# Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative
towers

Anwesh Ray  Department of Mathematics
University of British Columbia
Vancouver BC, Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

Let $E$ be an elliptic curve defined over a number field $F$ with good
ordinary reduction at all primes above $p$, and let $F_{\infty}$ be a finitely
ramified uniform pro-$p$ extension of $F$ containing the cyclotomic
$\mathbb{Z}_{p}$-extension $F_{\operatorname{cyc}}$. Set $F^{(n)}$ be the
$n$-th layer of the tower, and $F^{(n)}_{\operatorname{cyc}}$ the cyclotomic
$\mathbb{Z}_{p}$-extension of $F^{(n)}$. We study the growth of the rank of
$E(F^{(n)})$ by analyzing the growth of the $\lambda$-invariant of the Selmer
group over $F^{(n)}_{\operatorname{cyc}}$ as $n\rightarrow\infty$. This method
has its origins in work of A. Cuoco, who studied
$\mathbb{Z}_{p}^{2}$-extensions. Refined estimates for growth are proved that
are close to conjectured estimates. The results are illustrated in special
cases.

###### Key words and phrases:

Elliptic curves, Iwasawa invariants, Kida’s formula, towers of number fields,
asymptotic growth of ranks

###### 2010 Mathematics Subject Classification:

11G05, 11R23

##  1\. Introduction

The _Mordell–Weil Theorem_ states that given an elliptic curve $E$ defined
over a number field $F$, its $F$-rational points form a finitely generated
abelian group, i.e.,

$E(F)\simeq\mathbb{Z}^{r}\oplus E(F)_{\operatorname{tors}}$

where $r$ is a non-negative integer called the Mordell–Weil _rank_. In [21],
B. Mazur initiated the study of Selmer groups of elliptic curves in
$\mathbb{Z}_{p}$-extensions. A major application of Iwasawa theory is the
study of the growth of Mordell–Weil ranks of abelian varieties in towers of
number fields. Given an abelian variety defined over a number field $F$ with
good ordinary reduction at the primes above $p$, Mazur showed that the rank of
$A$ is bounded in the cyclotomic $\mathbb{Z}_{p}$-extension of $F$. K. Kato
and D. Rohlrich proved the analogous statement for all elliptic curves defined
over abelian number fields, see [14, 23]. Alongside further developments in
Iwasawa theory over larger $p$-adic Lie extensions, there has been significant
interest in analyzing the asymptotic growth of Mordell–Weil ranks in towers.
For ordinary primes, Mazur in [22] formulated a precise conjecture on the
growth of ranks in any $\mathbb{Z}_{p}$-extension of an imaginary quadratic
field $F$, called the _Growth number conjecture_. This question has been
studied in anticyclotomic extensions by C. Cornut [4] and V. Vatsal [24]. A
prototypical example of interest is the $\mathbb{Z}_{p}^{2}$-extension
$F_{\infty}$ of an imaginary quadratic field $F$. For every integer $n\geq 1$,
set $F^{(n)}$ to denote the $n$-th layer. In this setting, it is the extension
contained in $F_{\infty}$ such that
$\operatorname{Gal}(F^{(n)}/F)=\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{2}$.
For elliptic curves $E_{/F}$, asymptotic formulas for the growth of the rank
of $E(F^{(n)})$ as $n\rightarrow\infty$ have been proven by A. Lei and F.
Sprung in [18]. More recently, such growth questions are studied in admissible
uniform pro-$p$ extensions of number fields by D. Delbourgo and A. Lei in [6],
and by P. C. Hung and M. F. Lim in [13].

In this note, we employ a new strategy to study the growth of ranks in non-
commutative towers. Let $F^{(n)}_{\operatorname{cyc}}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $F^{(n)}$. We study the growth of the rank of
$E(F^{(n)})$ by analyzing the growth of the $\lambda$-invariant of the Selmer
group over $F^{(n)}_{\operatorname{cyc}}$ as $n\rightarrow\infty$ using a
generalizations of Kida’s formula due to Y. Hachimori $\&$ K. Matsuno [10],
and M. F. Lim [19]. The method has several advantages. First, it is a
straightforward application of Kida’s formula which a prioi allows one to
circumvent technicalities of noncommutative Iwasawa theory. In other words,
the proof is short and can be understood without familiarity with methods in
noncommutative Iwasawa theory, though it does build on previous work of M. F.
Lim [20] which uses nontrivial results in the subject. Given an elliptic curve
$E$ over a number field $F$, we do however require that the Selmer group of
$E$ over the cyclotomic $\mathbb{Z}_{p}$-extension of $F$ be cotorsion over
the Iwasawa algebra and impose the $\mathfrak{M}_{H}(G)$ conjecture (see [2,
19]).

