.

Iwasawa Theory of Elliptic Curves and Galois Module Structure A. Agboola^{1}

Mathematical Sciences Research Institute^{2} 1000 Centennial Drive Berkeley, CA 94720 U. S. A.

1 Partially supported by an NSF Postdoctoral Research Fellowship. 2 Current address: Department of Mathematics, University of California, Berke-

ley, CA 94720.

§0 Introduction

In this paper we apply techniques arising from Iwasawa theory to study the Galois module structure of principal homogeneous spaces constructed via points of infinite order on CM elliptic curves defined over a number field. This theory was introduced by M.J. Taylor in [T1] (see also [ST], [CN-S] and [CN-T]) and is motivated by the fact that such principal homogeneous spaces are very closely connected with certain rings of integers.

Let E be an elliptic curve with complex multiplication by 𝔒, the ring of integers of an imaginary quadratic field K. If α ∈ 𝔒, we shall often (but not always) write [α] for the corresponding endomorphism of E. Let F/K be a finite extension over which E is defined and acquires everywhere good reduction. We write Δ = Gal(F/K), and we assume that all endomorphisms of E are defined over F. For any field L, we write L^{c} for an algebraic closure of L, and we set Ω_{L} = Gal(L^{c}/L).

Let p be an odd rational prime which splits in 𝔒, with p𝔒 = 𝔭𝔭^{∗}. Asssume that p |̸|Δ|. Choose π ∈ 𝔭 with 𝔭^{h} = π𝔒 for some h ≥ 1 and write π^{∗} for the complex conjugate of π. Set q = ππ^{∗}.

Write G_{i} for the subgroup of elements of E(ℚ^{c}) which are killed by [π^{∗i}]. Let 𝔅_{i} denote the 𝔒_{F} -Hopf algebra which represents the 𝔒_{F} -group scheme of [π^{∗i}]- torsion on E, and let 𝔄_{i} be the Cartier dual of 𝔅_{i}. A detailed description of these algebras is given in [T1] (see also [ST]). There it is shown that 𝔅_{i} is an 𝔒_{F} -order in the algebra B_{i} = Map_{Ω_{F}} (G_{i}, ℚ^{c}) and 𝔄_{i} is an order in the algebra A_{i} = (F^{c}G_{i})^{Ω_{F}} (where here Ω_{F} acts upon both G_{i} and F^{c}). (Here and elsewhere we shall omit from our notation the dependence of our constructions upon the underlying field F unless there is some danger of ambiguity.)

Suppose that Q ∈ E(F), and write

G_{Q}(i) = {Q′ ∈ E(ℚ^{c}) : [π^{∗i}]Q′ = Q}.  (0.1)

Define the Kummer algebra F_{Q}(i) by

F_{Q}(i) = Map_{Ω_{F}} (G_{Q}(i), ℚ^{c}).  (0.2)

Then [F_{Q}(i) : F] = |G_{i}|, and A_{i} acts on F_{Q}(i) via

(f. ∑_{g∈G_{i}} a_{g}g)(Q′) = ∑_{g∈G_{i}} a_{g}f(Q′ + g)  (0.3)

for f ∈ F_{Q}(i) and ∑_{g∈G_{i}} a_{g}g ∈ A_{i}.

Let 𝔒_{Q}(i) denote the integral closure of 𝔒_{F} in F_{Q}(i). In general, 𝔒_{Q}(i) does not admit an action of 𝔄_{i}. We define the Kummer order ℭ_{Q}(i) to be the maximal 𝔄_{i}-stable submodule of 𝔒_{Q}(i), i.e.

ℭ_{Q}(i) = {x ∈ 𝔒_{Q}(i) | x.𝔄_{i} ⊆ 𝔒_{Q}(i)}.  (0.4)

It is shown in [T1] that ℭ_{Q}(i) is a principal homogeneous space (phs) of 𝔅_{i}, and that ℭ_{Q}(i) is a locally free 𝔄_{i}-module. Thus ℭ_{Q}(i) defines an element (ℭ_{Q}(i)) in the locally free classgroup Cl(𝔄_{i}) of 𝔄_{i}. It follows from the definition of ℭ_{Q}(i) that (ℭ_{Q}(i)) depends only upon the image of Q in E(F)/π^{∗i}E(F), and so we obtain a map

ψ_{i} : E(F)/π^{∗i}E(F) ⟶ Cl(𝔄_{i})  (0.5)

given by ψ_{i}(Q) = (ℭ_{Q}(i)). The map ψ_{i} is in fact a homomorphism since E/F has everywhere good reduction (see Theorem 1 of [T1]), and ψ_{i}(Q) = 0 if and only if ℭ_{Q}(i) is a globally free 𝔄_{i}-module.

Let M_{i} be the maximal order in A_{i} containing 𝔄_{i}. By composing ψ_{i} with the natural surjection e_{i} : Cl(𝔄_{i}) → Cl(M_{i}) given by extension of scalars, we obtain a homomorphism

φ_{i} : E(F)/π^{∗i}E(F) ⟶ Cl(M_{i})  (0.6)

given by φ_{i}(Q) = (\overline{ℭ_{Q}(i)}), where \overline{ℭ_{Q}(i)} = ℭ_{Q}(i).M_{i}.

The phs ℭ_{Q}(i) is very closely related to the ring of integers of the field obtained by adjoining the coordinates of the the elements of G_{Q}(i) to F. Thus a knowledge of the homorphisms ψ_{i} and φ_{i} yields important information on the Galois module structure of these rings of integers. (We refer the reader to [BT] for a good general account of applications of the theory of Hopf orders to the study of Galois module structure.)

While the behaviour of the homomorphism ψ_{i} on torsion points of E(F) is quite well-understood (see [ST], [CN-S]), less is known about the Galois structure of phs arising from points of infinite order. The only general results in this direction of which we are aware are those contained in [T3] and [T4]. In [T3], the methods of [T1] and [Gr] are applied to Heegner points of certain modular elliptic curves. (See also the article by Ph. Cassou-Noguès et al. in [CT] for a brief account of this work.) In [T5], the kernel of ψ_{1} is described (under suitable hypotheses) in terms of congruences satisfied by special values of certain p-adic L-functions attached to K. The purpose of this paper is to examine the behaviour of the homomorphisms φ_{i} on points of infinite order. For further results in this direction, see [AT].

We now give a brief description of the contents of this article. In §1 we show that we may take inverse limits of (0.6) to obtain a homomorphism

Ψ_{M} : E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ lim_{i^{⟵}} Cl(M_{i}).  (0.7)

We then show that (assuming certain standard conjectures about elliptic curves) Ψ_{M} is closely related to the algebraic p-adic height pairing

{., .}_{F,𝔭} : E(F) ⊗_{𝔒} 𝔒_{𝔭} × E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ K_{𝔭}  (0.8)

described in [PR1] and [PR2]. By exploiting this relationship, we are able to give a description of Ker(Ψ_{M}) up to ℤ_{p}-pseudo-isomorphism (see §6). One application on of this description is as follows.

Suppose that K is of classnumber 1, that E is defined over K, and that F/K is abelian. Let Δ̂ denote the group of characters of Δ, and let 𝔒_{1} (resp. 𝔒_{2}) denote the ring of integers in some finite extension of K_{𝔭} (resp. K_{𝔭^{∗}}) which contains all of the character values of Δ. Let \overline{χ} denote the contragredient of χ.

Define r_{χ} = rank_{𝔒_{1}}[(E(F) ⊗_{𝔒} 𝔒_{1})^{χ})], where the superscript χ denotes the χ-equivariant eigenspace for the action of Δ. Let � (F)(𝔭) be the 𝔭-primary component of the Tate-Shafarevitch group � (F) of E/F.

Theorem 0.1. Suppose that {., .}_{F,𝔭} is non-degenerate modulo torsion, and that � (F)(𝔭) is finite. Suppose also that r_{χ} ≥ 1. Then

rank_{𝔒_{2}}[(Ker(Ψ_{M}) ⊗_{𝔒} 𝔒_{2})\overline{χ}] = 1.

