# A positive density of elliptic curves are diophantine stable in certain
Galois Extensions

Anwesh Ray  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>and  Pratiksha
Shingavekar  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>
###### Abstract.

Let $p\in\\{3,5\\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We
show that there exists an effective positive density of elliptic curves $E$
defined over $\mathbb{Q}$, ordered by height, that are _diophantine stable_ in
$L$.

###### 2020 Mathematics Subject Classification:

##  1\. Introduction

###  1.1. Background and motivation

Let $E$ be an elliptic curve over a number field $K$ and $L$ be a finite
extension of $K$. Following Mazur and Rubin [MR18], $E$ is said to be
_diophantine stable_ in $L$ if the natural inclusion induces an equality
$E(K)=E(L)$. Mazur and Rubin show that if all $\bar{K}$-automorphisms of $E$
are defined over $K$, then there exist infinitely many prime-power cyclic
extensions $L/K$ in which $E$ is diophantine stable. We refer to [MR18,
Theorem 1.2 and Theorem 1.3] for further details. For elliptic curves over
$\mathbb{Q}$, there are precise heuristics for the growth of the rank, due to
David, Fearnley and Kisilevsky [DFK07]. These heuristics are based on the
relationship between modular symbols and the special values of the Hasse-Weil
$L$-function $\mathscr{L}_{E}(s,L)$. More precisely, let $L/\mathbb{Q}$ be a
Galois extension with cyclic Galois group
$G=\operatorname{Gal}(L/\mathbb{Q})$. Then the Hasse-Weil $L$-function of $E$
over $L$ decomposes as follows
$\mathscr{L}_{E}(s,L)=\prod_{\chi}\mathscr{L}_{E}(s,\chi)$, where $\chi$
ranges over all Dirichlet characters $\chi\mathrel{\mathop{\mathchar
58\relax}}\widehat{G}\rightarrow\mathbb{C}^{\times}$. The Birch and
Swinnerton–Dyer conjecture predicts that

$\operatorname{rank}E(L)=\operatorname{ord}_{s=1}\mathscr{L}_{E}(s,L)=\sum_{\chi\in\widehat{G}}\operatorname{ord}_{s=1}\mathscr{L}_{E}(s,\chi).$

Given a character $\chi$, set $\operatorname{cond}(\chi)$ to denote its
conductor. One has the following conjecture which predicts that the rank
_rarely jumps_ in prime cyclic extensions.

######  Conjecture 1.1 (Conjecture 1.2 of [DFK07]).

Let $p$ be an odd prime and fix an elliptic curve $E_{/\mathbb{Q}}$. Setting

$N_{E,p}(X)\mathrel{\mathop{\mathchar 58\relax}}=\\#\\{\chi\text{ of order
}p\mid\operatorname{cond}(\chi)\leq X\text{ and
}\mathscr{L}_{E}(1,\chi)=0\\},$

one has the following predictions.

  1. (1)

If $p=3$, then as $X\to\infty$,

$\log N_{E,p}(X)\sim\frac{1}{2}\log X.$

  2. (2)

If $p=5$, then as $X\to\infty$, $N_{E,p}(X)$ is unbounded but $N_{E,p}(X)\ll
X^{\epsilon}$ for any $\epsilon>0$.

  3. (3)

If $p\geq 7$, then $N_{E,p}(X)$ is bounded.

In particular, the conjecture predicts that when $p\geq 7$, there are only
finitely many $p$-cyclic extensions in which the rank increases. There has
been an increasing interest in the study of diophantine stability and rank
growth of a given elliptic curve in families of number field extensions, see
for instance [MR08, LOT21, SW23, BKR24, BRY24]. Such considerations also
motivate conjectures about the distribution of modular symbols associated with
a given elliptic curves, cf. [MR23].

###  1.2. Main result

In this article we formulate a new direction of enquiry. Instead of fixing an
elliptic curve and varying the number field extension in a family, we shall
fix an extension $L/\mathbb{Q}$ and count the number of elliptic curves
$E_{/\mathbb{Q}}$, ordered by height, such that $E(L)=E(\mathbb{Q})$. The
result is proven only for cyclic extensions $L/\mathbb{Q}$ with
$\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$, where
$p\in\\{3,5\\}$.

######  Theorem A (Theorem 3.8).

Let $p$ be a prime in $\\{3,5\\}$ and $L/\mathbb{Q}$ be a Galois extension of
$\mathbb{Q}$ with
$\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$. Take $Z$ to
denote the primes that ramify in $L$. Assume that

  1. (1)

$2,p\notin Z$,

  2. (2)

for all $\ell\in Z$, there exists an elliptic curve $\mathbb{E}$ over
$\mathbb{F}_{\ell}$, such that $\mathbb{E}(\mathbb{F}_{\ell})[p]=0$ and
$\mathbb{E}^{-1}(\mathbb{F}_{\ell})[p]=0$. Here, $\mathbb{E}^{-1}$ is the
quadratic twist of $\mathbb{E}$ by $-1$.

Then there is a positive density of elliptic curves $E_{/\mathbb{Q}}$ such
that

$E(L)=E(\mathbb{Q})=0.$

Evidently, these elliptic curves are _diophantine stable_ in $L$.

######  Remark 1.2.

At this point, several remarks are in order.

  * •

The results rely on the techniques of Bhargava and Shankar [BS15a, BS15b,
BS13], who study the average size of the $p$-Selmer group of elliptic curves
$E_{/\mathbb{Q}}$. This is only done for primes $p=2,3,5$, however, the case
when $p=2$ is omitted since our methods do not apply to this case.

  * •

The condition (2) can be dealt with via explicit calculation (cf. the code in
Remark 4.7). Indeed, when $p=5$ and $\ell$ is large enough, the condition (2)
is shown to be be satisfied using results of Howe [How93]. We refer to Lemma
4.6 for further details.

  * •

An explicit lower density is obtained for the curves $E_{/\mathbb{Q}}$ that
are diophantine stable in $L$. This expression is quite fascinating and
involves locally defined invariants associated to the extension
$L/\mathbb{Q}$, see the statement of Theorem 3.8 for details. For instance,
when $p=3$ and $L=\mathbb{Q}(\mu_{7})^{+}$, the lower density of elliptic
curves $E_{/\mathbb{Q}}$ that are diophantine stable in $L$ is

$\geq\frac{1}{4}\times\frac{1}{2^{21}}\times\left(1-\frac{1}{3}\right)\times\frac{12}{49}\times\prod_{\ell\neq
2,3,7}\left(1-\frac{2}{\ell^{2}}+\frac{1}{\ell^{3}}\right).$

###  1.3. Organization

Including the introduction, the article consists of four sections. In section
2, we study the stability of Selmer groups in prime cyclic extensions. In
greater detail, let $p$ be an odd prime number and $L/K$ be an extension with
$\operatorname{Gal}(L/K)\simeq\mathbb{Z}/p\mathbb{Z}$. Suppose that the
$p$-Selmer group of $E$ over $K$ is $0$. Then if certain additional local
condition are satisfied for $E_{/K}$, then, it follows that the $p$-Selmer
group of $E$ over $L$ is also $0$. The Galois cohomological arguments
presented in this section are essentially due to Česnavičius [Č17] and Brau
[Bra14]. Section 3 is preparatory in nature, and delineates the strategy that
we take to prove our main result. We introduce both local and global density
conditions on our families of elliptic curves that will play a role in our
sieve theoretic arguments. Section 4 is devoted to the proof of our main
result. In this section, we describe a suitable large family of elliptic
curves defined by local conditions and show that a positive proportion of
elliptic curves in this family are indeed diophantine stable in $L$. The
analysis leads to an explicit lower bound for the density of elliptic curves
that are diophantine stable in our fixed extension $L$. This is precisely the
step in which the aforementioned results of Bhargava and Shankar are applied.

### Data availability

No data was analyzed in proving the results in the article.

##  2\. Selmer groups and diophantine stability

In this section, we review the necessary background and notation required for
our examination of elliptic curves over number fields. Some of the results in
this section essentially follow arguments due to Česnavičius [Č17] and Brau
[Bra14]. Throughout, we denote $K$ as a number field. We use $\Omega(K)$
(resp. $\Omega_{\infty}(K)$) to represent the set of finite primes (resp.
archimedean places) of $K$. Given a prime number $\ell$, $\Omega_{\ell}(K)$
denotes the primes in $\Omega(K)$ that lie above $\ell$. Let $\overline{K}$ be
an algebraic closure of $K$, and $\operatorname{G}_{K}$ denote the absolute
Galois group $\operatorname{Gal}(\overline{K}/K)$. For each prime
$v\in\Omega(K)$, we select an embedding $\iota_{v}\mathrel{\mathop{\mathchar
58\relax}}\overline{K}\hookrightarrow\overline{K}_{v}$. Setting
$\operatorname{G}_{K_{v}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\overline{K}_{v}/K_{v})$, the inclusion
$\iota_{v}$ induces a Galois group inclusion
$\iota_{v}^{*}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{K}\hookrightarrow\operatorname{G}_{K_{v}}$.

