2.1 Uniform pro- $p$ extensions

Throughout, p will be a prime $\geq 5$ and F a number field. Let $F_\infty $ be an infinite Galois extension of F with pro-p Galois group $G:=\operatorname {Gal}(F_\infty /F)$ . The lower central p-series of G is recursively defined as follows:

$$ \begin{align*}G_0:=G\text{ and }G_{n+1}:=G_n^p[G_n, G].\end{align*} $$

Setting $d:=[G:G_1]$ , we observe that $[G:G_n]=p^{dn}$ . Assume that G has the structure of a p-adic Lie group. We say that $F_\infty $ is a strongly admissible p-adic Lie extension if

1. (1) only finitely many primes ramify in $F_\infty $ ,

2. (2) $F_\infty $ contains the cyclotomic $\mathbb {Z}_p$ -extension $F_{\operatorname {cyc}}$ of F, and

3. (3) the p-torsion subgroup of G is trivial.

We assume that $F_\infty $ is pro-p a strongly admissible p-adic Lie extension and G is uniform. Note that $G_{n}/G_{n+1}\simeq \left (\mathbb {Z}/p\mathbb {Z}\right )^d$ for $n\in \mathbb {Z}_{\geq 1}$ . It is well known that the dimension of G is equal to d and that $G_n=G^{p^n}$ , see [Reference Dixon, Du Sautoy, Mann and Segal8, Theorem 3.6]. The extension $F_\infty $ is filtered by a tower of number fields. Setting $F^{(n)}:=F_\infty ^{G_n}$ , consider the nonabelian tower

$$ \begin{align*}F=F^{(0)}\subset F^{(1)}\subset \cdots \subset F^{(n)}\subset \cdots, \end{align*} $$

and let $F^{(n)}_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of $F^{(n)}$ . We have thus filtered the extension $F_\infty $ into a tower of cyclotomic $\mathbb {Z}_p$ -extensions

$$ \begin{align*}F_{\operatorname{cyc}}=F^{(0)}_{\operatorname{cyc}}\subset F^{(1)}_{\operatorname{cyc}}\subset \cdots \subset F^{(n)}_{\operatorname{cyc}}\subset \cdots\end{align*} $$

Set $H:=\operatorname {Gal}(F_\infty /F_{\operatorname {cyc}})$ and $\Gamma :=G/H\simeq \mathbb {Z}_p$ . For $n\in \mathbb {Z}_{\geq 1}$ , we write $H_n$ (resp. $\Gamma _n$ ) for the descending central series of H (resp. $\Gamma $ ). We list a few useful facts.

As a result, we have that $F^{(n)}_{\operatorname {cyc}}=F_\infty ^{H_n}$ and hence,

$$ \begin{align*}\operatorname{Gal}(F^{(n+1)}_{\operatorname{cyc}}/F^{(n)}_{\operatorname{cyc}})=H_n/H_{n+1}\simeq \left(\mathbb{Z}/p\mathbb{Z}\right)^{d-1}.\end{align*} $$

Since $\Gamma _n=G_n/H_n$ , we have that $\Gamma _n=\operatorname {Gal}(F^{(n)}_{\operatorname {cyc}}/F^{(n)})$ . We now introduce the Iwasawa-algebra at the nth level, taken to be

$$ \begin{align*}\Lambda(\Gamma_n):=\varprojlim_L \mathbb{Z}_p[\operatorname{Gal}(L/F^{(n)})],\end{align*} $$

where L ranges over all number fields contained in between $F^{(n)}$ and $F^{(n)}_{\operatorname {cyc}}$ . Choose a topological generator $\gamma _n$ of $\Gamma _n$ and fix the isomorphism sending $\gamma _n-1$ to x.

More generally, if $\mathcal {G}$ is any pro-p group, set

$$ \begin{align*}\Lambda(\mathcal{G}):=\varprojlim_U \mathbb{Z}_p[\mathcal{G}/U],\end{align*} $$

where U ranges over all finite index normal subgroups of $\mathcal {G}$ . Given a number field F, we set $\Lambda _F:=\Lambda \left (\operatorname {Gal}(F_{\operatorname {cyc}}/F)\right )$ .

