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

A. AGBOOLA

Abstract. We study Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation.

1. Introduction

Let E/Q be an elliptic curve with complex multiplication by O_{K}, the ring of integers of an imaginary quadratic field K (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_{∞}^{∗}. Write K_{∞} (resp. K_{∞}^{∗}) for the unique Z_{p} extension of K unramified outside 𝔭 (resp. 𝔭^{∗}). Let O denote the completion of the ring of integers of the maximal unramified extension of Q_{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 X(L) (resp. X^{∗}(L)) for the Pontryagin dual of the 𝔭-primary Selmer group Sel(L, E_{π^{∞}}) (resp. the 𝔭^{∗}-primary Selmer group Sel(L, E_{π^{∗∞}})) of E/L.

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 [14, p. 325]. We write T (resp. T^{∗}) for the 𝔭-adic (resp. 𝔭^{∗}-adic) Tate module of E.

The two-variable Iwasawa main conjecture (proved by Rubin [16]) implies that X(𝔎_{∞}) is a torsion Λ(𝔎_{∞})-module whose characteristic ideal in Λ(𝔎_{∞})_{O} is generated by a twist of

Date: Final version. January 22, 2007.

2000 Mathematics Subject Classification. 11G05, 11R23, 11G16.

Key words and phrases. Rubin, p-adic L-function, Birch and Swinnerton-Dyer conjecture, elliptic curve, Mazur-Tate-Teitelbaum.

Partially supported by NSF grant DMS-0401319.

2 A. AGBOOLA

Katz’s two-variable p-adic L-function L_{𝔭} by the character ψ. The function L_{𝔭} satisfies a p-adic interpolation formula that may be described as follows (see [14, Theorem 7.1] for the version given here, and also [6, Theorem II.4.14]). 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 need not concern us here.

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 𝔭-adic Birch and Swinnerton-Dyer conjecture for E (see [1, pages 133–134], [12, Theorem V.8]) predicts that ord_{s=1} L_{𝔭}(s) is equal to the rank r of E(Q), and that

lim_{s→1} \frac{L_{𝔭}(s)}{(s − 1)^{r}} ∼ [log_{p}(ψ(γ_{1}))]^{r} · (1 − \frac{ψ(𝔭)}{p}) · (1 − \frac{ψ(𝔭^{∗})}{p}) · |� (K)(𝔭)| · R_{K,𝔭},

where γ_{1} is a topological generator of Gal(K_{∞}/K), � (K)(𝔭) is the 𝔭-primary component of the Tate-Shafarevich group � (K) of E/K, R_{K,𝔭} is the regulator associated to the algebraic 𝔭-adic height pairing

{ , }_{K,𝔭} : Sel(K, T^{∗}) × Sel(K, T) → O_{K,𝔭}

on E/K (see [10]), and the symbol ‘∼’ denotes equality up to multiplication by a p-adic unit.

On the other hand, the character ψ^{∗} 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 only results concerning L_{𝔭}^{∗}(s) of which the author is aware are due to Rubin (see [14], [15]). 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 a striking p-adic construction of a global point of infinite order in E(Q) directly from the special value of a p-adic L-function.

When r = 0, however, the above analysis breaks down, and the situation is less clear. The functional equation satisfied by L_{𝔭} (see [6, II §6]) shows that ord_{s=1} L_{𝔭}(s) and ord_{s=1} L_{𝔭}^{∗}(s) have opposite parity, and so when r = 0, one expects that ord_{s=1} L_{𝔭}^{∗}(s) is odd. This may perhaps be viewed as being an analogue of a similar exceptional zero phenomenon observed

BIRCH AND SWINNERTON-DYER CONJECTURE 3

in the work of Mazur, Tate and Teitelbaum concerning p-adic Birch and Swinnerton-Dyer conjectures for elliptic curves without complex multiplication (see [9], [8]). As Rubin points out (see [15, Remark on p. 74]), it is reasonable to guess that ord_{s=1} L_{𝔭}^{∗}(s) = 1. If this is so, then one would like to determine the value of lim_{s→1}[L_{𝔭}^{∗}(s)/(s − 1)].

In this paper we study an Iwasawa module naturally associated to L_{𝔭}^{∗}(s) via the twovariable main conjecture and, among other things, we prove that the above guess is indeed correct. The Iwasawa module in question is the Pontryagin dual X_{𝔭^{∗}}(K_{∞}^{∗}, W^{∗}) of a certain restricted Selmer group Σ_{𝔭^{∗}}(K_{∞}^{∗}, W^{∗}). 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_{∞}^{∗}, W^{∗}). The two-variable main conjecture implies that a characteristic power series H_{K} ∈ Λ(K_{∞}^{∗}) of X_{𝔭^{∗}}(K_{∞}^{∗}, W^{∗}) may be viewed as being an algebraic p-adic L-function corresponding to L_{𝔭}^{∗}(s). We study L_{𝔭}^{∗}(s) by analysing the behaviour of H_{K}.

A special case of our results may be described as follows. We define a compact restricted Selmer group Σ̌_{𝔭^{∗}}(K, T^{∗}) ⊆ H^{1}(K, T^{∗}). The O_{K,𝔭^{∗}}-module Σ̌_{𝔭^{∗}}(K, T^{∗}) is free of rank |r − 1|, and if r ≥ 1, then it lies in the usual Selmer group Sel(K, T^{∗}) associated to T^{∗}. The O_{K,𝔭^{∗}}-rank of Σ̌_{𝔭^{∗}}(K, T^{∗}) governs the order of vanishing of L_{𝔭}^{∗}(s) at s = 1 in the same way that the O_{K,𝔭}-rank of Sel(K, T) determines ord_{s=1} L_{𝔭}(s). We also define a similar group Σ̌_{𝔭}(K, T) ⊆ H^{1}(K, T), and we explain how to construct a p-adic height pairing

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

If r ≥ 1, then in fact Σ̌_{𝔭}(K, T) ⊆ Sel(K, T), Σ̌_{𝔭^{∗}}(K, T^{∗}) ⊆ Sel(K, T^{∗}), and, if the 𝔭^{∗}-adic Birch and Swinnerton-Dyer conjecture is true, then the p-adic height pairing [ , ]_{K,𝔭^{∗}} is nondegenerate. We conjecture that [ , ]_{K,𝔭^{∗}} is also non-degenerate when r = 0 (see Remark 6.6).

Define

� _{rel(𝔭)}(K) := Ker [H^{1}(K, E) → ∏_{v∤𝔭}H^{1}(K_{v}, E)] ,

and write � _{rel(𝔭)}(K)(𝔭^{∗}) for its 𝔭^{∗}-primary subgroup. Let � _{rel(𝔭)}(K)(𝔭^{∗})_{/ div} denote the quotient of � _{rel(𝔭)}(K)(𝔭^{∗}) by its maximal divisible subgroup. It may be shown that � _{rel(𝔭)}(K)(𝔭^{∗}) has O_{K,𝔭^{∗}}-corank one, and that � _{rel(𝔭)}(K)(𝔭^{∗})_{/ div} is finite.

Theorem A. Suppose that [ , ]_{K,𝔭^{∗}} is non-degenerate, and let γ be a topological generator of Gal(K_{∞}^{∗}/K). Then, if r = 0, we have ord_{s=1} L_{𝔭}^{∗}(s) = 1, and

lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{s − 1} ∼

log_{p}(ψ^{∗}(γ)) · (1 − ψ(𝔭^{∗})) · \frac{|� _{rel(𝔭)}(K)(𝔭^{∗})_{/ div}|}{[H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ_{𝔭}(K, T)]} · R_{K,𝔭^{∗}},

where R_{K,𝔭^{∗}} is a p-adic regulator associated to [ , ]_{K,𝔭^{∗}}.

We also obtain an exact (but much less explicit) formula for lim_{s→1} L_{𝔭}^{∗}(s)/(s − 1) by applying the methods of [14] in our present setting (see Theorem 9.5 below).

Suppose now that r ≥ 1, and assume that � (K)(p) is finite. Then E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}} is a free O_{K,𝔭^{∗}}-module of rank r, and the kernel of the localisation map

E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}} → E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}}

has O_{K,𝔭^{∗}}-rank r − 1. Let y_{1}, …, y_{r−1} be an O_{K,𝔭^{∗}}-basis of this kernel, and extend it to an O_{K,𝔭^{∗}}-basis y_{1}, …, y_{r−1}, y_{𝔭^{∗}} of E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}}. We write x_{1}, …, x_{r−1}, y_{𝔭} for a similarly constructed O_{K,𝔭}-basis of E(K) ⊗_{O_{K}} O_{K,𝔭}. The following result is a direct consequence of Rubin’s precise formula for lim_{s→1}[L_{𝔭}^{∗}(s)/(s − 1)^{r−1}] (see [14, Corollary 11.3]). We give a new proof of this result which is different from that contained in [14]. In particular, our proof gives an alternative way of viewing the somewhat unusual regulator R_{𝔭}^{∗} defined in [14, §11].

Theorem B. Suppose that r ≥ 1 and that [ , ]_{K,𝔭^{∗}} is non-degenerate. Then ord_{s=1} L_{𝔭}^{∗}(s) = r − 1, and

lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{(s − 1)^{r−1}} ∼
[log_{p}(ψ^{∗}(γ))]^{r−1} · p^{−2} · |� (K)(𝔭^{∗})| · log_{E,𝔭^{∗}}(y_{𝔭^{∗}}) · log_{E,𝔭}(y_{𝔭}) · R_{K,𝔭^{∗}},  (1.2)

where log_{E,𝔭^{∗}} (resp. log_{E,𝔭}) denotes the 𝔭^{∗}-adic (resp. 𝔭-adic) logarithm associated to E.

An outline of the contents of this paper is as follows. In Section 2 we recall some basic facts about twists of Iwasawa modules and derivatives of characteristic power series, and we apply these results to describe the relationship between L_{𝔭}^{∗}(s) and a characteristic power series H_{K} ∈ Λ(K_{∞}^{∗}) of X_{𝔭^{∗}}(K_{∞}^{∗}, W^{∗}). In Section 3 we define various Selmer groups, and we establish some of their properties. We describe how to construct an algebraic p-adic height pairing on restricted Selmer groups in Section 4. In Section 5 we calculate (under certain hypotheses) the leading term of a characteristic power series H_{F} ∈ Λ(F_{∞}^{∗}) of X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}), where F/K is any finite extension, and F_{∞}^{∗} := F K_{∞}^{∗}. In Section 6 we study restricted Selmer

BIRCH AND SWINNERTON-DYER CONJECTURE 5

groups over K, and we show that, under certain standard assumptions, ord_{s=1} L_{𝔭}^{∗}(s) = |r−1|. We then give the proof of Theorem A in Section 7, and that of Theorem B in Section 8. Finally, in Section 9, we explain how the methods of [14] may be used to give a formula for the exact value of lim_{s→1} L_{p}^{∗}(s)/(s − 1) when r = 0.