We remark that similar one-dimensional subspaces of the completed Mordell-Weil group have arisen in the work of several authors in different contexts (see [G1], [P], [R]).

As a consequence of Theorem 0.1, we are able to deduce

Theorem 0.2. Suppose that the hypotheses of Theorem 0.1 hold and that r_{χ} = 1 for some χ ∈ Δ̂. Then there is a point Q ∈ E(F) of infinite order such that φ_{i}(Q) = 0 for all i.

It is interesting to note that the corresponding version of Theorem 0.2 is false in the analogous situation for CM abelian varieties defined over global function fields (see [A]).

The results contained in this paper formed a portion of my Ph.D. thesis (Columbia University, 1991). It is with the greatest of pleasure that I thank my advisor, Professor T.Chinburg, for all of his help, kindness and encouragement. I also wish to extend my warmest thanks to Professor M.J. Taylor for a great deal of extremely generous advice, and to Professor K. Rubin for many very helpful conversations and suggestions. I would like to acknowledge with thanks the hospitality of the MSRI, Berkeley, where this paper was written. Finally, I am very grateful to the referee for pointing out some mistakes in the original version of the manuscript.

§1 Classgroups In this section we recall various facts we shall require concerning the Hom decsription of the locally free classgroups Cl(𝔄_{i}) and Cl(M_{i}) in terms of character maps.

We begin by giving a description of the group of characters Ĝ_{i} which will be particularly well-suited to certain calculations we shall carry out later. Recall that q = ππ^{∗}. Let

w_{i} : E_{π^{∗i}} × E_{π^{i}} ⟶ μ_{q^{i}}  (1.1)

denote the Weil pairing on E. For each R ∈ E_{π^{i}}, we define a character χ_{R}^{(i)} ∈ Ĝ_{i} by

χ_{R}^{(i)}(g) = w_{i}(g, R)  ∀g ∈ E_{π^{∗i}}.  (1.2)

This identifies Ĝ_{i} with E_{π^{i}}. If ω ∈ Ω_{F} and χ ∈ Ĝ_{i}, then ω acts on χ via χ^{ω}(g) = χ(g^{ω^{−1}})^{ω}. The identification (1.2) preserves this action since the Weil pairing is Ω_{F} -equivariant.

Let J(ℚ^{c}) denote the group of finite ideles of ℚ^{c} (i.e. the direct limit of the finite idele groups of all finite extensions of ℚ), and write U(ℚ^{c}) for the subgroup of finite unit ideles of ℚ^{c}. Set U(𝔄_{i}) = ∏_{𝔮<∞} 𝔄_{i,𝔮}^{∗}, where the direct product runs over all finite places of 𝔒_{F}. (Here and elsewhere we write ℭ_{Q}(i)_{𝔮} (resp. 𝔄_{i,𝔮}) for the semi-local completion of ℭ_{Q}(i) (resp. 𝔄_{i}) at 𝔮.)

Suppose that u ∈ U(𝔄_{i}). Then u determines a map Det(u) ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) which is defined by

Det(u)(R)_{𝔮} = χ_{R}^{(i)}(u_{𝔮}).  (1.3)

Using the identification (1.2) above in Fröhlich’s Hom-description of classgroups (see [F] Ch. I §2 or [T2] Ch. I §3), we obtain isomorphisms

Cl(𝔄_{i}) = \frac{Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c}))}{Det(U(𝔄_{i})).Map_{Ω_{F}} (E_{π^{i}}, ℚ^{c∗})}  (1.4)

and

Cl(M_{i}) = \frac{Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c}))}{Map_{Ω_{F}} (E_{π^{i}}, U(ℚ^{c})).Map_{Ω_{F}} (E_{π^{i}}, ℚ^{c∗})}.  (1.5)

We now recall a method for explicitly constructing a representing map for (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}) and (\overline{ℭ_{Q}(i)}) ∈ Cl(M_{i}).

For each a ∈ F_{Q}(i) and χ_{R}^{(i)} ∈ Ĝ_{i}, the χ_{R}^{(i)}-resolvent (a|χ_{R}^{(i)}) ∈ Map(G_{Q}(i), ℚ^{c})

of a is defined by

(a|χ_{R}^{(i)}) = ∑_{g∈G_{i}} χ_{R}^{(i)}(g^{−1})a^{g}.  (1.6)

Choose d_{Q}^{(i)} ∈ F_{Q}(i) such that F_{Q}(i) = d_{Q}^{(i)}.A_{i}, and for each prime 𝔮 of 𝔒_{F}, choose m_{Q,𝔮} ∈ ℭ_{Q}(i)_{𝔮} such that ℭ_{Q}(i)_{𝔮} = m_{Q,𝔮}.𝔄_{i,𝔮}. Then the map h_{Q}^{(i)} ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) given by

h_{Q}^{(i)}(R)_{𝔮} = (m_{Q,𝔮}|χ_{R}^{(i)})(d_{Q}^{(i)}|χ_{R}^{(i)})^{−1}  (1.7)

represents both (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}) and (\overline{ℭ_{Q}(i)}) ∈ Cl(M_{i}).

For any field L, we let L(Q; i) denote the field obtained by adjoining the coordinates of all points in G_{Q}(i) to L. We shall have need of the following result on the evaluation of local resolvents which is proved in [T1] (Theorem 3).

Proposition 1.1. Let 𝔮 be a prime of 𝔒_{F}, and choose m ∈ ℭ_{Q}(i) such that ℭ_{Q}(i)_{𝔮} = m.𝔄_{i,𝔮}. Then, for all Q′ ∈ G_{Q}(i), π^{∗−i} ∑_{g∈G_{i}} m(Q′ + g)g^{−1} is a unit in the ring 𝔄_{i}(F (Q; i))_{𝔮}.

We shall also require the following result regarding a change of basefield.

Proposition 1.2. Let L/F be a finite extension. Then the following diagrams commute:

\begin{matrix}E(L) & ⟶^{ψ_{i,L}} & Cl(𝔄_{i}(L)) \\ ^{T r_{L/F}} ↓ & & ↓^{Res} \\ E(F) & ⟶_{ψ_{i,F}} & Cl(𝔄_{i}(F)) \end{matrix}  (1.8)

\begin{matrix}E(L) & ⟶^{φ_{i,L}} & Cl(M_{i}(L)) \\ ^{T r_{L/F}} ↓ & & ↓^{Res} \\ E(F) & ⟶_{φ_{i,F}} & Cl(M_{i}(F)) \end{matrix}  (1.9)

Here T r_{L/F} is the usual trace map and Res is the restriction map on classgroups which is induced by the corestriction map

N : Map_{Ω_{L}}(E_{π^{i}}, J(ℚ^{c})) ⟶ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c}))  (1.10)

defined by

(N f)(R) = ∏_{i=1}^{n} f(R^{ω_{i}^{−1}})^{ω_{i}}  ∀R ∈ E_{π^{i}}  (1.11)

where {ω_{1}, …, ω_{n}} is a transversal of Ω_{L}\Ω_{F}.

Proof. See (1.2) of [T1] for a proof of (1.8). An identical argument also works for (1.9).

§2 Inverse Limits

In this section we shall construct the homomorphism Ψ_{M} of (0.7). To do this we shall first describe the relationship between (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}) and (ℭ_{Q}(j)) ∈ Cl(𝔄_{j}) for 0 < j < i using the methods of §2 of [ST].

We begin by noting that the natural surjection [π^{∗i−j}] : G_{i} → G_{j} induces a surjective homomorphism A_{i} → A_{j} of Hopf algebras (which we shall also denote by [π^{∗i−j}]) given by

[π^{∗i−j}]( ∑_{g∈G_{i}} α_{g}g) = ∑_{g∈G_{i}} α_{g}([π^{∗i−j}]g).  (2.1)

Similarly, the inclusion G_{j} → G_{i} induces an inclusion A_{j} → A_{i} of Hopf algebras. Passing to the integral level, we deduce that 𝔄_{j} may be viewed either as a quotient algebra or as a subalgebra of 𝔄_{i}.

