2 Preliminaries

Suppose $p \geq 5$ and G is a pro-p, compact open subgroup of $GL_2({\mathbb Z}_p)$ with center C. Let ${\mathrm {PG}}$ be the quotient $G/C$ . Then G and ${\mathrm {PG}}$ are p-torsion-free, and hence their Iwasawa algebras are Auslander regular local rings (cf. [Reference VenjakobV, Th. 3.26]) and so we have a dimension theory. Furthermore, the usual notion of rank of a module over these Iwasawa algebras (cf. [Reference VenjakobV, Def. 1.2]) coincides with the homological rank (see [Reference HowsonHo2]). The module is torsion if and only if its rank over the corresponding Iwasawa algebra is zero. Finally, note that the p-cohomological dimensions of G and ${\mathrm {PG}}$ are $4$ and $3$ , respectively.

Now, suppose G is not necessarily pro-p (still assuming $p>3$ ). Let $G^{\prime }$ be a compact open pro-p subgroup of G. Both $\Lambda (G)$ and $\Lambda (G^{\prime })$ are Auslander regular integral domains. (This result does not need the pro-p assumption (see [Reference VenjakobV, Th. 3.36]).) Suppose M is a finitely generated $\Lambda (G)$ -module. Then M is $\Lambda (G)$ -torsion if and only if

$$ \begin{align*} M^+:={\mathrm E}^0_{\Lambda(G)}(M)=\mathrm{Hom}_{\Lambda(G)}(M, \Lambda(G)) =0. \end{align*} $$

Given a finitely generated $\Lambda (G)$ -module M, there is an exact sequence

$$ \begin{align*} 0 \rightarrow {\mathrm E}^1_{\Lambda(G)}DM \rightarrow M \rightarrow (M^+)^+ \rightarrow {\mathrm E}^2_{\Lambda(G)}DM \rightarrow 0 \end{align*} $$

(see [Reference VenjakobV, Prop. 2.5]). This submodule ${\mathrm E}^1_{\Lambda (G)}DM$ is the $\Lambda (G)$ -torsion submodule of M (see [Reference VenjakobV, Def. 2.6]). Hence, the module M is said to be $\Lambda (G)$ -torsion if and only if

$$ \begin{align*} {\mathrm E}^1_{\Lambda(G)}DM=M. \end{align*} $$

It is easy to see that if $G^{\prime }$ is an open pro-p subgroup, then there are natural isomorphisms

$$ \begin{align*} {\mathrm E}^0_{\Lambda(G)}(M)\cong {\mathrm E}^0_{\Lambda(G^{\prime})}(M) \quad \text{and} \quad {\mathrm E}^1_{\Lambda(G)}DM \cong {\mathrm E}^1_{\Lambda(G^{\prime})}DM \end{align*} $$

(see [Reference VenjakobV, Prop. 2.7(ii)] and the discussion after Definition 2.6 of [Reference VenjakobV]). Consequently, M is torsion as a $\Lambda (G)$ -module if and only if M is torsion as a $\Lambda (G^{\prime })$ -module. Thus, in order to show that M is a torsion $\Lambda (G)$ -module, it suffices to show that the homological rank of M as a $\Lambda (G^{\prime })$ -module is zero. Since any compact p-adic analytic group contains an open characteristic subgroup which is uniform and extrapowerful pro-p group, one may study modules over $\Lambda (G^{\prime })$ by restriction of scalars (see [Reference VenjakobV, Rem. 3.23]).

Let E be an elliptic curve over a number field F without complex multiplication, and let $p\geq 5$ . Let us suppose that E has good ordinary reduction at the primes of F above p. Let S be a finite set of primes of F including those above $\{p,\infty \} $ and the primes where E has bad reduction. Suppose $F_S$ is the maximal extension of F unramified outside S. Assume that $H_{\infty }$ is an infinite Galois extension of F, contained in $F_S$ , whose Galois group ${\mathrm {Gal}}(H_{\infty }/F)$ is a pro-p, compact, p-torsion-free, p-adic Lie group of positive dimension.

For such a p-adic Lie extension $H_{\infty }$ of F, we can define the Selmer group of E over $H_{\infty }$ as a kernel of a natural global to local cohomological map defined by the following sequence:

(2.1) $$ \begin{align} 0 \rightarrow {\mathrm{Sel}}(E/H_{\infty}) \rightarrow H^1(F_S/H_{\infty}, E_{p^{\infty}}) \xrightarrow{\lambda_{H_{\infty}}} \oplus_{v \in S}J_v(H_{\infty}). \end{align} $$

Here, $J_v(H_{\infty })$ ’s are local cohomology groups defined as follows (cf. [Reference Coates and HowsonCo1, §3.1]). Let $H_{\infty }$ be the union of an increasing tower of finite extensions $H_n$ of F. Let $v \in S$ , and for each n, denote by $H_{n,\omega _n}$ the completion of $H_n$ with respect to a prime $\omega _n$ of $H_n$ such that $\omega _n$ divides v and such that $\omega _n$ form a compatible sequence of primes $\omega _{n+1}\mid \omega _n$ . Then

$$ \begin{align*} J_v(H_{\infty}):=\varinjlim_{n \rightarrow \infty}\oplus_{\omega_n \mid v}H^1(H_{n,\omega_n},E)(p), \end{align*} $$

where the limit is taken with respect to the restriction maps.

The Selmer group encodes several p-adic arithmetic information about the elliptic curve. It follows immediately from Kummer theory of E over $H_{\infty }$ that we have the following exact sequence:

where

is the Tate–Shafarevich group (cf. [Reference Coates and SujathaCS1, p. 15]).