###  2.1. Selmer groups associated to elliptic curves

Let $E$ be an elliptic curve defined over $K$. The Mordell-Weil group of $E$
over $K$, denoted $E(K)$, comprises the $K$-rational points on $E$. Choose an
algebraic closure $\overline{K}$ of $K$, and let $E[p^{n}]$ be the
$p^{n}$-torsion subgroup of $E(\overline{K})$. Let $\operatorname{G}_{K}$
denote the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$ and
consider the natural action of $\operatorname{G}_{K}$ on $E[p^{n}]$. The
Selmer group $\operatorname{Sel}_{p^{n}}(E/K)$ can be interpreted as a certain
subgroup of the cohomology group $H^{1}(K,E[p^{n}])$, which measures the
obstructions to representing $K$-rational points of $E$ using elements of
$E[p^{n}]$. It is defined as the kernel to the natural restriction map in
Galois cohomology

$\operatorname{Sel}_{p}(E/K)=\ker\left(H^{1}(K,E[p^{n}])\to\bigoplus_{v}H^{1}(K_{v},E)[p^{n}]\right),$

where $v$ runs over all primes of $K$.

The $p^{n}$-Selmer group is closely related to the Mordell–Weil group and
Tate–Shafarevich group. Indeed, there is a natural short exact sequence that
describes this relationship

$0\rightarrow
E(K)/p^{n}E(K)\rightarrow\operatorname{Sel}_{p^{n}}(E/K)\rightarrow\Sh(E/K)[p^{n}]\rightarrow
0.$

Thus in particular, $\operatorname{Sel}_{p^{n}}(E/K)$ is trivial if and only
if

  * •

$E(K)$ has rank $0$,

  * •

$E(K)[p]=0$,

  * •

$\Sh(E/K)[p^{\infty}]=0$.

###  2.2. Growth and stability of Selmer groups

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $K$ be a number field. Let $p$
be an odd prime number. Fix a finite Galois extension $L/K$ for which
$G\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(L/K)$ is a
$p$-group and assume that $G\simeq\mathbb{Z}/p\mathbb{Z}$. Set $k_{v}$ to
denote the residue field of $K$ at $v$. If $v$ is a prime of good reduction,
we set $\widetilde{E}(k_{v})$ to be the group of $k_{v}$-rational points of
the mod-$v$ reduction of $E$. Let $c_{v}(E/K)$ be the Tamagawa number of $E$
at $v$. Throughout, we impose the following assumption.

######  Assumption 2.1.

Throughout, we assume that the following conditions hold.

  1. (1)

The Selmer group $\operatorname{Sel}_{p}(E/K)=0$. In particular, this implies
that the rank of $E(K)$ is $0$ and both $E(K)$ and $\Sh(E/K)$ have no
nontrivial $p$-torsion.

  2. (2)

At all primes $v$ of $K$ that lie above $p$, assume that $E$ has good
reduction.

  3. (3)

If $v\in\Omega_{p}(K)$ is ramified in $L$, we additionally require that $E$
has ordinary reduction at $v$ and $\widetilde{E}(k_{v})[p]=0$.

  4. (4)

For all primes $v\nmid p$ that are ramified in $L$, we require that $E$ has
good reduction at $v$. Moreover, for such primes $v$, we have that
$\widetilde{E}(k_{v})[p]=0$.

  5. (5)

At all primes $v\nmid p$ at which $E$ has split multiplicative reduction, we
require that $p\nmid c_{v}(E/K)$.

Define $S$ as the set of primes $v\in\Omega(K)$ satisfying at least one of the
following conditions:

  * •

$v\in\Omega_{p}(K)$,

  * •

$E$ has bad reduction at $v$,

  * •

$v$ is ramified in $L$.

Let $S(L)$ denote the set of primes $w\in\Omega(L)$ that lie above some prime
$v\in S$. The natural restriction map

$\alpha\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Sel}_{p}(E/K)\rightarrow\operatorname{Sel}_{p}(E/L)^{G}$

fits within the following fundamental diagram

(2.1)
${0=\operatorname{Sel}_{p}(E/K)}$${H^{1}(K_{S}/K,E[p])}$${\bigoplus_{v\in
S}H^{1}(K_{v},E)[p]}$${0}$${0}$${\operatorname{Sel}_{p}(E/L)^{G}}$${H^{1}(L_{S}/L,E[p])^{G}}$${\left(\bigoplus_{w\in
S(L)}H^{1}(L_{w},E)[p]\right)^{G}.}$$\scriptstyle{\alpha}$$\scriptstyle{\Phi_{K}}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

Here, $\beta$, and $\gamma$ are restriction maps on global and local
cohomology respectively. Since it is assumed that
$\operatorname{Sel}_{p}(E/K)=0$, it follows from the Cassels-Poitou-Tate long
exact sequence that $\Phi_{K}$ is surjective. For $v\in S$, the restriction
map is denoted by

$\gamma_{v}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(K_{v},E)[p]\rightarrow\left(\bigoplus_{w|v}H^{1}(L_{w},E)[p]\right)^{G},$

where $w$ runs through all primes of $L$ above $v$. We note that $\gamma$ is
the direct sum $\bigoplus_{v\in S}\gamma_{v}$.

######  Lemma 2.2.

Consider a finite dimensional $\mathbb{F}_{p}$-vector space $V$ and the finite
$p$-group $G$ acting on $V$. If $V^{G}=0$, then it follows that $V=0$.

###### Proof.

The result follows via an easy application of Nakayama’s lemma. For further
details, cf. [MR24, Lemma 4.2]. ∎

######  Proposition 2.3.

With respect to notation above, assume that $\gamma_{v}$ is injective for all
$v\in S$. Then it follows that $\operatorname{Sel}_{p}(E/L)=0$, and
consequently, the rank of $E(L)$ is also 0.

###### Proof.

From (2.1), we arrive at the following long exact sequence

(2.2)
$0=\operatorname{ker}(\alpha)\rightarrow\operatorname{ker}(\beta)\rightarrow\operatorname{ker}(\gamma)\rightarrow\operatorname{cok}(\alpha)\rightarrow\operatorname{cok}(\beta)\rightarrow\operatorname{cok}(\gamma).$

Since we assume that $\operatorname{Sel}_{p}(E/K)=0$, it follows that
$E(K)[p]=0$. From the inflation-restriction sequence

$0\rightarrow H^{1}(L/K,E(L)[p])\rightarrow
H^{1}(K_{S}/K,E[p])\xrightarrow{\beta}H^{1}(K_{S}/L,E[p])^{G}\rightarrow
H^{2}(L/K,E(L)[p])$

it follows that $\beta$ is an isomorphism. This implies that
$\operatorname{ker}\gamma$ is isomorphic to the $\operatorname{cok}\alpha$.
Since the maps $\gamma_{v}$ are injective, so is $\gamma$. Hence,
$\operatorname{Sel}_{p}(E/L)^{G}=0$. The result then follows from Lemma 2.2,
which asserts that $\operatorname{Sel}_{p}(E/L)=0$. ∎

We are now left with studying the maps $\gamma_{v}$ for $v\in S$, and the
conditions under which they are injective. We first deal with the case in
which $v\nmid p$. It follows from the Assumption 2.1 that either

  * •

$v$ is totally ramified in $L$ and that $E$ has good reduction at $v$,

  * •

$v$ is unramified in $L$ and $E$ has nonsplit multiplicative or additive
reduction at $v$,

  * •

$v$ is unramified in $L$, $E$ has split multiplicative reduction at $v$ and
$p\nmid c_{v}(E/K)$.

######  Lemma 2.4.

Let $v\nmid p$ be a prime in $S$ at which $E$ has good reduction. Then
$\gamma_{v}$ is injective.

###### Proof.

It follows from the definition of $S$ that $v$ must be totally ramified in
$L$. Recall that Assumption 2.1 requires that $\widetilde{E}(k_{v})[p]=0$. The
result then follows from [PR24, Proposition 2.8]. ∎

######  Lemma 2.5.

Let $v\nmid p$ be a prime in $S$ at which $E$ has bad reduction. Then,
$\gamma_{v}$ is injective.

###### Proof.

First we consider the case in which $E$ has nonsplit multiplicative reduction
or additive reduction at $v$. In this case, the result follows from [Bra14,
Proposition 5.8]. On the other hand, consider the case in which $E$ has split
multiplicative reduction. Note that in light of Assumption 2.1, $v$ must be
unramified in $L$ and $p\nmid c_{v}(E/K)$. Then, the result follows from
[Bra14, Proposition 5.7, (ii)]. ∎

Next, we show that the maps $\gamma_{v}$ are injective for all primes $v|p$.

######  Lemma 2.6.

Let $v|p$ be a prime (in $S$), then the map $\gamma_{v}$ is injective.

###### Proof.

It follows from Assumption 2.1 that $E$ has good reduction at $v$. There are
three cases to consider.

  * •

First, assume that $v$ is unramified in $L$ and that $E$ has good ordinary
reduction at $v$. In this case, $\gamma_{v}$ is injective by [Bra14,
Proposition 5.9, (ii)].

  * •

Next consider the case when $v$ is ramified in $L$ and $E$ has good ordinary
reduction at $v$. In this case, it has been assumed that
$\widetilde{E}(k_{v})[p]=0$, by Assumption 2.1. In this case, $\gamma_{v}$ is
injective, according to [Bra14, Proposition 5.9, (i)].

  * •

Consider the case when $v$ is unramified in $L$ and $E$ has good supersingular
reduction at $v$. Then, $\gamma_{v}$ is injective by [Bra14, Proposition 5.10,
(ii)].

∎

######  Proposition 2.7.

Let $E$ be an elliptic curve satisfying the conditions of Assumption 2.1.
Then, we have that $\operatorname{Sel}_{p}(E/L)=0$.