Acknowledgements. I am very grateful indeed to Karl Rubin for extremely helpful conversations and correspondence. Parts of this paper were written while I was visiting the Université de Bordeaux I and the Centre de Recherches Mathématiques at the Université de Montreal. I thank these institutions for their hospitality and support.

Notation and conventions. For each integer n ≥ 1, we write

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

For each place v of K, we write k_{v} for the residue field of v, and Ẽ_{v}/k_{v} for the reduction of the elliptic curve E modulo v. We set W := E_{π^{∞}} and W^{∗} := E_{π^{∗∞}}.

Throughout this paper, F denotes a finite extension of K, and we set

F_{n} := F K_{n}, F_{∞} := F K_{∞}, F_{∞} := F K_{∞},
F_{n}^{∗} := F K_{n}^{∗}, F_{∞}^{∗} := F K_{∞}^{∗}, F_{∞}^{∗} := F K_{∞}^{∗},
𝔉_{∞} := F 𝔎_{∞}.

For any extension L/K we write M(L) (resp. M^{∗}(L)) for the maximal abelian pro-p extension of L which is unramified away from 𝔭 (resp. 𝔭^{∗}), and we set

X (L) := Gal(M(L)/L), X^{∗}(L) := Gal(M^{∗}(L)/L).

We let B(L) (resp. B^{∗}(L)) denote the maximal abelian pro-p extension of L which is unramified away from 𝔭 (resp. 𝔭^{∗}) and totally split at all places of L lying above 𝔭^{∗} (resp. 𝔭), and we write

Y(L) := Gal(B(L)/L), Y^{∗}(L) := Gal(B^{∗}(L)/L).

If M is any Z_{p}-module, then M_{div} denotes the maximal divisible submodule of M, and we set M_{/ div} := M/M_{div}. We write M_{tors} for the torsion submodule of M, and M^{∧} for the Pontryagin dual of M. If M is a torsion O_{K,𝔮}-module, with 𝔮 ∈ {𝔭, 𝔭^{∗}}, then we write T_{𝔮}(M) for the 𝔮-adic Tate module of M.

We set D_{𝔭} := K_{𝔭}/O_{K,𝔭} and D_{𝔭^{∗}} := K_{𝔭^{∗}}/O_{K,𝔭^{∗}}.

6 A. AGBOOLA

2. Twists and derivatives

In this section we shall recall some basic facts concerning twists of Iwasawa modules and derivatives of characteristic power series. We then apply these results to a twist of the Katz two-variable p-adic L-function L_{𝔭} by the character ψ^{∗}.

Let G_{F} := Gal(𝔉_{∞}/F), and suppose that ρ : G_{F} → Z_{p}^{×} is any character. Then we have a twisting map

Tw_{ρ} : Λ(G_{F}) → Λ(G_{F})

associated to ρ which is induced by the map g ↦ ρ(g)g for all g ∈ G_{F}. If M is a finitely generated Λ(G_{F})-module with characteristic power series f_{M}, then a routine computation shows that Tw_{ρ}(f_{M}) is a characteristic power series of M(ρ^{−1}) := M ⊗ ρ^{−1}.

Set H := Ker(ρ). Then there is a natural quotient map

Π_{G_{F} /H} : Λ(G_{F}) → Λ(G_{F} /H),

and Π_{G_{F} /H}(Tw_{ρ}(f_{M})) is a characteristic power series of the Λ(G_{F} /H)-module M(ρ^{−1})⊗_{Λ(G_{F})} Λ(G_{F} /H). If ρ_{1} : G_{F} → Z_{p}^{×} is any character which factors through G_{F} /H, then

[Tw_{ρ}(f_{M})](ρ_{1}) = [Π_{G_{F} /H}(Tw_{ρ}(f_{M}))](ρ_{1}),  (2.1)

and there is an isomorphism

M(ρ^{−1}) ⊗_{Λ(G_{F})} Λ(G_{F} /H) ≃ (M ⊗_{Λ(G_{F})} Λ(G_{F} /H))(ρ^{−1})

of Λ(G_{F} /H)-modules. Hence we may study the values of Tw_{ρ}(f_{M}) at characters ρ_{1} which factor through G_{F} /H by studying the values of Π_{G/H}(Tw_{ρ}(f_{M})) at such characters.

Suppose now that ρ is of infinite order, and let N be a finitely generated Λ(G_{F} /H)-module with characteristic power series f_{N} ∈ Λ(G_{F} /H). We may write

G_{F} /H ≃ Δ × G,

where |Δ| is prime to p, and G ≃ Z_{p}. Let γ be a fixed topological generator of G_{F} /H, and let Π_{G} : Λ(G_{F} /H) → Λ(G) be the natural quotient map. We identify Λ(G) with Z_{p}[[t]] in the usual way via the map Π_{G}(γ) ↦ 1 + t.

Let I_{G_{F} /H} denote the augmentation ideal of Λ(G_{F} /H), and suppose that n ≥ 0 is the largest integer such that f_{N} ∈ I_{G_{F} /H}^{n} and f_{N} ∉ I_{G_{F} /H}^{n+1}. It is not hard to check that Π_{G}(f_{N})(t) is a characteristic power series of the Λ(G)-module N^{Δ}, and that

((γ − 1)^{−n}f_{N})(1) = \frac{Π_{G}(f_{N})}{t^{n}} |_{t=0},  (2.2)

where 1 denotes the identity character of G_{F} /H.

BIRCH AND SWINNERTON-DYER CONJECTURE 7

For any character ν : G_{F} /H → Z_{p}^{×}, we set ϑ_{ν} := ν(γ)^{−1}γ − 1. Then if m ≥ 0 is any integer, it follows from the definitions that we have

(ϑ_{ν}^{−m}f_{N})(ν) = [(γ − 1)^{−m} Tw_{ν}(f_{N})](1),  (2.3)

where Tw_{ν} : Λ(G_{F} /H) → Λ(G_{F} /H) is the twisting map associated to ν.

We now recall how (2.3) is related to derivatives of certain p-adic analytic functions as described in [14, §7]. Write < ν >: G_{F} /H → Z_{p}^{×} for the composition of ν with the natural projection Z_{p}^{×} → 1 + pZ_{p}, and suppose that χ : G_{F} /H → Z_{p}^{×} is any character of order prime to p. The map from Z_{p} to C_{p} given by s ↦ f_{N}(νχ < ν >^{s−1}) defines an analytic function on Z_{p}. Define

ord_{νχ}(f_{N}) := ord_{s=1} f_{N}(νχ < ν >^{s−1}),

and set

D^{(m)}f_{N}(νχ) := \frac{1}{m!} (\frac{d}{ds})^{m} f_{N}(νχ < ν >^{s−1})|_{s=1}.

We write

f_{N}^{(m)}(νχ) := D^{(m)}f_{N}(νχ),

and we extend these definitions to Λ(G_{F}) via the quotient map Π_{G_{F} /H}. A routine calculation shows that we have

D^{(m)}(ϑ_{ν}^{m}(νχ)) = {log_{p}(ν(γ))}^{m},

and

D^{(m)}(ϑ_{ν}^{m}f_{N})(νχ) = {log_{p}(ν(γ))}^{m}f_{N}(νχ) = [{log_{p}(ν(γ))}^{m} Tw_{ν}(f_{N})](χ).  (2.4)

We can now see from (2.2), (2.3) and (2.4) that if n_{ν} := ord_{ν}(f_{N}), then we may write f_{N} = ϑ_{ν}^{n_{ν}}F_{ν} with F_{ν} ∈ Λ(G_{F} /H), and we have

f_{N}^{(n_{ν})}(ν) = lim_{s→1} \frac{f_{N}(ν < ν >^{s−1})}{(s − 1)^{n_{ν}}}
= D^{(n_{ν})}(ϑ_{ν}^{n_{ν}}F_{ν})(ν)
= [{log_{p}(ν(γ))}^{n_{ν}} Tw_{ν}(F_{ν})](1)
= {log_{p}(ν(γ))}^{n_{ν}} · Π_{G}(Tw_{ν}(F_{ν}))(0)
= \{log_{p}(ν(γ))\}^{n_{ν}} · \frac{Π_{G}(Tw_{ν}(f_{N}))}{t^{n_{ν}}} |_{t=0}.  (2.5)

We shall now apply the above discussion to the case in which F = K, M = X (𝔎_{∞}), ρ = ν = ψ^{∗}, H = Gal(𝔎_{∞}/K_{∞}^{∗}), G = Gal(K_{∞}^{∗}/K) and χ = 1.

Recall that the two-variable main conjecture asserts that X (𝔎_{∞}) is a torsion Λ(𝔎_{∞})- module, and that the Katz two-variable p-adic L-function L_{𝔭} is a characteristic power series

8 A. AGBOOLA

of X (𝔎_{∞}) in Λ(𝔎_{∞})_{O}. We therefore see that Tw_{ψ^{∗}}(L_{𝔭}) ∈ Λ(𝔎_{∞})_{O} is a characteristic power series of X (𝔎_{∞})(ψ^{∗−1}). Let I_{K_{∞}^{∗}} denote the kernel of the natural map Λ(𝔎_{∞}) → Λ(K_{∞}^{∗}). Fix any characteristic power series H_{K} ∈ Λ(K_{∞}^{∗}) of the Λ(K_{∞}^{∗})-module

X (𝔎_{∞})(ψ^{∗−1}) ⊗_{Λ(𝔎_{∞})} (Λ(𝔎_{∞})/I_{K_{∞}^{∗}}) ≃ X (𝔎_{∞})(ψ^{∗−1})/I_{K_{∞}^{∗}}X (𝔎_{∞})(ψ^{∗−1}).