Next we observe that G_{i} acts on Map(G_{i}, ℚ^{c}) via translations, i.e.

f^{g}(h) = f(g + h)  ∀f ∈ Map(G_{i}, ℚ^{c}), g, h ∈ G_{i}.  (2.2)

For each i ≥ j ≥ 0, there is a group isomorphism G_{i}/G_{j} ≃ G_{i−j}. This induces identifications

ℚ^{c}G_{i−j} = (ℚ^{c}G_{i})^{G_{j}}  (2.3)

and

Map(G_{i−j}, ℚ^{c}) = Map(G_{i}, ℚ^{c})^{G_{j}}.  (2.4)

(2.3) and (2.4) in turn induce isomorphisms of 𝔄_{i−j} with a subalgebra of 𝔄_{i} and 𝔅_{i−j} with a subalgebra of 𝔅_{i}. We also observe that the identification of of G_{j} with G_{i}/G_{i−j} induces an identification of Ĝ_{j} with E_{π^{j}} via the map R ↦ χ_{R}^{(i)}, R ∈ E_{π^{j}}.

Proposition 2.1. There are isomorphisms

F_{Q}(j) ≃ Σ_{i−j}F_{Q}(i)  (2.5)

as A_{j}-modules, and

ℭ_{Q}(j) ≃ \frac{Σ_{i−j}}{π^{∗i−j}} ℭ_{Q}(i)  (2.6)

as 𝔄_{j}-modules.

Here Σ_{i−j} = ∑_{g∈G_{i−j}} g viewed as an element of A_{i}, and A_{j} (resp. 𝔄_{j}) acts on the right hand side of (2.5) (resp. (2.6)) via the homomorphism [π^{∗i−j}]. Proof. See §2 of [ST], especially Proposition 1.

Lemma 2.2. The homomorphism

ρ′_{i,j} : Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) ⟶ Map_{Ω_{F}} (E_{π^{j}}, J(ℚ^{c}))

defined by ρ′_{i,j}(f) = f|E_{π^{j}} induces homomorphisms

Cl(𝔄_{i}) ⟶ Cl(𝔄_{j})  (2.7)(a)

and

Cl(M_{i}) ⟶ Cl(M_{j}).  (2.7)(b)

We denote both of these homomorphisms by ρ_{i,j}.

Proof. We shall just prove that (2.7)(a) holds, as the proof of (2.7)(b) is entirely similar.

We have to show that

ρ′_{i,j}(Det(U(𝔄_{i})).Map_{Ω_{F}} (E_{π^{i}}, ℚ^{c∗})) ⊆ Det(U(𝔄_{j})).Map_{Ω_{F}} (E_{π^{j}}, ℚ^{c∗}).  (2.8)

It is plain that ρ′_{i,j}[Map_{Ω_{F}} (E_{π^{i}}, ℚ^{c∗})] ⊆ Map_{Ω_{F}} (E_{π^{j}}, ℚ^{c∗}). Now suppose that u ∈ Det(U(𝔄_{j})) and R ∈ E_{π^{j}}. Then

χ_{R}^{(i)}(u_{𝔮}) = χ_{R}^{(j)}([π^{∗i−j}]u_{𝔮})  (2.9)

and [π^{∗i−j}]u_{𝔮} ∈ 𝔄_{j,𝔮}^{∗}. Hence ρ′_{i,j}(Det(u)) = Det([π^{∗i−j}]u) ∈ Det(U(𝔄_{j})), and this establishes the result.

Proposition 2.3. Suppose that f ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) is a representing map for (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}). Then f|E_{π^{j}} is a representing map for (ℭ_{Q}(j)) ∈ Cl(𝔄_{j}) for all 0 < j ≤ i.

Proof. Choose d_{Q}^{(i)} ∈ F_{Q}(i) such that F_{Q}(i) = d_{Q}^{(i)}.𝔄_{i}. Then (3.6) implies that F_{Q}(j) ≃ (d_{Q}^{(i)}.(Σ_{i−j}/π^{∗i−j})).A_{j}, (where A_{j} acts on the right-hand side of this last isomorphism via the homomorphism [π^{∗i−j}]).

Let H_{ij} be a collection of coset representatives of G_{i}/G_{i−j}, and suppose that R ∈ E_{π^{j}}. Then

∑_{g∈G_{j}} (d_{Q}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )^{g}χ_{R}^{(i)}(g) = ∑_{g∈G_{j}} (d_{Q}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )^{g}w_{i}(g, R)

= ∑_{h∈H_{ij}} (d_{Q}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )^{h}w_{i}(h, R)

= π^{∗j−i} ∑_{g∈G_{i}} d_{Q}^{(i)g}w_{i}(g, R)

= (d_{Q}^{(i)}|χ_{R}^{(i)})π^{∗j−i}

Similarly, if for each place 𝔮 of F we choose m_{Q,𝔮}^{(i)} ∈ ℭ_{Q}(i)_{𝔮} such that ℭ_{Q}(i)_{𝔮} = m_{Q,𝔮}^{(i)}.𝔄_{i,𝔮}, then we obtain

∑_{g∈G_{j}} (m_{Q,𝔮}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )χ_{R}^{(i)}(g) = (m_{Q,𝔮}^{(i)}|χ_{R}^{(i)})π^{∗j−i}.

Hence,

\begin{aligned}h_{Q}^{(j)}(R)_{𝔮} & = [ ∑_{g∈G_{j}} (m_{Q,𝔮}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )^{g} χ_{R}^{(i)}(g)] [ ∑_{g∈G_{j}} (d_{Q}^{(i)}.\frac{Σ_{i−j}}{π^{∗i−j}} )^{g} χ_{R}^{(i)}(g)]^{−1} \\ & = h_{Q}^{(i)}(R)_{𝔮}, \end{aligned}

and now the result follows from (1.6).

Thus, if red : E(F)/π^{∗i}E(F) → E(F)/π^{∗i−1}E(F) denotes the natural reduction homomorphism, then we have commutative diagrams

\begin{matrix}E(F)/π^{∗i}E(F) & ⟶^{ψ_{i}} & Cl(𝔄_{i}) \\ ^{red}↓ & & ^{↓^{ρ_{i,i−1}}} \\ E(F)/π^{∗i−1}E(F) & ⟶_{ψ_{i−1}} & Cl(𝔄_{i−1}) \end{matrix}  (2.10)(a)

and

\begin{matrix}E(F)/π^{∗i}E(F) & ⟶^{φ_{i}} & Cl(M_{i}) \\ ^{red}↓ & & ^{↓^{ρ_{i,i−1}}} \\ E(F)/π^{∗i−1}E(F) & ⟶_{φ_{i−1}} & Cl(M_{i−1}). \end{matrix}  (2.10)(b)

Taking inverse limits yields homomorphisms

Ψ : E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ lim_{i^{⟵}} Cl(𝔄_{i})  (2.11)(a)

and

Ψ_{M} : E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ lim_{i^{⟵}} Cl(M_{i}).  (2.11)(b)

§3 The homomorphism Ψ′_{M}

The purpose of this section is to describe certain technical results which will enable us to relate the homomorphism Ψ_{M} to the Iwasawa theory of E in §5.

We begin by establishing a useful property of a representing map for (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}) (resp. (\overline{ℭ_{Q}(i)}) ∈ Cl(M_{i}).

Proposition 3.1. Let h_{Q}^{(i)} ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) be as in (1.7). Then for each r ∈ ℕ, the Ω_{F} -map given by R ↦ h_{Q}^{(i)}(rR).h_{Q}^{(i)}(R)^{−r} represents the trivial class in Cl(𝔄_{i}) (resp. in Cl(M_{i})).

Proof. Suppose that Q′ ∈ G_{Q}(i), and let ω ∈ Ω_{F}. Write Q^{′ω} = Q′ + h_{ω}, with h_{ω} ∈ E_{π^{∗i}}. Choose d_{Q}^{(i)} ∈ F_{Q}(i) such that F_{Q}(i) = d_{Q}^{(i)}.A_{i}. Then for each R ∈ E_{π^{i}} and r ∈ ℕ, both ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)[r]g and ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)g lie in A_{i}(F (Q; i). We have

[( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)[r]g)( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)g)^{−r}]^{ω}

= ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q^{′ω} − g^{ω})[r]g^{ω}) × ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q^{′ω} − g^{ω})g^{ω})^{−r}

= ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g^{ω} + h_{ω})[r]g^{ω}) × ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g^{ω} + h_{ω})g^{ω})^{−r}

= ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g).[r](g + h_{ω})) × ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g).(g + h_{ω}))^{−r}

= ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)[r]g) × ( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)g)^{−r}.