###### Proof.

By Proposition 2.3, it suffices to show that the maps $\gamma_{v}$ are
injective for all $v\in S$. Lemma 2.4 (resp. 2.5) asserts that $\gamma_{v}$ is
injective when $v\nmid p$ is a prime in $S$ of good (resp. bad) reduction for
$E$. For primes $v|p$, $\gamma_{v}$ is injective by Lemma 2.6. The result thus
follows. ∎

##  3\. Asymptotics for elliptic curves

In this section, we shall fix a prime $p\in\\{3,5\\}$ and a
$\mathbb{Z}/p\mathbb{Z}$-extension $L/\mathbb{Q}$. Given an elliptic curve
$E_{/\mathbb{Q}}$, it is well known that up to isomorphism $E$ admits a short
Weierstrass equation of the form

$E_{A,B}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+Ax+B,$

where $(A,B)\in\mathbb{Z}^{2}$ are such that for all primes $\ell$, either
$\ell^{4}\nmid A$ or $\ell^{6}\nmid B$. Such an equation is _globally minimal_
and the pair $(A,B)$ is uniquely determined. The discriminant of $E_{A,B}$ is
given by

$\Delta_{A,B}=\Delta(E_{A,B})=-16(4A^{3}+27B^{2}),$

and the _$j$ -invariant_ is

$j_{A,B}=j(E_{A,B})=\frac{2^{8}3^{3}A^{3}}{4A^{3}+27B^{2}}.$

We shall denote by $\mathcal{C}$ the set of all pairs of integers $(A,B)$ such
that $\ell^{4}\nmid A$ or $\ell^{6}\nmid B$ for all primes $\ell$. The
association taking $(A,B)$ to the isomorphism class of $E_{A,B}$, gives a
natural parametrization of isomorphism classes of elliptic curves
$E_{/\mathbb{Q}}$.

This parametrization of elliptic curves allows us to formulate counting
questions. Given an elliptic curve $E_{A,B}$ the _naive height_ of $E_{A,B}$
is defined as follows

$h_{A,B}=h\left(E_{A,B}\right)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{|A|^{3},|B|^{2}\\}.$

The elliptic curves we shall consider shall be ordered according to their
height. Let us explain this further. Let $X>0$ be a real number and
$\mathcal{C}(X)$ to be the family of elliptic curves $E_{A,B}$ with height at
most $X$, i.e.,

$\mathcal{C}(X)\mathrel{\mathop{\mathchar
58\relax}}=\\{(A,B)\in\mathcal{C}\mathrel{\mathop{\mathchar
58\relax}}h\left(E_{A,B}\right)\leq X\\}.$

It is easy to see that the set $\mathcal{C}(X)$ is finite, in fact Brumer
proves a precise asymptotic.

######  Lemma 3.1.

Let $\mathcal{C}(X)$ be given as above. One has the following asymptotic
estimate for the growth of $\\#\mathcal{C}(X)$, as $X$ goes to $\infty$:

$\\#\mathcal{C}(X)=\frac{4X^{5/6}}{\zeta(10)}+O\left(\sqrt{X}\right).$

###### Proof.

For a proof of this result, see [Bru92, Lemma 4.3]. ∎

######  Definition 3.2.

Any set of isomorphism classes of elliptic curves over $\mathbb{Q}$ can be
identified with a subset $S$ of $\mathcal{C}$. Given $X>0$, set
$S(X)\mathrel{\mathop{\mathchar 58\relax}}=S\mathcal{A}p\mathcal{C}(X)$. The
density of $S$ (if it exists) is defined as follows

(3.1) $\mathfrak{d}(S)\mathrel{\mathop{\mathchar
58\relax}}=\lim_{X\rightarrow\infty}\frac{\\#S(X)}{\\#\mathcal{C}(X)}.$

When the limit in the above expression is replaced by $\limsup$ (resp
$\liminf$), we write $\overline{\mathfrak{d}}(S)$ (resp.
$\underline{\mathfrak{d}}(S)$) to denote the upper (resp. lower) density. The
set $S$ is said to have _positive density_ if $\underline{\mathfrak{d}}(S)>0$.

######  Remark 3.3.

It is worth noting that the density $\mathfrak{d}(S)$ is a quantity that need
not be defined when the limit (3.1) does not exist. However, the upper and
lower densities are always well defined.

The rank of an elliptic curve is a fundamental and subtle invariant. It is
natural to view the rank as a random function of the set of all elliptic
curves and study its distribution. Katz and Sarnak modeled the distribution of
ranks of elliptic curves and conjectured the following.

######  Conjecture 3.4 (Rank distribution conjecture).

The density of elliptic curves $E_{/\mathbb{Q}}$ with rank $0$ (resp. $1$) is
$1/2$ (resp. $1/2$).

The conjecture thus also predicts that the density of elliptic curves with
rank $\geq 2$ is $0$. Although this conjecture is open, significant progress
was made in seminal work of Bhargava and Shankar [BS15a, BS15b, BS13]. The
idea is to relate Selmer groups of elliptic curves to integral forms. The
integral forms are lattice points in certain fundamental regions in Euclidean
space and techniques from the geometry of numbers are applied to study their
distribution. Given a prime $p$, it is expected that the average size of the
$p$-Selmer group of $E_{/\mathbb{Q}}$ is $(p+1)$. This has been proven to be
the case for primes $p\leq 5$. We refer to work of Poonen and Rains [PR12] for
predictions on the average size of Selmer groups and Tate–Shafarevich groups.

In this section, we fix a Galois extension $L/\mathbb{Q}$ with Galois group
$\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$, where
$p\in\\{3,5\\}$. Let $E_{/\mathbb{Q}}$ be an elliptic curve. Following Mazur
and Rubin [MR18], $E$ is said to be _diophantine stable_ in $L$ if the natural
inclusion yields an equality $E(\mathbb{Q})=E(L)$. We assume throughout that
$2$ and $p$ are unramified in $L$, and show that there is an effective
positive density of elliptic curves $E_{/\mathbb{Q}}$ that are diophantine
stable in $L$. This is done by showing that

  * •

the set of elliptic curves $E_{/\mathbb{Q}}$ for which
$\operatorname{E}(L)_{\operatorname{tors}}=0$ has density $1$.

  * •

The set of elliptic curves $E_{/\mathbb{Q}}$ for which
$\operatorname{rank}E(L)=0$ has positive density.

It then follows that the set of elliptic curves $E_{/\mathbb{Q}}$ for which
both $E(\mathbb{Q})=E(L)=0$ has positive density. We shall obtain an effective
lower bound for the lower density of this set of elliptic curves.

###  3.1. Controlling the torsion

Consider the set $S_{1}\subset\mathcal{C}$, of all isomorphism classes of
elliptic curves $E_{/\mathbb{Q}}$ such that $E(L)_{\operatorname{tors}}=0$. In
this section, we show that $S_{1}$ has density $1$, i.e.,

$\lim_{X\rightarrow\infty}\frac{\\#S_{1}(X)}{\\#\mathcal{C}(X)}=1.$

An elliptic curve $E_{/\mathbb{Q}}$ gives rise to a family of Galois
representations. Choose an algebraic closure $\overline{\mathbb{Q}}$ of
$\mathbb{Q}$ and set $\operatorname{G}_{\mathbb{Q}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Given a
natural number $n$, let $E[n]$ be the $n$-torsion subgroup of
$E(\overline{\mathbb{Q}})$. Note that
$E[n]\simeq\left(\mathbb{Z}/n\mathbb{Z}\right)^{2}$ and that there is a
natural action of $\operatorname{G}_{\mathbb{Q}}$ on $E[n]$. The associated
Galois representation

$\rho_{E,n}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{Aut}(E[n])\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{Z}/n\mathbb{Z})$

encodes many arithmetic properties of $E$. Suppose that $E$ does not have
complex multiplication. Then, Serre’s Open image theorem implies that for all
but finitely many primes $\ell$, the mod-$\ell$ representation $\rho_{E,\ell}$
is surjective. On the other hand, a prime $\ell$ for which $\rho_{E,\ell}$
fails to be surjective is called an _exceptional prime_. In this context, we
set $S_{1}^{\prime}\subset\mathcal{C}$ to be the set of isomorphism classes of
elliptic curves with no exceptional primes.

######  Theorem 3.5 (Duke [Duk97]).

The set $S_{1}^{\prime}$ has density $1$, i.e., almost all isomorphism classes
of elliptic curves (over $\mathbb{Q}$) have no exceptional primes.

Thus, in order to show that $S_{1}$ has density $1$, it suffices to show that
$S_{1}^{\prime}$ is contained in $S_{1}$. In other words, if an elliptic curve
$E_{/\mathbb{Q}}$ has the property that it has no exceptional primes, then, it
follows that $E(L)_{\operatorname{tors}}=0$. This assertion follows from the
following result.

######  Lemma 3.6.

Let $L/\mathbb{Q}$ be a $\mathbb{Z}/p\mathbb{Z}$-extension (where $p=3$ or
$5$) and let $\ell$ be any prime number. Let $E_{/\mathbb{Q}}$ be an elliptic
curve for which the representation

$\rho_{E,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$

is surjective. Then, we have that $E(L)[\ell]=0$.

