# Watkins’s conjecture for elliptic curves with a rational torsion

Subham Bhakta  Indian Institute of Science Education and Research,
Thiruvananthapuram, India<EMAIL_ADDRESS>and  Srilakshmi
Krishnamoorthy  Indian Institute of Science Education and Research,
Thiruvananthapuram, India<EMAIL_ADDRESS>
###### Abstract.

Watkins’s conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the
rank of the group $E(\mathbb{Q})$ of rational points is bounded above by
$\nu_{2}(m_{E})$, where $m_{E}$ is the modular degree associated with $E$. In
[4, Thm. 2], it has been established that Watkins’s conjecture holds on
average. This article investigates the conjecture over certain thin families
of elliptic curves. For example, for prime $\ell$, we quantify the elliptic
curves featuring a rational $\ell$-torsion that satisfies Watkins’s
conjecture. Additionally, the study extends to a broader context,
investigating the inequality
$\mathrm{rank}(E(\mathbb{Q}))+M\leq\nu_{2}(m_{E})$ for any positive integer
$M$.

###### Key words and phrases:

Elliptic curves, Modular curves, Watkins’s conjecture

###### 2020 Mathematics Subject Classification:

Primary 11F30, 11L07; Secondary 11F52, 11F80

###### Contents

  1. 1 Introduction
  2. 2 Tools with modular degrees and ranks
  3. 3 Setting up the ground work for counting with heights
  4. 4 Watkins’s conjecture over ${\mathcal{E}}_{2}$ and ${\mathcal{E}}_{3}$
  5. 5 $M$-Watkins over ${\mathcal{E}}_{3},{\mathcal{E}}_{5}$ and ${\mathcal{E}}_{7}$

##  1\. Introduction

Let $X_{0}(N)$ be the modular curve that can be realized as the Riemann
surface obtained by the standard action of the congruence subgroup
$\Gamma_{0}(N)$ on the upper half place $\mathbb{H}$. Consider $E/\mathbb{Q}$
to be any arbitrary elliptic curve. We know that $E$ is modular by the work of
Wiles, Taylor-Wiles, et al. In other words, there exists a surjective morphism
(defined over $\mathbb{Q}$) $\phi_{E}:X_{0}(N)\to E$, which is called the
modular parametrization of $E$. Throughout the article, we shall assume that
$\phi_{E}$ has a minimal degree, sending the cusp $\infty$ of $X_{0}(N)$ to
the identity of $E$. We denote the minimal degree as $m_{E}$, the modular
degree of $E$. In this article, we mainly aim to study Watkins’s conjecture.

######  Conjecture 1 (Watkins).

For any elliptic curve $E/\mathbb{Q}$, we have
$\mathrm{rank}_{{\mathbb{Z}}}(E(\mathbb{Q}))\leq\nu_{2}(m_{E})$.

The conjecture is known to be true when the modular degree is odd. This
follows from the work of Calegari, Emerton [6], Yazdani [29] and Kazalicki,
Kohem [21]. When $E(\mathbb{Q})[2]$ is trivial, Dummigan [13] heuristically
proved that certain classes of elliptic curves satisfy Watkins’s conjecture,
under the assumption that $R\cong T$. This assumption is about a relation
between the tangent space of a suitable deformation ring that captures
relevant properties of the residual representation
$\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(E[2])$,
and a certain Selmer group which is large enough to capture the weak Mordell-
Weil group.

An alternative heuristic proof of Watkins’s conjecture is presented in [4,
Thm. 2], demonstrating its average truth. Specifically, the conjecture holds
true for nearly all elliptic curves under the Minimalist conjecture. This
follows as a consequence of the primary outcome in Dummigan and Krishnamoorthy
[14], which essentially establishes a lower bound on $\nu_{2}(m_{E})$ based on
the number of prime factors in the conductor of the elliptic curve $E$.
Recently Park, Poonen, Voight, and Wood [25] conjectured that, there are only
finitely many elliptic curves of rank $21$. With these heuristics, Watkins’s
conjecture is true for all but finitely many elliptic curves with a large
number of bad reductions.