It is easy to see that the dual Selmer group is a finitely generated module over the Iwasawa algebra

$$ \begin{align*} \Lambda(\Omega)= \varprojlim_W{\mathbb Z}_p[\Omega/W], \end{align*} $$

where W runs over all open normal subgroups of $\Omega ={\mathrm {Gal}}(H_{\infty }/F)$ . In the rest of the text, $H_{\infty }$ is either the cyclotomic ${\mathbb Z}_p$ -extension of F, or a pro-p noncommutative $GL(2)$ extension of F, or a pro-p noncommutative $PGL(2)$ extension of F (see below). Since we are assuming $p\geq 5$ , the corresponding Galois groups are p-torsion-free, pro-p, compact p-adic Lie groups. It is natural to study the structure of this Selmer group. In the classical case, when the group $\Omega ={\mathrm {Gal}}(F_{\mathrm {cyc}}/F)\cong {\mathbb Z}_p$ is commutative, the Iwasawa algebra $\Lambda (\Omega )$ is a commutative local ring. One may then use the well-known structure theorem of finitely generated modules over $\Lambda ({\mathrm {Gal}}(F_{\mathrm {cyc}}/F))$ in this setting [Reference Backhausz and ZábrádiBo, Chap. VII, §4]. In the noncommutative cases, when $\Omega $ is a compact pro-p, p-adic analytic group without p-torsion, the Iwasawa algebra $\Lambda (\Omega )$ is an Auslander regular local ring [Reference VenjakobV, Th. 3.26], and hence there is a dimension theory for modules in this setting. Suppose M is a finitely generated module over a noncommutative Iwasawa algebra $\Lambda (\Omega )$ . Then M is a torsion $\Lambda (\Omega )$ -module if $\dim M \leq \dim \Lambda (\Omega ) - 1 $ . The module M is pseudonull if $\dim M \leq \dim \Lambda (\Omega ) - 2$ (see [Reference VenjakobV, §3]). There is also a weak structure theorem [Reference CremonaCSS].

Set

$$ \begin{align*} F_n=F(E_{p^{n+1}}), \hspace{.5in} F_{\infty} = F(E_{p^{\infty}}), \end{align*} $$

and write

$$ \begin{align*} G_n = {\mathrm{Gal}}(F_{\infty}/F_n), \hspace{.5in} G={\mathrm{Gal}}(F_{\infty}/F). \end{align*} $$

By a well-known result of Serre [Reference SerreS], G is open in $GL_2({\mathbb Z}_p)$ for all primes and $G=GL_2({\mathbb Z}_p)$ for all but a finite number of primes. Note that $F_{\infty }$ contains $F_{\mathrm {cyc}}$ , the cyclotomic ${\mathbb Z}_p$ -extension over F with Galois group $\Gamma $ , a p-adic Lie group of dimension $1$ .

A deep conjecture of [Reference MazurMa] asserts the following conjecture.

This conjecture is known to hold in some cases (e.g., when $F={\mathbb Q}$ ), thanks to a celebrated result of Kato [Reference KatoK]. Coates and Howson in [Reference Coates, Fukaya, Kato, Sujatha and VenjakobCH1], [Reference Coates and GreenbergCH2], developed Iwasawa theory over the $GL(2)$ extension $F_{\infty }$ and provided conditions under which the Selmer group ${\mathrm {Sel}}(E/F_{\infty })$ is cotorsion as a module over the noncommutative Iwasawa algebra $\Lambda (G).$ In particular, they showed the following results (cf. Lemmas 4.7 and 4.8, Proposition 4.3, and Theorem 4.5 of [Reference Coates and HowsonCo1]).

Coates and Howson have also calculated an explicit formula for the Euler characteristic $\chi (G,{\mathrm {Sel}}(E/F_{\infty })$ (cf. [Reference Coates and GreenbergCH2, Th. 1.1]) under the assumption that ${\mathrm {Sel}}(E/F_{\infty })$ is cotorsion as a $\Lambda (G)$ -module. Their point of view was to understand how Iwasawa theory over the cyclotomic extension $F_{\mathrm {cyc}}$ influences the Iwasawa theory when one climbs up the tower to $F_{\infty }$ .

Let N be a module endowed with the discrete topology and a continuous action of a profinite group G. Let H be a closed normal subgroup of G. Recall the Hochschild–Serre spectral sequence

(2.2) $$ \begin{align} H^r(G/H,H^s(H,N)) \implies H^{r+s}(G,N). \end{align} $$

We shall make repeated use of this sequence with $H=C$ , the center of G.

3 Descent from $GL(2)$ Iwasawa theory to $PGL(2)$ Iwasawa theory

Let us make the following assumptions throughout the rest of this article. Assume that G is a pro-p group. Since $p \geq 5$ , we note that the compact p-adic pro-p Lie groups G, C, and ${\mathrm {PG}}$ are all p-torsion-free and hence have p-cohomological dimensions $4, 1$ , and $3$ , respectively. Hence, their Iwasawa algebras are all Auslander regular local rings by [Reference VenjakobV, Th. 3.26].

From the Hochschild–Serre spectral sequence (see (2.2)) with $G={\mathrm {Gal}}(F_{\infty }/F)$ , $H=C$ , and $G/H={\mathrm {PG}}$ , it is easy to see that we obtain the commutative diagram (3.1) with exact rows.

(3.1)

The vertical maps are given by restriction maps. Our first objective is to prove the following theorem.

Theorem 3.1. The vertical maps $\alpha , \beta , \gamma $ in the fundamental diagram (3.1) are all isomorphisms. In particular,

$$ \begin{align*}{\mathrm{Sel}}(E/K_{\infty}) \cong {\mathrm{Sel}}(E/F_{\infty})^C\end{align*} $$

as $\Lambda ({\mathrm {PG}})$ -modules.

Proof. From the Hochschild–Serre spectral sequence, we have the following exact sequence:

$$ \begin{align*}0 \rightarrow H^1(C,E_{p^{\infty}}) \rightarrow H^1(F_S/K_{\infty},E_{p^{\infty}}) \xrightarrow{\beta} H^1(F_S/F_{\infty}, E_{p^{\infty}})^C \rightarrow H^2(C,E_{p^{\infty}}).\end{align*} $$

Since the p-cohomological dimension of C is $1$ , $H^2(C,E_{p^{\infty }})=0$ , and hence the cokernel of the map $\beta $ is zero. By applying Shapiro’s lemma (cf. the proof of [Reference Coates and HowsonCo1, Lem. 3.1]), we have the following exact sequence:

(3.2) $$ \begin{align} 0 \rightarrow H^1(\Theta_{w}^C, E(F_{\infty, w}))(p) \rightarrow J_v(K_{\infty}) \xrightarrow{\gamma_v} \big(J_v(F_{\infty})\big)^C \rightarrow H^2(\Theta_{w}^C, E(F_{\infty, w}))(p), \end{align} $$

where w is a prime of $F_{\infty }$ above v and $\Theta _{w}^C$ is the decomposition group of C at w (see also [Reference Coates and GreenbergCH2, (19), (133), and (134)] for the corresponding exact sequence when $K_{\infty }$ is replaced by $F_{\mathrm {cyc}}$ ). As $\Theta _{w}^C$ is a closed subgroup of C which is of dimension $1$ as a p-adic Lie group, $H^2(\Theta _{w}^C, E(F_{\infty , w}))(p)=0,$ which proves that the cokernel of $\gamma _v$ is zero.