Secondly, (and perhaps more importantly) the method strengthens known results
and these estimates are closer to conjectured asymptotics. The error term in
the asymptotic formulas of Hung-Lim are removed in the process, and the main
term is sharper (see Remark 2.6 for further details). The significance of this
is illustrated for certain examples, namely, _$\mathbb{Z}_{p}^{d}$
-extensions_ and _false Tate-curve extensions_. It should be pointed out here
that for $\mathbb{Z}_{p}^{2}$-extensions, a similar question was studied by A.
Cuoco in [5], who studied the growth of Iwasawa invariants associated with
class group towers in families of $\mathbb{Z}_{p}$-extensions contained in the
composite of two $\mathbb{Z}_{p}$-extensions. The results can also be applied
to prove statistical results, see Corollaries 3.3 and 3.7.

The method employed in this paper shows that in any context in which a
satisfactory generalization of Kida’s formula is proved, it should be possible
to analyze the growth of $\lambda$-invariants in noncommutative towers. We
point out that analogs of Kida’s formula have been proven for fine Selmer
groups by D. Kundu in [16]. In this particular context, the number fields are
assumed to be totally real. Also, such results were proved by J. Hatley and A.
Lei in the supersingular setting, see [12].

### Acknowledgement

The author is grateful to Debanjana Kundu for pointing out an inaccuracy in
the previous draft in which it was assumed that the $\mu$-invariant is zero
over the cyclotomic extension.

##  2\. Growth of Iwasawa invariants in towers

In this section, we introduce some preliminary notions and prove the main
result of this paper.

###  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
$G\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(F_{\infty}/F)$. The
lower central $p$-series of $G$ is recursively defined as follows:

$G_{0}\mathrel{\mathop{\mathchar 58\relax}}=G\text{ and
}G_{n+1}\mathrel{\mathop{\mathchar 58\relax}}=G_{n}^{p}[G_{n},G].$

######  Definition 2.1.

The group $G$ is said to be _uniform_ if

  1. (1)

it is finitely generated,

  2. (2)

it is _powerful_ , i.e., $[G,G]\subseteq G^{p}$,

  3. (3)

$[G_{n}\mathrel{\mathop{\mathchar
58\relax}}G_{n+1}]=[G\mathrel{\mathop{\mathchar 58\relax}}G_{1}]$ for all
$n\in\mathbb{Z}_{\geq 1}$.

Setting $d\mathrel{\mathop{\mathchar 58\relax}}=[G\mathrel{\mathop{\mathchar
58\relax}}G_{1}]$, we observe that $[G\mathrel{\mathop{\mathchar
58\relax}}G_{n}]=p^{dn}$. Make the following standard assumptions on
$F_{\infty}$

  1. (1)

$G$ is uniform and pro-$p$,

  2. (2)

only finitely many primes ramify in $F_{\infty}$,

  3. (3)

$F_{\infty}$ contains the cyclotomic $\mathbb{Z}_{p}$-extension
$F_{\operatorname{cyc}}$ of $F$.

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 [7, Theorem 3.6]. Any $p$-adic analytic
group is a closed subgroup of $\operatorname{GL}_{m}(\mathbb{Z}_{p})$ for some
integer $m$ and contains an open uniform subgroup, see [7, Theorem II.8.32].
Hence, after replacing $F$ by a finite extension $F^{\prime}/F$, our results
apply for $p$-adic analytic Lie-extensions as well.