2.2 Iwasawa invariants

Let M be a cofinitely generated cotorsion

-module, i.e., the Pontryagin-dual $M^{\vee }:=\operatorname {Hom}(M, \mathbb {Q}_p/\mathbb {Z}_p)$ is a finitely generated and torsion

-module. Recall that a polynomial

is said to be distinguished if it is a monic polynomial whose nonleading coefficients are all divisible by p. Note that all height $1$ prime ideals of

are principal ideals $(a)$ , where $a=p$ or $a=f(x)$ , where $f(x)$ is an irreducible distinguished polynomial. According to the structure theorem for

-modules (see [Reference Washington27, Theorem 13.12]), $M^{\vee }$ is pseudo-isomorphic to a finite direct sum of cyclic

-modules, i.e., there is a map

with finite kernel and cokernel. Here, $\mu _i>0$ , $e_j>0$ , and $g_j(x)$ is an irreducible distinguished polynomial. Furthermore, the numbers $\mu _1, \dots , \mu _s$ and irreducible distinguished polynomials $g_1(x),\dots , g_t(x)$ are uniquely determined. The characteristic ideal of $M^\vee $ is (up to a unit) generated by

$$ \begin{align*} f_{M}^{(p)}(x) = f_{M}(x) := p^{\sum_{i} \mu_i} \prod_j g_j^{e_j}(x). \end{align*} $$

The $\mu $ -invariant of M is defined as the power of p in $f_{M}(x)$ . More precisely,

$$ \begin{align*} \mu_p(M):=\begin{cases} \sum_{i=1}^s \mu_i & \textrm{ if } s>0,\\ 0 & \textrm{ if } s=0. \end{cases} \end{align*} $$

The $\lambda $ -invariant of M is the degree of the characteristic element, i.e.,

$$ \begin{align*} \lambda_p(M) :=\begin{cases} \sum_{i=1}^t e_i\deg g_i & \textrm{ if } s>0,\\ 0 & \textrm{ if } s=0. \end{cases} \end{align*} $$

Since the numbers $\mu _i$ and polynomials $g_j(x)$ are uniquely determined by M, the $\mu $ and $\lambda $ -invariants determined above are well defined.

Let F be a number field and E an elliptic curve over F with good ordinary reduction at all primes of F above p. Denote by $\operatorname {Sel}_{p^\infty }(E/F_{\operatorname {cyc}})$ the p-primary Selmer group of E over $F_{\operatorname {cyc}}$ (see [Reference Coates and Sujatha1] for further details). Suppose $\operatorname {Sel}_{p^\infty }(E/F_{\operatorname {cyc}})$ that is a cotorsion $\Lambda _F$ -module, we set $\mu _p(E/F)$ and $\lambda _p(E/F)$ to denote the $\mu $ and $\lambda $ -invariant of $\operatorname {Sel}_{p^\infty }(E/F_{\operatorname {cyc}})$ respectively, when viewed as a module over $\Lambda _F$ . We fix a strongly admissible pro-p, uniform, p-adic Lie extension $F_\infty /F$ and let $\operatorname {Sel}_{p^\infty }(E/F_\infty )$ be the Selmer group of E over $F_\infty $ (see [Reference Lim21] for the definition). Throughout, we make the following assumption.

Assumption 2.3 With notation as above, assume that $\operatorname {Sel}_{p^\infty }(E/F_\infty )$ satisfies the $\mathfrak {M}_H(G)$ -conjecture. In greater detail, set $X(E/F_\infty )$ to be the Pontryagin dual of $\operatorname {Sel}_{p^\infty }(E/F_\infty )$ . We assume that

$$ \begin{align*}X_f(E/F_\infty):=\frac{X(E/F_\infty)}{X(E/F_\infty)[p^\infty]}\end{align*} $$

is finitely generated as a $\Lambda (H)$ -module.