The following argument shows that $\ker (\beta )= H^1(C, E_{p^{\infty }})$ is zero. The congruence subgroups of $GL_2({\mathbb Z}_p)$ form a base of neighborhood for its topology, and thus C must contain a scalar matrix

$$\begin{align*}x= \left(\!\! {\begin{array}{cc} 1+p^n & 0 \\ 0 & 1+p^n \\ \end{array} } \!\!\right) \end{align*}$$

for n sufficiently large. However, x lies in C, so $x-1$ annihilates $H^1(C,E_{p^{\infty }})$ (see [Reference MilneMi, Chap. I, Lem. 6.21]). Therefore, $p^nH^1(C,E_{p^{\infty }})=0.$ Now, consider the short exact sequence

(3.3) $$ \begin{align} 0 \rightarrow E_{p^n} \rightarrow E_{p^{\infty}} \xrightarrow{\times p^n} E_{p^{\infty}} \rightarrow 0. \end{align} $$

As $p^nH^1(C,E_{p^{\infty }})=0$ , the long exact sequence corresponding to (3.3) gives rise to the following short exact sequence:

(3.4) $$ \begin{align} 0 \rightarrow H^1(C, E_{p^{\infty}}) \rightarrow H^2(C,E_{p^n}) \rightarrow H^2(C,E_{p^{\infty}}) \rightarrow 0. \end{align} $$

Now, C has p-cohomological dimension $1$ , and hence $H^2(C,E_{p^n})=0$ . Therefore, $H^1(C, E_{p^{\infty }})=0$ by (3.4). This shows that the map $\beta $ is injective.

Writing $\gamma =\oplus _{v \in S} \gamma _v$ , (3.2) gives

(3.5) $$ \begin{align} {\mathrm{Ker}}(\gamma_v)=H^1(\Theta_{w}^C, E(F_{\infty, w}))(p), \end{align} $$

where w is a prime of $F_{\infty }$ above v and $\Theta _{w}^C$ is the decomposition group of C at w. In the following, it is shown that

(3.6) $$ \begin{align} H^1(\Theta_w^C,E_{p^{\infty}})=0. \end{align} $$

The decomposition subgroup $\Theta _w^C$ is a subgroup of C which is pro-p. Therefore, $\Theta _w^C$ is a subgroup of $1+p{\mathbb Z}_p$ and hence a procyclic group. Choose $\sigma $ to be a topological generator of $\Theta _w^C$ ; for instance, $\sigma =1+p^n$ for some n. By [Reference Neukirch, Schmidt and WingbergNS, Prop. 1.7.7],

$$ \begin{align*} H^1(\Theta_w^C,E_{p^{\infty}})=E_{p^{\infty}}/(\sigma-1)E_{p^{\infty}}=E_{p^{\infty}}/p^nE_{p^{\infty}}. \end{align*} $$

As $E_{p^{\infty }}$ is a p-divisible group, $E_{p^{\infty }}/p^nE_{p^{\infty }}=0$ , giving us $H^1(\Theta _w^C,E_{p^{\infty }})=0$ .

If $v \nmid p$ , by Kummer theory [Reference GreenbergG, §2], $H^1(\Theta _{w}^C, E(F_{\infty , w}))(p)=H^1(\Theta _w^C,E_{p^{\infty }})$ , which vanishes by (3.6). This proves that ${\mathrm {Ker}}(\gamma _v)=0$ when $v \nmid p$ .

Now, suppose that $v \mid p$ . Let u be the prime of $K_{\infty }$ such that $w\mid u$ and $u\mid v$ , and let $I_u$ be the inertia subgroup of $K_{\infty }$ over F at the prime u. Since $I_u$ is infinite, $K_{\infty , u}$ is deeply ramified (see [Reference Coates, Fieker and KohelCG]). It is also well known that $F_{\infty ,w}$ is deeply ramified.

Hence, [Reference Coates, Fieker and KohelCG, Prop. 4.8 and Th. 2.13] gives

(3.7) $$ \begin{align} H^1(K_{\infty,u},D) &\cong H^1(K_{\infty,u},E)(p), \end{align} $$

(3.8) $$ \begin{align} H^1(F_{\infty,w},D) &\cong H^1(F_{\infty,w},E)(p), \end{align} $$

where D can be identified with $\tilde {E}_{v,p^{\infty }}$ , the p-primary subgroup of the reduction of E modulo v. Note that E has good ordinary reduction at primes of F above p. By [Reference Coates, Fieker and KohelCG], the module D also satisfies the following exact sequence:

(3.9) $$ \begin{align} 0 \rightarrow C^{\prime} \rightarrow E_{p^{\infty}} \rightarrow D \rightarrow 0, \end{align} $$

where $C^{\prime }$ is divisible and D is the maximal quotient of $E_{p^{\infty }}$ by a divisible subgroup such that $I_v$ acts on D via a finite quotient. Note that the following sequence is exact:

$$ \begin{align*} 0 \rightarrow H^1(\Theta_w^C,D) \rightarrow H^1(K_{\infty,u},D) \rightarrow H^1(F_{\infty,w},D)^{\Theta_w^C}. \end{align*} $$

Hence, using (3.7) and (3.8) into the above exact sequence, it is easy to see that

$$ \begin{align*} \ker(\gamma_v) =H^1(\Theta_w^C,E(F_{\infty,w}))(p) \cong H^1(\Theta_w^C,D). \end{align*} $$

The exact sequence (3.9) gives that the following sequence

$$ \begin{align*} H^1(\Theta_w^C,E_{p^{\infty}}) \rightarrow H^1(\Theta_w^C,D) \rightarrow H^2(\Theta_w^C,C) \end{align*} $$

is exact. By (3.6), we know that $H^1(\Theta _w^C,E_{p^{\infty }}) =0$ , and since $\Theta _w^C$ is a subgroup of C which has p-cohomological dimension $1$ , we have $H^2(\Theta _w^C,C)=0$ . Therefore, it is clear that $H^1(\Theta _w^C,D)=0, $ which shows that $\ker (\gamma _v)=0$ for $v \mid p$ .

