ_{k}^{∗}, s) at s = 1 are related in a manner similar to [12, Theorem 10.1] (if L_{𝔭}(ϕ_{k}) = 0) or to Theorem A above (if L_{𝔭}(ϕ_{k}) ≠ 0). We shall discuss this more fully in a future article (see [2]).

The strategy of the proof of Theorem A is similar to that employed in [12]; however, because we work with restricted Selmer groups rather than true Selmer groups, the details are rather different. These differences mainly arise from the fact that the p-adic height pairing [ , ]_{K,𝔭^{∗}} on restricted Selmer groups (when r = 0) is somewhat more difficult to work with than is the p-adic height pairing on true Selmer groups (when r = 1). The basic ideas involved in the proof of Theorem A may be described as follows. Using elliptic units, we construct canonical elements

s_{𝔭} ∈ Σ̌_{𝔭}(K, T), s_{𝔭^{∗}} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}).

It follows from the proof of Theorem 8.4(a) below that s_{𝔭^{∗}} is of infinite order only if L′_{𝔭}(ψ^{∗}) ≠ 0. By analysing certain Kummer and cup product pairings, and using Wiles’s explicit reciprocity for formal groups, we prove that

exp_{𝔭}^{∗}(s_{𝔭^{∗}}) ≐ L_{𝔭}(ψ),  (1.2)

where the symbol “≐” denotes equality up to multiplication by a non-zero factor. Hence we see that if L_{𝔭}(ψ) ≠ 0, then s_{𝔭^{∗}} is indeed of infinite order, and so L_{𝔭}^{∗}(s) has a first-order zero at s = 1.

We then compute [s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}} using Kummer theory, Hilbert symbols, and Wiles’s explicit reciprocity law, and we see that

[s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}} ≐ L_{𝔭}(ψ) · L′_{𝔭}(ψ^{∗})  (1.3)

The proof of Theorem A is then completed by showing that if y ∈ Σ̌_{𝔭}(K, T) and y^{∗} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}) are both of infinite order, then

\frac{exp_{𝔭}^{∗}(y^{∗}) · exp_{𝔭^{∗}}^{∗}(y)}{[y, y^{∗}]_{K,𝔭^{∗}}} = \frac{exp_{𝔭}^{∗}(s_{𝔭^{∗}}) · exp_{𝔭^{∗}}^{∗}(s_{𝔭})}{[s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}}},

and so the desired result follows by applying (1.2) and (1.3).

An outline of the contents of this paper is as follows. In Section 2 we recall some basic properties of restricted Selmer groups. In Section 3 we describe a local decomposition of the p-adic height pairing [ , ]_{K,𝔭^{∗}} in terms of local Artin symbols. We recall some basic facts concerning the formal group Ê associated to E in Section 4, and we establish a number of

6 A. AGBOOLA

conventions for use in subsequent calculations. We discuss properties of various Kummer pairings in Section 5, and we use these pairings to compute the value of [ , ]_{K,𝔭^{∗}} on certain cohomology classes in restricted Selmer groups that are constructed using global units. In Section 6, we use the results of Section 5 to compute certain special values of the dual exponential map in terms of logarithmic derivatives of certain Coleman power series. In Section 7, we describe the construction of the canonical elements s_{𝔭} and s_{𝔭^{∗}} via elliptic units. Finally, in Section 8, we apply our previous results to these canonical elements, and we prove Theorem A.

We conclude this Introduction by remarking that the methods and results of this paper remain valid if we assume that E is defined over its field of complex multiplication K rather than over Q.

Notation and conventions. Throughout this paper, K denotes an imaginary quadratic field of class number one. If L is any field, we write L^{ab} for the maximal abelian extension of L, and \overline{L} for an algebraic closure of L.

For each integer n ≥ 1, we write

K_{n} := K(E_{π^{n}}), K_{n}^{∗} := K(E_{π^{∗n}}), 𝔎_{n} := K(E_{p^{n}}) = K_{n} · K_{n}^{∗},

and

K_{∞} := K(E_{π^{∞}}), K_{∞}^{∗} := K(E_{π^{∗∞}}), 𝔎_{∞} := K(E_{p^{∞}}).

We also put N_{n} := K_{n} · K_{∞,}^{∗}, and we write 𝔪_{n,𝔭} for the maximal ideal of the completion of the ring of integers of N_{n,𝔭}. The symbol O denotes the completion of the ring of integers of K_{∞,𝔭}^{∗}.

For any extension L/K we set Λ(L) := Λ(Gal(L/K)) := Z_{p}[[Gal(L/K)]], and Λ(L)_{O} := O[[Gal(L/K)]]. We write

I^{∗} := Ker(ψ^{∗} : Λ(K_{∞}^{∗}) → Z_{p}), I := Ker(ψ : Λ(K_{∞}) → Z_{p}),

and let ϑ^{∗} and ϑ be the generators of I^{∗} and I fixed in [12, §6]; so ϑ^{∗} = γψ^{∗}(γ^{−1}) − 1, where γ is any topological generator of Gal(K_{∞}^{∗}/K) satisfying log_{p}(ψ^{∗}(γ)) = p, and ϑ is defined analogously.

We set D_{𝔭} := K_{𝔭}/O_{K,𝔭} and D_{𝔭^{∗}} := K_{𝔭^{∗}}/O_{K,𝔭^{∗}}. If M is any Z_{p}-module, we write M^{∧} for the Pontryagin dual of M.

For each integer n ≥ 1, we let

e_{n} : E_{π^{n}} × E_{π^{∗n}} → μ_{p^{n}}

BIRCH AND SWINNERTON-DYER CONJECTURE II 7

denote the Weil pairing, normalised as decribed in [8, §3.1.2] (the reader should note that this is not the same normalisation as that used in [12]). This pairing satisfies the identities

e_{n}(π^{∗}ς_{n}, ς_{n}^{∗}) = e_{n}(ς_{n}, πς_{n})

for ς_{n} ∈ E_{π^{n}}, ς_{n}^{∗} ∈ E_{π^{∗n}}, and

e_{n+1}(ς_{n}, ς_{n+1}^{∗}) = e_{n}(ς_{n}, π^{∗}ς_{n+1}^{∗})

for ς_{n} ∈ E_{π^{n}}, ς_{n+1}^{∗} ∈ E_{π^{∗(n+1)}}.

We set W := E_{π^{∞}} and W^{∗} := E_{π^{∗∞}}. We write T and T^{∗} for the π-adic and π^{∗}-adic Tate modules of E respectively. Let w = [w_{n}] and w^{∗} = [w_{n}^{∗}] denote generators of T and T^{∗} respectively.

We use the following notation to denote various unit groups:

U_{n,𝔭} := units in K_{n,𝔭} congruent to 1 modulo 𝔭;
U_{n,𝔭^{∗}} := units in K_{n,𝔭^{∗}} congruent to 1 modulo 𝔭^{∗};
U_{∞,𝔭} := \underleftarrow{lim} U_{n,𝔭}, U_{∞,𝔭^{∗}} := \underleftarrow{lim} U_{n,𝔭^{∗}};
U_{n,𝔭}^{∗} := units in K_{n,𝔭}^{∗} congruent to 1 modulo 𝔭;
U_{n,𝔭^{∗}}^{∗} := units in K_{n,𝔭^{∗}}^{∗} congruent to 1 modulo 𝔭^{∗};
U_{∞,𝔭}^{∗} := \underleftarrow{lim} U_{n,𝔭}^{∗}, U_{∞,𝔭^{∗}}^{∗} := \underleftarrow{lim} U_{n,𝔭^{∗}},

where all inverse limits are taken with respect to the obvious norm maps. We also set

E_{n} := global units of K_{n}, E_{n}^{∗} := global units of K_{n}^{∗};
\overline{E}_{n} := the closure of the projection of E_{n} into U_{n,𝔭};
\overline{E}_{n}^{∗} := the closure of the projection of E_{n}^{∗} into U_{n,𝔭^{∗}}^{∗};
\overline{E}_{∞} := \underleftarrow{lim} \overline{E}_{n}, \overline{E}_{∞}^{∗} := \underleftarrow{lim} \overline{E}_{n}^{∗}.

Remark 1.1. Note that since the strong Leopoldt conjecture holds for all abelian extensions of K (see [4]), we have that

\overline{E}_{n} ≃ E_{n} ⊗_{Z} Z_{p}, \overline{E}_{n}^{∗} ≃ E_{n}^{∗} ⊗_{Z} Z_{p},

and so we may also view \overline{E}_{∞} as being a submodule of U_{∞,𝔭^{∗}} and \overline{E}_{∞}^{∗} as being a submodule of U_{∞,𝔭}^{∗}. We shall do this without further comment several times in what follows. □

Remark 1.2. It is important for the reader to bear in mind that every theorem or construction in this paper that depends upon a choice of prime 𝔭 of K lying above p also has a corresponding version in which the roles of 𝔭 and 𝔭^{∗} are interchanged. We shall sometimes make use of this fact without stating it explicitly. □

8 A. AGBOOLA

2. Restricted Selmer groups

In this section we shall recall some basic properties of restricted Selmer groups. We refer the reader to [1, Sections 3 and 4] for more complete details.

Suppose that F/K is any finite extension. For any place v of F, we define H_{f}^{1}(F_{v}, W) to be the image of E(F_{v}) ⊗ D_{𝔭} under the Kummer map

E(F_{v}) ⊗_{O_{K}} D_{𝔭} → H^{1}(F_{v}, W),

and we define H_{f}^{1}(F_{v}, W^{∗}) in a similar manner. Note that H_{f}^{1}(F_{v}, W) = 0 if v ∤ 𝔭. We also set

H_{f}^{1}(F_{v}, E_{π^{n}}) := Im[E(F_{v})/π^{n}E(F_{v}) → H^{1}(F_{v}, E_{π^{n}})],
H_{f}^{1}(F_{v}, E_{π^{∗n}}) := Im[E(F_{v})/π^{∗n}E(F_{v}) → H^{1}(F_{v}, E_{π^{∗n}})].