Then we deduce from (2.1), (2.2) and (2.5) that

ord_{s=1} L_{𝔭}^{∗}(s) = ord_{t=0} H_{K},  (2.6)

and if we set n_{ψ^{∗}} := ord_{s=1} L_{𝔭}^{∗}(s), then

L_{𝔭}^{(n_{ψ^{∗}})}(ψ^{∗}) = lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{(s − 1)^{n_{ψ^{∗}}}} ∼ \{log_{p}(ψ^{∗}(γ))\}^{n_{ψ^{∗}}} · \frac{H_{K}}{t^{n_{ψ^{∗}}}} |_{t=0},  (2.7)

where ‘∼’ denotes equality up to multiplication by a p-adic unit (in fact, in this case, we have equality up to multiplication by an element of O^{×}).

3. Selmer Groups

In this section we shall define various Selmer groups that we require, and establish some of their properties.

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}) ⊗ 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 strict Selmer group Sel_{str}(L, M) by

Sel_{str}(F, M) = {c ∈ Sel(F, M) | loc_{v}(c) = 0 for all v dividing p};

BIRCH AND SWINNERTON-DYER CONJECTURE 9

• the 𝔮-strict Selmer group Sel_{str(𝔮)}(F, M) by

Sel_{str(𝔮)}(F, M) = {c ∈ Sel(F, M) | loc_{v}(c) = 0 for all v dividing 𝔮};

• 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

Šel_{?}(F, T) := \underleftarrow{lim}_{n} Sel_{?}(F, E_{π^{n}}), Šel_{?}(F, T^{∗}) := \underleftarrow{lim}_{n} Sel_{?}(F, E_{π^{∗n}}),

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

Sel_{?}(L, M) = \underrightarrow{lim} Sel_{?}(L′, M), Σ_{𝔮}(L, M) = \underrightarrow{lim} Σ_{𝔮}(L′, M),
Šel_{?}(L, T) = \underrightarrow{lim} Šel_{?}(L′, T), Šel_{?}(L, T^{∗}) = \underrightarrow{lim} Šel_{?}(L′, T^{∗}),

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

For any extension L/K, we set

Sel_{?}(L, M)^{∧} = X_{?}(L, M), Σ_{𝔮}(L, M)^{∧} = X_{𝔮}(L, M).

Theorem 3.1. Let L be any field such that F_{∞}^{∗} ⊆ L ⊆ 𝔉_{∞}. Then there is an isomorphism

X_{𝔭^{∗}}(L, W^{∗}) ≃ X (L)(ψ^{∗−1})  (3.1)

of Λ(L)-modules.

Proof. This is simply the analogue for restricted Selmer groups of a well-known theorem of Coates concerning true Selmer groups (see [4, Theorem 12]). We first observe that, since F_{∞}^{∗} ⊆ L, we have isomorphisms of Λ(L)-modules

X (L)(ψ^{∗−1}) ≃ Hom(T^{∗}, X (L)), X (L)(ψ^{∗−1})^{∧} ≃ Hom(X (L), W^{∗}).

Hence, in order to establish the desired result, it suffices to show that there is a natural isomorphsim

Σ_{𝔭^{∗}}(L, W^{∗}) ⟶^{∼} Hom(X (L), W^{∗}).  (3.2)

This may be proved in exactly the same way as [4, Theorem 12]. □

The following result is a ‘control theorem’ for restricted Selmer groups.

Proposition 3.2. (a) Let I_{F_{∞}^{∗}} denote the kernel of the quotient map Π_{F_{∞}^{∗}} : Λ(𝔉_{∞}) → Λ(F_{∞}^{∗}). Then the kernel of the restriction map

Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}) → Σ_{𝔭^{∗}}(𝔉_{∞}, W^{∗})[I_{F_{∞}^{∗}}]

is finite. A characteristic power series in Λ(F_{∞}^{∗}) of the Pontryagin dual of the cokernel of this map is given by

e_{F} = (γ − ψ^{∗−1}(γ))^{−1} ∏_{v|𝔭^{∗}}(γ_{v} − ψ^{∗−1}(γ_{v})),

where γ is a topological generator of Gal(F_{∞}^{∗}/F), and, for each place v of F_{∞}^{∗} lying above 𝔭^{∗}, γ_{v} denotes a topological generator of Gal(F_{∞,v}^{∗}/F_{v}) ≤ Gal(F_{∞}^{∗}/F).

Hence if f ∈ Λ(𝔉_{∞}) is a characteristic power series of X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}), then e_{F}^{−1}Π_{F_{∞}^{∗}}(f) ∈ Λ(F_{∞}^{∗}) is a characteristic power series of X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}).

(b) Suppose that L is any field such that F ⊆ L ⊆ F_{∞}^{∗}, and write I_{L} for the kernel of the quotient map Λ(F_{∞}^{∗}) → Λ(L). Then the restriction map

Σ_{𝔭^{∗}}(L, W^{∗}) → Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})[I_{L}]

is an isomorphism.

Hence the dual of this restriction map is an isomorphism of Λ(L)-modules:

X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})/I_{L}X_{𝔭^{∗}}(F_{∞}, W^{∗}) ⟶^{∼} X_{𝔭^{∗}}(L, W^{∗}).

Proof. Let N denote the maximal extension of 𝔉_{∞} that is unramified away from all places of 𝔉_{∞} lying above p. Consider the following commutative diagram:

0 ⟶ Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}) ⟶ H^{1}(N /F_{∞}^{∗}, W^{∗}) ⟶^{loc_{𝔭^{∗}}} ∏_{v|𝔭^{∗}} H^{1}(N_{v}/F_{∞,v}^{∗}, W^{∗})
α↓  ↓  ↓
0 ⟶ Σ_{𝔭^{∗}}(𝔉_{∞}, W^{∗})[I_{F_{∞}^{∗}}] ⟶ H^{1}(N /𝔉_{∞}, W^{∗})[I_{F_{∞}^{∗}}] ⟶^{loc_{𝔭^{∗}}} ∏_{v|𝔭^{∗}} H^{1}(N_{v}/𝔉_{∞,v}, W^{∗})

in which the vertical arrows are the obvious restriction maps.

Applying the Snake Lemma (together with the inflation-restriction exact sequence) to this diagram yields the exact sequence

0 → Ker(α) → H^{1}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ⟶^{g_{1}} ∏_{v|𝔭^{∗}}H^{1}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) →

→ Coker(α) → H^{2}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ⟶^{g_{2}} ∏_{v|𝔭^{∗}}H^{2}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) → 0.  (3.3)

BIRCH AND SWINNERTON-DYER CONJECTURE 11

Now,

H^{1}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ≃ Hom(Gal(𝔉_{∞}/F_{∞}^{∗}), W^{∗}),
∏_{v|𝔭^{∗}}H^{1}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) ≃ ∏_{v|𝔭^{∗}}Hom(Gal(𝔉_{∞,v}/F_{∞,v}^{∗}), W^{∗}),  (3.4)

and, as Gal(𝔉_{∞}/F_{∞}^{∗}) ≃ Δ × Z_{p} with p ∤ Δ, we have

H^{2}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ≃ H^{0}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ≃ W^{∗},

∏_{v|𝔭^{∗}}H^{2}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) ≃ ∏_{v|𝔭^{∗}}H^{0}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) ≃ ∏_{v|𝔭^{∗}}W^{∗}.

We now deduce that g_{1} is non-zero, and therefore has finite kernel (since H^{1}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) is divisible), and that g_{2} is injective. It follows from (3.3) that Ker(α) is finite, and that there is an exact sequence

0 → Ker(α) → H^{1}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}) ⟶^{g_{1}} ∏_{v|𝔭^{∗}}H^{1}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}) → Coker(α) → 0. (3.5)

It follows from (3.4) that

Char_{Λ(F_{∞}^{∗})} (H^{1}(𝔉_{∞}/F_{∞}^{∗}, W^{∗}))^{∧} = γ − ψ^{∗−1}(γ);

Char_{Λ(F_{∞}^{∗})} (∏_{v|𝔭^{∗}}H^{1}(𝔉_{∞,v}/F_{∞,v}^{∗}, W^{∗}))^{∧} = ∏_{v|𝔭^{∗}}(γ_{v} − ψ^{∗−1}(γ_{v})).

Hence we deduce from (3.5) that

Char_{Λ(F_{∞}^{∗})}(Coker(α))^{∧} = e_{F} = (γ − ψ^{∗−1}(γ))^{−1} ∏_{v|𝔭^{∗}}(γ_{v} − ψ^{∗−1}(γ_{v})),

as asserted.

(b) In this case we consider the commutative diagram

0 ⟶ Σ_{𝔭^{∗}}(L, W^{∗}) ⟶ H^{1}(N /L, W^{∗}) ⟶^{loc_{𝔭^{∗}}} ∏_{v|𝔭^{∗}} H^{1}(N_{v}/L_{v}, W^{∗})
β_{1}↓  ^{β_{2}}↓  ^{β_{3}}↓
0 ⟶ Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})[I_{L}] ⟶ H^{1}(N /F_{∞}^{∗}, W^{∗}) ⟶^{loc_{𝔭^{∗}}} ∏_{v|𝔭^{∗}} H^{1}(N /F_{∞,v}^{∗}, W^{∗})

We have that

Ker(β_{2}) = H^{1}(F_{∞}^{∗}/L, W^{∗}) = 0,

Ker(β_{3}) = ∏_{v|𝔭^{∗}}H^{1}(F_{∞,v}^{∗}/L_{v}, W^{∗}) = 0,

Coker(β_{2}) = H^{2}(F_{∞}^{∗}/L, W^{∗}) = 0, (see [12, p. 40], for example), and so the Snake Lemma implies that β_{1} is an isomorphism, as claimed. □

Corollary 3.3. For any field L with F ⊆ L ⊆ F_{∞}^{∗}, we have an isomorphism

X_{𝔭^{∗}}(L, T^{∗}) ≃ X (F_{∞})(ψ^{∗−1})/I_{L}(X (F_{∞})(ψ^{∗−1})  (3.6)

of Λ(L)-modules.

Proof. This follows directly from Proposition 3.2 and Theorem 3.1. □

Remark 3.4. If we take F = K in Proposition 3.2, then it is easy to check that e_{K} ∈ Λ(K_{∞}^{∗})^{×}. We therefore see from Proposition 3.2(a) and Corollary 3.3 that the element H_{K} ∈ Λ(K_{∞}^{∗}) fixed in Section 2 is a characteristic power series of X_{𝔭^{∗}}(K_{∞}^{∗}, W^{∗}). □

Definition 3.5. For any finite extension F/K and any prime 𝔮 of K we define

� (F)_{rel(𝔮)} := Ker [H^{1}(F, E) → ∏_{v∤𝔮}H^{1}(F_{v}, E)] ,

and we set

E_{1,𝔮}(F) := Ker [E(F) ⊗_{O_{K}} O_{K,𝔮} → ∏_{v|𝔮}E(F_{v})] .