Thus, the maps $\beta $ and $\gamma $ are isomorphisms. The theorem now follows from the snake lemma applied to the fundamental diagram (3.1).

Proof. The assertion follows from the natural injection

$$ \begin{align*} H^1\Big(PG, \big({\mathrm{Sel}}(E/F_{\infty})\big)^C\Big)\hookrightarrow H^1(G, {\mathrm{Sel}}(E/F_{\infty})) \end{align*} $$

and Theorem 3.1.

The reader is referred to Theorem 2.3 in §2 for cases where $H^1(G,{\mathrm {Sel}}(E/F_{\infty }))$ is known to be finite.

3.3 Conditions when the Selmer over ${\mathrm {PGL}}(2)$ extension is cotorsion

The following theorems give several conditions when $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is torsion as a $\Lambda ({\mathrm {PG}})$ -module.

Theorem 3.4. Assume weak Leopoldt’s conjecture at $K_{\infty }$ , that is, $H^2(F_S/K_{\infty }, E_{p^{\infty }})=0$ . Then the dual Selmer group $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is $\Lambda ({\mathrm {PG}})$ -torsion if and only if the map $\lambda _{K_{\infty }}$ in (3.1) is surjective.

Proof. The proof follows from [Reference Shekhar and SujathaSS, Th. 7.2].

Theorem 3.5. The dual Selmer group $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is a torsion $\Lambda ({\mathrm {PG}})$ -module if any of the following conditions hold:

1. 1. The Selmer group ${\mathrm {Sel}}(E/F)$ is finite and $H_2({\mathrm {PG}}, \widehat {{\mathrm {Sel}}(E/K_{\infty })})$ is finite.

2. 2. The Selmer group ${\mathrm {Sel}}(E/F)$ is finite, $\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ is a torsion $\Lambda (G)$ -module, and $H_0({\mathrm {PG}}, H_1(C, \widehat {{\mathrm {Sel}}(E/F_{\infty })})$ is finite.

3. 3. $\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ is a torsion $\Lambda (G)$ -module and $H_1(C, \widehat {{\mathrm {Sel}}(E/F_{\infty })})=0$ .

Conversely, if $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is a torsion $\Lambda ({\mathrm {PG}})$ -module, then $(3)$ holds.

Proof. Let $M=\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ . We have $M_G \cong H_0({\mathrm {PG}},M_C)$ and $M_C \cong \widehat {{\mathrm {Sel}}(E/K_{\infty })}$ . Now, since ${\mathrm {Sel}}(E/F)$ is finite, by Theorem 2.3, it follows that $H_0({\mathrm {PG}}, M_C)$ is finite. By assumption, $H_2({\mathrm {PG}}, M_C)$ is also finite. Since

$$ \begin{align*} {\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M_C &=\sum_{k \geq 0}^{3}(-1)^k{\mathrm{rank}}_{{\mathbb Z}_p}H_k({\mathrm{PG}},M_C) \end{align*} $$

(see [Reference HowsonHo2, Th. 1.1]), we obtain ${\mathrm {rank}}_{\Lambda ({\mathrm {PG}})}M_C=0$ and hence $M_C$ is $\Lambda ({\mathrm {PG}})$ -torsion. This shows that $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is a torsion as a $\Lambda ({\mathrm {PG}})$ -module and hence proves that the condition (1) is sufficient.

To show that the condition in (2) is also sufficient, note that if M is $\Lambda (G)$ -torsion and ${\mathrm {Sel}}(E/F)$ is finite, then $H_1(G,M)$ is finite and $H_2(G,M)=0$ (see Theorem 2.3). By Hochschild–Serre spectral sequence, we conclude that $H_0({\mathrm {PG}}, H_1(C, M))$ is finite if and only if $H_2({\mathrm {PG}},M_C)$ is finite, and hence (2) follows from (1).

For (3), note that it follows from [Reference HowsonHo2, Th. 1.1] the Hochschild–Serre spectral sequence (use (2.2) with $H=C$ and $N=H_l(C,M)$ ) that

(3.10) $$ \begin{align} {\mathrm{rank}}_{\Lambda(G)}M &=\sum_{k \geq 0}(-1)^k{\mathrm{rank}}_{{\mathbb Z}_p}H_k(G,M) \hspace{9pc} \end{align} $$

(3.11) $$ \begin{align} &=\sum_{k,l \geq 0}(-1)^{k+l}{\mathrm{rank}}_{{\mathbb Z}_p}H_k({\mathrm{PG}}, H_l(C,M)) \end{align} $$

(3.12) $$ \begin{align} &=\sum_{l\geq 0}(-1)^l{\mathrm{rank}}_{\Lambda({\mathrm{PG}})}H_l(C,M) \end{align} $$

(3.13) $$ \begin{align} \hspace{4pc} ={\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M_C - {\mathrm{rank}}_{\Lambda({\mathrm{PG}})}H_1(C,M). \end{align} $$

Suppose (3) holds. Since $H_1(C,M)$ is precisely the $\Lambda (G)$ -submodule of M consisting of the elements in M annihilated by the augmentation ideal $I(C)=\langle c-1 \rangle $ , we have $H_1(C,M)=0$ . Hence, the conclusion follows from (3.13) and Theorem 3.1.

Finally, suppose if $M_C \cong \widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is a torsion $\Lambda ({\mathrm {PG}})$ -module. Then it follows from (3.13) that M is torsion as a $\Lambda (G)$ -module and $H_1(C,M)$ is torsion as a $\Lambda ({\mathrm {PG}})$ -module. Then $H_1(C,M)$ is pseudonull as a $\Lambda (G)$ -module. However, M has no nonzero pseudonull submodules (see [Reference Ochi and VenjakobOV, Th. 5.1]), and therefore $H_1(C,M)=0$ .

The following theorem gives a restatement of condition (3) in Theorem 3.5 using the structure theorem of dual Selmer groups over noncommutative Iwasawa algebras [Reference CremonaCSS]. By [Reference CremonaCSS], there is an injection of $\Lambda (G)$ -modules

$$ \begin{align*} \oplus_{i=1}^m\Lambda(G)/J_i \hookrightarrow M/M_0 \end{align*} $$

with pseudonull cokernel. Here, $J_i$ ’s are reflexive ideals in $\Lambda (G)$ which are pure of grade $1$ , and $M_0$ is the maximal pseudonull submodule of M. Recall that $M=\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ ; hence, M has no nontrivial pseudonull submodule and therefore $M_0=0.$ This gives the exact sequence

(3.14) $$ \begin{align} 0 \rightarrow \oplus_{i=1}^m\Lambda(G)/J_i \rightarrow M \rightarrow N \rightarrow 0, \end{align} $$

where N is a pseudonull $\Lambda (G)$ -module.