The extension $F_{\infty}$ is filtered by a tower of number fields. Setting
$F^{(n)}\mathrel{\mathop{\mathchar 58\relax}}=F_{\infty}^{G_{n}}$, consider
the nonabelian tower

$F=F^{(0)}\subset F^{(1)}\subset\dots\subset F^{(n)}\subset\dots,$

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

$F_{\operatorname{cyc}}=F^{(0)}_{\operatorname{cyc}}\subset
F^{(1)}_{\operatorname{cyc}}\subset\dots\subset
F^{(n)}_{\operatorname{cyc}}\subset\dots.$

Set $H\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(F_{\infty}/F_{\operatorname{cyc}})$ and
$\Gamma\mathrel{\mathop{\mathchar 58\relax}}=G/H\simeq\mathbb{Z}_{p}$. 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.

######  Lemma 2.2.

The following assertions hold:

  1. (1)

The normal subgroup $H$ is uniform with $(d-1)$ generators, and $H_{n}$ is
identified with $H\cap G_{n}$,

  2. (2)

$\Gamma_{n}$ is identified with $G_{n}/H_{n}$.

###### Proof.

See [7, Theorem 3.6] and [13, Lemma 2.6] for further details. ∎

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

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

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 $n$-th level, taken to be

$\Lambda(\Gamma_{n})\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{L}\mathbb{Z}_{p}[\operatorname{Gal}(L/F^{(n)})],$

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
$\Lambda(\Gamma_{n})\simeq\mathbb{Z}_{p}\llbracket x\rrbracket$ sending
$\gamma_{n}-1$ to $x$.

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

$\Lambda(\mathcal{G})\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{U}\mathbb{Z}_{p}[\mathcal{G}/U],$

where $U$ ranges over all finite index normal subgroups of $\mathcal{G}$.

###  2.2. Iwasawa invariants

Let $M$ be a cofinitely generated cotorsion $\mathbb{Z}_{p}\llbracket
x\rrbracket$-module, i.e., the Pontryagin-dual
$M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ is a finitely
generated and torsion $\mathbb{Z}_{p}\llbracket x\rrbracket$-module. Recall
that a polynomial $f(x)\in\mathbb{Z}_{p}\llbracket x\rrbracket$ is said to be
distinguished if it is a monic polynomial whose non-leading coefficients are
all divisible by $p$. Note that all height $1$ prime ideals of
$\mathbb{Z}_{p}\llbracket x\rrbracket$ 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 $\mathbb{Z}_{p}\llbracket
x\rrbracket$-modules (see [25, Theorem 13.12]), $M^{\vee}$ is pseudo-
isomorphic to a finite direct sum of cyclic $\mathbb{Z}_{p}\llbracket
x\rrbracket$-modules, i.e., there is a map

$M^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\mathbb{Z}_{p}\llbracket
x\rrbracket/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\mathbb{Z}_{p}\llbracket
x\rrbracket/(f_{j}(T))\right)$

with finite kernel and cokernel. Here, $\mu_{i}>0$ and $f_{j}(T)$ is a
distinguished polynomial. The characteristic ideal of $M^{\vee}$ is (up to a
unit) generated by

$f_{M}^{(p)}(T)=f_{M}(T)\mathrel{\mathop{\mathchar
58\relax}}=p^{\sum_{i}\mu_{i}}\prod_{j}f_{j}(T).$

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

$\mu_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{i=1}^{s}\mu_{i}&\textrm{ if }s>0\\\ 0&\textrm{
if }s=0.\end{cases}$

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

$\lambda_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{i=1}^{s}\deg f_{i}&\textrm{ if }s>0\\\
0&\textrm{ if }s=0.\end{cases}$

###  2.3. Main result

Let $E$ be an elliptic curve over $F$ with good ordinary reduction at all
primes above $p$, and
$\operatorname{Sel}_{p^{\infty}}(E/F^{(n)}_{\operatorname{cyc}})$ the
$p$-primary Selmer group over $F^{(n)}_{\operatorname{cyc}}$, see [1]. Let
$\mu_{p}(E/F^{(n)}_{\operatorname{cyc}})$ and
$\lambda_{p}(E/F^{(n)}_{\operatorname{cyc}})$ denote the Iwasawa $\mu$ and
$\lambda$-invariants of the $\mathbb{Z}_{p}\llbracket x\rrbracket$-module
$\operatorname{Sel}_{p^{\infty}}(E/F^{(n)}_{\operatorname{cyc}})$. Throughout,
we shall make the following assumption.