□

Lemma 3.6. Let F/K be any finite extension, and let 𝔮 ∈ {𝔭, 𝔭^{∗}}. Then Σ̌_{𝔮}(F, T_{𝔮}) is a free O_{K,𝔮}-module.

Proof. It follows from the definitions that Σ̌_{𝔮}(F, T_{𝔮})_{tors} ⊆ Šel(F, T_{𝔮}). The desired result now follows from the fact that the restriction of the localisation map

Šel(F, T_{𝔮}) → ∏_{v|𝔮}E(F_{v}) ⊗_{O_{K}} O_{K,𝔮}

to Šel(F, T_{𝔮})_{tors} is injective. □

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

In this section we shall explain how the methods described by Perrin-Riou in [10] and [12] may be used to construct a p-adic height pairing

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

We begin by describing the 𝔭-adic Leopoldt hypotheses with which we shall work.

BIRCH AND SWINNERTON-DYER CONJECTURE 13

Definition 4.1. Let M/K be any finite extension, and consider the diagonal injection

i_{M} : O_{M}^{×} → ∏_{v|𝔭}O_{M,v}^{×}.

Let \overline{i_{M}(O_{M}^{×})} denote the 𝔭-adic closure of i_{M}(O_{M}^{×}) in ∏_{v|𝔭} O_{M,v}^{×}, and set

δ(M) := rk_{Z}(O_{M}^{×}) − rk_{Z_{p}}(\overline{i_{M}(O_{M}^{×})}).

The weak 𝔭-adic Leopoldt hypothesis for F asserts that the numbers δ(L′) are bounded as L′ runs through all finite extensions of F contained in F_{∞}^{∗}. The strong 𝔭-adic Leopoldt hypothesis for F asserts that the numbers δ(L′) are all equal to zero.

We remark that the strong Leopoldt hypothesis is known to hold for all abelian extensions of K (see [2]). □

Recall that B(F_{∞}^{∗}) denotes the maximal abelian pro-p extension of F_{∞}^{∗} which is unramified away from 𝔭 and totally split at all places above 𝔭^{∗}, and that Y(F_{∞}^{∗}) = Gal(B(F_{∞}^{∗})/F_{∞}^{∗}). The main ingredient in the construction of [ , ]_{F,𝔭^{∗}} is the following result.

Theorem 4.2. If the weak 𝔭-adic Leopoldt hypothesis holds for F then there is a natural isomorphism

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

The proof of this theorem is very similar to that of [10, Théorème 3.2]. We shall therefore just describe the main outlines of the proof, and we refer the reader to [10] for some of the details which we omit.

In order to describe the proof of Theorem 4.2, we require a number of intermediary results.

Lemma 4.3. There is an isomorphism of Gal(F_{n}^{∗}/F)-modules

H^{1}(F_{n}^{∗}, E_{π^{n}}) ⟶^{∼} Hom(E_{π^{∗n}}, F_{n}^{∗×}/F_{n}^{∗×p^{n}}); f ↦ f̃.  (4.1)

For each place v of F_{n}^{∗}, there is also a corresponding local isomorphism

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

Proof. See [10, Lemme 3.8]. The isomorphism (4.1) is defined as follows. Let f ∈ H^{1}(F_{n}^{∗}, E_{π^{n}}), and write

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

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

σ ↦ w_{n}(f(σ), u)

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

14 A. AGBOOLA

Lemma 4.4. For each place v of F_{n}^{∗} with v ∤ 𝔭^{∗}, there is an isomorphism

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

Proof. See [10, Lemme 3.11]. □

Corollary 4.5. Suppose that h ∈ H^{1}(F_{n}^{∗}, E_{π^{n}}). Then h ∈ Σ_{𝔭}(F_{n}^{∗}, E_{π^{n}}) if and only if, for each u ∈ E_{π^{n}}, the following local conditions are satisfied:

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

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

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

Proof. This follows directly from Lemmas 4.3 and 4.4. □

In what follows, we set G_{n} := Gal(F_{n}^{∗}/F), and we write J_{n} for the group of finite ideles of F_{n}^{∗}. We let V_{n} denote the subgroup of J_{n} consisting of those elements whose components are equal to 1 at all places dividing 𝔭 and are units at all places not dividing 𝔭^{∗}. We set

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

and we note that the order of Ω_{n} is bounded as n varies.

Proposition 4.6. There is an exact sequence

Hom(E_{π^{∗n}}, Ω_{n})^{G_{n}} → Hom(E_{π^{∗n}}, C_{n})^{G_{n}} ⟶^{η_{n}} Σ_{𝔭}(F, E_{π^{n}}) → 0.

Proof. The proof of this Proposition is identical, mutatis mutandis, to that of [10, Proposition 3.13]. □

Now let η′_{n} be the map obtained from η_{n} via passage to the quotient by the kernel of η_{n}, and write C_{n}(p) for the p-primary part of C_{n}. Then it may be shown exactly as on [10, pp. 387–389] that passing to inverse limits over the maps η_{n}^{′−1} yields an isomorphism

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

(Here the inverse limit \underleftarrow{lim} C_{n}(p) is taken with respect to the norm maps F_{n}^{∗×} → F_{n−1}^{∗×}.) The proof of Theorem 4.2 is completed by the following result.

Proposition 4.7. If the weak 𝔭-adic Leopoldt hypothesis holds for F, then there is an isomorphism

Hom(T^{∗}, \underleftarrow{lim} C_{n}(p))^{Gal(F_{∞}^{∗}/F)} ≃ Hom(T^{∗}, Y(F_{∞}^{∗}))^{Gal(F_{∞}^{∗}/F)}.

Proof. This may be shown in the same way as [10, Lemme 3.18]. □

BIRCH AND SWINNERTON-DYER CONJECTURE 15

We now explain how the isomorphism Ψ_{F} may be used to construct a p-adic height pairing

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

We first recall (see Proposition 3.2(b)) that the restriction map

Σ_{𝔭^{∗}}(F, W^{∗}) → Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})  (4.2)

is injective, and that there is a natural isomorphism (see Theorem 3.1)

Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}) ⟶^{∼} Hom(X (F_{∞}^{∗}), W^{∗}).  (4.3)

It follows from the local conditions defining the restricted Selmer group Σ_{𝔭^{∗}}(F, W^{∗}) that (4.2) and (4.3) induce an injection

Σ_{𝔭^{∗}}(F, W^{∗}) → Hom(Y(F_{∞}^{∗}), W^{∗}),  (4.4)

and taking Pontryagin duals yields a surjection

Hom(T^{∗}, Y(F_{∞}^{∗})) → X_{𝔭^{∗}}(F, W^{∗}).  (4.5)

Composing this with the natural surjection

X_{𝔭^{∗}}(F, W^{∗}) → [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧}

and taking Gal(F_{∞}^{∗}/F)-invariants yields a homomorphism

β_{F} : Hom(T^{∗}, Y(F_{∞}^{∗}))^{Gal(F_{∞}^{∗}/F)} → [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧}.

Next, we observe that we have a canonical isomorphism

[Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧} ≃ Hom_{O_{K,𝔭^{∗}}}(T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})_{div}), O_{K,𝔭^{∗}})
= Hom_{O_{K,𝔭^{∗}}}(T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})), O_{K,𝔭^{∗}}),

where the last equality holds because

T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})_{div} = T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})).

Also, for each n ≥ 1, we have a surjective map

Σ_{𝔭^{∗}}(F, E_{π^{∗n}}) → Σ_{𝔭^{∗}}(F, W^{∗})_{π^{∗n}}

with finite kernel. Via passage to inverse limits, these yield a map

Σ̌_{𝔭^{∗}}(F, T^{∗}) → T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗}))

which is an isomorphism because Σ̌_{𝔭^{∗}}(F, T^{∗}) is O_{K,𝔭^{∗}}-free (see Lemma 3.6).

It follows from the above discussion that we may view β_{F} as a homomorphism

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

We thus obtain a map

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

and this yields the desired pairing

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

It is natural to conjecture that this pairing is always non-degenerate (see Remark 6.6).

If x_{1}, …, x_{m} is an O_{K,𝔭}-basis of Σ̌_{𝔭}(F, T) (resp. if y_{1}, …, y_{m} is an O_{K,𝔭^{∗}}-basis of Σ̌_{𝔭^{∗}}(F, T^{∗})), then we define the regulator R_{F,𝔭^{∗}} associated to [ , ]_{F,𝔭^{∗}} by

R_{F,𝔭^{∗}} := det([x_{i}, y_{j}]_{F,𝔭^{∗}}).  (4.6)

5. The leading term

We retain the notation of the previous section. Write Γ_{F} := Gal(F_{∞}^{∗}/F), fix a topological generator γ_{F} of Γ_{F}, and identify Λ(F_{∞}^{∗}) with the power series ring Z_{p}[[t]] via the map γ_{F} ↦ t + 1. Let H_{F} ∈ Λ(F_{∞}^{∗}) be a characteristic power series of X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}). In this section we shall calculate the leading coefficient of H_{F}, assuming that the strong Leopoldt hypothesis holds for F and that [ , ]_{F,𝔭^{∗}} is non-degenerate.

Proposition 5.1. Suppose that F satisfies the strong 𝔭-adic Leopoldt hypothesis. Then the Λ(F_{∞}^{∗})-module X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}) has no finite, non-trivial submodules.

Proof. It is straightforward to show that a slight modification of the arguments given in [7, §4] establishes the fact that if F satisfies the strong 𝔭-adic Leopoldt hypothesis, then the Λ(F_{∞}^{∗})-module X(F_{∞}^{∗}) has no finite, non-trivial submodules. For brevity, we omit the details. The desired result now follows from Proposition 3.2 and Theorem 3.1. □

Theorem 5.2. Let H_{F} ∈ Λ(F_{∞}^{∗}) be a characteristic power series of X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}). Assume that the strong 𝔭-adic Leopoldt hypothesis holds for F, and that [ , ]_{F,𝔭^{∗}} is non-degenerate. Set m := rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(F, T^{∗})). Then ord_{t=0} H_{F} = m, and

\frac{H_{F}}{t^{m}} |_{t=0} ∼ |Σ_{𝔭^{∗}}(F, W^{∗})_{/ div}| · R_{F,𝔭^{∗}}.  (5.1)

Proof. We begin by noting that there is a surjective homomorphism

X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗}) → [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧}.