Theorem 3.6. The Selmer group ${\mathrm {Sel}}(E/K_{\infty })$ is a cotorsion $\Lambda ({\mathrm {PG}})$ -module if and only if ${\mathrm {Sel}}(E/F_{\infty })$ is a cotorsion $\Lambda (G)$ -module and $H_1(C,\oplus _{i=1}^m\Lambda (G)/J_i)=0$ .

Proof. Since the center C has p-cohomological dimension $1$ , the sequence (3.14) gives the following exact sequence of $\Lambda ({\mathrm {PG}})$ -modules:

(3.15) $$ \begin{align} 0\rightarrow H_1(C, \oplus_{i=1}^m \Lambda(G)/J_i) &\rightarrow H_1(C,M) \rightarrow H_1(C,N) \end{align} $$

(3.16) $$ \begin{align} &\rightarrow (\oplus_{i=1}^m\Lambda(G)/J_i)_C \rightarrow M_C \rightarrow N_C \rightarrow 0. \end{align} $$

Let $\overline {J_i}$ be the image of $J_i$ under the natural projection $\Lambda (G) \rightarrow \Lambda (G/C)=\Lambda (G)/I(C)$ . Since $H_1(C,\Lambda (G)/J_i)=0$ for each $i=[1,m]$ by assumption, then $J_i \nsubseteq I(C)$ and hence $\overline {J_i}$ is nonzero. Therefore, we have

(3.17) $$ \begin{align} \dim_{\Lambda({\mathrm{PG}})}(\Lambda(G)/J_i)_C =\dim_{\Lambda({\mathrm{PG}})}(\Lambda(G/C)/\overline{J_i}) <\dim\Lambda({\mathrm{PG}})=4, \end{align} $$

which implies that $(\Lambda (G)/J_i)_C $ is $\Lambda ({\mathrm {PG}})$ -torsion. Now, as N is a pseudonull $\Lambda (G)$ -module,

$$ \begin{align*}\dim_{\Lambda(G)}N \leq \dim\Lambda(G)-2=\dim\Lambda({\mathrm{PG}})-1.\end{align*} $$

This implies that $N_C$ is a torsion $\Lambda ({\mathrm {PG}})$ -module, and the same holds for the module $M_C$ from (3.17) and (3.16).

Conversely, suppose that $M_C \cong \widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is torsion as a $\Lambda ({\mathrm {PG}})$ -module. Then, by Theorem 3.5, M is torsion as a $\Lambda (G)$ -module and $H_1(C, M)=0$ whence $H_1(C, \oplus _{i=1}^m \Lambda (G)/J_i)=0$ by (3.15).

Remark 3.7. We note that the ideals $J_i$ cannot be $I(C)$ for all $ i\geq 1$ because the center C cannot act trivially on the Selmer group at $F_{\infty }$ (see Proposition 3.8).

Suppose none of the ideals $J_i$ are contained in $I(C)$ . Then $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is torsion as a $\Lambda ({\mathrm {PG}})$ -module. On the other hand, if $J_i=I(C)$ for some i, then $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ cannot be torsion as a $\Lambda ({\mathrm {PG}})$ -module (see Theorem 3.6).

Suppose ${\mathrm {Sel}}(E/F_{\infty })$ is cotorsion as a $\Lambda (G)$ -module, and ${\mathrm {Sel}}(E/F)$ and $H_1(G,N)$ are finite. Then none of the ideals $J_i$ can be contained in $I(C)$ . This is because, from (3.14), we obtain the exact sequence

$$ \begin{align*} H_1(G,N) \rightarrow (\oplus_{i=1}^m\Lambda(G)/J_i)_G \rightarrow M_G, \end{align*} $$

whose first and last terms are finite. In this case, $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ is torsion as a $\Lambda ({\mathrm {PG}})$ -module.

In [Reference CremonaCSS, Prop. 8.10], for the elliptic curve $E=X_1(11): y^2+y=x^3-x^2$ of conductor $11$ and prime $p=5$ , the authors show that the center C cannot act trivially on ${\mathrm {Sel}}(E/F_{\infty })$ . The following proposition generalizes this for any elliptic curve without complex multiplication and with good ordinary reduction for the primes above p.

Proposition 3.8. Suppose $G \cong C \times {\mathrm {PG}}$ , and ${\mathrm {Sel}}(E/F)$ is finite. Then the center C cannot act trivially on ${\mathrm {Sel}}(E/F_{\infty })$ .

Proof. If C acts trivially on ${\mathrm {Sel}}(E/F_{\infty })$ , then

$$ \begin{align*} {\mathrm{Sel}}(E/F_{\infty}) ={\mathrm{Sel}}(E/F_{\infty})^C \cong {\mathrm{Sel}}(E/K_{\infty}). \end{align*} $$

Hoewever, ${\mathrm {Sel}}(E/K_{\infty })$ is cofinitely generated over $\Lambda ({\mathrm {PG}})$ and so with the above identification, ${\mathrm {Sel}}(E/F_{\infty })$ is cotorsion over $\Lambda (G)$ . As ${\mathrm {Sel}}(E/F_{\infty })$ is $\Lambda (G)$ -cotorsion and ${\mathrm {Sel}}(E/F)$ is finite, the cohomology groups $H^i(G,{\mathrm {Sel}}(E/F_{\infty }))$ are finite for all i. The degeneration of the Hochschild–Serre spectral sequence (see [Reference Neukirch, Schmidt and WingbergNS, Prop. 2.4.5]) gives an injection

$$ \begin{align*}H^i({\mathrm{PG}},{\mathrm{Sel}}(E/K_{\infty})) \hookrightarrow H^i(G,{\mathrm{Sel}}(E/F_{\infty})).\end{align*} $$