Suppose that M ∈ {W, W^{∗}, E_{π^{n}}, E_{π^{∗n}}} and that 𝔮 ∈ {𝔭, 𝔭^{∗}} . If c ∈ H^{1}(F, M), then we write loc_{v}(c) for the image of c in H^{1}(F_{v}, M). We define

• the true Selmer group Sel(F, M) by

Sel(F, M) = {c ∈ H^{1}(F, M) | loc_{v}(c) ∈ H_{f}^{1}(F_{v}, M) for all v};

• the relaxed Selmer group Sel_{rel}(F, M) by

Sel_{rel}(F, M) = {c ∈ H^{1}(F, M) | loc_{v}(c) ∈ H_{f}^{1}(F_{v}, M) for all v not dividing p};

• the 𝔮-restricted Selmer group (or simply restricted Selmer group for short when 𝔮 is understood) Σ_{𝔮}(F, M) by

Σ_{𝔮}(F, M) = {c ∈ Sel_{rel}(F, M) | loc_{v}(c) = 0 for all v dividing 𝔮} .

(The terminology ‘restricted Selmer group’ is meant to reflect a choice of a combination of relaxed and strict Selmer conditions at places above p.)

We also define

Σ̌_{𝔮}(F, T) := \underleftarrow{lim}_{n} Σ_{𝔮}(F, E_{π^{n}}), Σ̌_{𝔮}(F, T^{∗}) := \underleftarrow{lim}_{n} Σ_{𝔮}(F, E_{π^{∗n}}).

If L/K is an infinite extension, we define

Σ_{𝔮}(L, M) = \underrightarrow{lim} Σ_{𝔮}(L′, M),

where the direct limits are taken with respect to restriction over all subfields L′ ⊂ L finite over K.

We record the following standard cohomological result that will be used later.

BIRCH AND SWINNERTON-DYER CONJECTURE II 9

Lemma 2.1. Let n ≥ 0 be an integer, and suppose that L and M are fields with K ⊆ L ⊆ M ⊆ N_{n}. Then, for every integer m ≥ 1, the restriction maps

H^{1}(L, E_{π^{m}}) → H^{1}(M, E_{π^{m}}), H^{1}(L, μ_{p^{m}}) → H^{1}(M, μ_{p^{m}})

are injective and they induce isomorphisms

H^{1}(L, E_{π^{m}}) ≃ H^{1}(M, E_{π^{m}})^{Gal(M/L)}, H^{1}(L, μ_{p^{m}}) ≃ H^{1}(M, μ_{p^{m}})^{Gal(M/L)}.

A similar result holds if L and M are replaced by L_{𝔭} and M_{𝔭} with K_{𝔭} ⊆ L_{𝔭} ⊆ M_{𝔭} ⊆ N_{n,𝔭}.

Proof. This is quite standard, and may be proved via the argument given in [10, p.40], for example. □

We now explain how elements in restricted Selmer groups may be constructed by combining Kummer theory on E with Kummer theory on the multiplicative group (see Proposition 2.6 below). In order to do this, we require several preparatory lemmas. Our starting point is the following result of Perrin-Riou.

Lemma 2.2. There is an O_{K}-linear isomorphism of Gal(K_{n}^{∗}/K)-modules

H^{1}(K_{n}^{∗}, E_{π^{n}}) ⟶^{∼} Hom(E_{π^{∗n}}, K_{n}^{∗×}/K_{n}^{∗×p^{n}}); f ↦ f̃.  (2.1)

For each place v of K_{n}^{∗}, there is also a corresponding local O_{K}-linear isomorphism

H^{1}(K_{n,v}^{∗}, E_{π^{n}}) ⟶^{∼} Hom(E_{π^{∗n}}, K_{n,v}^{∗×}/K_{n,v}^{∗×p^{n}}).

Proof. See [8, Lemme 3.8]. The isomorphism (2.1) is defined as follows. Let f ∈ H^{1}(K_{n}^{∗}, E_{π^{n}}), and recall that

e_{n} : E_{π^{n}} × E_{π^{∗n}} → μ_{p^{n}}

denotes the Weil pairing. We identify K_{n}^{∗×}/K_{n}^{∗×p^{n}} with H^{1}(K_{n}^{∗}, μ_{p^{n}}) via Kummer theory. If ς^{∗} ∈ E_{π^{∗n}}, then f̃(ς^{∗}) ∈ H^{1}(K_{n}^{∗}, μ_{p^{n}}) is defined to be the element represented by the cocycle

σ ↦ e_{n}(f(σ), ς^{∗})

for all σ ∈ Gal(\overline{K}/K_{n}^{∗}). □

Corollary 2.3. There are isomorphisms

r_{n} : H^{1}(K, E_{π^{n}}) ⟶^{∼} Hom(E_{π^{∗n}}, K_{n}^{∗×}/K_{n}^{∗×p^{n}})^{Gal(K_{n}^{∗}/K)}

r_{n}^{∗} : H^{1}(K, E_{π^{∗n}}) ⟶^{∼} Hom(E_{π^{n}}, K_{n}^{×}/K_{n}^{×p^{n}})^{Gal(K_{n}/K)}.

Proof. This follows from Lemmas 2.1 and 2.2 (cf. also [12, Lemma 2.1] or [8, Lemme 3.8]). □

10 A. AGBOOLA

Lemma 2.4. For each place v of K_{n}^{∗} with v ∤ 𝔭^{∗}, there is an O_{K}-linear isomorphism

E(K_{n,v}^{∗})/π^{n}E(K_{n,v}^{∗}) ⟶^{∼} Hom(E_{π∗^{n}}, O_{K_{n,v}^{∗}}^{×}/O_{K_{n,v}^{∗}}^{×p^{n}}).

Proof. See [8, Lemme 3.11]. □

Corollary 2.5. Suppose that h ∈ H^{1}(K_{n}^{∗}, E_{π^{n}}). Then h ∈ Σ_{𝔭}(K_{n}^{∗}, E_{π^{n}}) if and only if, for each ς^{∗} ∈ E_{π^{∗n}}, the following local conditions are satisfied:

(a) h̃(ς^{∗}) ∈ K_{n,v}^{∗×p^{n}} for all v | 𝔭;

(b) p^{n} | v_{K_{n}^{∗}}(h̃(ς^{∗})) for all v ∤ 𝔭^{∗}.

(Note that we impose no local conditions at places lying above 𝔭^{∗}.)

Proof. This follows from Lemmas 2.2 and 2.4, using the definition of the isomorphism 2.1 (see also [8, Lemme 3.8]). □

Proposition 2.6. There are natural injections

ρ : Hom(T^{∗}, (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)} ↪ Σ̌_{𝔭}(K, T),
ρ^{∗} : Hom(T, (U_{∞,𝔭^{∗}} ⊗ Q)/\overline{E}_{∞})^{Gal(K_{∞}/K)} ↪ Σ̌_{𝔭^{∗}}(K, T^{∗})

Proof. The proof of this result is essentially the same, mutatis mutandis, as that of [12, Proposition 2.4]. The map ρ is defined as follows.

For any f ∈ Hom(T^{∗}, (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)} and any integer n ≥ 1, we define f_{n} ∈ Hom(E_{π^{∗n}}, E_{n}^{∗}/E_{n}^{∗p^{n}})^{Gal(K_{∞}^{∗}/K)} to be the image of f under the following composition of maps:

Hom(T^{∗}, (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)} → Hom(T^{∗}, (U_{n,𝔭}^{∗} ⊗ Q)/\overline{E}_{n}^{∗})^{Gal(K_{∞}^{∗}/K)}
→ Hom(E_{π^{∗n}}, E_{n}^{∗}/E_{n}^{∗p^{n}})^{Gal(K_{∞}^{∗}/K)},

where the first arrow is the map induced by the natural projection U_{∞,𝔭}^{∗} → U_{n,𝔭}^{∗}, and the second arrow is induced by raising to the p^{n}-th power in U_{n,𝔭}^{∗}.

We define

ρ(f) := [(p − 1)(π^{∗})^{n}r_{n}^{−1}(f_{n})] ∈ \underleftarrow{lim}_{n} H^{1}(K, E_{π^{n}}).  (2.2)

It follows from [8, Lemma 3.16] that ρ(f) does indeed lie in \underleftarrow{lim}_{n} H^{1}(K, E_{π^{n}}). It is not hard to check from the definition that ρ is injective. It follows from [1, Theorem 3.1, Proposition 3.2 and Corollary 3.3] that r_{n}^{−1}(f_{n}) ∈ Σ_{𝔭}(K, E_{π^{n}}) if and only if the restriction of r_{n}^{−1}(f_{n}) to H^{1}(𝔎_{∞}, E_{π^{n}}) is unramified outside 𝔭^{∗}. It may be shown via an argument very similar to that given in [12, Lemmas 2.1 and 2.3] that this in fact the case. □

We shall use Proposition 2.6 to produce canonical elements in restricted Selmer groups by applying ρ and ρ^{∗} to certain Galois-equivariant homomorphisms that are constructed using norm-coherent sequences of elliptic units (see Section 7 below).

BIRCH AND SWINNERTON-DYER CONJECTURE II 11

3. The p-adic height pairing on restricted Selmer groups

Our goal in this section is to describe a local decomposition of the p-adic height pairing

[ , ]_{K,𝔭^{∗}} : Σ̌_{𝔭}(K, T) × Σ̌_{𝔭^{∗}}(K, T^{∗}) → O_{K,𝔭^{∗}}.  (3.1)

in terms of Artin symbols; this is analogous to the local decomposition of the standard p-adic height pairing on true Selmer groups described in [8, Lemme 3.19]. This local decomposition will be used in Section 5 to determine certain values of [ , ]_{K,𝔭^{∗}} explicitly; these will in turn play a key rôle in showing that the pairing (3.1) is non-degenerate when r = 0 in Section 8.

We begin by recalling the outlines of the main steps in the construction of [ , ]_{K,𝔭^{∗}}. For more complete details, we refer the reader to [1, §4].