###### Proof.

By way of contradiction, assume that $E(L)[\ell]\neq 0$. Thus,
$\rho_{E,\ell}(\operatorname{G}_{L})$ is contained in the mirabolic subgroup
$M\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left({\begin{array}[]{cc}1&\ast\\\ 0&\ast\\\
\end{array}}\right)\right\\}$ of
$\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$. In particular, we find
that

$[\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}\rho_{E,\ell}(\operatorname{G}_{L})]\geq[\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}M]=\frac{(\ell^{2}-\ell)(\ell^{2}-1)}{(\ell-1)\ell}=\ell^{2}-1.$

On the other hand,

$[\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}\rho_{E,\ell}(\operatorname{G}_{L})]=[\rho_{E,\ell}(\operatorname{G}_{\mathbb{Q}})\mathrel{\mathop{\mathchar
58\relax}}\rho_{E,\ell}(\operatorname{G}_{L})]\leq[\operatorname{G}_{\mathbb{Q}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{L}]\leq 5.$

Thus, we find that $\ell^{2}-1\leq 5$, i.e., $\ell=2$.

Assume therefore that $\ell=2$. Note that
$\operatorname{GL}_{2}(\mathbb{Z}/2\mathbb{Z})\simeq S_{3}$, and $M\simeq
C_{2}$. Since $\operatorname{G}_{L}$ is a normal subgroup of
$\operatorname{G}_{\mathbb{Q}}$, it follows that $C_{2}$ is normal in $S_{3}$,
which is a contradiction. Thus, we have shown that $E(L)[\ell]=0$. ∎

######  Theorem 3.7.

The set $S_{1}$ has density $1$.

###### Proof.

Recall that Theorem 3.5 asserts that the density of the set $S_{1}^{\prime}$
is $1$. Hence, to prove the result, it suffices to show that
$S_{1}^{\prime}\subset S_{1}$. Let $E$ be an elliptic curve in
$S_{1}^{\prime}$. This means by definition, that for all primes $\ell$, the
Galois representation $\rho_{E,\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$
is surjective. Then by Lemma 3.6, we get that $E(L)[\ell]=0$ for all primes
$\ell$. This in turn implies that $E(L)_{\operatorname{tors}}=0$ and hence
$E\in S_{1}$. This completes the proof. ∎

###  3.2. Controlling the rank

In this section, we take $S_{2}\subset\mathcal{C}$ to consist of isomorphism
classes of elliptic curves $E_{/\mathbb{Q}}$ such that
$\operatorname{rank}E(L)=0$. Given a pair
$(\overline{A},\overline{B})\in(\mathbb{Z}/\ell\mathbb{Z})^{2}$ with
$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$, let
$E_{\overline{A},\overline{B}}\mathrel{\mathop{\mathchar
58\relax}}y^{2}=x^{3}+\overline{A}x+\overline{B}$. Let $\ell$ be a prime that
ramifies in $L$, then set $\mathfrak{A}_{\ell}$ to denote the number of pairs
$(\overline{A},\overline{B})\in(\mathbb{Z}/\ell\mathbb{Z})^{2}$ such that

  * •

$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$,

  * •

$E_{\overline{A},\overline{B}}(\mathbb{F}_{\ell})[p]=0$ and
$E_{\overline{A},-\overline{B}}(\mathbb{F}_{\ell})[p]=0$.

We show in this section that $S_{2}$ has positive density, and note that the
set $S_{1}\cap S_{2}$ consists of isomorphism classes of elliptic curves
$E_{/\mathbb{Q}}$ such that $E(L)=0$. Note that in particular,
$E(\mathbb{Q})=0$ and $E(L)=0$, hence $E$ is diophantine stable in the fixed
extension $L/\mathbb{Q}$ for all $E\in S_{1}\cap S_{2}$. Theorem 3.7 asserts
that $S_{1}$ has density $1$. Thus, if $S_{2}$ has positive density, then so
does $S_{1}\cap S_{2}$. This then yields the following result.

######  Theorem 3.8.

Let $p$ be a prime in $\\{3,5\\}$ and $L/\mathbb{Q}$ be a Galois extension of
$\mathbb{Q}$ with
$\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$. Take $Z$ to
denote the primes that ramify in $L$. Assume that

  1. (1)

$2,p\notin Z$,

  2. (2)

for all $\ell\in Z$, there exists an elliptic curve $\mathbb{E}$ over
$\mathbb{F}_{\ell}$, such that $\mathbb{E}(\mathbb{F}_{\ell})[p]=0$ and
$\mathbb{E}^{-1}(\mathbb{F}_{\ell})[p]=0$. Here, $\mathbb{E}^{-1}$ is the
quadratic twist of $\mathbb{E}$ by $-1$.

Then there is a positive density of elliptic curves $E_{/\mathbb{Q}}$ such
that

$E(L)=E(\mathbb{Q})=0.$

Evidently, these elliptic curves are _diophantine stable_ in $L$. Moreover,
the lower density of this set of elliptic curves is at least
$\eta_{p}\prod_{\ell}\delta_{\ell}$, where

$\eta_{p}\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}\frac{1}{4}&\text{
if }p=3;\\\ \frac{3}{8}&\text{ if }p=5;\end{cases}$

and

$\delta_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\frac{2}{3}&\text{ if }\ell\notin Z\cup\\{2,p\\}\text{
and }\ell=3;\\\ 1-\frac{2}{\ell^{2}}+\frac{1}{\ell^{3}}&\text{ if }\ell\notin
Z\cup\\{2,p\\}\text{ and }\ell\neq 3;\\\ 1-\frac{1}{\ell}&\text{ if
}\ell=p;\\\ \frac{1}{2^{21}}&\text{ if }\ell=2;\\\
\frac{\\#\mathfrak{A}_{\ell}}{\ell^{2}}&\text{ if }\ell\in Z.\end{cases}$

We note that (2) is a condition on the prime $\ell$, and it is shown that when
$p=5$ and $\ell\geq 5779$ (resp. $p=3$ and $\ell\equiv 1\pmod{4}$), this
condition is satisfied, see Lemma 4.6. We shall also give explicit lower
bounds for the density of the set of elliptic curves that are diophantine
stable in $L$ (in terms of $E$ and the set of primes that ramify in $L$). We
shall show that $\operatorname{Sel}_{p}(E/L)=0$ for a positive density of
elliptic curves $E_{/\mathbb{Q}}$. This is achieved by applying Proposition
2.7 by showing that Assumption 2.1 (for $K\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Q}$ and $p\in\\{3,5\\}$) is satisfied for a positive
density of elliptic curves. Recall that $p$ is unramified in $L$, and hence,
Assumption 2.1 (3) is satisfied.

######  Definition 3.9.

Given any prime $\ell$, let $\Sigma_{\ell}$ be a choice of a closed subset of
$\mathbb{Z}_{\ell}^{2}$ defined by a congruence condition. In other words,
there is a sufficiently large integer $n_{\ell}\geq 1$ and
$\overline{\Sigma}_{\ell}\subseteq\left(\mathbb{Z}/\ell^{n_{\ell}}\mathbb{Z}\right)^{2}$
such that $(A,B)\in\Sigma_{\ell}$ if and only if

$\left(A\pmod{\ell^{n_{\ell}}},B\pmod{\ell^{n_{\ell}}}\right)\in\left(\mathbb{Z}/\ell^{n_{\ell}}\mathbb{Z}\right)^{2}.$

Let $E_{A,B}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+Ax+B$ be an
elliptic curve such that either $\ell^{4}\nmid A$ or $\ell^{6}\nmid B$. Assume
that choice of $\Sigma_{\ell}$ is made for every prime number. Then, we set
$\Sigma$ to denote the tuple of conditions $(\Sigma_{\ell})_{\ell}$, where
$\ell$ ranges over all prime numbers. We associate the family
$\mathcal{F}_{\Sigma}$ of elliptic curves to the tuple
$(\Sigma_{\ell})_{\ell}$, where $E_{A,B}\in\mathcal{F}_{\Sigma}$ if
$(A,B)\in\Sigma_{\ell}$ for all $\ell$. The family $\mathcal{F}_{\Sigma}$ is
then said to be defined by congruence conditions.

Consider the following example. Given a prime $\ell$, we let
$\overline{\Pi}_{\ell}\subset(\mathbb{Z}/\ell^{6}\mathbb{Z})^{2}$ consist of
all tuples $(\overline{A},\overline{B})$ such that $\ell^{4}\nmid\overline{A}$
and $\overline{B}\neq 0$. Let $\Pi_{\ell}\subset\mathbb{Z}_{\ell}^{2}$ be the
set of all tuples $(A,B)$ such that
$\left(A\pmod{\ell^{6}},B\pmod{\ell^{6}}\right)\in\overline{\Pi}_{\ell}$.
Thus, the set

$\Pi_{\ell}=\\{(A,B)\in\mathbb{Z}_{\ell}^{2}\mid\text{ either }\ell^{4}\nmid
A\text{ or }\ell^{6}\nmid B\\}$

corresponds to the minimal Weierstrass models of elliptic curves over
$\mathbb{Z}_{\ell}$.

If $\mathcal{F}$ is a family of elliptic curves defined by congruence
conditions, then we denote by $\operatorname{Inv}(\mathcal{F})$ the subset of
$\mathbb{Z}_{\ell}^{2}$ defined by

$\operatorname{Inv}(\mathcal{F})\mathrel{\mathop{\mathchar
58\relax}}=\\{(A,B)\mathrel{\mathop{\mathchar
58\relax}}E_{A,B}\in\mathcal{F}\\}$

and let $\big{(}\operatorname{Inv}(\mathcal{F})\big{)}_{\ell}$ be its
$\ell$-adic closure in $\mathbb{Z}_{\ell}^{2}$.