It is noteworthy that any elliptic curve over $\mathbb{Q}$ can be expressed in
the minimal Weierstrass form

(1) $E_{A,B}:y^{2}=x^{3}+Ax+B,~{}A,B\in{\mathbb{Z}},~{}p^{4}|A\implies
p^{6}\nmid B.$

In the context of [4, Thm. 2], elliptic curves are arranged based on their
height, defined as

(2) $H(E_{A,B})=\max\Big{\\{}|A|^{3},|B|^{2}\Big{\\}}^{1/6}.$

Throughout the article, we shall arrange the elliptic curves with respect to
the height $H$, as defined above. Note that there are approximately $\asymp
X^{5}$ elliptic curves with heights not exceeding $X$. The arithmetic
statistics presented in [4, Thm. 2] do not address thin families, i.e.,
families that exclusively consist of $0\%$ of all elliptic curves. In this
article, our main focus revolves around the thin family given by the elliptic
curves equipped with at least one rational torsion point. In other words, we
work over the families of elliptic curves for which
$G=E(\mathbb{Q})_{\mathrm{tor}}$ is non-trivial. Due to the celebrated theorem
of Mazur [23], it is enough to study the following $14$ possibilities of $G$:

(3) $\displaystyle{\mathbb{Z}}/n{\mathbb{Z}}$ with $2\leq n\leq 10$ or $n=12$
$\displaystyle{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/n{\mathbb{Z}}$ with
$n=2,4,6,8$.

Note that Caro and Pasten [9], later Pasupulati and both of the authors of
this article in [4, Thm. 6], exhibited a family of elliptic curves satisfying
Watkins’s conjecture when $G$ is one of the $10$ groups that has an even
order. Following this, we set the following notations, that will be followed
throughout the whole article

######  Definition 1.

For any prime $\ell\in\\{2,3,5,7\\}$, we set

$\mathcal{E}=\\{E_{A,B}~{}\mathrm{minimal}:A,B\in{\mathbb{Z}}^{2}\\},~{}\mathcal{E}_{\ell}=\\{E\in\mathcal{E}:E(\mathbb{Q})[\ell]\neq~{}\mathrm{trivial}\\},$

and for the counting purpose, we set

$\mathcal{E}(X)=\\{E\in\mathcal{E}:H(E)\leq
X\\},~{}\mathcal{E}_{\ell}(X)=\\{E\in\mathcal{E}:H(E)\leq
X,~{}E(\mathbb{Q})[\ell]\neq~{}\mathrm{trivial}\\},$

i.e., $\mathcal{E}_{\ell}(X)=\mathcal{E}(X)\cap\mathcal{E}_{\ell}$.

In this article, we aim to quantify the proportion of elliptic curves over
each of the families in Definition 1, that satisfy Watkins’s conjecture. In
this regard, let us first denote $\mathcal{E}^{\prime}_{2}$ be the set of all
elliptic curves $E_{a,b}:=y^{2}=x^{3}+ax^{2}+bx,~{}a,b\in{\mathbb{Z}}$. Note
that any $E_{A,B}\in\mathcal{E}_{2}$ can be written in the Weierstrass form
$E_{-3b,~{}3b^{2}+a}:=y^{2}=x^{3}-3bx^{2}+(3b^{2}+a)x,~{}a,b\in{\mathbb{Z}}$.
This shows that $\mathcal{E}_{2}\subset\mathcal{E}^{\prime}_{2}$. We shall
arrange the elliptic curves in $\mathcal{E}^{\prime}_{2}$ with respect to the
function $H_{2}(E_{a,b})=\max\\{|a|^{2},|b|\\}^{1/2}$. We shall demonstrate in
Section 3.1 that the number of elliptic curves in $\mathcal{E}_{2}$ of height
(with respect to $H$) at most $X$, is asymptotically the number of elliptic
curves in $\mathcal{E}^{\prime}_{2}$ of height (with respect to $H_{2}$) at
most $X$. In this regard, we first prove the following.

######  Theorem 1.

When arranging the elliptic curves in $\mathcal{E}^{\prime}_{2}$ according to
the height $H_{2}$, then Watkins’s conjecture is true for a positive
proportion of elliptic curves in $\mathcal{E}^{\prime}_{2}$. Consequently,
Watkins’s conjecture is true for a positive proportion of elliptic curves in
$\mathcal{E}_{2}$, arranging with respect to height $H$.

Moreover, we leverage the reasoning outlined in [20] to construct a specific
sub-family within $\mathcal{E}_{2}$ by considering quadratic twists of an
elliptic curve, all of which confirms the Watkins’s conjecture. This finding
is documented in Corollary 4.2, and can be viewed as extensions of [16, Thm.
1.2] and [7, Thm. 1.2] for a specific family of elliptic curves.