Let Y(K_{∞}^{∗}) denote the Galois group over K_{∞}^{∗} of the maximal abelian pro-p extension of K_{∞}^{∗} that is unramified away from 𝔭 and totally split at all places of K_{∞}^{∗} lying above 𝔭^{∗}. The first step in the construction of the pairing [ , ]_{K,𝔭^{∗}} is the construction of an isomorphism

Ψ_{K} : Σ̌_{𝔭}(K, T) ⟶^{∼} Hom(T^{∗}, Y(K_{∞}^{∗}))^{Gal(K_{∞}^{∗}/K)}.  (3.2)

This isomorphism is constructed as follows. Write J_{n} for the group of finite ideles of K_{n}^{∗}, and let V_{n} denote the subgroup of J_{n} whose components are equal to 1 at all places dividing 𝔭 and are units at all places not dividing 𝔭^{∗}. Set

C_{n} := J_{n}/(V_{n} · K_{n}^{∗×}), Ω_{n} := ∏_{v|𝔭}μ_{p^{n}}(K_{n,v}^{∗}),

and write C_{n}(p) for the p-primary subgroup of C_{n}. Note that the order of Ω_{n} remains bounded as n varies. We view Ω_{n} as being a subgroup of C_{n}(p) via the obvious embedding of Ω_{n} into J_{n}.

Using Kummer theory (cf. Corollary 2.5 above), one constructs an exact sequence

0 → Hom(E_{π^{∗n}}, Ω_{n})^{Gal(K_{n}^{∗}/K)} → Hom(E_{π^{∗n}}, C_{n})^{Gal(K_{n}^{∗}/K)} ⟶^{η_{n}} Σ_{𝔭}(K, E_{π^{n}}) → 0

(see [1, Proposition 4.6]). Let η′_{n} denote the map obtained from η_{n} via passage to the quotient by Ker(η_{n}). It may be shown that passing to inverse limits over the maps η_{n}^{′−1} yields an isomorphism

Ξ_{K} : \underleftarrow{lim} Σ_{𝔭}(K, E_{π^{n}}) = Σ̌_{𝔭}(K, T) ⟶^{∼} Hom(T^{∗}, \underleftarrow{lim} C_{n}(p))^{Gal(K_{∞}^{∗}/K)},

where the inverse limit \underleftarrow{lim} C_{n}(p) is taken with respect to the norm maps K_{n}^{∗×} → K_{n−1}^{∗×}. One then shows via class field theory (along with the fact that the weak p-adic Leopoldt conjecture holds for K) that there is an isomorphism

Hom(T^{∗}, \underleftarrow{lim} C_{n}(p))^{Gal(K_{∞}^{∗}/K)} ≃ Hom(T^{∗}, Y(K_{∞}^{∗}))^{Gal(K_{∞}^{∗}/K)},  (3.3)

12 A. AGBOOLA

and composing this last isomorphism with with Ξ_{K} yields the desired isomorphism Ψ_{K}.

Next, by suitably interpreting restricted Selmer groups in terms of certain Galois groups (see [1, Theorem 3.1]), one shows that there is a natural homomorphism

β_{K} : Hom(T^{∗}, Y(K_{∞}^{∗}))^{Gal(K_{∞}^{∗}/K)} → Hom_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗}), O_{K,𝔭^{∗}}).

We thus obtain a map

β_{K} ∘ Ψ_{K} : Σ̌_{𝔭}(K, T) → Hom_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗}), O_{K,𝔭^{∗}}),

and this yields the p-adic height pairing pairing

[ , ]_{K,𝔭^{∗}} : Σ̌_{𝔭}(K, T) × Σ̌_{𝔭^{∗}}(K, T^{∗}) → O_{K,𝔭^{∗}}

on restricted Selmer groups.

In order to describe the local decomposition of [ , ]_{K,𝔭^{∗}}, we must introduce some further notation.

Suppose that

y = [y_{n}] ∈ Σ̌_{𝔭}(K, T), y^{∗} = [y_{n}^{∗}] ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}).

For each positive integer n, we define q_{n} to be the map

q_{n} : Σ̌_{𝔭}(K, T) ⟶^{Ψ_{K}} Hom(T^{∗}, Y(K_{∞}^{∗}))^{Gal(K_{∞}^{∗}/K)} → Hom(E_{π^{∗n}}, C_{n})^{Gal(K_{∞}^{∗}/K)},

where the second arrow is the natural quotient map afforded by the isomorphism (3.3).

For each ς^{∗} ∈ E_{π^{∗n}}, let n(ς^{∗}) denote the exact power of π^{∗} that kills ς^{∗}. Let S_{n,ς^{∗}}(y_{n}) denote any representative of η_{n}^{−1}(y_{n})(ς^{∗}) in J_{n}. For each finite place v of K, define {y, y^{∗}}_{n,v}^{(ς^{∗})} to be the unique element of O_{K}/π^{∗n(ς^{∗})}O_{K} such that

{y, y^{∗}}_{n,v}^{(ς^{∗})} · ς^{∗} = y_{n}^{∗}([S_{n,ς^{∗}}(y_{n})_{v}, K_{v}^{ab}/K_{n,v}]),

where [S_{n,ς^{∗}}(y_{n})_{v}, K_{v}^{ab}/K_{n,v}] ∈ Gal(K_{v}^{ab}/K_{n,v}) is the obvious local Artin symbol.

Proposition 3.1. (cf. [8, Lemma 3.19])

(a) For any ς^{∗} ∈ E_{π^{∗n}}, we have

[y, y^{∗}]_{K,𝔭^{∗}} · ς^{∗} = y_{n}^{∗}([q_{n}(y)(ς^{∗}), K^{ab}/K_{n}^{∗}]),  (3.4)

where [q_{n}(y)(ς^{∗}), K^{ab}/K_{n}^{∗}] ∈ Gal(K^{ab}/K_{n}^{∗}) is the obvious global Artin symbol.

(b) We have

[y, y^{∗}]_{K,𝔭^{∗}} ≡ ∑_{v}{y, y^{∗}}_{n,v}^{(ς^{∗})} (mod π^{∗n(ς^{∗})}O_{K,𝔭^{∗}}),  (3.5)

where the sum is over all finite places v of K_{n}^{∗}.

BIRCH AND SWINNERTON-DYER CONJECTURE II 13

Proof. (a) This follows immediately from the following commutative diagram:

Σ̌_{𝔭}(K, T) ⟶^{Ψ_{K}} Hom(T^{∗}, Y(K_{∞}^{∗}))^{Gal(K_{∞}^{∗}/K)} ⟶^{β_{K}} Hom_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗}), O_{K,𝔭^{∗}})
↓  ↓  ↓
Σ_{𝔭}(K, E_{π^{n}}) ⟶^{η_{n}^{′−1}} \frac{Hom(E_{π^{∗n}}, C_{n})^{Gal(K_{∞}^{∗}/K)}}{Ker(η_{n})} ⟶ Hom(Σ_{𝔭^{∗}}(K, E_{π^{∗n}}), O_{K}/π^{∗n}O_{K})

In this diagram, the second arrow on the bottom row is induced by the map

f ↦ (c ↦ {ς^{∗} ↦ c([f(ς^{∗}), K^{ab}/K_{n}^{∗}])}), f ∈ Hom(E_{π^{∗n}}, C_{n})^{Gal(K_{∞}^{∗}/K)}

and is well-defined because [z, K^{ab}/K_{n}^{∗}] = 0 for all z ∈ Ω_{n}. Note also that here we have canonically identified O_{K}/π^{∗n}O_{K} with Hom(E_{π^{∗n}}, E_{π^{∗n}}) via the map

β ↦ {ς^{∗} ↦ β · ς^{∗}}.

(b) This follows from the local decomposition of the global Artin symbol afforded via class field theory, viz. if α ∈ J_{n}, then

[α, K^{ab}/K_{n}^{∗}] = ∏_{v}[α_{v}, K_{v}^{ab}/K_{n,v}^{∗}],

where the product is over all finite places v of K_{n}^{∗}. □

4. Formal Groups

The purpose of this section is to recall a number of facts concerning formal groups, and to establish certain conventions that we shall use in Section 5.

We fix a minimal Weierstrass model of E over O_{K,𝔭}, and we write Ê for its associated formal group. Let Ĝ_{m} denote the formal group over O_{K,𝔭} associated to the multiplicative group G_{m}. If x is a point on G_{m} or on E, then we write x̂ for the corresponding value of the parameter on Ĝ_{m} or on Ê. We denote the formal group logarithm associated to Ê by λÊ(Z) = Z + (higher order terms) ∈ O_{K,𝔭}[[Z]], and we write log_{E,𝔭} for the corresponding 𝔭-adic logarithm associated to E. We denote the 𝔭-adic logarithm associated to G_{m} by log_{𝔭}.

Recall that O denotes the completion of the ring of integers of K_{∞,𝔭}^{∗}. Since Ê is a height one Lubin-Tate formal group, we may fix an isomorphism

η : Ĝ_{m} → Ê, η ∈ O[[Z]].

As explained in [12, §6], this choice of isomorphism then yields:

(a) A generator w^{∗} = [w_{n}^{∗}] of T^{∗} such that for every n ≥ 1 and ς ∈ E_{π^{n}}, we have

η({ \atop e_{n}(π^{∗−n}ς, w_{n}^{∗})}) = ς̂.  (4.1)

(b) A 𝔭-adic period Ω_{𝔭} := η′(0) ∈ O^{×} which is such that

Ω_{𝔭}^{σ} = ψ^{∗}(σ^{−1})Ω_{𝔭}

for every σ ∈ Gal(\overline{K}/K).

We fix a generator w = [w_{n}] of T, and for each n ≥ 0, we set

ζ_{n} := e_{n}(π^{∗−n}w_{n}, w_{n}^{∗});

so from (4.1) above, we have that

η(ζ̂_{n}) = ŵ_{n}.

We note that for each integer n ≥ 1, our choice of η induces an isomorphism

χ_{n} : μ_{p^{n}} ⟶^{∼} E_{π^{n}}; ζ_{n} ↦ w_{n}

which is Gal(\overline{K}/K_{n}^{∗})-equivariant; this in turn induces an isomorphism (which we denote by the same symbol)

χ_{n} : H^{1}(N_{n,𝔭}, μ_{p^{n}}) ⟶^{∼} H^{1}(N_{n,𝔭}, E_{π^{n}}).  (4.2)

Proposition 4.1. Recall that for each integer n ≥ 1, 𝔪_{n,𝔭} denotes the maximal ideal in the completion of the ring of integers of N_{n,𝔭}.