######  Definition 3.10.

A family $\mathcal{F}$ of elliptic curves defined by congruence conditions is
said to be large if the set
$\big{(}\operatorname{Inv}(\mathcal{F})\big{)}_{\ell}$ contains all pairs
$(A,B)\in\mathbb{Z}_{\ell}^{2}$ such that $\ell^{2}\nmid\Delta(E_{A,B})$, for
all but finitely many primes $\ell$.

Recall that the _root number_ $\omega(E)$ of an elliptic curve
$E_{/\mathbb{Q}}$ is the sign of the functional equation of the $L$-function
$L(E,s)$ of $E$. It is widely believed that when the elliptic curves are
ordered by height, the root numbers $+1$ and $-1$ occur equally often. The
following result of Bhargava and Shankar [BS15b, Theorem 41] then gives the
density of elliptic curves with rank $0$.

######  Theorem 3.11.

Let $p$ be a prime in $\\{3,5\\}$. Suppose $\mathcal{F}$ is a large family of
elliptic curves such that exactly $50\%$ of the curves in $\mathcal{F}$, when
ordered by height, have root number $+1$. Then if $p=3$ (resp. $p=5$) at least
$25\%$ (resp. $37.5\%$) of the curves $E_{A,B}$ in $\mathcal{F}$, when ordered
by height, have $\operatorname{Sel}_{p}(E_{A,B}/\mathbb{Q})=0$.

###### Proof.

The statement of the result above is a little stronger than that of [BS15b,
Theorem 41] (resp. [BS13, Proposition 40]) since it is asserted that
$\operatorname{Sel}_{p}(E_{A,B}/\mathbb{Q})=0$. The proof of the mentioned
Theorems rely on showing that this Selmer group is indeed $0$ for at least
$25\%$ (resp. $37.5\%$) of curves in $\mathcal{F}$. When $p=3$, we refer to p.
617, ll. 11-12 of [BS15b]. On the other hand, for $p=5$, see [BS13, p.29, l.
-7]. ∎

For an elliptic curve $E\mathrel{\mathop{\mathchar
58\relax}}=E_{A,B}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+Ax+B$ with
$A,B\in\mathbb{Z}$, denote by $E^{-1}\mathrel{\mathop{\mathchar
58\relax}}=E^{-1}_{A,B}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+Ax-B$
the twist by $-1$ of $E_{A,B}$. Note that $\Delta(E^{-1})=\Delta(E)$ and
$j(E^{-1})=j(E)$.

The global root number is a product of local root numbers

$\omega(E)=-\prod_{\ell}\omega_{\ell}(E).$

The local root number at a prime $\ell$ of multiplicative reduction is easy to
describe. One has that

(3.2) $\omega_{\ell}(E)=\begin{cases}+1&\text{ if }E\text{ has good reduction
or non-split multiplicative reduction at }\ell;\\\ -1&\text{ if }E\text{ has
split multiplicative reduction at }\ell.\end{cases}$

There is a simple criterion for this at a prime $\ell\geq 3$ of multiplicative
reduction. The elliptic curve $E=E_{A,B}$ has split multiplicative reduction
precisely when $\left(\frac{6B}{\ell}\right)=1$ (see [Won01, Lemma 10]). One
has that $\left(\frac{-1}{\ell}\right)=1$ if and only if $\ell\equiv
1\pmod{4}$. Therefore, we deduce that for any prime $\ell\geq 3$ at which $E$
has multiplicative reduction,

(3.3) $\omega_{\ell}(E^{-1})=\omega_{\ell}(E)\Leftrightarrow\ell\equiv
1\pmod{4}.$

######  Definition 3.12.

Let $Z$ be the set of primes $\ell$ of $\mathbb{Q}$ that ramify in $L$. Note
that $2,p\notin Z$ by assumption. Also, let $\mathcal{E}$ be the family of
elliptic curves defined by the following conditions

  1. (1)

$E$ has good reduction at $p$;

  2. (2)

$E$ has good reduction at every prime $\ell\in Z$. Moreover,
$\widetilde{E}(\mathbb{F}_{\ell})[p]=0$ and
$\widetilde{E}^{-1}(\mathbb{F}_{\ell})[p]=0$ for all $\ell\in Z$.

  3. (3)

Let $\Delta^{\prime}(E)$ be the positive prime to $2$ part of $\Delta(E)$
(i.e., $\Delta^{\prime}(E)\mathrel{\mathop{\mathchar
58\relax}}=\frac{|\Delta(E)|}{2^{v_{2}(\Delta(E))}}$). Then,
$\Delta^{\prime}(E)$ is squarefree and $\Delta^{\prime}(E)\equiv 1\pmod{4}$.

  4. (4)

Both $E$ and $E^{-1}$ have additive reduction at $2$ and $2\nmid j(E)$.

Let $\mathcal{E}^{0}$ be the subset of $\mathcal{E}$ consisting of isomorphism
classes of elliptic curves $E\in\mathcal{E}$ such that
$\operatorname{Sel}_{p}(E/\mathbb{Q})=0$.

######  Remark 3.13.

It follows from the conditions above that if $E$ is an elliptic curve in
$\mathcal{E}$, then $E^{-1}\in\mathcal{E}$ as well. This is simply because
$\Delta$, $\Delta^{\prime}$ and $j$ are the same for $E$ and $E^{-1}$.

######  Proposition 3.14.

Let $E$ be an elliptic curve in $\mathcal{E}^{0}$. Then, setting
$K\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{Q}$, the Assumption 2.1 is
satisfied for $\mathcal{E}^{0}$.

###### Proof.

All the conditions of Assumption 2.1 simply follow from the definition of
$\mathcal{E}^{0}$. ∎

######  Lemma 3.15.

For $E\in\mathcal{E}$, we have that $\omega(E^{-1})=-\omega(E)$.

###### Proof.

Recall that $\Delta(E^{-1})=\Delta(E)$ and $j(E^{-1})=j(E)$. Recall also that
the root number $\omega(E)=-\prod_{\ell}\omega_{\ell}(E)$. We study the
relationship between $\omega_{\ell}(E^{-1})$ and $\omega_{\ell}(E)$ for all
primes $\ell$.

  * •

We begin with primes $\ell$ not dividing $\Delta(E)$. Since
$\Delta(E^{-1})=\Delta(E)$, both $E$ and $E^{-1}$ have good reduction at
$\ell$. Thus, $\omega_{\ell}(E^{-1})=\omega_{\ell}(E)=1$.

  * •

Next, let $\ell\neq 2$ be a prime dividing $\Delta(E)$. Recall that condition
(3) in Definition 3.12 requires that $\Delta^{\prime}(E)$ is squarefree. In
particular, $\ell^{2}\nmid\Delta(E)$, and thus, $E$ has multiplicative
reduction at $\ell$. According to (3.3) we have that

$\omega_{\ell}(E^{-1})=\omega_{\ell}(E)\text{ if and only if }\ell\equiv
1\pmod{4}.$

Since it is assumed that $\Delta^{\prime}(E)\equiv 1\pmod{4}$ (by part (3) of
Definition 3.12), we must have that the number of primes $\ell\equiv
3\pmod{4}$ dividing $\Delta(E)$ is even. Hence,
$\omega_{\ell}(E^{-1})=-\omega_{\ell}(E)$ for an even number of primes
$\ell\equiv 3\pmod{4}$. Thus, we find that

$\prod_{\begin{subarray}{c}\ell\neq 2,\\\
\ell|\Delta(E)\end{subarray}}\omega_{\ell}(E^{-1})=\prod_{\begin{subarray}{c}\ell\neq
2,\\\ \ell|\Delta(E)\end{subarray}}\omega_{\ell}(E).$

  * •

Finally, let us consider $\ell=2$. By part (4) of Definition 3.12, $E$ has
additive reduction at $2$ and $2\nmid j(E)$. It follows from [Won01, Lemma 12]
that $\omega_{2}(E^{-1})=-\omega_{2}(E)$.

It follows from the analysis above that for $E\in\mathcal{E}$,

$\begin{split}\omega(E^{-1})=&-\prod_{\ell}\omega_{\ell}(E^{-1})\\\
=&-\prod_{\ell\nmid\Delta(E)}\omega_{\ell}(E^{-1})\times\prod_{\begin{subarray}{c}\ell\neq
2,\\\
\ell|\Delta(E)\end{subarray}}\omega_{\ell}(E^{-1})\times\omega_{2}(E^{-1})\\\
=&-\prod_{\ell\nmid\Delta(E)}\omega_{\ell}(E)\times\prod_{\begin{subarray}{c}\ell\neq
2,\\\
\ell|\Delta(E)\end{subarray}}\omega_{\ell}(E^{-1})\times\left(-\omega_{2}(E^{-1})\right)\\\
=&-\omega(E).\end{split}$

∎

Recall that the _height of_ $E$, denoted $h(E)$, is defined to be
$\operatorname{max}\\{|A|^{3},B^{2}\\}$. Thus, $h(E)$ remains unaltered under
the twist by $-1$, i.e. $h(E)=h(E^{-1})$. Hence, Lemma 3.15 implies that
exactly $50\%$ of the curves $E$ in $\mathcal{E}$ have root number $+1$, when
ordered by height. We note that there is a natural short exact sequence

$0\rightarrow
E(\mathbb{Q})/3E(\mathbb{Q})\rightarrow\operatorname{Sel}_{3}(E/\mathbb{Q})\rightarrow\Sh(E/\mathbb{Q})[3]\rightarrow
0,$

and thus $\mathcal{E}^{0}$ precisely consists of elliptic curves
$E\in\mathcal{E}$ for which

  1. (1)

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

  2. (2)

$E(\mathbb{Q})[3]=0$,

  3. (3)

$\Sh(E/\mathbb{Q})[3]=0$.