Additionally, we will delve into an extended version of Watkins’s conjecture
for further investigation.

######  Definition 2.

For any integer $M\geq 0$, we say that an elliptic curve $E$ is $M$-Watkins,
if $r+M\leq\nu_{2}(m_{E}).$

Certainly, Watkins’s conjecture suggests that any elliptic curve
$E/\mathbb{Q}$ is $0$-Watkins. In [4], several results regarding the counting
of $0$-Watkins elliptic curves were discussed. The main objective of the
article is to enumerate the generalized $M$-Watkins elliptic curves for any
integer $M$. In this setting, we have the following.

######  Theorem 2.

For any integer $M$, a positive proportion of elliptic curves in
$\mathcal{E}_{2}$ is $M$-Watkins when arranged with respect to height $H$.

Certainly, one can derive Theorem 1 from Theorem 2. However, the density we
achieve in this case is reduced by approximately a factor of
$\frac{1}{(2M)^{M}}$, as demonstrated in Lemma 6.

Subsequently, we investigate the $M$-Watkins property over $\mathcal{E}_{3}$
in both unconditional and conditional ways. For the unconditional case, the
key ingredient lies in studying the elliptic curves of Type-I. Specifically,
this means examining the elliptic curves of the form
$E_{a}:y^{2}=x^{3}+a,~{}a\in{\mathbb{Z}}$. In this context, we prove the
following result.

######  Theorem 3.

For the Type-I elliptic curves, we have the following estimates for any
integer $M$:

  1. (i)

almost all the elliptic curves over Type-I are $M$ Watkins. More precisely,

$\\#\Big{\\{}a\in{\mathbb{Z}}:|a|\leq
X,~{}E_{a}~{}\mathrm{is~{}not}~{}M-\mathrm{Watkins}\Big{\\}}=O_{M}\left(\frac{X}{\log\log
X}\right).$

  2. (ii)

For any integer $k$, consider the set of integers
$N_{k,m}=\\{kn^{2}:n\in{\mathbb{Z}}\\}$. Then, for any integer $M$, we have
$\\#\Big{\\{}a\in N_{k,m}:|a|\leq
X,~{}E_{a}~{}\mathrm{is}~{}M-\mathrm{Watkins}\Big{\\}}\gg_{k,M}\left(\frac{X}{\log
X}\right)^{1/2}$.

  3. (iii)

The size of $\mathcal{E}_{2}(X)$ is approximately $\asymp X^{3}$, and the
number of $M$-Watkins Type-I elliptic curves in $\mathcal{E}_{2}(X)$ has a
lower bound of order $\gg\frac{X}{(\log X)^{1/4}}$.

As an immediate application, we deduce the following consequence for
$\mathcal{E}_{3}$, taking $k=1$.

######  Corollary 1.1.

For any integer $M\geq 0$, infinitely many elliptic curves in
$\mathcal{E}_{3}$ are $M$-Watkins. More specifically, the size of
$\mathcal{E}_{3}(X)$ is approximately $\asymp X^{2}$, and the number of
$M$-Watkins elliptic curves in $\mathcal{E}_{3}(X)$ has a lower bound of order
$\frac{X}{(\log X)^{1/2}}$.

Furthermore, we aim to enhance Theorem 1.1 by proving that all elliptic curves
within $\mathcal{E}_{3}$, $\mathcal{E}_{5}$, and $\mathcal{E}_{7}$ adhere to
being $M$-Watkins, with $100\%$ certainty. The approach involves parametrizing
the points in $\mathcal{E}_{5}$, and $\mathcal{E}_{7}$ using appropriate
elliptic surfaces over $\mathbb{Q}(t)$. For the points in $\mathcal{E}_{3}$,
it turns out that, except for the Type-I curves of the form
$y^{2}=x^{3}+b^{2},~{}b\in{\mathbb{Z}}$, all other points can also be
parametrized by a suitable elliptic surface over $\mathbb{Q}(t)$. After
addressing such Type-I curves in the proof of part $(ii)$ in Theorem 1.1 or
Theorem 3, we turn to studying the ranks and conductors and ranks of
conductors of the elliptic curves, lying in a given $1$-parameter family of
elliptic curves. Proposition 5.2 is our main tool for studying the conductors,
while for ranks, we rely on some of the standard conjectures in the theory of
elliptic curves mentioned in [28]. More specifically, we shall prove the
following.