(a) With notation as above, the following diagram commutes:

H^{1}(N_{n,𝔭}, μ_{p^{n}})  ⟶_{∼}^{χ_{n}} H^{1}(N_{n,𝔭}, E_{π^{n}})
↑  ↑
\frac{Ĝ_{m}(𝔪_{n,𝔭})}{Ĝ_{m}(𝔪_{n,𝔭})^{p^{n}}} ≃ \frac{O_{N_{n,𝔭}}^{×}}{O_{N_{n,𝔭}}^{×p^{n}}} ⟶_{∼}^{η} \frac{Ê(𝔪_{n,𝔭})}{𝔭^{n}Ê(𝔪_{n,𝔭})}.

(Here the vertical arrows denote the natural maps afforded by Kummer theory on Ĝ_{m} and Ê.)

(b) If x̂ ∈ Ĝ_{m}(𝔪_{n,𝔭}), then

log_{E,𝔭}(η(x̂)) ≡ Ω_{𝔭} · log_{𝔭}(x̂) (mod 𝔪_{n,𝔭}^{p^{n}})

on Ĝ_{m}(𝔪_{n,𝔭})/Ĝ_{m}(𝔪_{n,𝔭})^{p^{n}}.

Proof. (a) This follows directly from the definitions of η and χ_{n}.

(b) See the proof of [12, Corollary 9.2]. □

BIRCH AND SWINNERTON-DYER CONJECTURE II 15

5. Kummer pairings and p-adic heights

In this section we shall compute the value of the p-adic height pairing [ , ]_{K,𝔭} on elements of restricted Selmer groups that are constructed via Proposition 2.6 (see Theorem 5.11 below). This is accomplished by using Proposition 3.1 to express these values in terms of certain Kummer pairings, and then applying Wiles’s explicit reciprocity law.

Definition 5.1. For each integer n ≥ 1, we define a pairing

( , )_{𝔭,E_{π^{∗n}}} : U_{n,𝔭}^{∗} × H^{1}(K_{𝔭}, E_{π^{∗n}}) → E_{π^{∗n}}  (5.1)

by

(u_{n}, c_{n}^{∗})_{𝔭,E_{π^{∗n}}} = c_{n}^{∗}([u_{n}, K_{𝔭}^{ab}/K_{n,𝔭}^{∗}]) =: α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) · w_{n}^{∗},

with α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) ∈ O_{K}/𝔭^{∗n}O_{K} ≃ Z_{p}/p^{n}Z_{p}.

The pairings (5.1) give a pairing

( , )_{𝔭,T^{∗}} : U_{∞,𝔭}^{∗} × H^{1}(K_{𝔭}, T^{∗}) → T^{∗}; (u, c^{∗}) ↦ α_{𝔭,T^{∗}}(u, c^{∗}) · w^{∗}  (5.2)

that is defined as follows. Suppose that u = [u_{n}] ∈ U_{∞,𝔭}^{∗} and

c^{∗} = [c_{n}^{∗}] ∈ \underleftarrow{lim} H^{1}(K_{𝔭}, E_{π^{∗n}}) ≃ H^{1}(K_{𝔭}, T^{∗}).

Then we set

(u, c^{∗})_{𝔭,T^{∗}} = [(u_{n}, c_{n}^{∗})_{𝔭,E_{π^{∗n}}}] = [α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗})] · w^{∗} =: α_{𝔭,T^{∗}}(u, c^{∗}) · w^{∗}.
□

The pairing (5.2) is related to the p-adic height pairing [ , ]_{K,𝔭^{∗}} in the following way. Recall from Proposition 2.6 above that there is a natural injection

ρ : Hom(T^{∗}, (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)} ↪ Σ̌_{𝔭}(K, T).

Proposition 5.2. Suppose that

ξ ∈ Hom(T^{∗}, U_{∞,𝔭}^{∗}/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)}.

Let \overline{ξ(w^{∗})} ∈ U_{∞,𝔭}^{∗} denote any lift of ξ(w^{∗}), and suppose that y^{∗} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}). Then we have

[ρ(ξ), y^{∗}]_{K,𝔭^{∗}} · w^{∗} = (\overline{ξ(w^{∗})}, loc_{𝔭}(y^{∗}))_{𝔭,T^{∗}},

and so it follows that

[ρ(ξ), y^{∗}]_{K,𝔭^{∗}} = [α_{𝔭,E_{π^{∗n}}}((\overline{ξ(w^{∗})})_{n}, loc_{𝔭}(y_{n}^{∗}))] = α_{𝔭,T^{∗}}(\overline{ξ(w^{∗})}, loc_{𝔭}(y^{∗})).

Proof. This follows directly from Proposition 3.1 and Definition 5.1. □

16 A. AGBOOLA

Our goal in this section is to evaluate [ρ(ξ), y^{∗}]_{K,𝔭^{∗}} (see Theorem 5.11 below). The difficulty in doing this arises from the fact that that the pairing (5.2) cannot be directly evaluated using explicit reciprocity laws. In order to overcome this obstacle, we shall first express the Kummer pairings (5.1) and (5.2) in terms of certain Hilbert symbols. This will then enable us to relate these pairings to two other pairings (namely (5.5) and (5.6) below) that can be evaluated using Wiles’s explicit reciprocity law for formal groups.

Definition 5.3. We define the Hilbert pairing

( , )_{𝔭,μ_{p^{n}}} : \frac{N_{n,𝔭}^{×}}{N_{n,𝔭}^{×p^{n}}} × \frac{N_{n,𝔭}^{×}}{N_{n,𝔭}^{×p^{n}}} → μ_{p^{n}}  (5.3)

by

(a, b)_{𝔭,μ_{p^{n}}} = \frac{(b^{1/p^{n}})^{σ_{a}}}{b^{1/p^{n}}} ,

where σ_{a} denotes the local Artin symbol [a, K_{𝔭}^{ab}/N_{n,𝔭}] and b^{1/p^{n}} is any p^{n}-th root of b in K_{𝔭}^{ab}.

We remark that it is a standard property of the Hilbert pairing (see e.g. [17, Chapter XIV, §2]) that

(a, b)_{𝔭,μ_{p^{n}}} = (b, a)_{𝔭,μ_{p^{n}}}^{−1}.  (5.4)
□

Lemma 5.4. Suppose that u ∈ U_{∞,𝔭}^{∗} and c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}). Then

(u_{n}, r_{n}^{∗}(c_{n}^{∗})(π^{∗−n}w_{n}))_{𝔭,μ_{p^{n}}} = α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) · ζ_{n}.

Proof. It follows from the definition of r_{n}^{∗} (see Corollary 2.3) that

(u_{n}, r_{n}^{∗}(c_{n}^{∗})(π^{∗−n}w_{n}))_{𝔭,μ_{p^{n}}} = e_{n}(π^{∗−n}w_{n}, c_{n}^{∗}([u_{n}, K_{𝔭}^{ab}/N_{n,𝔭}]))
= e_{n}(π^{∗−n}w_{n}, c_{n}^{∗}([u_{n}, K_{𝔭}^{ab}/K_{n,𝔭}^{∗}])) (since u_{n} ∈ K_{n,𝔭}^{∗})
= e_{n}(π^{∗−n}w_{n}, α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) · w_{n}^{∗})
= α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) · e_{n}(w_{n}, π^{−n}w_{n}^{∗})
= α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) · ζ_{n},

as claimed. □

Definition 5.5. For each integer n ≥ 1, we define a pairing

( , )_{𝔭,E_{π^{n}}} : N_{n,𝔭}^{×} × H^{1}(N_{n,𝔭}, E_{π^{n}}) → E_{π^{n}}  (5.5)

by

(ν, y)_{𝔭,E_{π^{n}}} = y([ν, K_{𝔭}^{ab}/N_{n,𝔭}]).

BIRCH AND SWINNERTON-DYER CONJECTURE II 17

It is not hard to check (using Lemma 2.1) that the pairings ( , )_{𝔭,E_{π^{n}}} combine to yield a pairing

( , )_{𝔭,T} : U_{∞,𝔭} × H^{1}(K_{𝔭}, T) → T; (β, c) ↦ α_{𝔭,T}(β, c) · w  (5.6)

that is defined as follows. Suppose that β = [β_{n}] ∈ U_{∞,𝔭}, and that

c = [c_{n}] ∈ \underleftarrow{lim} H^{1}(K_{𝔭}, E_{π^{n}}) ≃ H^{1}(K_{𝔭}, T).

Then we set

(β, c)_{𝔭,T} = [(β_{n}, c_{n})_{𝔭,E_{π^{n}}}] = [α_{𝔭,E_{π^{n}}}(β_{n}, c_{n})] · w =: α_{𝔭,T}(β, c) · w. □

The advantage of the pairings (5.5) and (5.6) is that they can be evaluated using explicit reciprocity laws. The following result gives the relationship between the pairings (5.1), (5.3) and (5.5).

Lemma 5.6. (a) Suppose that c ∈ H^{1}(K_{𝔭}, T), and that u ∈ U_{∞,𝔭}. Then

(u_{n}, r_{n}(c_{n})(π^{−n}w_{n}^{∗}))_{𝔭,μ_{p^{n}}} = α_{𝔭,E_{π^{n}}}(u_{n}, c_{n}) · ζ_{n}.

(b) Suppose that u ∈ U_{∞,𝔭}^{∗} and c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}). Then

(u_{n}, χ_{n}(r_{n}^{∗}(c_{n}^{∗}(π^{∗−n}w_{n}))))_{𝔭,E_{π^{n}}} = −(r_{n}^{∗}(c_{n}^{∗})(π^{∗−n}w_{n}), χ_{n}(u_{n}))_{𝔭,E_{π^{n}}} = α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}) · w_{n}.

(Recall that the isomorphism χ_{n} is defined in (4.2).)

Proof. Part (a) may be proved in exactly the same way as Lemma 5.4, while Part (b) is an immediate consequence of the same lemma. □

In order to evaluate the pairings in Lemma 5.6(b) above using explicit reciprocity laws (and thereby also compute the value of the [ρ(ξ), y^{∗}]_{K,𝔭^{∗}} in Proposition 5.2 above), we shall require the following result (cf. Corollary 2.3).

Lemma 5.7. There are injective homomorphisms

κ : H^{1}(K, T) → \underleftarrow{lim} H^{1}(K_{n}^{∗}, Z_{p}(1))  (5.7)
κ^{∗} : H^{1}(K, T^{∗}) → \underleftarrow{lim} H^{1}(K_{n}, Z_{p}(1)),  (5.8)

where the inverse limits are taken with respect to the obvious corestriction maps. A similar result holds if K is replaced by K_{𝔭} or K_{𝔭^{∗}}.