Thus

( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)[r]g)( ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g)g)^{−r} ∈ (ℚ^{c}G_{i})^{Ω_{F}} = A_{i},

and similar reasoning also shows that this expression is independent of the choice of Q′ ∈ G_{Q}(i).

For each place 𝔮 of 𝔒_{F} choose m_{Q,𝔮}^{(i)} ∈ ℭ_{Q}(i)_{𝔮} such that ℭ_{Q}(i)_{𝔮} = m_{Q,𝔮}^{(i)}𝔄_{i,𝔮}. Proposition 1.1 implies that

π^{∗−i} ∑_{g∈G_{i}} m_{Q,𝔮}^{(i)}(Q′ − g)g, π^{∗−i} ∑_{g∈G_{i}} m_{Q,𝔮}^{(i)}(Q′ − g)[r]g ∈ 𝔄_{i}(F (Q; i))_{𝔮}^{∗}.

Reasoning exactly as above, we conclude that

(π^{∗−i} ∑_{g∈G_{i}} m_{Q,𝔮}^{(i)}(Q′ − g)[r]g)(π^{∗−i} ∑_{g∈G_{i}} m_{Q,𝔮}^{(i)}(Q′ − g)g)^{−r} ∈ [𝔄_{i}(F (Q; i))_{𝔮}^{∗}]^{Ω_{F}} ⊆ 𝔄_{𝔮}^{∗}.

Thus, it now follows from (1.4) (resp. (1.5)) that the map R ↦ h_{Q}^{(i)}([r]R).h_{Q}^{(i)}(R)^{−r} represents the trivial class in Cl(𝔄_{i}) (resp. Cl(M_{i}).

Remark 3.2. It may in fact easily be shown that if f ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})) is any representing map for (ℭ_{Q}(i)) ∈ Cl(𝔄_{i}), then f = h_{Q}^{(i)} for suitable choices of d_{Q}^{(i)} and m_{Q,𝔮}^{(i)}.

For each R ∈ E_{π^{i}}, we let F [R] denote the field obtained by adjoining the coordinates of R to F (with similar notation F [R′] for R′ ∈ E_{π^{∗i}}). We write F_{i} = ⋃_{R∈E_{π^{i}}} F [R] and F′_{i} = ⋃_{R′∈E_{π^{∗i}}} F [R′]. We shall now give a description of Cl(M_{i}) in terms of certain ideal classgroups.

Lemma 3.3. Let E_{π^{i}}\Ω_{F} denote a set of representatives of Ω_{F} -orbits of E_{π^{i}}. Then we have isomorphisms

(a) A_{i} ≃ ∏_{R∈E_{π^{i}} \Ω_{F}} F [R]

(b) M_{i} ≃ ∏_{R∈E_{π^{i}} \Ω_{F}} 𝔒_{F [R]}
(c) Cl(M_{i}) ≃ ∏_{R∈E_{π^{i}} \Ω_{F}} Cl(𝔒_{F [R]}).

Proof. (b) and (c) follow immediately from (a), which we shall now prove. Let {ω_{1}, …, ω_{n}} be a transversal of Ω_{F} \Ω_{F′_{i}}. then A_{i} = (ℚ^{c}G_{i})^{Ω_{F}} = (F′_{i}G_{i})^{Ω_{F}} is generated over F by all elements of the form ∑_{ω} l^{ω}g^{ω}, where l ∈ F′_{i} and g ∈ G_{i}. Suppose that R ∈ E_{π^{i}} and λ ∈ Ω_{F}. Then,

( ∑_{ω} l^{ω}χ_{R}^{(i)}(g^{ω}))^{λ} = ( ∑_{ω} l^{ω}w_{i}(R, g^{ω}))^{λ}

= ∑_{ω} l^{ωλ}w_{i}(R^{λ}, g^{ωλ})

= ∑_{ω} l^{ω}χ_{R^{λ}}^{(i)}(g^{ω})

Hence (F′_{i}G_{i})^{Ω_{F}} ⊆ ∏_{R∈E_{π^{i}} \Ω_{F}} F [R]. Since both sides of the inclusion have the same dimension over F as F -vector spaces, it follows that (a) holds.

If α ∈ Cl(M_{i}) is represented by f ∈ Map_{Ω_{F}} (E_{π^{i}}, J(ℚ^{c})), then the image of α under the isomorphism 3.3(b) is obtained by evaluating f on elements of E_{π^{i}}\Ω_{F} and then taking ideal content. Thus every element of Cl(M_{i}) may be represented by a map R ↦ I_{R}, with I_{R} ∈ Cl(𝔒_{F [R]}) satisfying I_{R^{ω}} = I_{R}^{ω} for all ω ∈ Ω_{F}. Conversely, every such map defines an element of Cl(M_{i}).

Lemma 3.4. Choose d_{Q}^{(i)} ∈ F_{Q}(i) such that F_{Q}(i) ∈ d_{Q}^{(i)}.A_{i}. Suppose that Q′ ∈ G_{Q}(i), and let R ∈ E_{π^{r}} with 0 < r ≤ i. Then

(a) ∑_{g∈G_{i}}[d_{Q}^{(i)g^{−1}}(Q′)χ_{R}^{(i)}(g)] ∈ F [R](Q; i)
(b) ( ∑_{g∈G_{i}}[d_{Q}^{(i)g^{−1}}(Q′)χ_{R}^{(i)}(g)])^{q^{r}} ∈ F [R].

(Recall that F [R](Q; i) denotes the field obtained by adjoining the coordinates of all points in G_{Q}(i) to F [R].)

Proof. We shall just prove (b), since the proof of (a) is very similar. Let ω ∈ Ω_{F [R]}, and write Q^{′ω} = Q′ + h_{ω}, with h_{ω} ∈ E_{π^{∗i}}. Recall that q = ππ^{∗}. Then,

([ ∑_{g∈G_{i}} d_{Q}^{(i)g^{−1}}(Q′)χ_{R}^{(i)}(g)]^{q^{r}})^{ω} = [ ∑_{g∈G_{i}} d_{Q}^{(i)}(Q^{′ω} − g^{ω}).w_{i}(g^{ω}, R^{ω})]^{q^{r}}

= [w_{i}(h_{ω}, R^{ω}) ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g).w_{i}(g, R^{ω})]^{q^{r}}

= [ ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g).w_{i}(g, R)]^{q^{r}}.

Hence [ ∑_{g∈G_{i}} d_{Q}^{(i)}(Q′ − g).w_{i}(g, R)]^{q^{r}} ∈ F [R], and is independent of the choice of Q′.

Now Proposition 1.1 implies that

c(h_{Q}^{(i)}(R)).𝔒_{F [R](Q;i)} = (d_{Q}^{(i)}|χ_{R}^{(i)})(Q′).π^{∗−1}.𝔒_{F [R](Q;i)}  (3.1),

where c denotes ideal content. If d_{Q}^{(i)} is chosen to be an 𝔄_{i,𝔮}-basis of ℭ_{Q}(i)_{𝔮} for all primes 𝔮|π^{∗} then (from Proposition 1.1 again), (d_{Q}^{(i)}|χ_{R}^{(i)})(Q′).π^{∗−i} is a unit at all such 𝔮. Since we have assumed that E/F has everywhere good reduction, it follows from the criterion of Néron-Ogg-Shafarevitch that F [R](Q; i)/F [R] is unramified at all primes away from 𝔭^{∗}. As (d_{Q}^{(i)}|χ_{R}^{(i)})(Q′)^{q^{r}} ∈ F [R] (see Lemma 3.4(b)), it follows that (d_{Q}^{(i)}|χ_{R}^{(i)})(Q′).𝔒_{F [R](Q;i)} is an ambiguous ideal, i.e. that we have

(d_{Q}^{(i)}|χ_{R}^{(i)})(Q′).𝔒_{F [R](Q;i)} = ϑ_{R}.𝔒_{F [R](Q;i)}  (3.2)

where ϑ_{R} is an 𝔒_{F [R]}-ideal.