2.3 An analogue of Kida’s formula

Hachimori and Matsuno in [Reference Hachimori and Matsuno11] proved an analogue of Kida’s formula for Selmer groups of elliptic curves. We recall this result and a refinement due to Lim. Let F be a number field and $E_{/F}$ an elliptic curve. Let $L/F$ be a finite Galois extension such that $\operatorname {Gal}(L/F)$ is a p-group. Let $P_1(E, L_{\operatorname {cyc}})$ (resp. $P_2(E, L_{\operatorname {cyc}})$ ) be the set of primes $\eta \nmid p$ of $L_{\operatorname {cyc}}$ that are ramified in the extension $L_{\operatorname {cyc}}/K_{\operatorname {cyc}}$ , at which E has split multiplicative reduction (resp. E has good reduction and $E(L_{\operatorname {cyc},\eta })[p]\neq 0$ ). Given a prime $\eta $ of $L_{\operatorname {cyc}}$ , set $e_{L_{\operatorname {cyc}}/K_{\operatorname {cyc}}}(\eta )$ to denote the ramification index of $\eta $ with respect to the extension $L_{\operatorname {cyc}}/K_{\operatorname {cyc}}$ .

2.4 Main result

Let E be an elliptic curve over a number field F with good ordinary reduction at all primes above p. Throughout, we shall make the following assumption.

We introduce some further notation. Let $Q_1=Q_1(E,F_\infty )$ (resp. $Q_2=Q_2(E,F_\infty )$ ) be the set of primes $w\nmid p$ of $F_{\operatorname {cyc}}$ that are ramified in $F_\infty $ , at which E has split multiplicative reduction (resp. E has good reduction and $E(F_{\operatorname {cyc},w})[p]\neq 0$ ). We stress here that $Q_1$ and $Q_2$ consist of subsets of primes of $F_{\operatorname {cyc}}$ and not $F_\infty $ . Recall that it is stipulated that only finitely many primes ramify in $F_\infty $ , and since all primes are finitely decomposed in $F_{\operatorname {cyc}}$ , it follows that $Q_1$ and $Q_2$ are finite. For $i=1,2$ , we set $q_i:=\#Q_i$ .

Theorem 2.6 Let n be a positive integer. Suppose that the conditions of Assumption 2.3 hold, then, $\operatorname {Sel}_{p^\infty }(E/F^{(n)}_{\operatorname {cyc}})$ is a cotorsion $\Lambda _{F^{(n)}}$ -module, with

$$ \begin{align*}\mu_p(E/F^{(n)})=p^{nd}\mu_p(E/F).\end{align*} $$

Furthermore, we have that

$$ \begin{align*}p^{n(d-1)}\lambda_p(E/F)\leq \lambda_p(E/F^{(n)})\leq p^{n(d-1)}\lambda_p(E/F)+\left(p^{n(d-1)}-p^{n(d-2)}\right)(q_1+2q_2).\end{align*} $$

Proof According to Theorem 2.4, $\operatorname {Sel}_{p^\infty }(E/F^{(n)}_{\operatorname {cyc}})$ is a cotorsion module over $\Lambda _{F^{(n)}}$ and the $\mu $ -invariant is given by

$$ \begin{align*}\mu_p(E/F^{(n)})=[F^{(n)}:F]\mu_p(E/F)=p^{nd}\mu_p(E/F).\end{align*} $$

Furthermore, the $\lambda $ -invariant is

$$ \begin{align*}\lambda_p(E/F^{(n)})=[F^{(n)}_{\operatorname{cyc}}:F_{\operatorname{cyc}}]\lambda_p(E/F)+\sum_{w\in P_1} \left(e(w)-1\right)+2\sum_{w\in P_2} \left(e(w)-1\right),\end{align*} $$

where $e(w)$ is the ramification index of w in $F^{(n)}_{\operatorname {cyc}}/F_{\operatorname {cyc}}$ , $P_1$ and $P_2$ are the set of primes of $F^{(n)}_{\operatorname {cyc}}$ defined as follows:

$$ \begin{align*} & P_1=\{w\mid w\nmid p, E\text{ has split multiplicative reduction at }w\},\\ & P_2=\{w\mid w\nmid p, E\text{ has good reduction at }w\text{ and }E(\mathcal{F}_{n,w})\text{ has a point of order }p\}. \end{align*} $$