######  Assumption 2.3.

The Pontryagin dual of
$\operatorname{Sel}_{p^{\infty}}(E/F_{\operatorname{cyc}})$ is cotorsion as a
$\mathbb{Z}_{p}\llbracket x\rrbracket$-module. The
$\mathfrak{M}_{H}(G)$-conjecture is satisfied for
$\operatorname{Sel}_{p^{\infty}}(E/F_{\infty})$. In greater detail, set

$\mathscr{X}(E/F_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\operatorname{cnts}}\left(\operatorname{Sel}_{p^{\infty}}(E/F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$

and

$\mathscr{X}_{f}(E/F_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mathscr{X}(E/F_{\infty})}{\mathscr{X}(E/F_{\infty})[p^{\infty}]},$

then, it is conjectured that

$\operatorname{rank}_{\Lambda(H)}\left(\mathscr{X}_{f}(E/F_{\infty})\right)<\infty.$

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
$E(F_{\operatorname{cyc},w})[p]\neq 0$). 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 $q_{i}\mathrel{\mathop{\mathchar
58\relax}}=\\#Q_{i}$.

######  Theorem 2.4.

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 cofinitely generated and cotorsion $\mathbb{Z}_{p}\llbracket
x\rrbracket$-module, with

$\mu_{p}(E/F^{(n)}_{\operatorname{cyc}})=p^{n(d-1)}\mu_{p}(E/F_{\operatorname{cyc}}).$

Furthermore, we have that

$p^{n(d-1)}\lambda_{p}(E/F_{\operatorname{cyc}})\leq\lambda_{p}(E/F^{(n)}_{\operatorname{cyc}})\leq
p^{n(d-1)}\lambda_{p}(E/F_{\operatorname{cyc}})+\left(p^{n(d-1)}-p^{n(d-2)}\right)(q_{1}+2q_{2}).$

###### Proof.

First, we explain the result in the case when
$\mu_{p}(E/F_{\operatorname{cyc}})=0$. In this case, the result is obtained
via a direct application of the Riemann-Hurwitz formula for
$\lambda$-invariants due to Y. Hachimori and K. Matsuno, see [10]. We find
that

$\lambda_{p}(E/F^{(n)}_{\operatorname{cyc}})=[F^{(n)}_{\operatorname{cyc}}\mathrel{\mathop{\mathchar
58\relax}}F_{\operatorname{cyc}}]\lambda_{p}(E/F_{\operatorname{cyc}})+\sum_{w\in
P_{1}}\left(e(w)-1\right)+2\sum_{w\in P_{2}}\left(e(w)-1\right),$

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:

$\begin{split}&P_{1}=\\{w\mid w\nmid p,E\text{ has split multiplicative
reduction at }w\\},\\\ &P_{2}=\\{w\mid w\nmid p,E\text{ has good reduction at
}w\text{ and }E(\mathcal{F}_{n,w})\text{ has a point of order
}p\\}.\end{split}$

Since $p$ is odd, the primes $P_{i}$ lie above $Q_{i}$, therefore,

$\sum_{w\in P_{i}}\left(e(w)-1\right)=\sum_{v\in Q_{i}}\left(\sum_{w\mid
v}\left(e(w)-1\right)\right).$

Choose a prime $w_{0}\in P_{i}$ above $v$. Since $e(w)$ is the same for all
primes $w|v$, we have that

$\sum_{w\mid v}\left(e(w)-1\right)=\left(1-e(w_{0})^{-1}\right)\sum_{w\mid
v}e(w)\leq\left(1-e(w_{0})^{-1}\right)[F^{(n)}_{\operatorname{cyc}}\mathrel{\mathop{\mathchar
58\relax}}F_{\operatorname{cyc}}].$

According to Lemma 2.2, $H$ is uniform with $d-1$ generators. Note that
$[F^{(n)}_{\operatorname{cyc}}\mathrel{\mathop{\mathchar
58\relax}}F_{\operatorname{cyc}}]=[H\mathrel{\mathop{\mathchar
58\relax}}H_{n}]=p^{(d-1)n}$. 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 in the case when $\mu_{p}(E/F_{\operatorname{cyc}})=0$.