This implies that the cohomology groups $H^i({\mathrm {PG}},{\mathrm {Sel}}(E/K_{\infty }))$ are finite for all i. Hence,

$$ \begin{align*}{\mathrm{rank}}_{\Lambda({\mathrm{PG}})}\widehat{{\mathrm{Sel}}(E/K_{\infty})}=\sum_{i \geq 0}(-1)^i{\mathrm{rank}}_{{\mathbb Z}_p}H_i({\mathrm{PG}}, \widehat{{\mathrm{Sel}}(E/K_{\infty})})=0,\end{align*} $$

whereby ${\mathrm {Sel}}(E/K_{\infty }) $ is $\Lambda ({\mathrm {PG}})$ -cotorsion. We deduce that

$$ \begin{align*}\dim_{\Lambda(G)}\widehat{{\mathrm{Sel}}(E/F_{\infty})} \leq \dim\Lambda({\mathrm{PG}}) -1 = \dim\Lambda(G)-2.\end{align*} $$

This would imply that $\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ is a pseudonull $\Lambda (G)$ -module. However, $\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ has no nonzero pseudonull submodules. Hence, $\widehat {{\mathrm {Sel}}(E/F_{\infty })}=0$ . On the other hand, it is known that $\widehat {{\mathrm {Sel}}(E/F_{\infty })}$ is infinite dimensional as a ${\mathbb Q}_p$ -vector space (see [Reference Coates and HowsonCo1, Th. 1.5]). This gives us a contradiction.

Examples 3.9. Here are some examples of elliptic curves E such that $G={\mathrm {Gal}}(F_{\infty }/F)$ is a direct product of its center C and ${\mathrm {PG}}$ . We follow the nomenclature from Cremona tables [Reference Coates, Schneider and SujathaCr].

1. 1. Let E be the elliptic curve $X_1(11)$ , namely E is the curve $y^2+y=x^3-x^2$ and prime $p=5$ . This is a curve of conductor $11$ defined over ${\mathbb Q}$ , but we consider it over $F={\mathbb Q}(\mu _5)$ . Put $F_{\infty }=F(E_{5^{\infty }})$ . Then $G={\mathrm {Gal}}(F_{\infty }/F)$ has the form $G=C \times PG$ [Reference CremonaCSS, Exam. 8.7, p. 104].

2. 2. Let E be the elliptic curve $X_0(11)$ , namely E is the curve $y^2+y=x^3-x^2-10x-20$ and $p=5$ . This is a curve of conductor $11$ defined over ${\mathbb Q}$ , but we consider it over $F={\mathbb Q}(\mu _5)$ . Put $F_{\infty }=F(E_{5^{\infty }})$ . Then the Galois group $G={\mathrm {Gal}}(F_{\infty }/F)$ is a subgroup of the first congruence kernel of $GL_2({\mathbb Z}_5)$ [Reference FisherF, (3), p. 586]. Hence, G is of the form $C \times {\mathrm {PG}}$ .

3. 3. For any general elliptic curve E over F without complex multiplication and with good ordinary reduction for the primes above p, we can always find an integer k large enough such that, over the base field $F[E_{p^k}]$ , the Galois group $G={\mathrm {Gal}}(F_{\infty }/F[E_{p^k}])$ lies inside the first congruence kernel of $GL_2({\mathbb Z}_p)$ and hence can be written in the form $C \times {\mathrm {PG}}$ . The center C can be identified with a subgroup of diagonal matrices of a certain form (see [Reference VenjakobV, Rem. 4.11]).

4 Applications

Suppose $G \cong {\mathrm {PG}} \times C$ . Let $C_n=C^{p^n}$ and $G_n={\mathrm {PG}} \times C_n$ . Note that $G_n \neq G^{p^n}$ . Let $M = \widehat {{\mathrm {Sel}}(E/F_{\infty })}$ .

Proof. As $G_n$ is of finite index in G, M is also a finitely generated module over $\Lambda (G_n)$ . As $C_n \cong {\mathbb Z}_p$ , we can identify $M^{C_n}$ with $ H_1(C_n,M)$ and then they both are finitely generated over $\Lambda ({\mathrm {PG}})$ . The Hochschild–Serre spectral sequence $H^r({\mathrm {PG}},H^s(C_n,M)) \implies H^{r+s}(G,M)$ and the argument as in (3.10)–(3.13) give

$$ \begin{align*}{\mathrm{rank}}_{\Lambda(G_n)}M={\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M_{C_n} -{\mathrm{rank}}_{\Lambda({\mathrm{PG}})}H_1(C_n,M).\end{align*} $$

Identifying $M^{C_n}$ with $ H_1(C_n,M)$ , we deduce

$$ \begin{align*} {\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M_{C_n} &= {\mathrm{rank}}_{\Lambda(G_n)}M + {\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M^{C_n}\\[3pt] &=p^n{\mathrm{rank}}_{\Lambda(G)}M + {\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M^{C_n}\\[3pt] &={\mathrm{rank}}_{\Lambda({\mathrm{PG}})}M^{C_n}. \end{align*} $$

(The second equality follows since $G_n$ is of index $p^n$ in G, and the third equality follows as M is a torsion $\Lambda (G)$ -module by assumption.) Note that $C_n$ is in the center, and hence abelian, and therefore $M^{C_n}$ is a $\Lambda (G)$ -submodule of M. However, M is a finitely generated module over $\Lambda (G)$ , and hence M is a Noetherian module and satisfies the ascending chain condition on its submodules. Hence, the chain

$$ \begin{align*} M^{C_0=C} \subset M^{C_1} \subset \cdots M^{C_n} \cdots \end{align*} $$

stabilizes and so ${\mathrm {rank}}_{\Lambda ({\mathrm {PG}})}M^{C_n}$ is a constant independent of n, for all sufficiently large n.

Consider the following fundamental diagram:

(4.1)

Let ${\mathrm {Coker}}(\lambda _{K_{\infty }}^{\mathrm {PG}})$ be the cokernel of the map $\lambda _{K_{\infty }}^{\mathrm {PG}}$ in (4.1).

Proof. Since $H^1(C, E_{p^{\infty }})=0$ , using Hochschild–Serre spectral sequence and noting that the cohomology groups

$$ \begin{align*} H^i({\mathrm{PG}}, E_{p^{\infty}}(K_{\infty}))=H^i(G,E_{p^{\infty}}) \end{align*} $$

are finite for $i\geq 1$ , it is easy to see that the kernel and the cokernel of g are finite.

Next, we decompose h into local components $h =\oplus _{v \in S}h_v$ and analyze the kernel of cokernel of the map $h_v$ .