Proof. We shall just explain the construction of κ^{∗}, since that of κ is very similar. Recall that w = [w_{n}] denotes a generator of T. Suppose that

c^{∗} = [c_{n}^{∗}] ∈ \underleftarrow{lim} H^{1}(K_{𝔭}, E_{π^{∗n}}) ≃ H^{1}(K_{𝔭}, T^{∗}),

18 A. AGBOOLA

and consider the composition of maps

H^{1}(K_{n+1,𝔭}, μ_{p^{n+1}}) → H^{1}(K_{n,𝔭}, μ_{p^{n+1}}) → H^{1}(K_{n,𝔭}, μ_{p^{n}}),  (5.9)

where the first arrow is given by corestriction and the second arrow is induced by the natural map μ_{p^{n+1}} → μ_{p^{n}}. Recall from Corollary 2.3 that there is an isomorphism

r_{n}^{∗} : H^{1}(K, E_{π^{∗n}}) ⟶^{∼} Hom(E_{π^{n}}, K_{n}^{×}/K_{n}^{×p^{n}})^{Gal(K_{n}/K)}.  (5.10)

It is not hard to check that (5.9) maps {r_{n+1}^{∗}(c_{n+1}^{∗})}(π^{∗−(n+1)}w_{n+1}) to {r_{n}^{∗}(c_{n}^{∗})}(π^{∗−n}w_{n}). We may therefore define

c̃^{∗}(w) := [\{r_{n}^{∗}(c_{n}^{∗})\}(π^{∗−n}w_{n})] ∈ \underleftarrow{lim} H^{1}(K_{n,𝔭}, μ_{p^{n}}),

where the second inverse limit is taken with respect to the maps (5.9). It is a standard fact (see e.g. [15, Appendix B, Section B.3]) that

\underleftarrow{lim} H^{1}(K_{n,𝔭}, μ_{p^{n}}) ≃ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)),

where the right-hand inverse limit is taken with respect to the obvious corestriction maps. We may therefore view c̃^{∗}(w) as being an element of \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)), and we write κ^{∗}(c^{∗}) = [κ^{∗}(c^{∗})_{n}] ∈ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)) for this element. We remark that it follows from the construction of κ^{∗} that, for each integer n ≥ 1, we have:

κ^{∗}(c^{∗})_{n} ≡ {r_{n}^{∗}(c_{n}^{∗})}(π^{∗−n}w_{n}) (mod K_{n,𝔭}^{×p^{n}})  (5.11)

in H^{1}(K_{n,𝔭}, μ_{p^{n}}) ≃ K_{n,𝔭}^{×}/K_{n,𝔭}^{×p^{n}}. This implies that κ^{∗} is injective. □

Suppose now that u ∈ U_{∞,𝔭}^{∗} and that c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}) with κ^{∗}(c^{∗}) ∈ U_{∞,𝔭} ⊆ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)). Lemma 5.6(b) implies that, for each integer n ≥ 1, α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) may be determined by calculating the value of (κ^{∗}(c^{∗})_{n}, χ_{n}(u_{n}))_{𝔭,E_{𝔭^{n}}}; this in turn may be done by using Wiles’s explicit reciprocity law applied to the formal group Ê associated to E, as we shall now describe.

For any β = [β_{n}] ∈ U_{∞,𝔭}, let g_{β,w}(Z) ∈ O_{K,𝔭}[[Z]] denote the Coleman power series of β; so, for every integer n > 0, we have

g_{β,w}(ŵ_{n}) = β_{n}.  (5.12)

Set

δg_{β,w}(Z) := \frac{g′_{β,w}(Z)}{g_{β,w}(Z) · λ′ ; Ê(Z)},  (5.13)

and write

δ_{w}(β) = δg_{β,w}(0) := \frac{g′_{β,w}(0)}{g_{β,w}(0) · λ′ ; Ê(0)} = \frac{g′_{β,w}(0)}{g_{β,w}(0)},  (5.14)

(where for the last equality we have used the fact that λÊ(Z) = Z + (higher order terms)).

BIRCH AND SWINNERTON-DYER CONJECTURE II 19

Remark 5.8. Suppose that c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}) with κ^{∗}(c^{∗}) ∈ U_{∞,𝔭} ⊆ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)). In Proposition 6.2, we shall express δ_{w}(κ^{∗}(c^{∗})) in terms of exp_{𝔭}^{∗}(c^{∗}), and in Proposition 8.2(a), we shall show that, for suitably chosen β, we have that δ_{w}(β) is equal to a non-zero multiple of L_{𝔭}(ψ). This will enable us to deduce that the canonical class s_{𝔭^{∗}} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}) is of infinite order when L_{𝔭}(ψ) ≠ 0 (see Theorem 8.3). □

Proposition 5.9. Suppose that u ∈ U_{∞,𝔭}^{∗} and that c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}) with κ^{∗}(c^{∗}) ∈ U_{∞,𝔭} ⊆ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)). Then, with notation as above, we have

α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) ≡ (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(κ^{∗}(c^{∗})) · Ω_{𝔭}
× ∑_{σ∈Gal(K_{n,𝔭}^{∗}/K_{𝔭})} ψ^{∗}(σ^{−1}) log_{𝔭}(u_{n}^{σ}) (mod 𝔭^{n}),  (5.15)

where Ω_{𝔭} ∈ O^{×} is the 𝔭-adic period described in Section 4.

Hence we have (u, c^{∗})_{𝔭,T^{∗}} = α_{𝔭,T^{∗}}(u, c^{∗}) · w^{∗}, where

α_{𝔭,T^{∗}}(u, c^{∗}) = (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(κ^{∗}(c^{∗})) · Ω_{𝔭}
× lim_{n→∞} { ∑_{σ∈Gal(K_{n,𝔭}^{∗}/K_{𝔭})} ψ^{∗}(σ^{−1}) log_{𝔭}(u_{n}^{σ})} .  (5.16)

Proof. Applying Wiles’s explicit reciprocity law (see [5, Chapter I, Theorem 4.2]) to evaluate (κ^{∗}(c^{∗})_{n}, χ_{n}(u_{n}))_{𝔭,E_{π^{n}}}, we see (using Lemma 5.6) that the value of α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) is given by:

α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) ≡ − Tr_{K_{n,𝔭}^{∗}/K_{𝔭}} \{π^{−n} Tr_{𝔎_{n,𝔭}/K_{n,𝔭}^{∗}} (log_{E,𝔭}({ \atop χ_{n}(u_{n})})) · δg_{κ^{∗}(c^{∗}),w}(ŵ_{n}))\} (mod 𝔭^{n}).

(5.17)

Since u_{n} ∈ U_{n,𝔭}^{∗} and χ_{n} is Gal(\overline{K}/K_{n}^{∗})-equivariant, it follows that log_{E,𝔭}({ \atop χ_{n}(u_{n})}) ∈ K_{n,𝔭}^{∗}. Also, we have that π^{−n} Tr_{𝔎_{n,𝔭}/K_{n,𝔭}^{∗}}(δg_{κ^{∗}(c^{∗}),w}(ŵ_{n})) ∈ K_{𝔭} because ŵ_{n} ∈ K_{n,𝔭}. Hence (5.17) implies that

α_{𝔭,E_{π^{∗n}}}(u_{n}, c_{n}^{∗}) ≡ − \{π^{−n} Tr_{𝔎_{n,𝔭}/K_{n,𝔭}^{∗}}(δg_{κ^{∗}(c^{∗})}(ŵ_{n}))\} · Tr_{K_{n,𝔭}^{∗}/K_{𝔭}}(log_{E,𝔭}({ \atop χ_{n}(u_{n})})) (mod 𝔭^{n}).

(5.18)

From Proposition 4.1(b), we see that

log_{E,𝔭}({ \atop χ_{n}(u_{n})}) ≡ Ω_{𝔭} · log_{𝔭}(u_{n}) (mod 𝔭^{n}),

and this implies that

Tr_{K_{n,𝔭}^{∗}/K_{𝔭}}(log_{E,𝔭}({ \atop χ_{n}(u_{n})}) ≡ Ω_{𝔭} ∑_{σ∈Gal(K_{n,𝔭}^{∗}/K_{𝔭})} ψ^{∗}(σ^{−1} log_{𝔭}(u_{n}^{σ})) (mod 𝔭^{n+1}).  (5.19)

We also observe that [5, Chapter II, Proposition 4.5(iii)] implies that

π^{−n} Tr_{𝔎_{n,𝔭}/K_{n,𝔭}^{∗}}(δg_{κ^{∗}(c^{∗}),w}(ŵ_{n})) = (1 − \frac{ψ^{∗}(𝔭)}{p}) · δ_{w}(κ^{∗}(c^{∗})),  (5.20)

which is independent of n. The congruence (5.15) now follows from (5.18), (5.19) and (5.20). The equality (5.16) is a direct consequence of (5.15). □

Suppose now that

ξ ∈ Hom(T^{∗}, U_{∞,𝔭}^{∗}/\overline{E_{∞}^{∗}})^{Gal(K_{∞}^{∗}/K)}, ξ^{∗} ∈ Hom(T, U_{∞,𝔭^{∗}}/\overline{E_{∞}})^{Gal(K_{∞}/K)},

and set

c := ρ(ξ) ∈ Σ̌_{𝔭}(K, T), c^{∗} := ρ^{∗}(ξ^{∗}) ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}).

Then the definition of ρ^{∗} implies that c^{∗} = [c_{n}^{∗}] = [(p − 1)π^{n}r_{n}^{∗−1}(ξ_{n}^{∗})] ∈ \underleftarrow{lim}_{n} H^{1}(K, E_{π^{n}}). For any ς ∈ E_{π^{n}}, a routine computation shows that

r_{n}^{∗}(c_{n}^{∗})(ς) = ξ_{n}^{∗}((p − 1)π^{∗n}ς),

and so setting ς = π^{∗−n}w_{n}, we see that

r_{n}^{∗}(c_{n}^{∗})(π^{∗−n}w_{n}) = ξ_{n}^{∗}(w_{n})^{p−1}.