Hence, in the notation of the remarks following Lemma 3.3, we deduce that (\overline{ℭ_{Q}(i)}) ∈ Cl(M_{i}) is represented by the map

R ↦ (ϑ_{R})^{−1}  (3.3)

where (ϑ_{R}) denotes the class of the ideal ϑ_{R} in 𝔒_{F [R]}.

For each R ∈ E_{π^{i}}, it follows from (3.1) and (3.2), together with Proposition 3.1, that

(ϑ_{R})^{q} = (ϑ_{qR}𝔒_{F [R]})  (3.4)

in Cl(𝔒_{F [R]}). Now [F [R] : F [qR]] = [F [R] : F [πR]] = q if the order of R is sufficiently large. Thus, if [F [R] : F [qR]] = q, then

(ϑ_{qR}) = N_{F [R]/F [qR]}(ϑ_{R})  (3.5)

in Cl(𝔒_{F [qR]}), where N_{F [R]/F [qR]} is the norm from F [R] to F [qR].

We now observe that F [R] ⊆ F_{i}, and so via the natural homomorphism Cl(𝔒_{F [R]}) → Cl(𝔒_{F_{i}}), (ϑ_{R}) ∈ Cl(𝔒_{F [R]}) defines an element of Cl(𝔒_{F_{i}}). Thus φ_{i} induces a homomorphism

φ_{i}^{(e)} : E(F)/π^{∗i}E(F) ⟶ Map_{Ω_{F}} (E_{π^{i}}, Cl(𝔒_{F_{i}}))  (3.6)

defined by

φ_{i}^{(e)}(Q)(R) = (ϑ_{R}.𝔒_{F_{i}}).  (3.6)(a)

Now Proposition 3.1 implies that c(h_{Q}^{(i)}(rR)h_{Q}^{(i)}(R)^{−r}) is a principal ideal for each r ∈ ℕ. Hence

(ϑ_{rR}.𝔒_{F_{i}}) = (ϑ_{R}.𝔒_{F_{i}})^{r}

and so we in fact have

φ_{i}^{(e)} : E(F)/π^{∗i}E(F) ⟶ Hom_{Ω_{F}} (E_{π^{i}}, Cl(𝔒_{F_{i}})_{p})  (3.7)

where Cl(𝔒_{F_{i}})_{p} denotes the p-primary part of Cl(𝔒_{F_{i}}).

We next observe that the restriction of [π^{∗}] to E_{π^{i}} is an automorphism. For each Q ∈ E(F), define φ′_{i}(Q) ∈ Hom_{Ω_{F}} (E_{π^{i}}, Cl(𝔒_{F_{i}})_{p}) by

φ′_{i}(Q)(R) = φ_{i}^{(e)}(Q)([π^{∗}]^{−i}R).  (3.8)

Let T r_{F_{i}/F_{i−1}} denote the trace map from F_{i} to F_{i−1}. Since T r_{F_{i}/F_{i−1}}(R) = [q]R for all sufficiently large i, it follows that

N_{F_{i}/F_{i−1}}(φ′_{i}(Q)(R)) = φ′_{i−1}(Q)([π]R)  (3.9)

(for i sufficiently large), where N_{F_{i}/F_{i−1}} is the norm from F_{i} to F_{i−1}. In other words, the following diagram commutes

E(F)/π^{∗i}E(F) ⟶^{φ′_{i}} Hom_{Ω_{F}} (E_{π^{i}}, Cl(𝔒_{F_{i}})_{p})
red↓ ↓^{N_{i/i−1}}
E(F)/π^{∗i−1}E(F) ⟶_{φ′_{i−1}} Hom_{Ω_{F}} (E_{π^{i−1}}, Cl(𝔒_{F_{i−1}})_{p}).

(3.10)

Here N_{i/i−1} is defined by
N_{i/i−1}(φ′_{i}(Q)(R)) = N_{F_{i}/F_{i−1}}(φ′_{i}(Q)(R′))  (3.11)

where R′ is any element of E_{π^{i}} satisfying [π]R′ = R. It is easily checked that this is well-defined.

Taking inverse limits of (3.10), we obtain a homomorphism

Ψ′_{M} : E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ Hom_{Ω_{F}} (T_{π}, lim_{i^{⟵}} Cl(𝔒_{F_{i}})_{p})  (3.12)

where T_{π} is the π-adic Tate module of E and the inverse limit lim_{i^{⟵}} Cl(𝔒_{F_{i}})_{p} is taken with respect to the norm maps N_{F_{i}/F_{i−1}}.

It is clear that the homomorphism Ψ′_{M} completely determines Ψ_{M}. In particular, we have

Lemma 3.5. Ker(Ψ′_{M}) = Ker(Ψ_{M}).

§4. Selmer Groups

In this section we recall various results we shall require concerning Selmer groups.

Set F_{∞} = ⋃_{i} F_{i} and F′_{∞} = ⋃_{i} F′_{i}. Let N_{∞}/F be the unique ℤ_{p}-extension contained in F_{∞}/F, and write Γ = Gal(N_{∞}/F). We may identify Γ with Gal(F_{∞}/F (E_{𝔭})), and Gal(F (E_{𝔭})/F) with Gal(N_{∞}/F_{∞}). Let χ_{∞} : Gal(F_{∞}/F) → 𝔒_{𝔭}^{∗} denote the character giving the action of Gal(F_{∞}/F) on E_{π^{∞}}. We have

Gal(F_{∞}/F) ≃ Gal(F_{∞}/F (E_{𝔭})) × Gal(F (E_{𝔭})/F)
≃ Γ × Gal(F (E_{𝔭})/F).

Write ε (resp. κ) for the restriction of χ_{∞} to Gal(F (E_{𝔭})/F) (resp. Γ).

Choose a topological generator γ of Γ, and write Λ for the power series ring ℤ_{p}[[T]]. Then we may identify Λ with the completed group ring ℤ_{p}[[Γ]] in the usual manner via the map γ ↦ T + 1.

For each i, let H_{i}/F_{i} be the maximal everwhere unramified abelian pro-p extension of F_{i}, and let X_{i}/F_{i} be the maximal abelian pro-p extension of F_{i} which is unramified away from primes dividing 𝔭. Set H_{i} = Gal(H_{i}/F_{i}), X_{i} = Gal(X_{i}/F_{i}), and write H_{∞} (resp. X_{∞}) for the inverse limit of the H_{i} (resp. X_{i}).

If 𝔮 is a prime of F, we write k_{𝔮} for the residue field of F at 𝔮, and we let Ẽ(k_{𝔮}) denote the reduction of E(F) at 𝔮. Write E_{1,𝔮}(F) for the kernel of reduction of E(F) at 𝔮, and define E_{1,𝔭}(F) = E_{1}(F) via exactness of the sequence

0 ⟶ E_{1}(F) ⟶ E(F) ⟶ ∏_{𝔮|𝔭} Ẽ(k_{𝔮}).  (4.1)

We now recall the definitions of the Selmer and the Tate-Shafarevitch groups of E. Let L be any extension of K over which E is defined. The Selmer group S(L)^{π^{∗i}} is defined to be the kernel of the natural homomorphism

H^{1}(Ω_{L}, E_{π^{∗i}}) ⟶ ∏_{𝔮 of L} H^{1}(Ω_{L_{𝔮}}, E).  (4.2)

The enlarged Selmer group S′(L)^{π^{∗i}} is the kernel of the homomorphism

H^{1}(Ω_{L}, E_{π^{∗i}}) ⟶ ∏_{𝔮 |̸𝔭^{∗}} H^{1}(Ω_{L_{𝔮}}, E).  (4.3)

The Selmer groups S(L)^{π^{i}} and S′(L)^{π^{i}} are defined similarly. We set S(L) = lim_{i^{⟶}} S(L)^{π^{i}} and we write Y (L) for the Pontryagin dual of S(L)

The Tate-Shafarevitch group � (L) of E/L is defined to be the kernel of the map

H^{1}(Ω_{L}, E) ⟶ ∏_{𝔮 of L} H^{1}(Ω_{L_{𝔮}}, E).  (4.4)

Define Σ(F)^{(π^{∗i})} to be the subgroup of S(F)^{π^{∗i}} which makes the following sequence exact

0 ⟶ Σ(F)^{(π^{∗i})} ⟶ S(F)^{π^{∗i}} ⟶ ∏_{v|𝔭} H^{1}(Ω_{F_{v}}, E_{π^{∗i}}),

and set Σ(F) = lim_{i^{⟵}} Σ(F)^{π^{∗i}}. Then there is a natural injection E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ Σ(F). The following result is shown in Corollaire 3.3 of [PR1].