Let w be a prime of $F_{\infty }$ above $v \in S$ , and let u be a prime of $K_{\infty }$ below w. If one denotes $\Theta _u$ (resp., $\Delta _w$ ) the decomposition group of ${\mathrm {PG}}$ at u (resp., the decomposition group of G at w), then one has an isomorphism $\Theta _u=\Delta _w/C\cap \Delta _w$ .

By Shapiro’s lemma, $H^i({\mathrm {PG}},J_v(K_{\infty })) \cong H^i(\Theta _u,H^1(K_{\infty ,u},E)(p)).$

By Hochschild–Serre spectral sequence, the following sequence is exact:

$$ \begin{align*} 0 \rightarrow H^1(\Theta_u,E(K_{\infty,u}))(p) &\rightarrow H^1(F_v,E)(p) \xrightarrow{h_v}H^1(K_{\infty,u},E)(p)^{\Theta_u}\\ &\rightarrow H^2(\Theta_u,E(K_{\infty,u}))(p). \end{align*} $$

Therefore, to prove that $\ker (h_v)$ and $\mathrm {cokernel}(h_v)$ are finite, it is sufficient to show that

1. 1. $H^1(\Theta _u,E(K_{\infty ,u}))(p)$ is finite, and

2. 2. $H^2(\Theta _u,E(K_{\infty ,u}))(p)$ is finite.

Consider the following exact sequence:

(4.2) $$ \begin{align} 0 \rightarrow H^1(\Theta_u,E(K_{\infty,u}))(p)& \rightarrow H^1(\Delta_w,E(F_{\infty,w}))(p) \rightarrow H^1(C\cap \Delta_w,E(F_{\infty,w}))(p)^{\Theta_u} \end{align} $$

(4.3) $$ \begin{align} & \rightarrow H^2(\Theta_u,E(K_{\infty,u}))(p) \rightarrow H^2(\Delta_w,E(F_{\infty,w}))(p). \end{align} $$

With $\Theta _w^C=C \cap \Delta _w$ , the third term of the above exact sequence is

$$ \begin{align*} H^1(C\cap \Delta_w,E(F_{\infty,w}))(p)^{\Theta_u}=H^1(\Theta_w^C,E(F_{\infty,w}))(p)^{\Theta_u}={\mathrm{Ker}}(\gamma_v)^{\Theta_u}. \end{align*} $$

The last equality is due to (3.5), which is zero by Theorem 3.1.

Therefore, by the exact sequence (4.2), it suffices to show that

1. 1. $H^1(\Delta _w,E(F_{\infty ,w}))(p)$ is finite, and

2. 2. $H^2(\Delta _w,E(F_{\infty ,w}))(p)$ is finite.

However, $H^1(\Delta _w,E(F_{\infty ,w}))(p)$ and $H^2(\Delta _w,E(F_{\infty ,w}))(p)$ are the kernel and cokernel of the map $\delta _v$ , the local map of the following fundamental diagram which is known to be finite by the work of Coates and Howson (cf. proof of [Reference Coates and GreenbergCH2, Prop. 3.3] or Proposition $5.21$ of [Reference HowsonHo1]):

(4.4)

Therefore, the kernel and cokernel of h are finite. By the snake lemma, we deduce that the same is true for f.

If ${\mathrm {Sel}}(E/F)$ is finite, ${\mathrm {Coker}}(\lambda _F)$ is finite (cf. [Reference Coates and SujathaCS1, p. 35]). This implies that ${\mathrm {Coker}}(h \circ \lambda _F)$ is finite. However, ${\mathrm {Coker}}(h \circ \lambda _F)={\mathrm {Coker}}(\lambda _{K_{\infty }}^{\mathrm {PG}} \circ g)$ , and hence ${\mathrm {Coker}}(\lambda _{K_{\infty }}^{\mathrm {PG}})$ is finite.

Let $\mathcal {M}$ be the category of all finitely generated $\Lambda (G)$ -modules, let $\mathcal {C}$ be the full subcategory of all pseudonull modules in $\mathcal {M}$ , and let $q:\mathcal {M} \rightarrow \mathcal {M}/\mathcal {C}$ denote the quotient functor. From [Reference ArdakovA, §1.3], recall that an object $q(M)$ in the quotient category $\mathcal {M}/\mathcal {C}$ is said to be completely faithful if $\text {Ann}(N)=0$ for any $N \in \mathcal {M}$ such that $q(N)$ is isomorphic to a nonzero subquotient of $q(M)$ .

The Euler characteristic of the Selmer group of the $PGL(2)$ extension has been computed in Zerbes’ dissertation [Reference ZerbesZ, Chap. 8] and Howson’s dissertation [Reference HowsonHo1, Th. 5.34], but we include a simpler proof here.

Let $\chi ({\mathrm {PG}}, {\mathrm {Sel}}(E/K_{\infty }))$ be the Euler characteristic of Selmer group of the ${\mathrm {PGL}}(2)$ extension, and let

(4.5) $$ \begin{align} \xi_p(E/F)=\rho_p(E/F) \times \prod_{v \in B}(1/L_v(E,1))^{(p)} \end{align} $$

be defined as in [Reference Coates and SujathaCS1, §3.14].

Theorem 4.5. Suppose ${\mathrm {Sel}}(E/K_{\infty })$ is $\Lambda ({\mathrm {PG}})$ -cotorsion and ${\mathrm {Sel}}(E/F)$ is finite. Then ${\mathrm {Sel}}(E/K_{\infty })$ has finite Euler characteristic given by

(4.6) $$ \begin{align} \chi({\mathrm{PG}}, {\mathrm{Sel}}(E/K_{\infty}))=\chi(G,{\mathrm{Sel}}(E/F_{\infty}))=\xi_p(E/F), \end{align} $$

where $\xi _p(E/F)$ is as in (4.5).

Proof. Since ${\mathrm {Sel}}(E/K_{\infty })$ is $\Lambda ({\mathrm {PG}})$ -cotorsion, it implies that ${\mathrm {Sel}}(E/F_{\infty })$ is $\Lambda (G)$ -torsion and $H^1(C,{\mathrm {Sel}}(E/F_{\infty }))=0$ (cf. Theorems 3.5 and 3.6). The Hochschild–Serre spectral sequence

$$ \begin{align*}H^i({\mathrm{PG}},H^j(C,{\mathrm{Sel}}(E/F_{\infty}))) \implies H^{i+j}(G,{\mathrm{Sel}}(E/F_{\infty}))\end{align*} $$

gives

$$ \begin{align*}\chi(G, {\mathrm{Sel}}(E/F_{\infty}))=\sum_{i\geq 0}(-1)^i\chi({\mathrm{PG}}, H^i(C,{\mathrm{Sel}}(E/F_{\infty})))=\chi({\mathrm{PG}}, {\mathrm{Sel}}(E/K_{\infty})).\end{align*} $$

Under the assumption that ${\mathrm {Sel}}(E/F)$ is finite, $\chi (G, {\mathrm {Sel}}(E/F_{\infty }))$ equals $\xi _p(E/F)$ by [Reference Coates and SujathaCS1, Th. 3.16].