##  4\. Density results for diophantine stability in $L$

This section is devoted to the key technical arguments of the article, finally
leading up to the proofs of our main results.

###  4.1. A suitable large family of elliptic curves

We wish to show that the family $\mathcal{E}$ (cf. Definition 3.12) has
positive lower density. In order to do this, we introduce a large family
$\mathcal{F}$ contained in $\mathcal{E}$ and estimate the density of
$\mathcal{F}$.

######  Definition 4.1.

Let $\mathcal{F}\subset\mathcal{C}$ consist of all elliptic curves
$E=E_{A,B}$, where $(A,B)$ is a minimal pair such that

  1. (1)

$p\nmid 4A^{3}+27B^{2}$,

  2. (2)

for all $\ell\in Z$, we require that $\ell\nmid 4A^{3}+27B^{2}$ and
$\widetilde{E}_{A,B}(\mathbb{F}_{\ell})[p]=0$ and
$\widetilde{E}_{A,-B}(\mathbb{F}_{\ell})[p]=0$,

  3. (3)

$A=4A^{\prime}$ and $B=16B^{\prime}$, where $A^{\prime}\equiv 189\pmod{256}$
and $B^{\prime}\equiv\pm 1\pmod{256}$.

  4. (4)

For all primes $\ell\neq 2$, we insist that $\ell^{2}\nmid\Delta_{A,B}$.

Clearly, the set $\mathcal{F}$ is a large family, and at the primes
$T\mathrel{\mathop{\mathchar 58\relax}}=Z\cup\\{2,p\\}$, we have associated
congruence conditions.

######  Lemma 4.2.

With respect to notation above, $\mathcal{F}$ is a subset of $\mathcal{E}$.

###### Proof.

It is clear from the definition of $\mathcal{F}$ that the conditions (1) and
(2) in Definition 3.12 are satisfied. Also, $\Delta^{\prime}(E)$ is clearly
squarefree for $E\in\mathcal{F}$. It thus suffices to show that

  * •

$E$ and $E^{-1}$ have additive reduction at $2$,

  * •

$\Delta^{\prime}(E)\equiv 1\pmod{4}$,

  * •

$2\nmid j(E)$

for all $E\in\mathcal{F}$. That $E$ and $E^{-1}$ have additive reduction at
$2$ follows from Tate’s algorithm (see [Sil94, step 7, p. 367]). Since it is a
standard procedure, we omit these details. The $j$-invariant is given as
follows

$j(E)=1728\frac{4A^{3}}{4A^{3}+27B^{2}}=\frac{27\times 2^{6}\times A^{\prime
3}}{A^{\prime 3}+27B^{\prime 2}}.$

We find that

$A^{\prime 3}+27B^{\prime 2}\equiv 189^{3}+27\equiv 64\pmod{256},$

and deduce that $j(E)$ is a $2$-adic unit. Lastly, note that

$\Delta(E)=4A^{3}+27B^{2}=2^{8}(A^{\prime 3}+27B^{\prime 2}),$

and

$A^{\prime 3}+27B^{\prime 2}=2^{6}(4n+1),$

where $n$ is a positive natural number. Thus, $\Delta^{\prime}(E)=4n+1$ and is
congruent to $1\pmod{4}$. ∎

Let us now spell out the congruence conditions that define $\mathcal{F}$ as a
subset of $\mathcal{C}$.

  * •

For $\ell\notin T$, we let $\mathcal{F}_{\ell}$ to consist of all Weierstrass
models $E_{A,B}$ (not necessarily minimal) such that $\Delta_{A,B}\not\equiv
0\pmod{\ell^{2}}$.

  * •

For $\ell\in Z$, $\mathcal{F}_{\ell}$ consist of $E_{A,B}$ such that
$\Delta_{A,B}\not\equiv 0\pmod{\ell}$ and $\widetilde{E}_{A,\pm
B}(\mathbb{F}_{\ell})[p]=0$.

  * •

For $\ell=p$, $\mathcal{F}_{p}$ is defined by $\Delta_{A,B}\not\equiv
0\pmod{p}$.

  * •

For $\ell=2$, $\mathcal{F}_{2}$ is defined by requiring that $A=4A^{\prime}$
and $B=16B^{\prime}$ such that $(A^{\prime},B^{\prime})\equiv(189,\pm
1)\pmod{256}$.

Thus, we find that $\mathcal{F}=\cap_{\ell}\mathcal{F}_{\ell}$, where $\ell$
ranges over all prime numbers. It is clear from our definition that for
$\ell\neq 2$, the elliptic curves $E\in\mathcal{F}_{\ell}$ are minimal at
$\ell$. On the other hand, for $\ell=2$, minimality at $2$ follows from Tate’s
algorithm. Thus, we find that $\mathcal{F}$ consists of elliptic curves that
are globally minimal (i.e., minimal at all primes $\ell$). In other words,
$\mathcal{F}$ is a subset of $\mathcal{C}$. We wish to compute the density of
$\mathcal{F}$. This is done in 2 steps.

  1. (1)

Given $X>0$, we set $\mathcal{F}_{\ell}(X)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(A,B)\in\mathcal{F}_{\ell}\mid\operatorname{max}\\{|A|^{3},B^{2}\\}\leq
X\right\\}$. We note that not all pairs $(A,B)\in\mathcal{F}_{\ell}$ are
minimal, but the pairs $(A,B)\in\mathcal{F}$ are. We derive an asymptotic
formula for $\\#\mathcal{F}_{\ell}(X)$ as $X\rightarrow\infty$.

  2. (2)

Next, we show that the asymptotic formulae for $\\#\mathcal{F}_{\ell}(X)$ can
be used to give an asymptotic formula for $\\#\mathcal{F}(X)$. Using this
formula, we derive an expression for the density of $\mathcal{F}$. Since
$\mathcal{F}$ is contained in $\mathcal{E}$, this gives us lower bound for the
lower density of $\mathcal{E}$.

In order to complete step (1) in an effective manner, we introduce the residue
classes $\overline{\mathcal{F}}_{\ell}\subset(\mathbb{Z}/\ell^{n_{\ell}})^{2}$
associated to the local condition $\mathcal{F}_{\ell}$ for all primes. We then
set $\delta_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}$. The
asymptotic for $\\#\mathcal{F}_{\ell}(X)$ is then given by

$\\#\mathcal{F}_{\ell}(X)\sim\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}\times\\#\\{(A,B)\in\mathbb{Z}^{2}\mid\operatorname{max}\\{|A|^{3},B^{2}\\}\leq
X\\}\sim 4\delta_{\ell}X^{5/6}.$

######  Definition 4.3.

The set $\overline{\mathcal{F}}_{\ell}$ is defined as follows.

  * •

For $\ell\notin T$, we set $n_{\ell}=2$ and
$\overline{\mathcal{F}}_{\ell}\subset(\mathbb{Z}/\ell^{2}\mathbb{Z})^{2}$ to
consist of pairs $(\overline{A},\overline{B})$ for which
$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$.

  * •

For $\ell\in Z$, take $n_{\ell}=1$ and $\overline{\mathcal{F}}_{\ell}$ consist
of $(\overline{A},\overline{B})$ such that
$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$ and
$E_{\overline{A},\pm\overline{B}}(\mathbb{F}_{\ell})[p]=0$.

  * •

For $\ell=p$, take $n_{\ell}=1$ and $\overline{\mathcal{F}}_{\ell}$ consist of
$(\overline{A},\overline{B})$ such that
$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$.

  * •

For $\ell=2$, we set $n_{\ell}=12$ and
$\overline{\mathcal{F}}_{\ell}\subset(\mathbb{Z}/2^{12}\mathbb{Z})^{2}$ to
consist of pairs $(\overline{A},\overline{B})$ where
$(\overline{A},\overline{B})=(756,\pm 16)$.

For primes $\ell\notin Z$, we can calculate $\delta_{\ell}$ explicitly.

######  Lemma 4.4.

Let $\ell$ be a prime, we find that $\\#\mathcal{F}_{\ell}(X)\sim
4\delta_{\ell}X^{5/6}$, where

$\delta_{\ell}=\begin{cases}\frac{2}{3}&\text{ if }\ell\notin T\text{ and
}\ell=3;\\\ 1-\frac{2}{\ell^{2}}+\frac{1}{\ell^{3}}&\text{ if }\ell\notin
T\text{ and }\ell\neq 3;\\\ 1-\frac{1}{\ell}&\text{ if }\ell=p;\\\
\frac{1}{2^{21}}&\text{ if }\ell=2.\\\ \end{cases}$

###### Proof.

Each of these conditions is described by congruence conditions on
$(A,B)\in\mathbb{Z}^{2}$. In other words, there is an integer $n_{\ell}>0$ and
a set of congruence classes

$\overline{\mathcal{F}}_{\ell}\subset(\mathbb{Z}/\ell^{n_{\ell}})^{2}$

such that $\mathcal{F}_{\ell}$ consists of all pairs $(A,B)$ such that

$(A\pmod{\ell^{n_{\ell}}},B\pmod{\ell^{n_{\ell}}})\in\overline{\mathcal{F}}_{\ell}.$

It is thus clear that

$\\#\mathcal{F}_{\ell}(X)\sim\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}\right)\times
4X^{5/6}.$

We count $\\#\overline{\mathcal{F}}_{\ell}$ in all cases.