Next, consider the other case, when $\mu_{p}(E/F_{\operatorname{cyc}})\neq 0$.
This follows from the formula of M. F. Lim, see [20, p.1141]. This warrants
some explanation, we refer to the notation used in loc. cit. The result is
applied to the extension $F^{(n)}/F^{\prime}$, where
$F^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=F^{(n)}\cap
F_{\operatorname{cyc}}$. Since $E$ has good ordinary reduction at all primes
above $p$, the set $M(F^{\prime}_{\operatorname{cyc}})$ is empty. Note that
since $F^{\prime}_{\operatorname{cyc}}=F_{\operatorname{cyc}}$, it is easy to
see that the Iwasawa-invariants coincide:

$\mu_{p}(E/F^{\prime}_{\operatorname{cyc}})=\mu_{p}(E/F_{\operatorname{cyc}})\text{
and
}\lambda_{p}(E/F^{\prime}_{\operatorname{cyc}})=\lambda_{p}(E/F_{\operatorname{cyc}}).$

Since the index

$[F^{(n)}\mathrel{\mathop{\mathchar
58\relax}}F^{\prime}]=[F^{(n)}_{\operatorname{cyc}}\mathrel{\mathop{\mathchar
58\relax}}F_{\operatorname{cyc}}]=p^{n(d-1)},$

the same calculation as in the previous paragraph goes through. ∎

######  Theorem 2.5.

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
$\mathbb{Z}_{p}\llbracket x\rrbracket$-module. Then, we have the following
bound

$\operatorname{rank}E(F^{(n)})\leq
p^{n(d-1)}\lambda_{p}(E/F_{\operatorname{cyc}})+\left(p^{n(d-1)}-p^{n(d-2)}\right)(q_{1}+2q_{2}).$

###### Proof.

The result immediately follows from Theorem 2.4 and the inequality

$\operatorname{rank}E(F^{(n)})\leq\lambda_{p}(E/F^{(n)}_{\operatorname{cyc}}),$

see [9, Theorem 1.9]. ∎

######  Remark 2.6.

The estimate above is stronger than [13, Theorem 3.1]. The error term is
$O(p^{n(d-2)})$ and their method used relies on the work of M. Harris. In
greater detail, [11, Theorem 1.10] is the key result used in the estimate in
[13, Lemma 3.3]. Note however, that the error estimate of Hung-Lim is known to
be 0 under certain additional constraints. Namely, if certain cohomology
groups vanish and the $p$-torsion group $E(F_{\infty})(p)$ is finite, see [13,
Theorem 3.2] and the remark following it. If $F_{\infty}$ contains the
extension $F(E[p^{\infty}])$ generated by the $p$-primary torsion of $E$, then
the error term of Hung-Lim is non-zero, even under additional assumptions.
Also, even when the error term of Hung-Lim is 0, the the estimate above is
strictly better when $q_{1}$ or $q_{2}$ is non-zero.

The improvement in the bound has some non-trivial consequences, which we shall
explain in the next section. The following is a Corollary to Theorem 2.4 and
is entirely unconditional.

######  Corollary 2.7.

Let $E$ and $F_{\infty}$ be as in Theorem 2.5. Assume that $q_{1}=q_{2}=0$ and

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

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

###### Proof.

Since the $\mu$-invariant $\mu_{p}(E/F_{\operatorname{cyc}})=0$, it follows
that $\mathfrak{M}_{H}(G)$ is satisfied for $F_{\infty}$, see [2, Theorem
2.1]. By Theorem 2.4, it follows that the $\mu$ and $\lambda$-invariants of
$\operatorname{Sel}_{p^{\infty}}(E/F^{(n)}_{\operatorname{cyc}})=0$ are $0$,
hence, $\operatorname{Sel}_{p^{\infty}}(E/F^{(n)}_{\operatorname{cyc}})$ is
finite. On the other hand, this Selmer does not contain any finite index
submodules (see [8, Proposition 4.14]), hence, must be 0. ∎

##  3\. Special Cases

In this section, we study special cases of Theorem 2.5. Assume throughout that
the $\mathfrak{M}_{H}(G)$-conjecture is satisfied.