BIRCH AND SWINNERTON-DYER CONJECTURE 17

This implies that H_{F} is divisible by t^{m}. If we write Z_{∞} for the kernel of this map, then the Snake Lemma yields the following exact sequence:

0 → (Z_{∞})^{Γ_{F}} → X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})^{Γ_{F}} ⟶^{ξ_{F}} [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧} →
→ (Z_{∞})_{Γ_{F}} → X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})_{Γ_{F}} → [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧} → 0.

The kernel of the last map

X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})_{Γ_{F}} → [Σ_{𝔭^{∗}}(F, W^{∗})_{div}]^{∧}

is dual to the cokernel of the map

Σ_{𝔭^{∗}}(F, W^{∗})_{div} → Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})^{Γ_{F}} .

Since Σ_{𝔭^{∗}}(F, W^{∗}) ≃ Σ_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})^{Γ_{F}} (via Proposition 3.2(b)), it follows that this cokernel is isomorphic to Σ_{𝔭^{∗}}(F, W^{∗})_{/ div}, which is finite.

We therefore deduce that the multiplicity of t in H_{F} is equal to m if and only if (Z_{∞})_{Γ_{F}} is finite, which in turn is the case if and only if the cokernel of ξ_{F} is finite. Recall (see Theorem 3.1)

X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})^{Γ_{F}} ≃ Hom(T^{∗}, X (F_{∞}^{∗}))^{Gal(F_{∞}^{∗}/F)},

and that the homomorphism ξ_{F} may be written as the following composition of maps

Hom(T^{∗}, X (F_{∞}^{∗}))^{Gal(F_{∞}^{∗}/F)} → Hom(T^{∗}, Y(F_{∞}^{∗}))^{Gal(F_{∞}^{∗}/F)} → Σ_{𝔭^{∗}}(F, W^{∗})^{∧} → [Σ_{𝔭^{∗}}(F, W^{∗})_{/ div}]^{∧}

(see (4.4), (4.5)). Hence the cokernel of ξ_{F} is finite if and only if the p-adic height pairing [ , ]_{F,𝔭^{∗}} is non-degenerate.

We now see that if [ , ]_{F,𝔭^{∗}} is non-degenerate, then (Z_{∞})_{Γ_{F}} is finite. This implies that (Z_{∞})^{Γ_{F}} is also finite, whence it follows via Proposition 5.1 that (Z_{∞})^{Γ_{F}} = 0. Hence we have

\frac{H_{F}}{t^{m}} |_{t=0} ∼ |(Z_{∞})_{Γ_{F}} | ∼ |Σ_{𝔭^{∗}}(F, W^{∗})_{/ div}| · | Coker(ξ_{F})|.

Now

| Coker(ξ_{F})| = [(Σ_{𝔭^{∗}}(F, W^{∗})_{div})^{∧} : ξ_{F} (X_{𝔭^{∗}}(F_{∞}^{∗}, W^{∗})^{Γ_{F}})]
= [T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})) : Ψ_{F} (Σ̌_{𝔭}(F, T))]
= R_{F,𝔭^{∗}} · [Ker(Σ̌_{𝔭^{∗}}(F, T^{∗}) → T_{𝔭^{∗}}(Σ_{𝔭^{∗}}(F, W^{∗})))]
= R_{F,𝔭^{∗}}.

Hence

\frac{H_{F}}{t^{m}} |_{t=0} ∼ |Σ_{𝔭^{∗}}(F, W^{∗})_{/ div}| · R_{F,𝔭^{∗}},

as claimed. □

6. Restricted Selmer groups over K

In this section we shall analyse various properties of restricted Selmer groups over K. The main tool for doing this is the Poitou-Tate exact sequence (see e.g. [5, Theorem 1.5] or [11, Proposition 4.1.1]).

We write S_{F} for the set of places of F lying above p, and G_{F,S_{F}} for the Galois group over F of the maximal abelian extension of F that is unramified away from all places in S_{F}.

Proposition 6.1. There are isomorphisms

Šel_{str}(F, T^{∗}) ≃ H^{2}(G_{F,S_{F}} , W)^{∧},  Šel_{str}(F, T) ≃ H^{2}(G_{F,S_{F}} , W^{∗})^{∧}.

Proof. The middle of the Poitou-Tate exact sequence yields

0 → Sel_{str}(F, E_{π^{∗n}})^{∧} → H^{2}(G_{F,S_{F}} , E_{π^{n}}) → ⨁_{v∈S_{F}} H^{2}(F_{v}, E_{π^{n}}).

Dualising, and using the fact that, via Tate local duality, we have H^{2}(F_{v}, E_{π^{n}})^{∧} ≃ H^{0}(F_{v}, E_{π^{∗n}}) for each place v of F gives

⨁_{v∈S_{F}} H^{0}(F_{v}, E_{π^{∗n}}) → H^{2}(G_{F,S_{F}} , E_{π^{n}})^{∧} → Sel_{str}(F, E_{π^{∗n}}) → 0.

By passing to limits we obtain

⨁_{v∈S_{F}} H^{0}(F_{v}, T^{∗}) → H^{2}(G_{F,S_{F}} , W)^{∧} → Šel_{str}(F, T^{∗}) → 0,

and this establishes the first isomorphism, since the first term of this last sequence is equal to zero.

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

Recall that r = rk_{O_{K}}(E(K)).

Proposition 6.2. Suppose that r ≥ 1. Then

rk_{O_{K,𝔭^{∗}}}(Šel_{str}(K, T^{∗})) = rk_{O_{K,𝔭^{∗}}}(Šel_{str(𝔭^{∗})}(K, T^{∗}))
= rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})) − 1.

Proof. Since r ≥ 1, the image of the localisation map

Sel(K, T^{∗}) → E(K_{𝔭^{∗}}) ⊗ O_{K,𝔭^{∗}}

is infinite. The result now follows from the fact that

rk_{O_{K,𝔭^{∗}}}[E(K_{𝔭^{∗}}) ⊗ O_{K,𝔭^{∗}}] = rk_{O_{K,𝔭^{∗}}} [∏_{v|p} E(K_{v}) ⊗ O_{K,𝔭^{∗}} ] = 1.

□

Lemma 6.3. (a) The cohomology group H_{f}^{1}(K_{𝔭^{∗}}, T) is finite, and

|H_{f}^{1}(K_{𝔭^{∗}}, T)| ∼ |Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})| ∼ 1 − ψ(𝔭^{∗})

in Z_{p}.

(b) We have

H_{f}^{1}(K_{𝔭^{∗}}, T) = H^{1}(K_{𝔭^{∗}}, T)_{tors},

and H^{1}(K_{𝔭^{∗}}, T)/H_{f}^{1}(K_{𝔭^{∗}}, T) is O_{K,𝔭^{∗}}-free of rank one.

Proof. Part (a) follows directly from [4, Lemma 1].

To prove part (b), we observe that, via Tate local duality, the dual of H^{1}(K_{𝔭^{∗}}, T)/H_{f}^{1}(K_{𝔭^{∗}}, T) is equal to E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}}, and this last group is divisible of O_{K,𝔭^{∗}}-corank one. □

Proposition 6.4. (a) Suppose that r ≥ 1. Then

rk_{O_{K,𝔭^{∗}}}(Šel_{rel}(K, T^{∗})) = rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})),

and

[Šel_{rel}(K, T^{∗}) : Šel(K, T^{∗})] ∼ |Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})|.

(b) Suppose that r = 0. Then

rk_{O_{K,𝔭^{∗}}}(Šel_{rel}(K, T^{∗})) = 1.

Proof. The Poitou-Tate exact sequence yields

0 → Šel(K, T^{∗}) → Šel_{rel}(K, T^{∗}) ⟶^{α} ⨁_{v|p} \frac{H^{1}(K_{v}, T^{∗})}{H_{f}^{1}(K_{v}, T^{∗})} → Sel(K, W)^{∧}.  (6.1)

The cokernel of α is the Pontryagin dual of the image of the localisation map

Sel(K, W) → ⨁_{v|p} H_{f}^{1}(K_{v}, W),

and so has O_{K,𝔭^{∗}}-rank one if r ≥ 1 and rank zero if r = 0. As

rk_{O_{K,𝔭^{∗}}}[⊕_{v|p}(H^{1}(K_{v}, T^{∗})/H_{f}^{1}(K_{v}, T^{∗}))] = 1,

we therefore deduce that rk_{O_{K,𝔭^{∗}}}(Šel_{rel}(K, T^{∗})) is equal to rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})) if r ≥ 1, and is equal to one if r = 0. In particular, we have that Šel_{rel}(K, T^{∗})/Šel(K, T^{∗}) is finite if r ≥ 1.

Now suppose that r ≥ 1. As H^{1}(K_{𝔭}, T^{∗})/H_{f}^{1}(K_{𝔭}, T^{∗}) is O_{K,𝔭^{∗}}-free of rank one (Lemma 6.3(b)) and Šel_{rel}(K, T^{∗})/Šel(K, T^{∗}) is finite, (6.1) implies that there is an exact sequence

0 → \frac{Šel_{rel}(K, T^{∗})}{Šel(K, T^{∗})} → \frac{H^{1}(K_{𝔭^{∗}}, T^{∗})}{H_{f}^{1}(K_{𝔭^{∗}}, T^{∗})} ⟶^{α′} Sel(K, W)^{∧}.

Since E(K_{𝔭^{∗}})⊗D_{𝔭} = 0, it follows that α′ is the zero map. The dual of H^{1}(K_{𝔭^{∗}}, T^{∗})/H_{f}^{1}(K_{𝔭^{∗}}, T^{∗}) is isomorphic to H_{f}^{1}(K_{𝔭^{∗}}, T), and Lemma 6.3(a) implies that

|H_{f}^{1}(K_{𝔭^{∗}}, T)| ∼ |Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})|.

Hence [Šel_{rel}(K, T^{∗}) : Šel(K, T^{∗})] ∼ |Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})|, as claimed. □

Proposition 6.5. Suppose that r ≥ 1. Then

Σ̌_{𝔭^{∗}}(K, T^{∗}) = Šel_{str(𝔭^{∗})}(K, T^{∗}).