Lemma 4.1. If |� (F)(𝔭^{∗})| < ∞, then E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} = Σ(F).

Proposition 4.2. (J.Coates). There are Λ-module isomorphisms

(a) S(N_{∞}) ≃ Hom(X_{∞}^{(ε)}, E_{π^{∞}}).

(b) Y (N_{∞}) ≃ Hom(T_{π}, X_{∞}^{(ε)})

Proof. (a) See Theorems 12 and 9 of [Co].

(b) Write D_{𝔭} = K_{𝔭}/𝔒_{𝔭}. Choosing a generator t of T_{π} over 𝔒_{𝔭} is equivalent to fixing an isomorphism 𝔒_{𝔭} → T_{π} such that 1 ↦ t. This in turn induces an isomorphism ϕ : D_{𝔭} → E_{π^{∞}}. Via such a choice of t, we may define a pairing

(., .) : Hom(T_{π}, X_{∞}^{(ε)}) × Hom(X_{∞}^{(ε)}, E_{π^{∞}}) ⟶ D_{𝔭}  (4.5)

by (f_{1}, f_{2}) = ϕ^{−1}(f_{2}(f_{1}(t))). It is easy to check that (., .) is independent of the choice of t, is non-degenerate, and is Ω_{F} -equivariant, which gives the result.

We conclude this section with a description of the weak 𝔭-adic Leopoldt conjecture for F. Let L/K be a finite extension, and for each prime 𝔮 of L, let U(L_{𝔮}) denote the group of units of the local completion L_{𝔮} of L at 𝔮. Write U(L) for the group of global units of L. Then there is a natural injection

i_{L} : U(L) ⟶ ∏_{𝔮|𝔭} U(L_{𝔮})  (4.6)

given by the diagonal embedding i(x) = (x, x, …, x).

Let \overline{i_{L}(U(L))} denote the closure of i_{L}(U(L)) with respect to the 𝔭-adic topology. Write δ(L) for the difference between the ℤ_{p}-rank of U(L) and the ℤ_{p}-rank of \overline{i_{L}(U(L))}. Then the weak 𝔭-adic Leopoldt conjecture for F asserts that the numbers δ(L) are bounded independently of L as L runs through the set of all finite extensions of F which are contained in F_{∞}.

We shall assume for the rest of this paper that the weak 𝔭-adic Leopoldt conjecture holds for F. This is known to be the case if F/K is abelian, via work of A. Brumer (see [Br]). We shall also assume that |� (F)(𝔭^{∗})| < ∞, and so in particular that Lemma 4.1 holds.

§5 The algebraic 𝔭-adic height pairing. In this section we describe the 𝔭-adic height pairing on E/F and explain its relationship to the homomorphism Ψ′_{M}. We refer the reader to §3 of [PR1] or Chapter IVof [PR2] for full details of the results we use concerning 𝔭-adic heights.

Let J(F_{i}) denote the ideles of F_{i}. Write U_{i}^{(𝔭)} for the subgroup of J(F_{i}) consisting of ideles which are equal to 1 at all places above 𝔭, and which are units elsewhere. Set C_{i} = J(F_{i})/U_{i}^{(𝔭)}F_{i}^{∗}, and let W_{i} = ∏_{𝔮|𝔭} μ_{q^{i}}(F_{i,𝔮}). It is shown in §3.2 of [PR1] that there is an isomorphism

η_{i}^{−1} : Σ(F)^{(π^{∗i})} ⟶ \frac{Hom(E_{π^{i}}, C_{i}(p))^{Γ_{1}}}{Hom(E_{π^{i}}, W_{i})^{Γ_{1}}}  (5.1)

where C_{i}(p) denotes the p-primary part of C_{i} and Γ_{1} = Gal(F_{∞}/F).

Suppose that T ∈ E_{1,𝔭}(F)⊗_{𝔒} 𝔒_{𝔭^{∗}}, and let T_{i} denote the image of T in Σ(F)^{π^{∗i}} under projection. Then it follows from the construction of η_{i} given in §3.2 of [PR] that

(c(η_{i}^{−1}(T_{i})(R))) = φ_{i}^{(e)}(T_{i})(R).  (5.2)

Define

Ξ_{i} : Σ(F)^{(π^{∗i})} ⟶ \frac{Hom(E_{π^{i}}, C_{i}(p))^{Γ_{1}}}{Hom(E_{π^{i}}, W_{i})^{Γ_{1}}}  (5.3)

by Ξ_{i}(s)(R) = η_{i}(s)(π^{∗−i}R). For each i, the global Artin map yields a surjection

(−, X_{i}/F_{i}) : C_{i}(p) ⟶ X_{i},

and this induces a homomorphism

γ_{i} : Σ(F)^{π^{∗i}} ⟶ Hom(E_{π^{∗i}}, X_{i})^{Γ_{1}}  (5.4)

which is defined by γ_{i}(s) = (−, X_{i}/F_{i}) ∘ Ξ_{i}(s).

It is shown in §3.2 of [PR1] that we may take inverse limits of (5.4) to obtain an isomorphism

Φ_{F} : Σ(F) ⟶ Hom(T_{π}, X_{∞})^{Γ_{1}}.  (5.5)

(The reader should note that our Φ_{F} is slightly different from the Φ_{F} of [PR1], because there Ξ_{i} is defined by Ξ_{i}(s)(R) = Ξ_{i}(s)(R) = η_{i}(s)(δ^{−1}π^{∗−i}R), where δ = [F (E_{𝔭}) : F].)

We shall now use the isomorphism Φ_{F} to construct a pairing

{., .}′_{F,𝔭} : E(F) ⊗_{𝔒} 𝔒_{𝔭} × E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ 𝔒_{𝔭}  (5.6)

as follows.

Suppose that S ∈ E_{1,𝔭}(F)⊗_{𝔒}𝔒_{𝔭^{∗}} and P ∈ E(F)⊗_{𝔒}𝔒_{𝔭}. Let t be a generator of of T_{π} (as an 𝔒_{𝔭}-module). Set x_{S} = Φ_{F} (S)(t) ∈ X_{∞}. For each j ∈ ℕ, let P_{j} ∈ E(F) be such that the image of P_{j} under the projection E(F) → E(F)/π^{j}E(F) is equal to the image of P under the projection E(F) ⊗_{𝔒} 𝔒_{𝔭} → E(F)/π^{j}E(F). Choose any P′_{j} ∈ E(F^{c}) such that [π^{j}]P′_{j} = P_{j}. Then the map π^{−j} ↦ ϕ^{−1}(P_{j}^{′x_{S}} − P′_{j}) induces an element τ ∈ Hom(D_{𝔭}, D_{𝔭}). There is an isomorphism

ξ : 𝔒_{𝔭} ⟶ Hom(D_{𝔭}, D_{𝔭})

given by ξ(a)(α) = a.α, where a ∈ 𝔒_{𝔭} and α ∈ D_{𝔭}. We set {P, S}′_{F,𝔭} = ξ^{−1}(τ). As [E(F) : E_{1}(F)] < ∞, {., .}′_{F,𝔭} may be extended to a pairing

{., .}_{F,𝔭} : E(F) ⊗_{𝔒} 𝔒_{𝔭} × E(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} ⟶ K_{𝔭}.  (5.7)

{., .}_{F,𝔭} is the algebraic 𝔭-adic height pairing on E/F and is conjectured to be non-degenerate modulo torsion. One of the main results of [PR1] (p.370; see also Chapter IV, Thèoréme 22 of [PR2]) asserts that if this is the case, and if in addition |� (F)(𝔭^{∗})| < ∞, then the 𝔭-adic L-function L_{𝔭}(E/F, s) associated to E/F vanishes to order rank_{𝔒}(E(F)) at s = 1.