  * •

First consider $\ell\notin T$. Recall that for $\ell\notin T$,
$\mathcal{F}_{\ell}$ consists of all pairs $(A,B)\in\mathbb{Z}^{2}$ such that
$\ell^{2}\nmid\Delta_{A,B}$. Thus, $\overline{\mathcal{F}}_{\ell}$ consists of
all pairs $(\overline{A},\overline{B})\in(\mathbb{Z}/\ell^{2})^{2}$ such that
$-4\overline{A}^{3}\neq 27\overline{B}^{2}$. Note that $\ell\neq 2$ since
$2\in T$. First assume that $\ell=3$ (this case potentially arises when
$p=5$). In this case, the condition becomes $3\nmid\overline{A}$, and thus,
$\\#\overline{\mathcal{F}}_{\ell}=2\times 3^{3}$ and thus,

$\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}\right)=\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{4}}\right)=\frac{2}{3}.$

Next, we move on the case when $\ell\neq 3$. Set
$(C,D)\mathrel{\mathop{\mathchar 58\relax}}=(-\overline{A}/3,\overline{B}/2)$
and count the number of pairs for which $C^{3}=D^{2}$. We consider two further
cases. In the first case, $\ell\nmid C$, we set
$(U,V)\mathrel{\mathop{\mathchar 58\relax}}=(C,D/U)$ and the equation becomes
$U=V^{2}$. Thus, in this case, there are $(\ell^{2}-\ell)$ values of $V$ and
$U$ is determined by $V$. On the other hand, if $\ell|C$ then $\ell|D$ as well
and there are $\ell^{2}$ choices in this case. Thus, we get

$\\#\overline{\mathcal{F}}_{\ell}=\ell^{4}-(\ell^{2}-\ell)-\ell^{2}=\ell^{4}-2\ell^{2}+\ell.$

From the above, we find that

$\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}\right)=\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{4}}\right)=1-\frac{2}{\ell^{2}}+\frac{1}{\ell^{3}}.$

  * •

Next, consider the case when $\ell=p$. In this case, the condition is that
$\ell\nmid\Delta_{A,B}$. Arguing as we did in the previous case, it is easy to
see that

$\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2n_{\ell}}}\right)=\left(\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2}}\right)=\frac{(\ell^{2}-\ell)}{\ell^{2}}=1-\frac{1}{\ell}.$

  * •

Recall that when $\ell=2$, the condition required that $A=4A^{\prime}$ and
$B=16B^{\prime}$, where $A^{\prime}\equiv 189\pmod{256}$ and
$B^{\prime}\equiv\pm 1\pmod{256}$. Thus, we find that

$\frac{\\#\mathcal{F}_{\ell}}{\ell^{2n_{\ell}}}=\frac{1}{2^{21}}.$

∎

######  Lemma 4.5.

Let $\mathcal{F}_{\ell}^{\prime}$ be the complement of $\mathcal{F}_{\ell}$ in
$\mathbb{Z}^{2}$. There is an absolute constant $C_{1}>0$ (independent of
$\ell$) such that for all large enough values of $X$,

$\\#\mathcal{F}_{\ell}^{\prime}(X)\leq\frac{C_{1}}{\ell^{2}}X^{5/6}$

for all $\ell\notin T$.

###### Proof.

Assume without loss of generality that $\ell\geq 5$. We refer to the first
part of the proof of Lemma 4.4. Thus, $\mathcal{F}_{\ell}^{\prime}$ is the set
of $(A,B)\in\mathbb{Z}^{2}$ such that $\ell^{2}|(4A^{3}+27B^{2})$. We may
write
$\mathcal{F}_{\ell}^{\prime}=\mathcal{B}_{1}\cup\mathcal{B}_{2}\cup\mathcal{B}_{3}$,
where

  * •

$\mathcal{B}_{1}$ consists of pairs $(A,B)$ for which $\ell|A$ and $\ell|B$,

  * •

$\mathcal{B}_{2}$ consists of pairs for which $\ell\nmid A,B$ and
$-4A^{3}\equiv 27B^{2}\pmod{\ell^{2}}$, and $4A^{3}+27B^{2}\neq 0$.

  * •

Finally, let $\mathcal{B}_{3}$ consist of the pairs for which
$4A^{3}+27B^{2}=0$.

It is clear that

$\begin{split}\\#\mathcal{B}_{1}(X)=&\\{(A,B)\in\mathcal{B}_{1}\mid\operatorname{max}\\{|A|^{3},B^{2}\\}\leq
X\\}\\\
\leq&4\left\lfloor\frac{X^{1/3}}{\ell}\right\rfloor\left\lfloor\frac{X^{1/2}}{\ell}\right\rfloor\leq\frac{4X^{5/6}}{\ell^{2}}.\end{split}$

For each $A$ there are at most $2$ residue classes of $B$ modulo $\ell$ which
satisfy $-4A^{3}\equiv 27B^{2}\pmod{\ell}$. Each residue class $B\pmod{\ell}$
lifts to a unique residue class modulo $\ell^{2}$ that satisfies
$-4A^{3}\equiv 27B^{2}\pmod{\ell^{2}}$, by Hensel’s lemma. This requires that
$B\not\equiv 0\pmod{\ell}$. The number of such $B$s in the interval
$[-X^{1/2},X^{1/2}]$ in a given reside class modulo $\ell^{2}$ is
$\frac{2X^{1/2}}{\ell^{2}}+O(1)$. Thus, in all, there are
$2X^{1/3}(\frac{2X^{1/2}}{\ell^{2}}+O(1))$ values in $\\#\mathcal{B}_{2}(X)$.
Hence, $\\#\mathcal{B}_{2}(X)\leq\frac{4X^{5/6}}{\ell^{2}}+O(X^{1/3})$.
Finally, it is easy to see that $\\#\mathcal{B}_{3}(X)=o(X^{5/6})$ since the
value of $A$ determines $B$ for $(A,B)\in\mathcal{B}_{3}(X)$. Thus, taking any
constant $C_{1}>8$ will work. ∎

Next, we treat primes $\ell\in Z$, i.e., primes that ramify in $L$. By
assumption, $\ell\notin\\{2,p\\}$. Recall that $\overline{\mathcal{F}}_{\ell}$
is the subset of $(\mathbb{Z}/\ell\mathbb{Z})^{2}$ consisting of pairs
$(\overline{A},\overline{B})$ such that

  * •

$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$,

  * •

$E_{\overline{A},\overline{B}}(\mathbb{F}_{\ell})[p]=0$ and
$E_{\overline{A},-\overline{B}}(\mathbb{F}_{\ell})[p]=0$.

Note that

$\\#\mathcal{F}_{\ell}(X)\sim 4\delta_{\ell}X^{5/6},$

where $\delta_{\ell}=\frac{\\#\overline{\mathcal{F}}_{\ell}}{\ell^{2}}$. We
estimate the size of $\\#\overline{\mathcal{F}}_{\ell}$ for $\ell\in Z$.

######  Lemma 4.6.

With respect to notation above, the following assertions hold.

  1. (1)

For $\ell\in Z$, suppose that $p=5$. Then for $\ell\geq 5779$, we have that

$\\#\overline{\mathcal{F}}_{\ell}\geq\left(\frac{\ell^{2}-\ell}{2}\right)\left(\frac{\sqrt{\ell}-76}{\sqrt{\ell}}\right).$

  2. (2)

Suppose that $p=3$ and that the only primes $\ell$ that ramify in $L$ are of
the form $\ell\equiv 1\pmod{4}$. Then for all primes $\ell\geq 233$, we have
that

$\\#\overline{\mathcal{F}}_{\ell}\geq\left(\frac{\ell^{2}-\ell}{2}\right)\left(\frac{\sqrt{\ell}-15.18}{\sqrt{\ell}}\right).$

###### Proof.

The complement of $\overline{\mathcal{F}}_{\ell}$ in
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$ can be expressed as a union
$\mathcal{A}_{1}\cup\mathcal{A}_{2}\cup\mathcal{A}_{3}$. Here,
$\mathcal{A}_{1}$ consists of pairs
$(\overline{A},\overline{B})\in(\mathbb{Z}/\ell\mathbb{Z})^{2}$ such that
$\Delta_{\overline{A},\overline{B}}=0$ and $\mathcal{A}_{2}$ (resp.
$\mathcal{A}_{3}$) consists of pairs for which
$E_{\overline{A},\overline{B}}(\mathbb{F}_{\ell})[p]\neq 0$ (resp.
$E_{\overline{A},-\overline{B}}(\mathbb{F}_{\ell})[p]\neq 0$). Thus, we find
that

$\\#\mathcal{F}_{\ell}\geq\ell^{2}-\\#\mathcal{A}_{1}-\\#\mathcal{A}_{2}-\\#\mathcal{A}_{3}.$

It is clear that

$\\#\mathcal{A}_{2}=\\#\mathcal{A}_{3}.$

As was discussed in the proof of Lemma 4.4, we have that
$\\#\mathcal{A}_{1}=\ell$. Then it follows from [How93] that

$\\#\mathcal{A}_{2}\leq\left(\frac{\ell^{2}-\ell}{p-1}\right)\left(1+2.53\frac{p(p+1)}{\sqrt{\ell}}\right).$

Therefore, we have that

$\\#\mathcal{F}_{\ell}\geq\ell^{2}-\ell-2\left(\frac{\ell^{2}-\ell}{p-1}\right)\left(1+2.53\frac{p(p+1)}{\sqrt{\ell}}\right).$

Consider the case when $p=5$. In this case, we get

$\\#\mathcal{F}_{\ell}\geq\left(\frac{\ell^{2}-\ell}{2}\right)\left(\frac{\sqrt{\ell}-76}{\sqrt{\ell}}\right).$

This bound is nontrivial when $\sqrt{\ell}>76$, i.e., $\ell\geq 5779$.