Hence it follows from (5.11) that

κ^{∗}(c^{∗})_{n} ≡ ξ_{n}^{∗}(w_{n})^{p−1} (mod K_{n,𝔭}^{×p^{n}})  (5.21)

in H^{1}(K_{n,𝔭}, μ_{p^{n}}) ≃ K_{n,𝔭}^{×}/K_{n,𝔭}^{×p^{n}}.

Recall from Section 1 that ϑ is a certain generator of the ideal I of Λ(K_{∞}). We now observe that as ξ^{∗} is Gal(K_{∞}/K)-equivariant, it follows that ξ^{∗}(w)^{ϑ} ∈ \overline{E}_{∞}/I\overline{E}_{∞}. The following result describes the relationship between κ^{∗}(c^{∗}) and ξ^{∗}(w)^{ϑ}.

Proposition 5.10. With the above notation, we have

δ_{w}(κ^{∗}(c^{∗})) = δ_{w}(ξ^{∗}(w)^{ϑ}),

where δ_{w}(ξ^{∗}(w)^{ϑ}) denotes δ_{w} applied to any lift in \overline{E}_{∞} of ξ^{∗}(w)^{ϑ} ∈ \overline{E}_{∞}/I\overline{E}_{∞}.

Proof. Suppose that ξ^{∗}(w) = α = [α_{n}], with α_{n} ∈ U_{n,𝔭^{∗}}/\overline{E}_{n}, and set ξ^{∗}(w)^{ϑ} = β = [β_{n}], with β_{n} ∈ \overline{E}_{n}/I\overline{E}_{n}. It follows from the definition of ξ_{n}^{∗} (see Proposition 2.6) that

ξ_{n}^{∗}(w_{n}) = α^{p^{n}} ∈ \overline{E}_{n}/\overline{E}_{n}^{p^{n}} = E_{n}/E_{n}^{p^{n}}.

For each n ≥ 1, let ϑ_{n} denote the projection of ϑ to Z_{p}[Gal(K_{n}/K)]. It follows from [12, Lemma 6.3] that

ϑ_{n} ∑_{σ∈Gal(K_{n}/K)} ψ^{−1}(\overline{σ})σ ≡ −(p − 1)p^{n} (mod p^{n}IZ_{p}[Gal(K_{n}/K)]),  (5.22)

where \overline{σ} denotes any lift of σ ∈ Gal(K_{n}/K) to Gal(K_{∞}/K).

Hence we have (using additive notation for Galois action):

ϑξ_{n}^{∗}(w_{n}) = p^{n}β_{n}

≡ −(p − 1)^{−1}ϑ ( ∑_{σ∈Gal(K_{n}/K)} ψ^{−1}(\overline{σ})σ) β_{n} (mod p^{n}ϑ\overline{E}_{n}).

Since \overline{E}_{n} has no ϑ-torsion (see [5, Chapter III, Proposition 1.3]), we can divide this relation by ϑ and apply (5.21) to obtain

(p − 1)ξ_{n}^{∗}(w_{n}) ≡ ( ∑_{σ∈Gal(K_{n}/K)} ψ^{−1}(\overline{σ})σ) β_{n} (mod p^{n}\overline{E}_{n})
≡ κ^{∗}(c^{∗})_{n} (mod p^{n}\overline{E}_{n}).

It now follows from [5, Chapter II, Lemma 4.8] that

δ_{w}(κ^{∗}(c^{∗})) = δ_{w}(β),

and this implies the desired result. □

Let \overline{ξ(w^{∗})} ∈ U_{∞,𝔭}^{∗} be any lift of ξ(w^{∗}).

Theorem 5.11. (a) For any y^{∗} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}), we have

[ρ(ξ), y^{∗}]_{K,𝔭^{∗}} = (p − 1) · (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(κ^{∗}(y^{∗})) · Ω_{𝔭}

× lim_{n→∞} \{ ∑_{σ∈Gal(K_{n,𝔭}^{∗}/K_{𝔭})} ψ^{∗}(σ^{−1}) log_{𝔭}((\overline{ξ(w^{∗})})_{n}^{σ})\} .

(b) We have

[ρ(ξ), ρ^{∗}(ξ^{∗})]_{K,𝔭^{∗}} = (p − 1) · (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(ξ^{∗}(w)^{ϑ}) · Ω_{𝔭}

× lim_{n→∞} \{ ∑_{σ∈Gal(K_{n,𝔭}^{∗}/K_{𝔭})} ψ^{∗}(σ^{−1}) log_{𝔭}((\overline{ξ(w^{∗})})_{n}^{σ})\} .

Proof. This result follows directly from Propositions 5.2, 5.9 and 5.10. □

6. The Dual Exponential Map

Suppose that c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}) with κ^{∗}(c^{∗}) ∈ U_{∞,𝔭} ⊆ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)), and let

exp_{𝔭}^{∗} : H^{1}(K_{𝔭}, T^{∗}) → Q_{p}

denote the Bloch-Kato dual exponential map (see e.g. [16, Section 5] for an account of the dual exponential map associated to an elliptic curve). In this section we shall evaluate exp_{𝔭}^{∗}(c^{∗}) in terms of the δ_{w}(κ^{∗}(c^{∗})) (cf. Remark 5.8 above and Proposition 6.2 below).

We begin by describing the relationship between the cup product pairing

∪ : H^{1}(K_{𝔭}, T) × H^{1}(K_{𝔭}, T^{∗}) → Z_{p}  (6.1)

and the Kummer pairing ( , )_{𝔭,T} of the previous section. This result is probably well known, but we are not aware of a written reference, and so we include a proof here.

Proposition 6.1. Suppose that c = [c_{n}] ∈ H^{1}(K_{𝔭}, T) and c^{∗} = [c_{n}^{∗}] ∈ H^{1}(K_{𝔭}, T^{∗}). Then

(κ^{∗}(c^{∗}), c)_{𝔭,T} = −(c ∪ c^{∗}) · w,

i.e.

α_{𝔭,T}(κ^{∗}(c^{∗}), c) = [α_{𝔭,E_{π^{n}}}(κ^{∗}(c^{∗})_{n}, c_{n})] = −(c ∪ c^{∗}).

Proof. Let

∪ : H^{1}(K_{𝔭}, E_{π^{n}}) × H^{1}(K_{𝔭}, E_{π^{∗n}}) → Z/p^{n}Z

denote the cup product pairing ‘at level n’ afforded by (6.1). To prove the desired result, it suffices to show that, for each integer n ≥ 1, we have

(r_{n}^{∗}(c_{n}^{∗})(π^{−∗n}w_{n}), c_{n})_{𝔭,E_{π^{n}}} = −(c_{n} ∪ c_{n}^{∗}) · w_{n}.

In order to do this, we first recall (see e.g. [17, Chapter XIV]) that the Hilbert pairing ( , )_{𝔭,μ_{p^{n}}} may be identified with a cup product pairing

∪ : H^{1}(N_{n,𝔭}, μ_{p^{n}}) × H^{1}(N_{n,𝔭}, μ_{p^{n}}) → Z/p^{n}Z

so that

(a, b)_{𝔭,μ_{p^{n}}} = (a ∪ b) · ζ_{n}.

We also note that the Gal(\overline{K}/N_{n})-equivariant isomorphisms

E_{π^{n}} ⟶^{∼} μ_{p^{n}}; w_{n} ↦ e_{n}(w_{n}, π^{−n}w_{n}^{∗}),

E_{π^{∗n}} ⟶^{∼} μ_{p^{n}}; w_{n}^{∗} ↦ e_{n}(π^{∗−n}w_{n}, w_{n}^{∗})

BIRCH AND SWINNERTON-DYER CONJECTURE II 23

induce isomorphisms of cohomology groups

κ_{n} : H^{1}(N_{n,𝔭}, E_{π^{n}}) ⟶^{∼} H^{1}(N_{n,𝔭}, μ_{p^{n}}); y_{n} ↦ r_{n}(y_{n})(π^{−n}w_{n}^{∗});

κ_{n}^{∗} : H^{1}(N_{n,𝔭}, E_{π^{∗n}}) ⟶^{∼} H^{1}(N_{n,𝔭}, μ_{p^{n}}); y_{n}^{∗} ↦ r_{n}^{∗}(y_{n}^{∗})(π^{∗−n}w_{n}),

and via functoriality of cup product pairings, we have

c_{n} ∪ c_{n}^{∗} = κ_{n}(c_{n}) ∪ κ_{n}^{∗}(c_{n}^{∗}) = −(κ_{n}^{∗}(c_{n}^{∗}) ∪ κ_{n}(c_{n})).

Hence, from Lemma 5.6(b), we see that

α_{𝔭,E_{π^{n}}}(r_{n}^{∗}(c_{n}^{∗})(π^{−∗n}w_{n}), c_{n}) = α_{𝔭,μ_{p^{n}}}(r_{n}^{∗}(c_{n}^{∗})(π^{−∗n}w_{n}), r_{n}(c_{n})(π^{−n}w_{n}^{∗}))
= κ_{n}^{∗}(c_{n}) ∪ κ_{n}(c_{n})
= −(κ_{n}(c_{n}) ∪ κ_{n}^{∗}(c_{n}^{∗}))
= −(c_{n} ∪ c_{n}^{∗}),

as required. □

We can now state the main result of this section.

Proposition 6.2. Suppose that c^{∗} ∈ H^{1}(K_{𝔭}, T^{∗}) with κ^{∗}(c^{∗}) ∈ U_{∞,𝔭} ⊆ \underleftarrow{lim} H^{1}(K_{n,𝔭}, Z_{p}(1)). Then

exp_{𝔭}^{∗}(c^{∗}) = (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(κ^{∗}(c^{∗})).

Proof. It follows from the definition of the dual exponential map (see [16, Section 5]) that

x ∪ c^{∗} = log_{E,𝔭}(x) · exp_{𝔭}^{∗}(c^{∗})

for all x ∈ H^{1}(K_{𝔭}, T). (Here we have extended the logarithm map log_{E,𝔭} from E(K_{𝔭}) ⊗ Z_{p} to H^{1}(K_{𝔭}, T) via linearity.)

From Wiles’s explicit reciprocity law (see [5, Chapter IV, §2.5] for the version we require), we have

(κ^{∗}(c^{∗}), x)_{𝔭,T} = \{log_{E,𝔭}(x) · (1 − \frac{ψ^{∗}(𝔭)}{p}) · δ_{w}(κ^{∗}(c^{∗}))\} · w.