In particular, we have

rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) = rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})) − 1.

Proof. From Proposition 6.4(a), we have

rk_{O_{K,𝔭^{∗}}}(Šel_{rel}(K, T^{∗})) = rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})).

This implies that

rk_{O_{K,𝔭^{∗}}}(Σ_{𝔭^{∗}}(K, T^{∗})) = rk_{O_{K,𝔭^{∗}}}(Šel_{str(𝔭^{∗})}(K, T^{∗}))
= rk_{O_{K,𝔭^{∗}}}(Šel(K, T^{∗})) − 1.  (6.2)

It follows from the definitions of Σ̌_{𝔭^{∗}}(K, T^{∗}) and Šel_{str(𝔭^{∗})}(K, T^{∗}) that we have the following exact sequence

0 → Šel_{str(𝔭^{∗})}(K, T^{∗}) → Σ̌_{𝔭^{∗}}(K, T^{∗}) ⟶^{β} \frac{H^{1}(K_{𝔭^{∗}}, T^{∗})}{H_{f}^{1}(K_{𝔭}, T^{∗})} → Coker(β) → 0,

where β is induced by the obvious localisation map. From (6.2), we see that Σ̌_{𝔭^{∗}}(K, T^{∗})/Šel_{str(𝔭^{∗})}(K, T^{∗}) is finite. Hence, as H^{1}(K_{𝔭}, T^{∗})/H_{f}^{1}(K_{𝔭}, T^{∗}) is O_{K,𝔭^{∗}}-free of rank one (see Lemma 6.3(b)), it follows that β is the zero map. This implies that

Σ̌_{𝔭^{∗}}(K, T^{∗}) = Šel_{str(𝔭^{∗})}(K, T^{∗})

as claimed.

The final assertion of the Proposition is a direct consequence of Proposition 6.2. □

BIRCH AND SWINNERTON-DYER CONJECTURE 21

Remark 6.6. Suppose that r ≥ 1. Then it follows from Proposition 6.5, together with the definition of [ , ]_{K,𝔭^{∗}} that the pairing [ , ]_{K,𝔭^{∗}} is simply the restriction of Perrin-Riou’s algebraic p-adic height pairing { , }_{K,𝔭^{∗}} to Šel_{str(𝔭^{∗})}(K, T^{∗}) × Šel_{str(𝔭)}(K, T). Hence, if r ≥ 1 and { , }_{K,𝔭^{∗}} is non-degenerate, then so is [ , ]_{K,𝔭^{∗}}. We conjecture that the pairing [ , ]_{K,𝔭^{∗}} is also non-degenerate when r = 0. □

Proposition 6.7. Suppose that r = 0. Then

rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) = 1.

Proof. We have an injection

0 → Σ̌_{𝔭^{∗}}(K, T^{∗}) → Šel_{rel}(K, T^{∗}),

and we know that rk_{O_{K,𝔭^{∗}}}(Šel_{rel}(K, T^{∗})) = 1 (Proposition 6.4(b)). Hence rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) is either zero or one.

Suppose that rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) = 0. Then the proof of Theorem 5.2 shows that the characteristic power series H_{K} ∈ Λ(K_{∞}^{∗}) of X_{𝔭^{∗}}(K, W^{∗}) does not vanish at t = 0. This implies that ord_{s=1} L_{𝔭}^{∗}(s) = 0 (see (2.6)). On the other hand, it follows from the functional equation satisfied by the two-variable p-adic L-function L_{𝔭} (see [6, Chapter II, §6]) that the orders of the zeros at s = 1 of L_{𝔭}(s) and L_{𝔭^{∗}}(s) have opposite parity. Since r = 0, the order of � (K) is known to be finite (see [13]), and so

ord_{s=1} L_{𝔭}(s) = rk_{O_{K,𝔭^{∗}}}(Sel(K, T^{∗})) = 0.

This implies that ord_{s=1} L_{𝔭}^{∗}(s) ≥ 1, which is a contradiction.

It therefore follows that rk_{O_{K,𝔭^{∗}}}(Σ̌_{𝔭^{∗}}(K, T^{∗})) = 1 as claimed. □

Corollary 6.8. Assume that [ , ]_{K,𝔭^{∗}} is non-degenerate.

(a) If r ≥ 1 and � (K)(𝔭^{∗}) is finite, then

ord_{s=1} L_{𝔭}^{∗}(s) = r − 1.

(b) If r = 0, then

ord_{s=1} L_{𝔭}^{∗}(s) = 1.

Proof. This follows directly from Propositions 6.5 and 6.7, and (2.6). □

Remark 6.9. Corollary 6.8(b) confirms the expectation expressed in [15, Remark on p.74] (see also [14, §11, Remarks(2)]). It would be interesting to know if there is any way of showing that rk_{O_{K,𝔭^{∗}}}(Σ_{𝔭^{∗}}(K, T^{∗})) = 1 when r = 0 without appealing to the functional equation satisfied by L_{𝔭}. □

Proposition 6.10. (a) Suppose that r ≥ 1, and assume that � (K)(𝔭^{∗}) is finite. Then � _{rel(𝔭)}(K)(𝔭^{∗}) is also finite, and we have

|� _{rel(𝔭)}(K)(𝔭^{∗})| = |� (K)(𝔭)| · [E(K_{𝔭}) ⊗ O_{K,𝔭} : loc_{𝔭}(Sel(K, T))].

(b) Suppose that r = 0. Then � _{rel(𝔭)}(K)(𝔭^{∗}) has O_{K,𝔭^{∗}}-corank one.

Proof. (a) For each n ≥ 1, we define B_{n} via exactness of the sequence

0 → � (K)_{π^{∗n}} → H^{1}(K, E)_{π^{∗n}} → ∏_{v} H^{1}(K_{v}, E)_{π^{∗n}} → B_{n} → 0.

Then there exists a map h_{n} : H^{1}(K_{𝔭}, E)_{π^{∗n}} → B_{n}, and the sequence

0 → � (K)_{π^{∗n}} → � _{rel(𝔭)}(K)_{π^{∗n}} → H^{1}(K_{𝔭}, E)_{π^{∗n}} ⟶^{h_{n}} B_{n}  (6.3)

is exact. Passing to direct limits over n in (6.3) yields the sequence

0 → � (K)(𝔭^{∗}) → � _{rel(𝔭)}(K)(𝔭^{∗}) → H^{1}(K_{𝔭}, E)(𝔭^{∗}) ⟶^{⟶^{lim} h_{n}} \underrightarrow{lim} B_{n}.  (6.4)

It follows from a theorem of Cassels (see [3, p.198]) that the dual of B_{n} is isomorphic to Sel(K, E_{π^{n}}). Tate local duality implies that the dual of H^{1}(K_{𝔭}, E)_{π^{∗n}} is isomorphic to E(K_{𝔭})/π^{n}E(K_{𝔭}) and that the kernel of \underrightarrow{lim} h_{n} is isomorphic to the dual of the cokernel of the localisation map

loc_{𝔭} : Šel(K, T) → E(K_{𝔭}) ⊗ O_{K,𝔭}.

If r ≥ 1, then this cokernel is finite, and we therefore deduce that

[� _{rel(𝔭)}(K)(𝔭^{∗}) : � (K)(𝔭^{∗})] = [E(K_{𝔭}) ⊗ O_{K,𝔭} : loc_{𝔭}(Šel(K, T))].

Hence, we have

|� _{rel(𝔭)}(K)(𝔭^{∗})| = |� (K)(𝔭^{∗})| · [E(K_{𝔭}) ⊗ O_{K,𝔭} : loc_{𝔭}(Šel(K, T))]

as claimed.

(b) If r = 0, then Šel(K, T) is trivial, because � (K) is known to be finite, and E(K)(𝔭) = 0. This implies that Coker(loc_{𝔭}) = E(K_{𝔭}) ⊗ O_{K,𝔭} is O_{K,𝔭}-free of rank one. It now follows from (6.4) that � _{rel(𝔭)}(K)(𝔭^{∗}) has O_{K,𝔭^{∗}}-corank one. □

Proposition 6.11. Suppose that r ≥ 1, and assume that � (K)(𝔭^{∗}) is finite. Then

|Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| = |� _{rel(𝔭)}(K)(𝔭^{∗})| · [E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(Šel(K, T^{∗}))].

BIRCH AND SWINNERTON-DYER CONJECTURE 23

Proof. Let y_{1}, …, y_{r−1} be an O_{K,𝔭^{∗}}-basis of E_{1,𝔭^{∗}}(K), and extend it to an O_{K,𝔭^{∗}}-basis y_{1}, …, y_{r−1}, y_{𝔭^{∗}} of E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}}. There is an exact sequence

0 → O_{K,𝔭^{∗}} · y_{𝔭^{∗}} → E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} → U → 0,

with

|U| = [E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}})]
= [E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(Šel(K, T^{∗}))].

Tensoring this sequence with D_{𝔭^{∗}} yields an exact sequence

0 → V → (O_{K,𝔭^{∗}} · y_{𝔭^{∗}}) ⊗_{O_{K}} D_{𝔭^{∗}} → E(K_{𝔭^{∗}}) ⊗_{O_{K}} D_{𝔭^{∗}} → 0,

with |U| = |V |. As

E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}} ≃ E_{1,𝔭^{∗}}(K) ⊕ (O_{K,𝔭^{∗}} · y_{𝔭^{∗}}),

it follows that the kernel of the localisation map

E(K) ⊗_{O_{K}} D_{𝔭^{∗}} → E(K_{𝔭^{∗}}) ⊗_{O_{K}} D_{𝔭^{∗}}

is isomorphic to (E_{1,𝔭^{∗}}(K) ⊗_{O_{K}} D_{𝔭^{∗}}) ⊕ V.

Define

� (K)_{rel} := Ker [H^{1}(K, E) → ∏_{v∤p}H^{1}(K_{v}, E)];

then we have an exact sequence

0 → E(K) ⊗ D_{𝔭^{∗}} → Sel_{rel}(K, W^{∗}) → � _{rel}(K)(𝔭^{∗}) → 0.