Since p is odd, the primes $P_i$ lie above $Q_i$ , therefore,

$$ \begin{align*}\sum_{w\in P_i} \left(e(w)-1\right)=\sum_{v\in Q_i} \left(\sum_{w\mid v} \left(e(w)-1\right)\right).\end{align*} $$

Choose a prime $w_0\in P_i$ above v. Since $e(w)$ is the same for all primes $w|v$ , we have that

$$ \begin{align*}\sum_{w\mid v} \left(e(w)-1\right)=\left(1-e(w_0)^{-1}\right)\sum_{w\mid v}e(w)\leq \left(1-e(w_0)^{-1}\right)[F^{(n)}_{\operatorname{cyc}}:F_{\operatorname{cyc}}].\end{align*} $$

According to Lemma 2.2, H is uniform with $d-1$ generators. Note that $[F^{(n)}_{\operatorname {cyc}}:F_{\operatorname {cyc}}]=[H:H_n]=p^{(d-1)n}$ . Since pro-p tame inertia is generated by a single element, it follows that $e(w_0)\leq p^{n}$ . Putting it all together, the result follows.▪

Theorem 2.7 Let E be an elliptic curve defined over a number field F and $F_\infty $ a uniform pro-p extension of F satisfying aforementioned conditions and suppose that the Selmer group over $F_{\operatorname {cyc}}$ is cotorsion as a -module. Then, we have the following bound

$$ \begin{align*}\operatorname{rank} E(F^{(n)})\leq p^{n(d-1)}\lambda_p(E/F)+\left(p^{n(d-1)}-p^{n(d-2)}\right)(q_1+2q_2).\end{align*} $$

Proof The result immediately follows from Theorem 2.6 and the inequality

$$ \begin{align*}\operatorname{rank} E(F^{(n)})\leq \lambda_p(E/F^{(n)}),\end{align*} $$

see [Reference Greenberg10, Theorem 1.9].▪

The improvement in the bound has some nontrivial consequences, which we shall explain in the next section. The following is a Corollary to Theorem 2.6 and is entirely unconditional.

Corollary 2.9 Let E and $F_\infty $ be as in Theorem 2.7. Assume that $q_1=q_2=0$ and

$$ \begin{align*}\operatorname{Sel}_{p^\infty}(E/F_{\operatorname{cyc}})=0.\end{align*} $$

Then, $\operatorname {Sel}_{p^\infty }(E/F^{(n)}_{\operatorname {cyc}})=0$ for all n.

3.1 $\mathbb {Z}_p^d$ -extensions

Throughout this subsection, F will be an abelian number field and $E_{/\mathbb {Q}}$ an elliptic curve with good ordinary reduction at p. Let $F_\infty $ be the composite of all $\mathbb {Z}_p$ -extensions of F, note that $G=\operatorname {Gal}(F_\infty /F)\simeq \mathbb {Z}_p^d$ , where $d=r_2(F)+1$ . For instance, when F is an imaginary quadratic field, then, this gives a $\mathbb {Z}_p^2$ -extension of F. To emphasize the dependence on the prime p, we denote the extension by $F_\infty (p)$ . On the other hand, it follows from results of Kato and Rohlrich [Reference Hachimori and Matsuno11, Theorem 2.2] that the Selmer group $\operatorname {Sel}_{p^\infty }(E/F_\infty )$ is cotorsion as a -module. It is well known that any $\mathbb {Z}_p$ -extension is unramified away from p (see [Reference Washington27]), hence, the composite of such extensions has the same property. Let us state a few Corollaries to Theorem 2.7, the first of which gives a simple criterion for the rank to be zero throughout the $\mathbb {Z}_p^d$ -tower.