###  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 K. Kato and D. Rohlrich [10,
Theorem 2.2] that the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/F_{\infty})$ is cotorsion as a
$\mathbb{Z}_{p}\llbracket x\rrbracket$-module. It is well known that any
$\mathbb{Z}_{p}$-extension is unramified away from $p$ (see [25]), hence, the
composite of such extensions has the same property. Let us state a few
Corollaries to Theorem 2.5, 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,

$\operatorname{rank}E(F^{(n)})=0$

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

$\mu_{p}(E/F^{(n)}_{\operatorname{cyc}})=p^{n(d-1)}\mu_{p}(E/F_{\operatorname{cyc}}).$

###### Proof.

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

$\operatorname{rank}E(F)\leq\lambda_{p}(E/F_{\operatorname{cyc}}),$

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 2.5 are both
equal to 0. ∎

######  Remark 3.2.

When $F$ is an imaginary quadratic field and $E$ is a CM elliptic curve over
$F$, the result of Hung-Lim in the above context shall imply that the rank in
_bounded_ in the tower, however, not identically 0. In the more general case,
their result implies that the growth is $O(p^{n(d-2)})$ unless certain
homology groups are known to vanish, see the discussion after [13, Theorem
3.2].

_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
$\mu_{p}(E/\mathbb{Q}_{\operatorname{cyc}})$ does not vanish. For instance,
let’s pick the elliptic curve of smallest conductor with cremona label $\href
https://www.lmfdb.org/EllipticCurve/Q/11a2/$. We find that
$\mu_{5}(E/\mathbb{Q}_{\operatorname{cyc}})=2$ and
$\lambda_{5}(E/\mathbb{Q}_{\operatorname{cyc}})=0$. Suppose that there is an
imaginary quadratic field $F/\mathbb{Q}$ in which $\operatorname{rank}E(F)=0$
and $\lambda_{5}(E/F_{\operatorname{cyc}})=0$ as well. Then, indeed, since
$\mu_{5}(E/F_{\operatorname{cyc}})\geq 2$, the above result implies that

$\mu_{5}(E/F_{\operatorname{cyc}}^{(n)})\geq 2p^{n}$

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

  1. (1)

$\operatorname{rank}E(F)=0$,

  2. (2)

$E$ does not have complex multiplication.

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

$\operatorname{rank}E(F_{\infty}(p))=0.$

###### Proof.

In her thesis [15, Theorem 5.1.1], D. Kundu generalized a result of R.
Greenberg to show that the proportion of primes $p$ such that

  1. (1)

$E$ has good ordinary reduction at the primes of $F$ above $p$,

  2. (2)

$\operatorname{Sel}_{p^{\infty}}(E/F_{\operatorname{cyc}})=0$,

is $100\%$. The result follows from this and Corollary 3.1. ∎

The following is a special case of [13, Conjecture 1].

######  Conjecture 3.4.

[13, Conjecture 1’] Let $E$ be an elliptic curve over an imaginary quadratic
field $F$ satisfying the above conditions, then,

$\operatorname{rank}E(F^{(n)})\leq\operatorname{rank}E(F_{\operatorname{cyc}})p^{n}.$

######  Corollary 3.5.

Consider the setting of the above conjecture. Under Assumption 2.3, the
Theorem 2.5 specializes to give that

$\operatorname{rank}E(F^{(n)})\leq\lambda_{p}(E/F_{\operatorname{cyc}})p^{n}.$

Thus, the Conjecture is true when

$\lambda_{p}(E/F_{\operatorname{cyc}})=\operatorname{rank}E(F_{\operatorname{cyc}}).$

######  Remark 3.6.

Note that
$\lambda_{p}(E/F_{\operatorname{cyc}})\geq\operatorname{rank}E(F^{(n)})$ for
all $n$ and hence,
$\lambda_{p}(E/F_{\operatorname{cyc}})\geq\operatorname{rank}E(F_{\operatorname{cyc}})$.
Indeed, it can be expected that
$\lambda_{p}(E/F_{\operatorname{cyc}})=\operatorname{rank}E(F)$ for $100\%$ of
primes above which $E$ has good ordinary reduction. There is much evidence
pointing towards this expectation for elliptic curves defined over the
rationals, see [17]. We do expect that similar arguments do carry over to
elliptic curves over imaginary quadratic fields.