On the other hand, consider the case when $p=3$ and assume that all primes
that ramify in $L$ are $\equiv 1\pmod{4}$. In this case,
$E_{\overline{A},\overline{B}}\simeq E_{\overline{A},-\overline{B}}$ over
$\mathbb{F}_{\ell}$. Hence, we find that $\mathcal{A}_{2}=\mathcal{A}_{3}$.
Therefore,
$\mathcal{A}_{1}\cup\mathcal{A}_{2}\cup\mathcal{A}_{3}=\mathcal{A}_{1}\cup\mathcal{A}_{2}$
and we have that

$\begin{split}\\#\mathcal{F}_{\ell}\geq&\ell^{2}-\\#\mathcal{A}_{1}-\\#\mathcal{A}_{2},\\\
\geq&\ell^{2}-\ell-\left(\frac{\ell^{2}-\ell}{2}\right)\left(1+\frac{15.18}{\sqrt{\ell}}\right)\\\
=&\left(\frac{\ell^{2}-\ell}{2}\right)\left(\frac{\sqrt{\ell}-15.18}{\sqrt{\ell}}\right).\end{split}$

This bound is nontrivial when $\sqrt{\ell}>15.18$, i.e., $\ell\geq 233$. ∎

######  Remark 4.7.

To avoid confusion, we reiterate that in Theorem 3.8, it is assumed that for
all $\ell\in Z$, there exists an elliptic curve $\mathbb{E}$ satisfying
required properties. This in particular implies that $\delta_{\ell}>0$.
However, the result above shows that this condition need only be checked for
primes $\ell\geq 5779$ when $p=5$. If $p=3$ and additionally, one assumes that
all primes $\ell\in Z$ are $1\pmod{4}$, then this condition needs to be
checked only for $\ell\geq 233$. This has been checked by us using the
following Sagemath code:

    P=Primes();
        for l in range(5,234):
            if l in P:
                c=0;
                for A in GF(l):
                    for B in GF(l):
                        if (4*A^3+27*B^2)!=0:
                            E=EllipticCurve([A,B]);
                            E1=EllipticCurve([A,-B]);
                            if (E.abelian_group().order())%3 != 0:
                                if (E1.abelian_group().order())%3 != 0:
                                    c=c+1;
                print(l,c)

In fact, the code gives the exact values of
$\\#\overline{\mathcal{F}}_{\ell}$.

For $z>0$, let $\mathcal{F}^{z}(X)$ be the intersection $\cap_{\ell\leq
z}\mathcal{F}_{\ell}(X)$ where $\ell$ ranges over prime numbers that are $\leq
z$. It is easy to see that since there are only finitely many congruence
conditions,

(4.1) $\mathcal{F}^{z}(X)\sim 4\prod_{\ell\leq z}\delta_{\ell}X^{5/6}.$

######  Proposition 4.8.

With respect to notation above, the set $\mathcal{F}$ has positive density
given by $\mathfrak{d}(\mathcal{F})\sim\prod_{\ell}\delta_{\ell}$.

###### Proof.

We prove our result by showing that

(4.2)
$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\leq\prod_{\ell}\delta_{\ell}.$

and

(4.3)
$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\geq\prod_{\ell}\delta_{\ell}.$

We have that $\mathcal{F}^{z}(X)\sim 4\prod_{\ell\leq z}\delta_{\ell}X^{5/6}$.
Since $\mathcal{F}(X)$ is contained in $\mathcal{F}^{z}(X)$, we find that

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\leq\prod_{\ell\leq
z}\delta_{\ell},$

and letting $z\rightarrow\infty$, one has that

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\leq\prod_{\ell}\delta_{\ell}.$

Since

$\mathcal{F}^{z}(X)\subseteq\mathcal{F}(X)\cup\left(\bigcup_{\ell>z}\mathcal{F}_{\ell}^{\prime}(X)\right),$

where we recall that $\mathcal{F}_{\ell}^{\prime}$ is the complement of
$\mathcal{F}_{\ell}$. Lemma 4.5 implies that there is an absolute constant
$C_{1}>0$ for which

$\begin{split}\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\geq&\lim_{X\rightarrow\infty}\frac{\\#\mathcal{F}^{z}(X)}{4X^{5/6}}-C_{1}\sum_{\ell>z}\frac{1}{\ell^{2}},\\\
=&\prod_{\ell\leq
Z}\delta_{\ell}-C_{1}\sum_{\ell>z}\frac{1}{\ell^{2}}.\end{split}$

Letting $z\rightarrow\infty$, we find that

(4.4)
$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{F}(X)}{4X^{5/6}}\geq\prod_{\ell}\delta_{\ell}.$

This completes the proof of the result. ∎

###  4.2. Proof of the main result

In this section, we provide the proof of our main result, Theorem 3.8.

######  Proposition 4.9.

Setting

$\eta_{p}\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}\frac{1}{4}&\text{
if }p=3;\\\ \frac{3}{8}&\text{ if }p=5;\end{cases}$

we find that

$\liminf_{X\rightarrow\infty}\frac{\\#\mathcal{E}^{0}(X)}{\\#\mathcal{E}(X)}\geq\eta_{p}.$

###### Proof.

The set $\mathcal{E}$ is a large family and by Lemma 3.15, exactly half the
curves in $\mathcal{E}$ have root number $1$. The result follows from Theorem
3.11. ∎

###### Proof of Theorem 3.8.

Let $E$ be an elliptic curve in the set $\mathcal{E}^{0}$. By Proposition
3.14, the Assumption 2.1 is satisfied for $E$ (and $K=\mathbb{Q}$). It follows
from Proposition 2.7 that $\operatorname{Sel}_{p}(E/L)=0$. In particular, the
rank of $E(L)$ is $0$. Let $S_{3}$ be the intersection $\mathcal{E}^{0}\cap
S_{1}$. Given $E\in S_{3}$, we find that $\operatorname{rank}E(L)=0$ and
$E(L)_{\operatorname{tors}}=0$, hence, $E(L)=0$.

Theorem 3.7 asserts that $S_{1}$ has density $1$. We note that for $\ell\in
Z$, the cardinality of $\overline{\mathcal{F}}_{\ell}$ is
$\mathfrak{A}_{\ell}$ by definition. Thus, we find that

$\underline{\mathfrak{d}}(S_{3})=\underline{\mathfrak{d}}(\mathcal{E}^{0}).$

Then it follows from Proposition 4.9 that

$\underline{\mathfrak{d}}(\mathcal{E}^{0})\geq\eta_{p}\underline{\mathfrak{d}}(\mathcal{E}).$

By Lemma 4.2, $\mathcal{F}$ is contained in $\mathcal{E}$, and thus,

$\eta_{p}\underline{\mathfrak{d}}(\mathcal{E})\geq\eta_{p}\underline{\mathfrak{d}}(\mathcal{F}).$

Finally, note that according to Proposition 4.8,

$\mathfrak{d}(\mathcal{F})=\prod_{\ell}\delta_{\ell}.$

Condition (2) implies that $\delta_{\ell}>0$ for all primes $\ell\in Z$. On
the other hand, that $\delta_{\ell}>0$ for all primes $\ell\notin Z$ follows
from Lemma 4.4. In conclusion,
$\underline{\mathfrak{d}}(S_{3})\geq\eta_{p}\prod_{\ell}\delta_{\ell}$, thus,
the density of elliptic curves $E_{/\mathbb{Q}}$ that are diophantine stable
in $L$ have positive lower density $\geq\eta_{p}\prod_{\ell}\delta_{\ell}$. ∎

###  4.3. An example

We illustrate Theorem 3.8 via an example. We fix $p=3$, and
$L=\mathbb{Q}(\mu_{7})^{+}$ to be the maximal real subfield of
$\mathbb{Q}(\mu_{7})$. Then $L/\mathbb{Q}$ is a cubic Galois extension with
Galois group $G\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(L/K)\simeq\mathbb{Z}/3\mathbb{Z}$. The only
prime that ramifies in $L$ is $7$, and thus $Z=\\{7\\}$ and $T=\\{2,3,7\\}$.
Recall that $\mathfrak{A}_{7}$ is the number of
$(\overline{A},\overline{B})\in(\mathbb{Z}/7\mathbb{Z})^{2}$ such that

  * •

$4\overline{A}^{3}+27\overline{B}^{2}\neq 0$,

  * •

$E_{\overline{A},\overline{B}}(\mathbb{F}_{7})[3]=0$ and
$E_{\overline{A},-\overline{B}}(\mathbb{F}_{7})[3]=0$.

Computations on Sagemath show that $\mathfrak{A}_{7}=12$. Thus, we find that
the lower density of elliptic curves $E_{/\mathbb{Q}}$ that are diophantine
stable in $L$ is

$\geq\eta_{3}\prod_{\ell}\delta_{\ell}=\frac{1}{4}\times\frac{1}{2^{21}}\times\left(1-\frac{1}{3}\right)\times\frac{12}{49}\times\prod_{\ell\neq
2,3,7}\left(1-\frac{2}{\ell^{2}}+\frac{1}{\ell^{3}}\right).$

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