Corollary 3.1 Let E be as above and assume that $\lambda _p(E/F_{\operatorname {cyc}})=0$ . Then,

$$ \begin{align*}\operatorname{rank} E(F^{(n)})=0\end{align*} $$

for all $n\in \mathbb {Z}_{\geq 1}$ and

$$ \begin{align*}\mu_p(E/F^{(n)})=p^{nd}\mu_p(E/F).\end{align*} $$

Proof Note that since E has good ordinary reduction at the primes above p and

$$ \begin{align*}\operatorname{rank} E(F)\leq \lambda_p(E/F),\end{align*} $$

the Mordell–Weil rank of E is 0. Since $F_\infty $ is unramified at all primes $w\nmid p$ , the quantities $q_1$ and $q_2$ in Theorem 2.7 are both equal to 0.▪

Example: Picking an elliptic curve $E_{/\mathbb {Q}}$ at random, there are typically some primes at which $E[p]$ is residually reducible as a Galois module. At these primes, it is possible that the $\mu $ -invariant $\mu _p(E/\mathbb {Q})$ does not vanish. For instance, let’s pick the elliptic curve of smallest conductor with cremona label 11a2. We find that E has good ordinary reduction at $5$ with $\mu _5(E/\mathbb {Q})=2$ and $\lambda _5(E/\mathbb {Q})=0$ . Suppose that there is an imaginary quadratic field $F/\mathbb {Q}$ in which $\operatorname {rank} E(F)=0$ and $\lambda _5(E/F)=0$ as well. Let $F_\infty $ be the $\mathbb {Z}_p^2$ -extension of F. Then, indeed, since $\mu _5(E/F)\geq 2$ , the above result implies that

$$ \begin{align*}\mu_5(E/F^{(n)})\geq 2p^{2n}\end{align*} $$

for all $n\geq 1$ , however, the rank of $E(F_n)$ remains $0$ throughout. Unfortunately, the author is not aware of any existing computer packages that can compute the $\lambda $ -invariant over an imaginary quadratic field.

Corollary 3.3 Let $E_{/\mathbb {Q}}$ be an elliptic curve and F an abelian number field satisfying

1. (1) $\operatorname {rank} E(F)=0$ and

2. (2) E does not have complex multiplication.

Then, for $100\%$ of primes p at which E has good ordinary reduction,

$$ \begin{align*}\operatorname{rank} E(F_\infty(p))=0.\end{align*} $$

The following is a special case of [Reference Hung and Lim14, Conjecture 1].

Corollary 3.5 Consider the setting of the above conjecture. Under Assumption 2.3, the Theorem 2.7 specializes to give that

$$ \begin{align*}\operatorname{rank} E(F^{(n)})\leq \lambda_p(E/F) p^{n}.\end{align*} $$

Thus, the Conjecture is true when

$$ \begin{align*} \lambda_p(E/F)=\operatorname{rank} E(F_{\operatorname{cyc}}).\end{align*} $$

3.2 False-Tate curve extensions

Let $\ell $ be a finite set of prime numbers that are coprime to p and let $F_\infty $ be the False-Tate curve extension of $F=\mathbb {Q}(\mu _p)$ , given by

$$ \begin{align*}F_\infty:=\mathbb{Q}(\mu_{p^\infty},\ell^{\frac{1}{p^{\infty}}}).\end{align*} $$

In other words, it is the extension obtained by adjoining all p-power roots of $1$ and $\ell $ . It is easy to see that $F_\infty /F$ is a uniform pro-p extension of F of dimension $d=2$ . Thus Theorem 2.7 specializes to give us that

$$ \begin{align*}\operatorname{rank} E(F^{(n)})\leq \lambda_p(E/F)p^n+(p^n-1)\left(q_1+2q_2\right).\end{align*} $$

Let us compute the values of $q_1$ and $q_2$ for a given example. We note that it is difficult to compute $\lambda _p(E/F)$ due to the base change to $F=\mathbb {Q}(\mu _p)$ .