The result now follows from Proposition 6.1. □

7. Elliptic units and canonical elements

We shall now explain how elliptic units may be used, following the methods described in [12], to construct canonical elements

s_{𝔭} ∈ Σ̌_{𝔭}(K, T), s_{𝔭^{∗}} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗})

24 A. AGBOOLA

when r = 0. These are the analogues in the present situation of the elements x_{𝔭}^{(1)} ∈ Šel(K, T) and x_{𝔭^{∗}}^{(1)} ∈ Šel(K, T^{∗}) constructed in [12, §4] when r = 1.

Let C_{∞} ⊆ E_{∞} and C_{∞}^{∗} ⊆ E_{∞}^{∗} denote the norm-coherent systems of elliptic units constructed in [12, §3], and write \overline{C}_{∞} and \overline{C}_{∞}^{∗} for the closures of C_{∞} in \overline{E}_{∞} and C_{∞}^{∗} in \overline{E}_{∞}^{∗} respectively. Recall that 𝔣 ⊆ O_{K} denotes the conductor of the Grossencharacter associated to E, and fix B ∈ E_{𝔣}/ Gal(\overline{K}/K) of exact order 𝔣 as described in [12, §6]. For each generator w = [w_{n}] of T and w^{∗} = [w_{n}^{∗}] of T^{∗}, let

θ_{B}(w) = [θ_{B}(w_{n})] ∈ \overline{C}_{∞} ⊆ U_{∞,𝔭} ⊗ Q
θ_{B}(w^{∗}) = [θ_{B}(w_{n}^{∗})] ∈ \overline{C}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q

denote the norm-coherent sequences of elliptic units constructed in [12, §3]. In particular, we have that

θ_{B}(w)^{σ} = θ_{B}(w^{σ}),  θ_{B}(w^{∗})^{σ} = θ_{B}(w^{∗σ})  (7.1)

for all σ ∈ Gal(\overline{K}/K), and for each n > 0, we have that

\overline{C}_{n} = Z_{p}[Gal(K_{n}/K)] · θ_{B}(w_{n}),  \overline{C}_{n}^{∗} = Z_{p}[Gal(K_{n}^{∗}/K)] · θ_{B}(w_{n}^{∗})  (7.2)

(see [13, Propositions 3.1 and 3.2]).

The following result is due to Rubin (see [13, Theorem 7.2]) and it provides a crucial link between elliptic units and p-adic L-functions. It is proved via a careful analysis of the construction of the Katz two-variable p-adic L-function L_{𝔭} using p-adic measures arising from elliptic units as described in [5, Chapter II, §4]. We refer the reader to [13, §7] for the details of the proof.

Theorem 7.1. Suppose that w is any generator of T. Then there are natural Galoisequivariant injections

ι_{w} : U_{∞,𝔭} → Λ(K_{∞})_{O}, ι^{∗} : U_{∞,𝔭}^{∗} → Λ(K_{∞}^{∗})_{O},

(where ι_{w} depends upon the choice of w, but ι^{∗} does not), satisfying the following properties. Suppose that w^{∗} is a generator of T^{∗} fixed as in Section 4. Then

(a) We have that

(1 − Fr_{𝔭^{∗}})ι_{w}(θ_{B}(w)) = L_{𝔭}|_{K_{∞}},  ι^{∗}(θ_{B}(−N(𝔣)^{−1}w^{∗})) = L_{𝔭}|_{K_{∞}^{∗}},  (7.3)

where Fr_{𝔭^{∗}} denotes the Frobenius of 𝔭^{∗} in Gal(K_{∞}/K), and L_{𝔭}|_{K_{∞}} and L_{𝔭}|_{K_{∞}^{∗}} denote the images of L_{𝔭} under the natural quotient maps Λ(𝔎_{∞})_{O} → Λ(K_{∞})_{O} and Λ(𝔎_{∞})_{O} → Λ(K_{∞}^{∗})_{O} respectively.

(b) For every β ∈ U_{∞,𝔭}, we have

ι_{w}(β) = (1 − \frac{ψ^{∗}(𝔭)}{p}) · Ω_{𝔭} · δ_{w}(β).  (7.4)

(c) For every u = [u_{n}] ∈ U_{∞,𝔭}^{∗}, we have

ι^{∗}(u)(ψ^{∗}) = (1 − \frac{ψ^{∗}(𝔭)}{p}) · lim_{n→∞} \{ ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ) log_{𝔭}(u_{n}^{τ})\} .  (7.5)
□

Now suppose that r = 0. Then L_{𝔭}(1) ≠ 0, and so we see from (7.3) that

\overline{C}_{∞} ⊆ \overline{E}_{∞} ⊂ U_{∞,𝔭} ⊗ Q and \overline{C}_{∞} ⊈ I(U_{∞,𝔭} ⊗ Q).

In particular, we have that \overline{C}_{∞} ⊈ I\overline{E}_{∞} ⊆ U_{∞,𝔭} ⊗ Q. Similar remarks imply that also \overline{C}_{∞}^{∗} ⊈ I^{∗}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭^{∗}}^{∗} ⊗ Q. Applying Remark 1.1, we deduce that

\overline{C}_{∞}^{∗} ⊈ I^{∗}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q.  (7.6)

Also, if L_{𝔭}(1) ≠ 0, then this implies that L_{𝔭}^{∗}(1) = 0, and so again from (7.3), it follows that we have

\overline{C}_{∞}^{∗} ⊆ I^{∗}(U_{∞,𝔭}^{∗} ⊗ Q).  (7.7)

Recall from Section 1 that ϑ^{∗} is a certain generator of the ideal I^{∗} of Λ(K_{∞}^{∗}).

Proposition 7.2. There exists a unique homomorphism σ_{𝔭} ∈ Hom(T^{∗}, (U_{∞,𝔭}^{∗}⊗Q)/\overline{E}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)} such that

σ_{𝔭}(w^{∗})^{ϑ^{∗}} = θ_{B}(w^{∗})

in \overline{E}_{∞}^{∗}/I^{∗}\overline{E}_{∞}^{∗}.

Proof. The proof of this result is very similar to that of [12, Theorem 4.2]. We first observe that it follows from (7.1) and (7.2) that Gal(K_{∞}^{∗}/K) acts on T^{∗} and \overline{C}_{∞}^{∗}/I^{∗}\overline{C}_{∞}^{∗} via the same character ψ^{∗}. Since Gal(K_{∞}^{∗}/K) also acts transitively on T^{∗} − pT^{∗}, it follows that the map w^{∗} ↦ θ_{B}(w^{∗}) induces a well-defined homomorphism σ_{𝔭}^{(0)} ∈ Hom(T^{∗}, \overline{C}_{∞}^{∗}/I^{∗}\overline{C}_{∞}^{∗})^{Gal(K_{∞}^{∗}/K)}.

Next, we note that it follows from [5, Chapter III, Proposition 1.3] that U_{∞,𝔭}^{∗} contains no ϑ^{∗}-torsion elements, and so, if α ∈ \overline{C}_{∞}^{∗}, then ϑ^{−1}α is a well-defined element of U_{∞,𝔭}^{∗} ⊗ Q. It is not hard to show that the image of ϑ^{−1}α in (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗} depends only upon the image of α in \overline{C}_{∞}^{∗}/I^{∗}\overline{C}_{∞}^{∗}. The map σ_{𝔭} is defined by σ_{𝔭} := ϑ^{−1}σ_{𝔭}^{(0)}. □

Definition 7.3. We set

s_{𝔭} := ρ(σ_{𝔭}) ∈ Σ̌_{𝔭}(K, T), s_{𝔭^{∗}} := ρ^{∗}(σ_{𝔭^{∗}}) ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}),

where of course the definition σ_{𝔭^{∗}} ∈ Hom(T, (U_{∞,𝔭^{∗}} ⊗ Q)/\overline{E}_{∞})^{Gal(K_{∞}/K)} is the same, mutatis mutandis, as that of σ_{𝔭}. □

Remark 7.4. We shall see from the proof of Theorem 8.4(a) below that s_{𝔭^{∗}} is of infinite order only if L′_{𝔭}(ψ^{∗}) ≠ 0. □

8. Proof of Theorem A

In this section we shall establish a number of properties of the cohomology classes s_{𝔭} and s_{𝔭^{∗}} constructed in Section 7, and thereby prove Theorem A of the Introduction.

We begin by recalling some basic facts concerning derivatives of elements in Iwasawa algebras (see [13, Section 7] or [1, Section 2], for example). Suppose that F ∈ Λ(𝔎_{∞})_{O}, and consider the function

s ↦ F (ψ^{∗}⟨ψ^{∗}⟩^{s}), s ∈ Z_{p};

this is a p-adic analytic function on Z_{p}. For any integer m ≥ 0, define

D^{∗(m)}F (ψ^{∗}) = \frac{1}{m!} \frac{d}{ds}F (ψ^{∗}⟨ψ^{∗}⟩^{s})|_{s=0};

then, by definition, we have

L′_{𝔭}(ψ^{∗}) = D^{∗(1)}L_{𝔭}(ψ^{∗}).

It is easy to check that

D^{∗(m)}ϑ^{∗m}(ψ^{∗}) = p^{m}

for every integer m ≥ 1.

The following result is [13, Lemma 7.3]. It is proved via a routine computation which we shall omit.

Lemma 8.1. With notation as above, we have

D^{∗(m)}(ϑ^{∗m}F)(ψ^{∗}) = p^{m}F (ψ^{∗})

for every integer m ≥ 1. □

We now turn to the precise relationship between the canonical elements s_{𝔭} and s_{𝔭^{∗}}, and certain special values of the p-adic L-function L_{𝔭}.