######  Theorem 4.

The sizes of $\mathcal{E}_{5}(X)$ and $\mathcal{E}_{7}(X)$ are approximately
$\asymp X$ and $\asymp X^{1/2}$, respectively. Assume that the conjectures
$B,C$, and $D$ in [28] hold. Then for any integer $M$, and any prime
$\ell\in\\{3,5,7\\}$, almost all the elliptic curves in $\mathcal{E}_{\ell}$
are $M$-Watkins.

###  1.1. Overview and strategies

Section 2 provides a brief overview of the rank bounds, highlighting the
relationships between modular degrees and ranks. Specifically, Corollary 2.1
boils down the study of Watkins’s conjecture to understanding the ranks and
conductors of elliptic curves. To get good enough bounds on the ranks over
$\mathcal{E}_{2}$ and $\mathcal{E}_{3}$, we employ classical methods such as
Descent via $2$ and $3$-isogenies. Specifically, Corollary 2.2 provides a
sufficiently nice bound in terms of the conductors of the elliptic curves over
$\mathcal{E}_{2}$, and in Section 4.1, we further refine this through a
sieving process in Lemma 2, demonstrating that the rank bound in Corollary 2.2
holds in almost all cases. To count the number of elliptic curves in
$\mathcal{E}_{2}$, we use Lipschitz’s lattice point counting criteria in
Section 3.1. Furthermore, to understand $M$-Watkins property over
$\mathcal{E}_{2}$, we study the quadratic twists of $E_{0}:y^{2}=x^{3}-1$ in
Section 4.4, using the techniques outlined in [20] to determine when the ranks
are at most $1$.

To study $M$-Watkins over $\mathcal{E}_{3}$, we employ both conditional and
unconditional approaches. First, we count the number of elliptic curves in
$\mathcal{E}_{3}$ in Section 3.2. For the unconditional approach to
$M$-Watkins, we rely on the rank bounds for the family
$y^{2}=x^{3}+a,~{}a\in{\mathbb{Z}}$, using the classical Descent via
$3$-isogeny, along with the main result of [5], which shows that a typical
elliptic curve of Type-I has a small rank. For the approach through
$3$-isogeny, we follow the rank bound discussed in Section 2.4. Specifically,
the observation at Lemma 8 produces a lot of quadratic fields, whose
$3$-torsion rank of class group is significantly smaller than the number of
prime factors of the discriminant. Finally, we accomplish our goal through the
unconditional approach in Section 4.5.

For the conditional approach over $\mathcal{E}_{3}$, which is also our
approach towards $\mathcal{E}_{5}$ and $\mathcal{E}_{7}$, we rely on the
existence of a universal elliptic curve over parametrizing them. We shall
briefly discuss this in Section 3.3. Now the heart of the study involves two
steps: first, showing that the average rank of the elliptic curves lying in a
non-split elliptic surface is bounded. To prove this, we need to rely on
various classical conjectures and extend the main result of Silverman in [28],
as briefly discussed in Section 5.2. The second step involves studying the
typical number of prime factors of the square-free parts of polynomial values,
which we address in Proposition 5.2. Note the normal order of $\omega(n)$ is
$\log\log n$, and this normal order is consistent for both $\omega(n)$ and
$\omega(s(n))$. Following this principle, we extend the normal order of
polynomial values from [10, Thm. 3.2.3]. The entire conditional approach is
carried out in Section 5.

###  1.2. Notations

We use the symbol $X$ as a quantifying parameter and $x$ for one of the
coordinates of an elliptic curve. The symbol $r$ is used for rationals, and
$t$ serves as an indeterminate for the polynomial rings.

By $O_{S_{1},S_{2},\cdots,S_{k}}(B)$ we mean a quantity with absolute value at
most $cB$ for some positive constant positive constant $c$ depending on
$S_{1},\cdots,S_{K}$ only; if the subscripts are omitted, the implied constant
is absolute. We write $A\ll_{S_{1},\cdots,S_{K}}B$ for
$A=O_{S_{1},S_{2},\cdots,S_{k}}(B)$. We write $A=o(B)$ for $A/B\to 0$, and
$A\asymp B$ for $A/B\to c$, for some absolute constant $c$.