Now consider the following commutative diagram, in which the vertical arrows are the obvious localisation maps:

0 ⟶ E(K) ⊗ D_{𝔭^{∗}} ⟶ Sel_{rel}(K, W^{∗}) ⟶ � _{rel}(K)(𝔭^{∗}) ⟶ 0
↓ ↓ ↓
0 ⟶ E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} ⟶ H^{1}(K_{𝔭^{∗}}, W^{∗}) ⟶ H^{1}(K_{𝔭^{∗}}, E)(𝔭^{∗}) ⟶ 0

Applying the Snake Lemma to this diagram yields the exact sequence

0 → (E_{1,𝔭^{∗}}(K) ⊗ D_{𝔭^{∗}}) ⊕ V → Σ_{𝔭^{∗}}(K, W^{∗}) → � _{rel(𝔭)}(K)(𝔭^{∗}) → 0.

As � _{rel}(K)(𝔭^{∗}) is finite (see Proposition 6.10) and E_{1,𝔭^{∗}}(K) ⊗_{O_{K}} D_{𝔭^{∗}} is divisible, it follows that

Σ_{𝔭^{∗}}(K, W^{∗})_{/ div} = |� _{rel}(K)(𝔭^{∗})| · |V |
= |� _{rel(𝔭)}(K)(𝔭^{∗})| · [E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(Šel(K, T^{∗}))],

as asserted. □

7. Proof of Theorem A

Proposition 7.1. Suppose that r = 0. Then

|Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| ∼ (1 − ψ(𝔭^{∗})) · \frac{|� (K)_{rel(𝔭)}(𝔭^{∗})_{/ div}|}{[H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ_{𝔭}(K, T))]}.

Proof. Consider the following diagram in which all columns are exact and f_{1}, f_{2} are the obvious localisation maps:

0  ⟶ Σ_{𝔭^{∗}}(K, W^{∗}) ⟶ � _{rel(𝔭)}(K)(𝔭^{∗})
↓ ↓ ↓
0 ⟶ E(K) ⊗ D_{𝔭^{∗}} = 0 ⟶ Sel_{rel}(K, W^{∗}) ⟶ � _{rel}(K)(𝔭^{∗}) ⟶ 0
↓ ↓^{f_{1}} ↓^{f_{2}}
0 ⟶ E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} ⟶ H^{1}(K_{𝔭^{∗}}, W^{∗}) ⟶ H^{1}(K_{𝔭^{∗}}, E)(𝔭^{∗}) ⟶ 0
↓ ↓ ↓
E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} ⟶ Coker(f_{1}) ⟶ Coker(f_{2})

Applying the Snake Lemma to this diagram yields an exact sequence

0 → Σ_{𝔭^{∗}}(K, W^{∗}) → � _{rel(𝔭)}(K)(𝔭^{∗}) → E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} → Coker(f_{1}) → Coker(f_{2}) → 0. (7.1)

Let us first determine Coker(f_{1}). The Poitou-Tate exact sequence gives

0 → Σ_{𝔭^{∗}}(K, W^{∗}) → Sel_{rel}(K, W^{∗}) ⟶^{f_{1}} H^{1}(K_{𝔭^{∗}}, W^{∗}) → Σ̌_{𝔭}(K, T)^{∧} → H^{2}(G_{K,S_{K}}, W^{∗}),

where G_{K,S_{K}} denotes the Galois group over K of the maximal extension of K that is unramified away from p. Since r = 0, Propositions 6.1 and 6.2 imply that H^{2}(G_{K,S_{K}}, W^{∗}) = 0, and so we have

Coker(f_{1}) ≃ Σ̌_{𝔭}(K, T)^{∧}.  (7.2)

In particular, it follows from Lemma 3.6 and Proposition 6.7 that Coker(f_{1}) is divisible of O_{K,𝔭^{∗}}-corank one.

In order to determine Coker(f_{2}), we observe that E(K_{𝔭^{∗}})⊗D_{𝔭^{∗}} is divisible of O_{K,𝔭^{∗}}-corank one, and the kernel of the map

E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} → Coker(f_{1})

in (7.1) is isomorphic to � _{rel(𝔭)}(K)(𝔭^{∗})/Σ_{𝔭^{∗}}(K, W^{∗}). This last group is finite, because both � _{rel(𝔭)}(K)(𝔭^{∗}) and Σ_{𝔭^{∗}}(K, W^{∗}) have O_{K,𝔭^{∗}}-corank one (see Propositions 6.10(b) and 6.7). It therefore follows that Coker(f_{2}) = 0.

From (7.1) and (7.2), we obtain the sequence

0 → \frac{� _{rel(𝔭)}(K)(𝔭^{∗})}{Σ_{𝔭^{∗}}(K, W^{∗})} → E(K_{𝔭^{∗}}) ⊗ D_{𝔭^{∗}} → Σ̌_{𝔭}(K, T)^{∧} → 0.  (7.3)

Dualising this sequence yields

0 → Σ̌_{𝔭}(K, T) → \frac{H^{1}(K_{𝔭^{∗}}, T)}{H_{f}^{1}(K_{𝔭^{∗}}, T)} → [\frac{� _{rel(𝔭)}(K)(𝔭^{∗})}{Σ_{𝔭^{∗}}(K, W^{∗})}]^{∧} → 0.

We therefore have

|[\frac{� _{rel(𝔭)}(K)(𝔭^{∗})}{Σ_{𝔭^{∗}}(K, W^{∗})}]^{∧}| = |\frac{� _{rel(𝔭)}(K)(𝔭^{∗})}{Σ_{𝔭^{∗}}(K, W^{∗})}|

= |\frac{� _{rel(𝔭)}(K)(𝔭^{∗})_{/ div}}{Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}} |

= [H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ̌_{𝔭}(K, T))] · |H_{f}^{1}(K_{𝔭^{∗}}, T)|^{−1},

which in turn implies that

|Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| = \frac{|� _{rel(𝔭)}(K)(𝔭^{∗})_{/ div}|}{[H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ̌_{𝔭}(K, T))]} · |H_{f}^{1}(K_{𝔭^{∗}}, T)|.

Since

|H_{f}^{1}(K_{𝔭^{∗}}, T)| ∼ 1 − ψ(𝔭^{∗})

(see Lemma 6.3), we finally obtain

|Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| ∼ (1 − ψ(𝔭^{∗})) · \frac{|� (K)_{rel(𝔭)}(𝔭^{∗})_{/ div}|}{[H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ_{𝔭}(K, T))]},

as claimed. □

Proof of Theorem A. We first note that, as [ , ]_{K,𝔭^{∗}} is non-degenerate (by hypothesis), we have ord_{s=1} L_{𝔭}^{∗}(s) = 1 (Corollary 6.8(b)). Hence from (5.1), (2.7), Proposition 7.1 and Remark 3.4, we have

lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{s − 1} ∼ log_{p}(ψ^{∗}(γ)) · \frac{H_{K}}{t} |_{t=0}

∼ log_{p}(ψ^{∗}(γ)) · |Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| · R_{K,𝔭^{∗}}

∼ log_{p}(ψ^{∗}(γ)) · (1 − ψ(𝔭^{∗}) · \frac{|� _{rel(𝔭)}(K)(𝔭^{∗})_{/ div}|}{[H^{1}(K_{𝔭^{∗}}, T) : loc_{𝔭^{∗}}(Σ_{𝔭}(K, T))]} · R_{K,𝔭^{∗}}.

This completes the proof of Theorem A. □

26 A. AGBOOLA

8. Proof of Theorem B

Suppose now that r ≥ 1. Then E(K)⊗O_{K,𝔭^{∗}} is a free O_{K,𝔭^{∗}}-module of rank r. Proposition 6.2 implies that the kernel of the localisation map

loc_{𝔭^{∗}} : E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}} → E(K_{𝔭^{∗}}) ⊗ O_{K,𝔭^{∗}}

has O_{K,𝔭^{∗}}-rank r − 1. Let y_{1}, …, y_{r−1} be an O_{K,𝔭^{∗}}-basis of this kernel, and extend it to an O_{K,𝔭^{∗}}-basis y_{1}, …, y_{r−1}, y_{𝔭^{∗}} of E(K) ⊗ O_{K,𝔭^{∗}}.

Proposition 8.1. With the above assumptions and notation, we have

[E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}})] ∼ p^{−1} log_{E,𝔭^{∗}}(y_{𝔭^{∗}}),

where log_{E,𝔭^{∗}} denotes the 𝔭^{∗}-adic logarithm associated to E. Similarly, we also have

[E(K_{𝔭}) ⊗_{O_{K}} O_{K,𝔭} : loc_{𝔭}(E(K) ⊗_{O_{K}} O_{K,𝔭})] ∼ p^{−1} log_{E,𝔭}(y_{𝔭}),

when y_{𝔭} ∈ E(K_{𝔭}) ⊗_{O_{K}} O_{K,𝔭} is defined analogously to y_{𝔭^{∗}}.

Proof. We give the proof of the first assertion; that of the second is of course essentially identical.

We first observe that, from the definitions, we have

[E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(E(K) ⊗_{O_{K}} O_{K,𝔭^{∗}})] = [E(K_{𝔭^{∗}}) ⊗ O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(O_{K,𝔭^{∗}} · y_{𝔭^{∗}})].

Let E_{0} denote the kernel of reduction modulo 𝔭^{∗} of E, so we have an exact sequence

0 → E_{0}(K_{𝔭^{∗}}) → E(K_{𝔭^{∗}}) → Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}}) → 0.

Set

Z := O_{K,𝔭^{∗}} · y_{𝔭^{∗}}, Z_{0} := loc_{𝔭^{∗}}(Z) ∩ E_{0}(K_{𝔭^{∗}}), C := loc_{𝔭^{∗}}(Z)/Z_{0}.

Write λ_{𝔭^{∗}} for the restriction of loc_{𝔭^{∗}} to Z. We have the following commutative diagram:

0 ⟶  Z_{0}  ⟶  Z  ⟶ C ⊗_{O_{K}} O_{K,𝔭^{∗}} ⟶ 0
↓^{ρ}  ↓^{λ_{𝔭^{∗}}}  ↓^{ρ′}
0 ⟶ E_{0}(K_{𝔭}^{∗}) ⊗_{O_{K}} O_{K,𝔭^{∗}} ⟶ E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} ⟶ Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} ⟶ 0

Observe that ρ is injective since λ_{𝔭^{∗}} is injective, and that Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} = 0 because Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})(p) = Ẽ_{𝔭^{∗}}(k_{𝔭^{∗}})(𝔭) (see e.g. [12, p. 28]). Applying the Snake Lemma to the diagram yields the exact sequence

0 → Ker(ρ′) → Coker(ρ) → Coker(λ_{𝔭^{∗}}) → 0,

and so we have

| Coker(λ_{𝔭^{∗}})| = |C ⊗_{O_{K}} O_{K,𝔭^{∗}}|^{−1} · | Coker(ρ)|.