The quantity $\xi _p(E/F)$ in (4.5) is related to the exact formula of the p-part of Birch and Swinnerton-Dyer conjecture, which in turn gives the value of the p-part of the leading coefficient of the complex L-value at $1$ .

Suppose that $\widehat {{\mathrm {Sel}}(E/F_{\mathrm {cyc}})}$ is $\Lambda (\Gamma )$ -torsion (e.g., $F={\mathbb Q}$ ) and ${\mathrm {Sel}}(E/F)$ is finite. It is well known that $\chi (\Gamma , {\mathrm {Sel}}(E/F_{\mathrm {cyc}}))=\rho _p(E/F) $ (cf. [Reference Coates and SujathaCS1, Th. 3.3]). From (4.5) and (4.6), we obtain

$$ \begin{align*}\chi({\mathrm{PG}}, {\mathrm{Sel}}(E/K_{\infty}))=\chi(G,{\mathrm{Sel}}(E/F_{\infty}))=\chi(\Gamma, {\mathrm{Sel}}(E/F_{\mathrm{cyc}})) \times \prod_{v \in B}(1/L_v(E,1))^{(p)}.\end{align*} $$

Then the Iwasawa Main Conjecture predicts that the characteristic ideal of $\widehat {{\mathrm {Sel}}(E/F_{\mathrm {cyc}})}$ is generated by a p-adic L-function $f_E(T)\in \Lambda (\Gamma )$ . Since ${{\mathrm {Sel}}(E/F_{\mathrm {cyc}})}$ is $\Lambda (\Gamma )$ -cotorsion, we deduce that the leading term of $f_E(T)$ is nonzero and

$$ \begin{align*}f_E(0)\sim \chi(\Gamma, {\mathrm{Sel}}(E/F_{\mathrm{cyc}})).\end{align*} $$

Here, $a \sim b$ means that a and b both have the same p-adic valuation (cf. [Reference GreenbergG, Lem. 4.2]). This gives a conjectural connection of $f_E(T)$ with the value of the Hasse–Weil complex L-function $L(E/F, s)$ at $s=1$ . This is based on the Birch and Swinnerton-Dyer conjecture for E over F. The conjecture then asserts that $L(E/F, 1) \neq 0$ , and for a suitably defined period $\Omega (E/F)$ , the value $L(E/F, 1)/\Omega (E/F)$ is rational. One would then expect

$$ \begin{align*}f_E(0) \sim \Big(\prod_{v \mid p}(1-\beta_vN(v)^{-1})^2\Big)L(E/F, 1)/\Omega(E/F),\end{align*} $$

where $(1-\beta _vN(v)^{-1})^2$ is a certain Euler factor as in [Reference GreenbergG, p. 91].

The nonexistence of nontrivial pseudonull submodules of $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ has been dealt with in Zerbes’ dissertation [Reference ZerbesZ, Chap. 9, §3]. The referee suggested that we include a proof of this assertion. This is done in the following theorem. Our proof is different to that of [Reference ZerbesZ, Chap. 9, §3]. The proof uses a standard result from Iwasawa theory and is originally due to Greenberg for commutative Iwasawa algebras.

Proof. Let $S_p$ be the set of places of S above the prime p. As ${\mathrm {Sel}}(E/K_{\infty })$ is $\Lambda ({\mathrm {PG}})$ -cotorsion, Theorem 3.4 gives the exact sequence

(4.7) $$ \begin{align} 0\rightarrow \underset{ v \in S_p}{\oplus}\widehat{J_v(K_{\infty})} \bigoplus \underset{v \in S\backslash S_p}{\oplus} \widehat{J_v(K_{\infty})} \rightarrow \widehat{H^1(F_S/K_{\infty}, E_{p^{\infty}})} \rightarrow \widehat{{\mathrm{Sel}}(E/K_{\infty})} \rightarrow 0. \end{align} $$

By Theorem 3.1, $\widehat {J_v(K_{\infty })}=\widehat {J_v(F_{\infty })}_C $ for all $v \in S$ .

For $v \in S_p$ , the corresponding decomposition subgroup of the Galois group of the trivializing extension $F_{\infty }$ is of dimension $3$ (see [Reference Coates and GreenbergCH2, Lem. 5.1]). Hence, $\widehat {J_v(F_{\infty })}$ is free as a $\Lambda (G)$ -module (see [Reference Ochi and VenjakobOV, Lem. 5.4(i)]), whence $\widehat {J_v(K_{\infty })}$ is free as a $\Lambda ({\mathrm {PG}})$ -module.

For $v \in S\backslash S_p$ , $J_v(F_{\infty })=0$ (see [Reference Coates and GreenbergCH2, Lem. 5.4]), and hence $J_v(K_{\infty })=0$ .

Since $H^2(F_S/K_{\infty }, E_{p^{\infty }})=0$ , [Reference Ochi and VenjakobOV, Th. 4.7] allows us to conclude that $\widehat {H^1(F_S/K_{\infty }, E_{p^{\infty }})} $ has no nonzero pseudonull submodule.

Now, (4.7) gives an exact sequence where the first term is $\Lambda ({\mathrm {PG}})$ -free, and the middle term has no nonzero pseudonull submodule. An analogue of the argument by Greenberg referred to above (see [Reference Hachimori and OchiaiHO, Prop. 3.5]) allows us to conclude that $\widehat {{\mathrm {Sel}}(E/K_{\infty })}$ has no nontrivial pseudonull submodule.

It seems to us that the full strength of the structure theorem has not been exploited. Specifically, the ideals in the summands are reflexive and pure of grade $1$ . It might be possible to use this effectively to study the homology groups $H_j({\mathrm {PG}}, \Lambda (G)/J_i).$ This would then yield finer results on the structure of the dual Selmer group. We hope to return to this line of investigation later.