Proposition 8.2. Let \overline{σ_{𝔭}(w^{∗})} ∈ U_{∞,𝔭}^{∗} be any lift of σ_{𝔭}(w^{∗}). We have

L_{𝔭}(ψ) = (1 − \frac{1}{ψ(𝔭^{∗})}) · (1 − \frac{1}{ψ(𝔭)}) · Ω_{𝔭} · δ_{w}(θ_{B}(w)).  (8.1)

and

L′_{𝔭}(ψ^{∗}) = −p · N(𝔣)^{−1} · (1 − \frac{ψ^{∗}(𝔭)}{p}) · lim_{n→∞} \{ ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ) log_{𝔭}(\overline{σ_{𝔭}(w^{∗})}_{n}^{τ})\} . (8.2)

Proof. The equality (8.1) follows directly from (7.3) and (7.4). To prove (8.2), we first observe that, since there is no ϑ^{∗}-torsion in U_{∞,𝔭}^{∗} (see [5, Chapter III, Proposition 1.3]), there exists a unique β = [β_{n}] ∈ U_{∞,𝔭}^{∗} ⊗_{Z} Q such that

β^{ϑ^{∗}} = θ_{B}(w^{∗}).

Applying Theorem 7.1(a), we see that

−ϑ^{∗}N(𝔣)^{−1}ι^{∗}(β) = ι^{∗}(θ_{B}(N(𝔣)^{−1}w^{∗})
= L_{𝔭}|_{K_{∞}^{∗}}.

It therefore follows from Lemma 8.1 and Theorem 7.1(c) that

L′_{𝔭}(ψ^{∗}) = −N(𝔣)^{−1}ι^{∗}(β)(ψ^{∗})
= −p · N(𝔣)^{−1} · (1 − \frac{ψ^{∗}(𝔭)}{p}) · lim_{n→∞} \{ ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ) log_{𝔭}(β_{n}^{τ})\} .  (8.3)

Now suppose that \overline{σ_{𝔭}(w^{∗})} ∈ U_{∞,𝔭}^{∗} is any lift of σ_{𝔭}(w^{∗}). It follows from Proposition 7.2 that we have

σ_{𝔭}(w^{∗})^{ϑ^{∗}} = θ_{B}(w^{∗}) = β^{ϑ^{∗}}  (8.4)

in \overline{E}_{∞}^{∗}/I^{∗}\overline{E}_{∞}^{∗}. For each integer n ≥ 1, let ϑ_{n}^{∗} denote the projection of ϑ to Z_{p}[Gal(cK_{n}^{∗}/K)]. It is shown in [13, Lemma 6.3] that

ϑ_{n}^{∗} ∑_{τ∈Gal(K_{n}/K)} ψ^{−1}(τ)τ ≡ −(p − 1)p^{n} (mod p^{n}IZ_{p}[Gal(K_{n}^{∗}/K)]).

We therefore deduce from (8.4) that, for each n ≥ 1, we have (writing Galois action additively)

ϑ^{∗} ( ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ)τ) β_{n} ≡ ϑ^{∗} ( ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ)τ) \overline{σ_{𝔭}(w^{∗})} (mod p^{n}ϑ^{∗}\overline{E}_{n}^{∗}).

Since \overline{E}_{n}^{∗} has no ϑ^{∗}-torsion (see [13, Chapter III, Proposition 1.3]), we can divide both sides of this last relation by ϑ^{∗}, and then take 𝔭-adic logarithms and let n → ∞ to deduce that

lim_{n→∞} \{ ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ) log_{𝔭}(β_{n}^{τ})\} = lim_{n→∞} \{ ∑_{τ∈Gal(K_{n}^{∗}/K)} ψ^{∗−1}(τ) log_{𝔭}(\overline{σ_{𝔭}(w^{∗})}_{n}^{τ})\} . (8.5)

The desired equality (8.2) now follows from (8.3) and (8.5). □

Theorem 8.3. We have

exp_{𝔭}^{∗}(s_{𝔭^{∗}}) = (\frac{ψ^{∗}(𝔭)}{p} − 1) · δ_{w}(θ_{B}(w))

= (\frac{ψ^{∗}(𝔭)}{p} − 1) · (1 − \frac{1}{ψ(𝔭^{∗})})^{−1} · (1 − \frac{1}{ψ(𝔭)})^{−1} · \frac{L_{𝔭}(ψ)}{Ω_{𝔭}}.

Hence, if L_{𝔭}(ψ) ≠ 0, then s_{𝔭^{∗}} is of infinite order.

Proof. This follows from Propositions 5.10, 6.2 and 8.2. □

We can now prove the first part of Theorem A.

Theorem 8.4. Suppose that L_{𝔭}(ψ) ≠ 0.

(a) We have L′_{𝔭}(ψ^{∗}) ≠ 0.

(b) The height pairing [ , ]_{K,𝔭} is non-degenerate, and we have

[s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}} = (p − 1) · p^{−1} · N(𝔣)^{−1} · (1 − \frac{1}{ψ(𝔭^{∗})})^{−1} · (1 − \frac{1}{ψ(𝔭)})^{−1} · L_{𝔭}(ψ) · L′_{𝔭}(ψ^{∗}).

Proof. (a) Suppose that L′_{𝔭}(ψ^{∗}) = 0, and set t = ord_{s=1} L_{𝔭}^{∗}(s), so t ≥ 2. Then it follows from (7.3) that

\overline{C}_{∞}^{∗} ⊆ I^{∗t}(U_{∞,𝔭}^{∗} ⊗ Q).

Hence we have that σ_{𝔭^{∗}}(w^{∗}) ∈ I^{∗(t−1)}(U_{∞,𝔭}^{∗} ⊗ Q) · \overline{E}_{∞}^{∗}, and this in turn implies that s_{𝔭^{∗}} = 0. This contradicts Theorem 8.3 because by hypothesis L_{𝔭}(ψ) ≠ 0. It therefore follows that L′_{𝔭}(ψ^{∗}) ≠ 0, as claimed.

(b) The equality follows directly from Theorem 5.11(b), Proposition 8.2, and Theorem 8.3. As [s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}} ≠ 0, we deduce that [ , ]_{K,𝔭^{∗}} is non-degenerate because Σ̌_{𝔭}(K, T) and Σ̌_{𝔭^{∗}}(K, T^{∗}) are both free, rank one Z_{p}-modules (see [1, Lemma 3.6 and Proposition 6.7]). □

The second part of Theorem A is a direct consequence of the following result.

Theorem 8.5. (cf. [12, Theorem 10.1])

BIRCH AND SWINNERTON-DYER CONJECTURE II 29

Suppose that L_{𝔭}(ψ) ≠ 0, and that y ∈ Σ̌_{𝔭}(K, T) and y^{∗} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗}) are both non-zero. Then

\frac{exp_{𝔭}^{∗}(y^{∗}) · exp_{𝔭^{∗}}^{∗}(y)}{[y, y^{∗}]_{K,𝔭^{∗}}} = (p − 1)^{−1} · p · N(𝔣) · (\frac{ψ^{∗}(𝔭)}{p} − 1) · (\frac{ψ(𝔭^{∗})}{p} − 1)

× \frac{L_{𝔭}(ψ)}{(1 − ψ^{∗−1}(𝔭)) · (1 − ψ^{−1}(𝔭^{∗})) · Ω_{𝔭^{∗}} · Ω_{𝔭} · L′_{𝔭}(ψ^{∗})}.

Proof. Since L_{𝔭}(ψ) ≠ 0, it follows that

Hom(Σ̌_{𝔭}(K, T) ⊗_{Z_{p}} Σ̌_{𝔭^{∗}}(K, T^{∗}), Z_{p})

is a free Z_{p}-module of rank one (see [1, Lemma 3.6 and Proposition 6.7]), and that both exp_{𝔭^{∗}}^{∗}( ) · exp_{𝔭}^{∗}( ) and [ , ]_{K,𝔭^{∗}} are non-zero elements of this module. Hence, since s_{𝔭} and s_{𝔭^{∗}} are of infinite order, we have that

\frac{exp_{𝔭^{∗}}^{∗}(y) · exp_{𝔭}^{∗}(y^{∗})}{[y, y^{∗}]_{K,𝔭^{∗}}} = \frac{exp_{𝔭^{∗}}^{∗}(s_{𝔭}) · exp_{𝔭}^{∗}(s_{𝔭^{∗}})}{[s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}}}.

Applying Theorems 8.3 and 8.4, we see that

\frac{exp_{𝔭^{∗}}^{∗}(s_{𝔭}) · exp_{𝔭}^{∗}(s_{𝔭^{∗}})}{[s_{𝔭}, s_{𝔭^{∗}}]_{K,𝔭^{∗}}} = (p − 1)^{−1} · p · N(𝔣) · (\frac{ψ^{∗}(𝔭)}{p} − 1) · (\frac{ψ(𝔭^{∗})}{p} − 1)

× \frac{L_{𝔭^{∗}}(ψ^{∗})}{(1 − ψ^{∗−1}(𝔭)) · (1 − ψ^{−1}(𝔭^{∗})) · Ω_{𝔭^{∗}} · Ω_{𝔭} · L′_{𝔭}(ψ^{∗})}.

The desired result now follows from the fact that L_{𝔭}(ψ) = L_{𝔭^{∗}}(ψ^{∗}) (cf. Remark 1.2). □

This completes the proof of Theorem A.

References

[1] A. Agboola, On Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture, Comp. Math. 143 (2007), 1374–1398.

[2] A. Agboola, On certain special values of the Katz two-variable p-adic L-function, in preparation.

[3] D. Bernardi, C. Goldstein, N. Stephens, Notes p-adiques sur les courbes elliptiques, Crelle 351 (1985), 129–170.

[4] A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121–124.

[5] E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication, Academic Press (1987).

[6] R. Greenberg, Trivial zeros of p-adic L-functions, In: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 149–174, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994.

[7] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.

[8] B. Perrin-Riou, Déscent infinie et hauteurs p-adiques sur les courbes elliptiques à multiplication complexe, Invent. Math. 70 (1983), 369–398.

[9] B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137–185.

[10] B. Perrin-Riou, Arithmétique des courbes elliptiques et théorie d’Iwasawa, Memoire de la Société Mathématique de France, 17 (1984).

[11] K. Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987), 527–560.

[12] K. Rubin, p-adic L-functions and rational points on elliptic curves with complex multiplication, Invent. Math. 107 (1992), 323–350.

[13] K. Rubin, p-adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. In: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 71–80, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994.

[14] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 109 (1992), 25–68.

[15] K. Rubin, Euler Systems, Princeton University Press (2000).

[16] K. Rubin, Euler systems and modular elliptic curves. In: Galois representations in arithmetic geometry, A.J. Scholl, R. L. Taylor (Eds.), CUP 1998.

[17] J.-P. Serre, Local Fields, Springer Verlag (1979).

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

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