Let

Φ_{M} : Σ(F) ⟶ Hom(T_{π}, H_{∞})^{Γ_{1}}  (5.8)

be the map obtained by composing Φ_{F} with the projection Hom(T_{π}, X_{∞})^{Γ_{1}} → Hom(T_{π}, H_{∞})^{Γ_{1}}. Write Φ′′ for the map obtained by composing Ψ′_{M} with the natural map lim_{i^{⟵}} Cl(𝔒_{F_{i}})(p) → H_{∞} arising from classfield theory. The link between the Galois structure of phs and the 𝔭-adic height pairing on E/F is given by the following lemma.

Lemma 5.1. Ker(Φ′′|Σ(F)) = Ker(Φ_{M}).

Proof. This follows immediately from the definition of Ψ′_{M} together with (5.2) and the definition of Φ_{F}.

§6. Unramified points In this section we shall obtain a description of KerΦ_{M} in terms of the pairing is {., .}_{F,𝔭}. We begin by introducing some notation. We shall say that two ℤ_{p}- modules A, B are pseudo-isomorphic (as ℤ_{p}-modules), and write A ∼ B, if there is a homomorphism A → B with finite kernel and cokernel. Thus, if A and B are finitely generated, then A ∼ B if and only if rank_{ℤ_{p}}(A) = rank_{ℤ_{p}}(B); in particular, A ∼ B if and only if B ∼ A.

Let Z_{∞} = Gal(X_{∞}/H_{∞}). Then Hom(T_{π}, X_{∞})^{Γ_{1}} = Hom(T_{π}, X_{∞}^{(χ_{∞})}), and so Φ_{M} is simply the composition of Φ_{F} with the natural projection

Hom(T_{π}, X_{∞}^{(χ_{∞})}) → Hom(T_{π}, (X_{∞}/Z_{∞})^{(χ_{∞})}).

The following proposition immediately implies that this projection is in fact surjective up to finite index. I would like to thank Karl Rubin for showing me the proof of this result.

Proposition 6.1. X_{∞}^{(χ_{∞})}/Z_{∞}^{(χ_{∞})} ∼ (X_{∞}/Z_{∞})^{(χ_{∞})}.

Proof. The exact sequence

0 ⟶ Z_{∞} ⟶ X_{∞} ⟶ X_{∞}/Z_{∞} ⟶ 0  (6.1)

yields

0 ⟶ Z_{∞}^{(χ_{∞})} ⟶ X_{∞}^{(χ_{∞})} ⟶ (X_{∞}/Z_{∞})^{(χ_{∞})}  (6.2)

and

0 ⟶ Z_{∞}^{(ε)} ⟶ X_{∞}^{(ε)} ⟶ (X_{∞}/Z_{∞})^{(ε)}  (6.2)(a)

Since Δ is a finite group, we deduce from (6.2)(a) that

rank_{ℤ_{p}}(X_{∞}^{(ε)}) = rank_{ℤ_{p}}(X_{∞}/Z_{∞})^{(ε)} + rank_{ℤ_{p}}(Z_{∞}^{(ε)}).

We claim that the right-hand arrow of (6.2) is surjective up to finite index. For, if this were not the case, then we would have

rank_{ℤ_{p}}(X_{∞}/Z_{∞})^{(χ_{∞})} + rank_{ℤ_{p}}(Z_{∞}^{(χ_{∞})}) > rank_{ℤ_{p}}(X_{∞}^{(χ_{∞})}).  (6.3)

Now Hom(T_{π}, X_{∞}^{(χ_{∞})}) ≃ E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}} (from (5.5) and Lemma 4.1), and so

rank_{ℤ_{p}}(X_{∞}^{(χ_{∞})}) = rank_{ℤ_{p}}(E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}}) = n_{E/F} ,  (6.4)

say. (Here we have identified 𝔒_{𝔭^{∗}} with ℤ_{p}.) Hence we obtain

rank_{ℤ_{p}}(X_{∞}^{(ε)}) = rank_{ℤ_{p}}(X_{∞}/Z_{∞})^{(ε)} + rank_{ℤ_{p}}(Z_{∞}^{(ε)})
≥ rank_{ℤ_{p}}(X_{∞}/Z_{∞})^{(χ_{∞})} + rank_{ℤ_{p}}(Z_{∞}^{(χ_{∞})})
> rank_{ℤ_{p}}(X_{∞}^{(χ_{∞})})
= n_{E/F} .

This implies that the 𝔭-adic L-function L_{𝔭}(E/F, s) associated to E/F vanishes to order strictly greater than n_{E/F} at s = 1. Since we have assumed that |� (F)(𝔭)| < ∞ and {., .}_{F,𝔭} is non-degenerate modulo torsion, this is a contradiction (see [PR1] p.370).

We now introduce a certain subgroup U of E(F) ⊗_{𝔒} 𝔒_{𝔭} which we shall call the group of unramified points of E(F) ⊗_{𝔒} 𝔒_{𝔭}. Suppose that P ∈ E(F) ⊗_{𝔒} 𝔒_{𝔭}, and for each j ∈ ℕ, choose P_{j} ∈ E(F) such that the image of P_{j} under the projection E(F) → E(F)/π^{j}E(F) is equal to the image of P under the projection E(F) ⊗_{𝔒} 𝔒_{𝔭} → E(F)/π^{j}E(F). Choose any P′_{j} ∈ E(F^{c}) such that [π^{j}]P′_{j} = P_{j}. Then we may define a homomorphism

h_{j} : X_{∞}^{(χ_{∞})} ⟶ E_{π^{∞}}  (6.5)

by h_{j}(ω) = P_{j}^{′ω} − P′_{j}. We say that P is an unramified point if h_{j} vanishes on Z_{∞}^{(χ_{∞})} for all j ∈ ℕ.

Proposition 6.3. The orthogonal complement [Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})})]^{⊥} of Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})}) with respect to the pairing {., .}_{F,𝔭} is equal to U.

Proof. It is clear from the definitions that

U ⊆ [Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})})]^{⊥}.

Suppose conversely that P ∈ E(F) ⊗_{𝔒} 𝔒_{𝔭} is orthogonal to all points S which lie in Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})}) ⊆ E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭^{∗}}. Then, (using the notation of §5)

P_{j}^{′x_{S}} − P′_{j} = 0 ∀j ∈ ℕ, ∀S ∈ [Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})})].

Let ω ∈ Z_{∞}. Since Φ_{F} is an isomorphism, it follows that (again using the notation of §5)

x_{S} = Φ_{F} (S)(t) = ω

for some S ∈ [Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})})]. Thus we deduce that

P_{j}^{′ω} − P′ = 0 ∀j ∈ ℕ, ∀ω ∈ Z_{∞}^{(χ_{∞})}.

Hence P ∈ U, and this establishes the result.

Corollary 6.4. Ker(Ψ_{M}) ∼ U^{⊥} and (Ker(Ψ_{M}))^{⊥} ∼ U.

Proof. We deduce from Lemmas 3.5, 5.1, and the remarks immediately preceding Proposition 6.1 that KerΨ_{M} = [Φ_{F}^{−1}(Hom(T_{π}, Z_{∞}^{(χ_{∞})})]. The result now follows from Proposition 6.3 together with the fact that {., .}_{F,𝔭} is non-degenerate modulo torsion.