Example: We pick an elliptic curve and prime at random. Let $E=$ 11a2 in Cremona label and $p=7$ . The elliptic curve is defined over $\mathbb {Q}$ and we consider its base change to $F=\mathbb {Q}(\mu _7)$ . It follows from Assumption 2.3 and from the proof of Theorem 2.4 that $\operatorname {Sel}_{7^\infty }(E/\mathbb {Q}(\mu _{7^\infty }))$ is cotorsion over $\Lambda _F$ . Note that since F is an abelian extension of $\mathbb {Q}$ , the cotorsion property of the Selmer group (over $\Lambda _F$ ) also follows from results due to Kato, see [Reference Kato15]. The image of the residual representation at $p=7$ contains $\operatorname {SL}_2(\mathbb {F}_7)$ , as stated in the link provided. Hence, after base change to $\mathbb {Q}(\mu _7)$ the image of the residual representation will still contain $\operatorname {SL}_2(\mathbb {F}_7)$ . It is thus reasonable to expect that the $\mu $ -invariant $\mu _7(E/F)=0$ , however, this is difficult to prove and needs to be assumed. Consider the False Tate extension

$$ \begin{align*}F_\infty:=\mathbb{Q}(\mu_{7^\infty},11^{\frac{1}{7^{\infty}}}).\end{align*} $$

Then, E has split multiplicative reduction at $11$ , hence $q_2=0$ , however, $q_1>0$ . Since $11^3\equiv 1\mod {7}$ and $11^3\not \equiv 1\mod {49}$ , there are precisely two primes above $11$ in $F_{\operatorname {cyc}}=\mathbb {Q}(\mu _{7^\infty })$ . It follows that $q_1=2$ . Putting it all together, we find that

$$ \begin{align*}\operatorname{rank} E(F^{(n)})\leq \lambda_7(E/F)7^n+2(7^n-1).\end{align*} $$

Next, we prove a statistical result.

Corollary 3.7 Consider the elliptic curve $E=11a2$ from the example above and set $p=7$ . There is a positive density set of primes $\ell $ such that

$$ \begin{align*}\operatorname{rank}E\left(\mathbb{Q}(\mu_{7^{n+1}}, \ell^{\frac{1}{7^n}})\right)\leq \lambda_7(E/F)7^n\end{align*} $$

for any integer $n\geq 1$ .

Proof For each prime number $\ell $ such that $\ell \equiv 1\mod {7}$ , let $F_\infty ^{(\ell )}$ denote the extension $\mathbb {Q}(\mu _{7^\infty }, \ell ^{\frac {1}{7^\infty }})$ . Note that $\ell $ splits in $\mathbb {Q}(\mu _7)$ and the only prime other than $7$ that ramifies in $F_\infty ^{(\ell )}$ is $\ell $ . Since E has bad reduction at only finitely many primes, we deduce that $q_1=0$ for all extensions $F_\infty ^{(\ell )}$ except for finitely many choices of $\ell $ . Recall that $Q_2$ consists of the primes $w\nmid 7$ of $F_{\operatorname {cyc}}$ that are ramified in $F_\infty $ , such that E has good reduction at w and $E(F_{\operatorname {cyc},w})[7]\neq 0$ . Since the formal group of E at w is pro- $\ell $ , $E(F_{\operatorname {cyc},w})[7]\simeq E(k_w)[7]$ , where $k_w$ is the residue field of $F_{\operatorname {cyc},w}$ . Since $k_w$ is a $7$ -extension of $\mathbb {F}_\ell $ , it follows that $E(k_w)[7]\neq 0$ if and only if $E(\mathbb {F}_\ell )[7]\neq 0$ . Thus, the prime w lies in $Q_2$ if the following conditions are satisfied:

1. (1) $w|\ell $ ,

2. (2) E has good reduction at $\ell $ , and

3. (3) $E(\mathbb {F}_\ell )[7]\neq 0$ .