BIRCH AND SWINNERTON-DYER CONJECTURE 27

Set k = [Z : Z_{0}] = |C ⊗ O_{K,𝔭^{∗}}|; then ky_{𝔭^{∗}} is an O_{K,𝔭^{∗}}-generator of Z_{0}. Since there is an isomorphism

log_{E,𝔭^{∗}} : E_{0}(K_{𝔭^{∗}}) ⟶^{∼} 𝔭^{∗}O_{K,𝔭^{∗}},

it follows that we have

| Coker(ρ)| ∼ p^{−1} log_{E,𝔭^{∗}}(ky_{𝔭^{∗}}) = kp^{−1} log_{E,𝔭^{∗}}(y_{𝔭^{∗}}).

Therefore

| Coker(λ_{𝔭^{∗}})| ∼ p^{−1} log_{E,𝔭^{∗}}(y_{𝔭^{∗}}),

and this establishes the desired result. □

Corollary 8.2. Suppose that r ≥ 1 and assume that � (K)(𝔭^{∗}) is finite. Then

|� _{rel(𝔭)}(K)(𝔭^{∗})| = p^{−1} · |� (K)(𝔭^{∗})| · log_{E,𝔭}(y_{𝔭}).

Proof. This follows directly from Propositions 6.10(a) and 8.1. □

Proof of Theorem B. By hypothesis, [ , ]_{K,𝔭^{∗}} is non-degenerate, r ≥ 1, and � (K)(p) is finite; hence we have that ord_{s=1} L_{𝔭}^{∗}(s) = r − 1 (Corollary 6.8(a)). Proposition 6.11 and Corollary 8.2 imply that

|Σ_{𝔭^{∗}}(K, W^{∗})_{/ div}| = |� _{rel(𝔭)}(K)(𝔭^{∗})| · [E(K_{𝔭^{∗}}) ⊗_{O_{K}} O_{K,𝔭^{∗}} : loc_{𝔭^{∗}}(Šel(K, T^{∗}))]
∼ p^{−2} · |� (K)(𝔭^{∗})| · log_{E,𝔭^{∗}}(y_{𝔭^{∗}}) · log_{E,𝔭}(y_{𝔭}).

We therefore deduce from (5.1), (2.7) and Remark 3.4 that

lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{(s − 1)^{r−1}} ∼
[log_{p}(ψ^{∗}(γ))]^{r−1} · p^{−2} · |� (K)(𝔭^{∗})| · log_{E,𝔭^{∗}}(y_{𝔭^{∗}}) · log_{E,𝔭}(y_{𝔭}) · R_{K,𝔭^{∗}},

as asserted.

This completes the proof of Theorem B. □

9. Canonical elements in restricted Selmer groups

The goal of this section is to explain how the methods of [14] may be used to produce an exact formula for lim_{s→1} L_{𝔭}^{∗}(s)/(s − 1) when r = 0 (see Theorem 9.5 below). The arguments involved are quite similar to those of [14], and so, in what follows, we assume that the reader has a copy of [14] and is willing to refer to it from time to time for some of the details we omit.

28 A. AGBOOLA

We begin by introducing the following notation (some of which differs from that of [14]):

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 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 9.1. Note that since the strong Leopoldt conjecture holds for all abelian extensions of K (see [2]), we have that

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

Proposition 9.2. 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 [14, 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}^{∗}.

BIRCH AND SWINNERTON-DYER CONJECTURE 29

Recall that, for each n ≥ 1, there is an isomorphism

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

(see e.g. [14, Lemma 2.1] or [10, Lemme 12]). We define

ρ(f) := [(p − 1)(π^{∗})^{2n}ρ_{n}^{−1}(f_{n})] ∈ \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 Theorem 3.1, Proposition 3.2, and Corollary 3.3 that ρ_{n}^{−1}(f_{n}) ∈ Σ_{𝔭}(K, E_{π^{n}}) if and only if the restriction of ρ_{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 [14, Lemmas 2.1 and 2.3] that this in fact the case. □

We shall now explain how elliptic units may be used (following [14]) to construct canonical elements

s_{𝔭}^{(1)} ∈ Σ̌_{𝔭}(K, T), s_{𝔭^{∗}}^{(1)} ∈ Σ̌_{𝔭^{∗}}(K, T^{∗})

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 [14] when r = 1.

Let C_{∞} ⊆ E_{∞} and C_{∞}^{∗} ⊆ E_{∞}^{∗} denote the norm-coherent systems of elliptic units constructed in [14, §3], and write \overline{C}_{∞} and \overline{C}_{∞}^{∗} for the closure of C_{∞} in \overline{E}_{∞} and C_{∞}^{∗} in E_{∞}^{∗} respectively. Set

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

and let ϑ^{∗} be the generator of J^{∗} fixed in [14, §6] (so ϑ^{∗} = γψ^{∗}(γ^{−1}) − 1, where γ is any topological generator of Gal(K_{∞}^{∗}/K) satisfying log_{p}(ψ^{∗}(γ)) = p). Write 𝔣 ⊆ O_{K} for the conductor of the Grossencharacter associated to E, and let N(𝔣) denote the norm of this ideal. Fix B ∈ E_{𝔣}/ Gal(\overline{K}/K), and generators w of T and w^{∗} of T^{∗} according to the recipe described in [14, §6]. Let

θ_{B}(N(𝔣)^{−1}w^{∗}) ∈ \overline{C}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q

denote the elliptic unit constructed in [14, §3].

Suppose that t is a positive integer such that

\overline{C}_{∞}^{∗} ⊆ I^{t−1}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q and \overline{C}_{∞}^{∗} ⊆ I^{t}(U_{∞,𝔭}^{∗} ⊗ Q).

Proposition 9.3. There exists a unique homomorphism σ_{𝔭}^{(t)} ∈ Hom(T^{∗}, (U_{∞,𝔭}^{∗} ⊗ Q)/\overline{E}_{∞}^{∗}) such that

σ_{𝔭}^{(t)}(w^{∗})^{ϑ^{∗t}} = θ_{B}(−N(𝔣)^{−1}w^{∗})

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

Proof. Theorem 7.2(i) of [14] implies that U_{∞,𝔭}^{∗} contains no ϑ^{∗}-torsion elements. The existence of σ_{𝔭}^{(t)} therefore follows via an argument very similar to that of [14, Theorem 4.2]. □

30 A. AGBOOLA

We set

s_{𝔭}^{(t)} := ρ(σ_{𝔭}^{(t)}), s_{𝔭^{∗}}^{(t)} := ρ^{∗}(σ_{𝔭^{∗}}^{(t)}),

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

Remark 9.4. In fact the only non-zero values of s_{𝔭}^{(t)} and s_{𝔭^{∗}}^{(t)} occur when r = 0 and t = 1:

(a) Suppose that r = 0. Then L_{𝔭}(1) ≠ 0, and so we have (via [14, Theorem 7.2(i)], for example):

\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 9.1, we deduce that

\overline{C}_{∞}^{∗} ⊈ I^{∗}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭^{∗}}^{∗} ⊗ Q.  (9.1)

Now suppose in addition that [ , ]_{K,𝔭^{∗}} is non-degenerate. Then Theorem A implies that ord_{s=1} L_{𝔭}^{∗}(s) = 1, and so from [14, Theorem 7.2(i)], we have

\overline{C}_{∞}^{∗} ⊆ I^{∗}(U_{∞,𝔭}^{∗} ⊗ Q).  (9.2)

We now deduce from (9.1) and (9.2) and the definition of ρ that s_{𝔭}^{(1)} ≠ 0.

A similar argument shows that s_{𝔭^{∗}}^{(1)} ≠ 0 also.

(b) Suppose now that r ≥ 1. Assume that � (K)(p) is finite, and that the height pairing [ , ]_{K,𝔭^{∗}} is non-degenerate. Then Theorem B (or [14, Corollary 11.3]) implies that ord_{s=1} L_{𝔭}^{∗}(s) = r − 1, and so it follows from [14, Theorem 7.2(i)] that

\overline{C}_{∞}^{∗} ⊆ I^{∗r−1}(U_{∞,𝔭}^{∗} ⊗ Q).  (9.3)

On the other hand, Theorem 4.2 and Proposition 4.4 of [14] imply that

\overline{C}_{∞}^{∗} ⊆ I^{∗r−1}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭^{∗}}^{∗} ⊗ Q, \overline{C}_{∞}^{∗} ⊈ I^{∗r}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭^{∗}}^{∗} ⊗ Q,

and so applying Remark 9.1, we deduce that

\overline{C}_{∞}^{∗} ⊆ I^{∗r−1}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q, \overline{C}_{∞}^{∗} ⊈ I^{∗r}\overline{E}_{∞}^{∗} ⊆ U_{∞,𝔭}^{∗} ⊗ Q.  (9.4)

It now follows from (9.3) and (9.4) that s_{𝔭}^{(t)} = 0 for 1 ≤ t ≤ r − 2 and that s_{𝔭}^{(t)} is not defined for t ≥ r − 1.

(c) Suppose that r = 0, but that ord_{s=1} L_{𝔭}^{∗}(s) > 1 (so, in particular, the pairing [ , ]_{K,𝔭^{∗}} is degenerate, which we expect never to happen). Then an argument similar to that given in (b) above shows that s_{𝔭}^{(1)} = 0, and that s_{𝔭}^{(t)} is not defined for t > 1. □

Theorem 9.5. Suppose that r = 0 and that [ , ]_{K,𝔭^{∗}} is non-degenerate, so ord_{s=1} L_{𝔭}^{∗}(s) = 1. Then

lim_{s→1} \frac{L_{𝔭}^{∗}(s)}{s − 1} = N(𝔣)^{−1}(p − 1) (1 − \frac{ψ^{∗}(𝔭)}{p}) lim_{n→∞} log_{𝔭}(σ_{𝔭,n}^{(1)}(w^{∗})).

Proof. This may be shown in exactly the same way as [14, Proposition 9.4(ii)]. □

Remark 9.6. The precise relationship between Theorem A and Theorem 9.5 is not clear, and it would be interesting to obtain a better understanding of this. □

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

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