We shall now give another description of the group U up to pseudo-isomorphism. For each prime 𝔓 of F lying above 𝔭, we write E_{1}(F_{𝔓}) for the kernel of reduction of E(F_{𝔓}). Then E_{1}(F_{𝔓}) is an 𝔒_{𝔭}-module, and we have a natural homomorphism

f_{𝔓} : E_{1,𝔭}(F) ⊗_{𝔒} 𝔒_{𝔭} ⟶ E_{1}(F_{𝔓})  (6.6)

which induces

f : E_{1,𝔭}(F) ⊗_{𝔒} 𝔒_{𝔭} ⟶ ∏_{𝔓|𝔭} E_{1}(F_{𝔓})  (6.7).

Proposition 6.5. U ∼ Ker(f).

Proof. We shall show that (U ∩ E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭}) ∼ Ker(f). (This will suffice since [E(F)⊗_{𝔒}𝔒_{𝔭} : E_{1}(F)⊗_{𝔒}𝔒_{𝔭}] < ∞.) Suppose that P ∈ E_{1}(F)⊗_{𝔒}𝔒_{𝔭}. For each i ∈ ℕ and 𝔓|𝔭, let F_{∞,𝔓}(\frac{1}{π^{i}} f_{𝔓}(P)) be the field obtained by adjoining π^{i}th roots of f_{𝔓}(P) (taken with respect to the formal group law on E) to F_{∞,𝔓}. Then P is unramified if and only if the extension F_{∞,𝔓}(\frac{1}{π^{i}} f_{𝔓}(P))/F_{∞,𝔓} is unramified for all 𝔓|𝔭 and all i ∈ ℕ. As we are only concerned with modules up to pseudo-isomorphism, we may assume that either f_{𝔓}(P) = 0 or that f_{𝔓}(P) is of infinite order. However, if f_{𝔓}(P) is of infinite order, then F_{∞,𝔓}(\frac{1}{π^{i}} f_{𝔓}(P))/F_{∞,𝔓} is non-trivial and ramified for all sufficiently large i (see [CW] Th. 11), and so f(P) ∉ U. The result follows.

Let us now give an application of this result. Suppose now that rank_{𝔒}E(F) = 1. Then rank_{𝔒_{𝔭}}(Im(f)) = 1, and so from Proposition 6.5 implies that the group U is finite. It therefore follows from Corollary 6.4 (together with the non-degeneracy of the pairing {., .}_{F,𝔭}) that rank_{𝔒_{𝔭^{∗}}} (Ker(Ψ_{M})) = 1, which in turn implies that the image of Ψ_{M} is finite. Hence we have

Theorem 6.6. Suppose that rank_{𝔒}E(F) = 1. Then Im(Ψ_{M}) is finite, and so if Q ∈ |Im(Ψ_{M})|.E(F) is a point of infinite order, then φ_{i}(Q) = 0 for all i ≥ 1. Hence \overline{ℭ_{Q}(i)} is a globally free M_{i}-module for all i ≥ 1.

We remark that Theorem 6.6 does not hold in the analogous situation over global function fields (see [A]).

Let L/F be any finite extension. It is shown in Theorem 2 of [ST] that if E(F) is finite and P ∈ E(L), then ℭ_{P} (i; L) ≃ 𝔄_{i}(L) as 𝔄_{i}(F)-modules for all i. We have a similar result in our situation if rank_{𝔒}E(F) = 1.

Theorem 6.7. Suppose that rank_{𝔒}E(F) = 1, and let P ∈ |Im(Ψ_{M,F})|.E(L). Then there is an isomorphism

\overline{ℭ_{P} (i; L)(L)} ≃ M_{i}(L)  (6.8)

of M_{i}(F)-modules for all i ≥ 1.

Proof. This follows immediately from Theorem 6.6 together with the trace-restriction diagram (1.9).

§7. Galois action on Ker(Ψ_{M})

We shall now give a proof of Theorem 0.1. Hence, in this section we assume that K has classnumber 1 and that E is defined over K.

Write E_{1,𝔭}(F) = ∏_{𝔓|𝔭} E_{1}(F_{𝔓}), and recall that we have a natural homomorphism f : E_{1}(F) ⊗_{𝔒} 𝔒_{𝔭} → E_{1,𝔭}(F). It is clear that f is a Δ-homomorphism. For each character χ ∈ Δ̂, set r_{χ} = rank_{𝔒_{1}}(E(F) ⊗_{𝔒} 𝔒_{1})^{χ}. Suppose that χ_{1} ∈ Δ̂ with r_{χ_{1}} ≥ 1. Let {χ_{1}, …, χ_{m}} be the set of all characters in Δ̂ lying in the Ω_{K}-orbit of χ_{1}. Then r_{χ_{j}} = r_{χ_{1}} ≥ 1 for all 1 ≤ j ≤ m, and so it follows that E(F) ∩ (⊕_{j=1}^{m}(E_{1}(F) ⊗_{𝔒} 𝔒_{1})^{χ_{j}}) contains a point Q of infinite order. Since f is a Δhomomorphism and f(Q) is of infinite order, we deduce that ∏_{j=1}^{m}(E_{1,𝔭}(F)⊗_{𝔒}𝔒_{1})^{χ_{j}} has 𝔒_{1}-rank at least 1. But this implies that

s_{χ_{j}} = rank_{𝔒_{1}}(E_{1,𝔭}(F) ⊗_{𝔒} 𝔒_{1})^{χ_{j}} ≥ 1  (7.1)

for all 1 ≤ j ≤ m, since s_{χ_{j}} = s_{χ_{1}} for all such j.

However in fact s_{χ} = 1 for all χ ∈ Δ̂. This latter assertion holds because E_{1,𝔭}(F) contains a subgroup of finite index which is isomorphic to ∏_{𝔓|𝔭} 𝔒_{F_{𝔓}} (see e.g. [Si] Chapter VII, Proposition 6.3). It follows from the above remarks that we have shown

rank_{𝔒_{1}}(Ker(f) ⊗_{𝔒} 𝔒_{1})^{χ} = rank_{𝔒_{1}}(U ⊗_{𝔒} 𝔒_{1})^{χ}
= r_{χ} − 1

(7.2)

for all χ ∈ Δ̂ with r_{χ} ≥ 1.

Now since we have assumed that E is defined over K, it may be shown that the pairing {., .}_{F,𝔭} is Ω_{K}-equivariant. Hence (7.1) implies that if r_{χ} ≥ 1, then

rank_{𝔒_{2}}(U^{⊥} ⊗_{𝔒} 𝔒_{2})\overline{χ} = 1.  (7.3)

But U^{⊥} ∼ Ker(Ψ_{M}) (Corollary 6.4), and now the result follows.

We remark that the proof of Theorem 0.1 in fact shows that if rank_{𝔒_{2}}[(E(F)⊗_{𝔒} 𝔒_{2})^{χ}] ≥ 1, then rank_{𝔒_{2}}[(Ker(Ψ_{M}) ⊗_{𝔒} 𝔒_{2})^{χ}] = 1.

Corollary 7.1. Suppose that rank_{𝔒_{2}}[(E(F) ⊗_{𝔒} 𝔒_{2})^{χ}] = 1 for some χ ∈ Δ̂. Then there is a point Q ∈ E(F) of infinite order with Ψ_{M}(Q) = 0.

Proof. Let {χ = χ_{1}, …, χ_{m}} be the set of all characters lying in the Ω_{K} orbit of χ. Then

rank_{𝔒_{2}}[(E(F) ⊗_{𝔒} 𝔒_{2})^{χ_{j}}] = 1 1 ≤ j ≤ m.

Hence E(F) ∩ ⊕_{j=1}^{m}(E(F) ⊗_{𝔒} 𝔒_{2})^{χ_{j}} contains a point of infinite order.

Theorem 0.1 implies that

[⊕_{j=1}^{m}(E(F) ⊗_{𝔒} 𝔒_{2})^{χ_{j}} : ⊕_{j=1}^{m}(Ker(Ψ_{M}) ⊗_{𝔒} 𝔒_{2})^{χ_{j}}] < ∞.

Therefore E(F) ∩ ⊕_{j=1}^{m}(Ker(Ψ_{M}) ⊗_{𝔒} 𝔒_{2})^{χ_{j}} also contains a point of infinite order, which proves the result.

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