###  3.2. False-Tate curve extensions

et $\ell$ 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

$F_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Q}(\mu_{p^{\infty}},\ell^{\frac{1}{p^{\infty}}}).$

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 2.5 specializes to give us
that

$\operatorname{rank}E(F^{(n)})\leq\lambda_{p}(E/F_{\operatorname{cyc}})p^{n}+(p^{n}-1)\left(q_{1}+2q_{2}\right).$

Let us compute the values of $q_{1}$ and $q_{2}$ for a given example. We note
that it is difficult to compute $\lambda_{p}(E/F_{\operatorname{cyc}})$ due to
the base change to $F=\mathbb{Q}(\mu_{p})$.

_Example:_ We pick an elliptic curve and prime at random. Let $E=\href
https://www.lmfdb.org/EllipticCurve/Q/11a2/$ in Cremona label and $p=7$. The
elliptic curve is defined over $\mathbb{Q}$ and we consider its base change to
$F=\mathbb{Q}(\mu_{7})$. Since $F$ is an abelian extension of $\mathbb{Q}$,
the Selmer group
$\operatorname{Sel}_{7^{\infty}}(E/\mathbb{Q}(\mu_{7^{\infty}}))$ is cotorsion
as a $\mathbb{Z}_{p}\llbracket x\rrbracket$-module. The residual
representation at $p=7$ is surjective, hence, after base change to
$\mathbb{Q}(\mu_{7})$ it will surject onto
$\operatorname{PSL}_{2}(\mathbb{F}_{7})$, in particular, it will remain
irreducible. In this setting, it is reasonable to expect that the
$\mu$-invariant $\mu_{7}(E/F_{\operatorname{cyc}})=0$, however, this needs to
be assumed. Let $\ell=11$ and consider the False Tate extension

$F_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Q}(\mu_{7^{\infty}},11^{\frac{1}{7^{\infty}}}).$

Then, $E$ has split multiplicative reduction at $\ell$, 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 $7$ in
$F_{\operatorname{cyc}}=\mathbb{Q}(\mu_{7^{\infty}})$. It follows that
$q_{1}=2$. Putting it all together, we find that

$\operatorname{rank}E(F^{(n)})\leq\lambda_{7}(E/F_{\operatorname{cyc}})p^{n}+2(p^{n}-1).$

In this context, we prove a statistical result.

######  Corollary 3.7.

Consider the elliptic curve $E=11a2$ from the example above and set $p=7$.
Given any integer $n\geq 1$, there is a positive density set of primes $\ell$
such that

$\operatorname{rank}E\left(\mathbb{Q}(\mu_{7^{n+1}},\ell^{\frac{1}{7^{n}}})\right)\leq\lambda_{7}(E/F_{\operatorname{cyc}})p^{n}.$

###### Proof.

For each prime number $\ell\neq 7$ 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 $E$ is ramified 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 $\ell$, 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 $\ell$,

  3. (3)

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

This latter condition is satisfied for $\frac{1}{7}$ of all primes $\ell$, see
[3, section 2] for further details. 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 $0$. ∎

###  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 the dimension of $G$ is $d=4$. In this setting, $F=\mathbb{Q}(E[p])$, and
$F^{(n)}=\mathbb{Q}(E[p^{n+1}])$. We find that

$\operatorname{rank}E(F_{n})\leq\lambda_{p}\left(E/\mathbb{Q}(\mu_{p^{\infty}})\right)p^{3n}+q_{1}(p^{3n}-p^{2n}),$

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
[13, Remark after Theorem 3.2] the result of Hung-Lim [13, Theorem 3.2] gives

$\operatorname{rank}E(F_{n})\leq(\lambda_{p}\left(E/\mathbb{Q}(\mu_{p^{\infty}})\right)+q_{1})p^{3n}+8,$

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

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