This latter condition is satisfied for $\frac {1}{7}$ of all primes $\ell $ , see [Reference Cojocaru3, Section 2] for further details. We show that a similar application of the Chebotarev density theorem shows that for a positive proportion of the primes $\ell $ , both $q_1$ and $q_2$ are $0$ . Note that if a prime $\ell $ splits in $\mathbb {Q}(\mu _7)$ precisely when $\ell \equiv 1\mod {7}$ . Let $\bar {\rho }:\operatorname {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})\rightarrow \operatorname {GL}_2(\mathbb {F}_7)$ be the Galois representation on the $7$ -torsion points $E[7]$ . Denote by $\mathbb {Q}(\bar {\rho })$ the number field fixed by the kernel of $\bar {\rho }$ . Note that $\operatorname {det}\bar {\rho }$ is the mod- $7$ cyclotomic character, which we denote by $\bar {\chi }$ . Therefore, the field $\mathbb {Q}(\bar {\rho })$ contains $\mathbb {Q}(\mu _7)$ , which is the field fixed by the kernel of $\bar {\chi }$ . Let $\ell \neq 7$ be a prime at which E has good reduction, note that $\bar {\rho }$ is unramified at $\ell $ . Let $\sigma _\ell \in \operatorname {Gal}\left (\mathbb {Q}(\bar {\rho })/\mathbb {Q}\right )$ denote the Frobenius element at $\ell $ , it is well known that the characteristic polynomial of $\bar {\rho }(\sigma _\ell )$ is given by

$$ \begin{align*}\operatorname{det}\left(\operatorname{Id}\cdot x-\bar{\rho}(\sigma_\ell)\right)=x^2-(\ell+1-\#E(\mathbb{F}_\ell))x+\ell.\end{align*} $$

Thus, if $\ell $ is a prime such that

(3.1) $$ \begin{align} \operatorname{det}\bar{\rho}(\sigma_\ell)=1\text{ and }\operatorname{tr} \bar{\rho}(\sigma_\ell)\neq 2,\end{align} $$

then, $\ell $ splits in $\mathbb {Q}(\mu _7)$ and $E(\mathbb {F}_\ell )[7]= 0$ . We show that there is a positive density set of primes $\ell $ satisfying the above conditions (3.1). According to the LMFDB Database [Reference Cremona, Jones, Sutherland and Voight5], the image of $\bar {\rho }$ contains $\operatorname {SL}_2(\mathbb {F}_7)$ . Thus there is $\sigma \in \operatorname {Gal}(\mathbb {Q}(\bar {\rho })/\mathbb {Q})$ such that $\bar {\rho }(\sigma )= \left ( {\begin {array}{cc} 2 & 1 \\ 1 & 1 \\ \end {array} } \right )$ , and thus satisfies (3.1). According to the Chebotarev density theorem, there is a positive proportion of primes such that $\sigma _\ell =\sigma $ , and thus, $\bar {\rho }(\sigma _\ell )= \left ( {\begin {array}{cc} 2 & 1 \\ 1 & 1 \\ \end {array} } \right )$ . As a result, there is a positive density set of primes $\ell $ such that $q_1$ and $q_2$ are both zero, and this completes the proof.▪

3.3 The field generated by torsion points

We come to the example in which $F_\infty $ is the field $\mathbb {Q}(E[p^\infty ])$ , i.e., the field generated by the p-primary torsion points of E. Assume that E does not have complex multiplication. Then, by Serre’s Open image theorem (see [Reference Serre25, Section 4 and Theorem 3]), G is a finite index subgroup of $\operatorname {GL}_2(\mathbb {Z}_p)$ and it follows from this that the dimension of G is $4$ . In this setting, $F=\mathbb {Q}(E[p])$ , and $F^{(n)}=\mathbb {Q}(E[p^{n+1}])$ . We find that

$$ \begin{align*}\operatorname{rank} E(F_n)\leq \lambda_p(E/F)p^{3n}+q_1(p^{3n}-p^{2n}),\end{align*} $$

where $q_1$ is simply the number of primes $\ell \neq p$ at which E has split multiplicative reduction, and $q_2=0$ . On the other hand, according to [Reference Hung and Lim14, Remark after Theorem 3.2] the result of Hung–Lim [Reference Hung and Lim14, Theorem 3.2] gives

$$ \begin{align*}\operatorname{rank} E(F_n)\leq (\lambda_p(E/F)+q_1)p^{3n}+8, \end{align*} $$

when the whole dual Selmer group $\mathscr {X}(E/F_\infty )$ is finitely generated over $\Lambda (H)$ .