Any natural number $n$ can be written as $n=n^{\prime}\times\Box$, where
$n^{\prime}$ is the square-free part of $n$ with $(n^{\prime},\Box)=1$. Then,
we denote $s(n)$ to be the square-free part of $n$. For any integer $a$, we
denote $\omega(a)$ and $s(a)$ respectively be the number of distinct prime
factors, and the square-free part of $|a|$. Additionally, use the standard
notation $\mu(a)$ to indicate the Möbius function of $|a|$. For any rational
number $r=\frac{a}{b}$ in the lowest terms, we set
$\omega(r)=\omega(a)+\omega(b)$, and $s(r)$ as $s(ab)$. Furthermore, for any
finite set of primes $S$, we denote $\omega_{S}(r)$ to be the number of prime
factors of $r$, that are not in $S$. Moreover, for any integer $n$, we denote
$\mathbb{P}^{1}(n)=\mathbb{P}^{1}(\mathbb{Z}/n\mathbb{Z})$, the projective
line over the ring of integers modulo $n$.

Given an elliptic curve $E/\mathbb{Q}$, we denote $\Delta(E)$ and $N(E)$ as
the discriminant and conductor of $E$, respectively. Moreover, we denote
$H(E)$ to be the height of the elliptic curves as in 2.

### Acknowledgements

We extend our sincere gratitude to IISER Thiruvananthapuram for providing an
excellent research environment that significantly contributed to the success
of this project. We are deeply thankful to the institute for their invaluable
support. SB is supported by the institute fellowship of IISER
Thiruvananthapuram.

A part of this study was conducted during the workshop Rational points on
Modular curves at the International Centre for Theoretical Sciences.
Furthermore, some parts of study was also conducted during the first author’s
visit to the Chennai Mathematical Institute, Mathematisches Forschungsinstitut
Oberwolfach, and the University of Kiel. We are grateful to these institutions
for their exceptional hospitality and support.

We would like to acknowledge CS Rajan for posing a thought-provoking question
during the second author’s presentation at ICTS, which inspired us to delve
into the study of $M$-Watkins elliptic curves.

Our appreciation also extends to Loïc Merel and Sunil Kumar Pasupulati for
their insightful discussions and contributions to this work.

##  2\. Tools with modular degrees and ranks

In this section, we will discuss the necessary materials to help count the
elliptic curves satisfying Watkins’s conjecture.

###  2.1. Dummigan, Krishnamoorthy’s lower bound

For each $p^{\alpha}\mid\mid N$, consider the Atkin-Lehner involution $W_{p}$
on $X_{0}(N)$. The set $\left\\{W_{p}~{}:~{}d\mid\mid N\right\\}$ forms an
abelian subgroup of automorphisms of rank equal to $\omega(N)$. For every
prime divisor $p\mid N$, there exists $w_{p}(f_{E})\in\\{\pm 1\\}$ such that
$W_{p}(f_{E})=w_{p}(f_{E})f_{E}$.

We now consider the homomorphism
$({\mathbb{Z}}/2{\mathbb{Z}})^{\omega(N)}\to\\{\pm 1\\}$ defined by,
$W_{d}\mapsto w_{d}$. Let $W^{\prime}$ denote the kernel of this map.
According to Dummigan and Krishnamoorthy [14, Prop. 2.1], there exists a
homomorphism $W^{\prime}\to E(\mathbb{Q})[2]$ with kernel $W^{\prime\prime}$,
and $m_{E}$ divides the order of $W^{\prime\prime}$.

The modular parametrization gives a map $\pi:J_{0}(n)\to E$. The key
observation here is that if $w\in W^{\prime}$, then $\pi(w(D))=\pi(D)$ for any
$D\in J_{0}(N)$. Taking $P_{0}\in X_{0}(N)(\mathbb{Q})$ and
$D=P_{0}-\omega(P_{0})\in J_{0}(N)$, we have $\pi(w(D))\in E(\mathbb{Q})[2]$.
This yields a map $W^{\prime}\to E[2]$ with kernel $W^{\prime\prime}$.
Consequently, considering $X^{\prime\prime}=X_{0}(N)/W^{\prime\prime}$ and
$J^{\prime\prime}=J(X^{\prime\prime})$, the map $\pi$ induces a map
$\pi^{\prime\prime}:J^{\prime\prime}\to E$ such that
$\deg(\pi)=\\#W^{\prime\prime}\deg(\pi^{\prime\prime})$. Following this,
Dummigan and Krishnamoorthy [14, Prop. 2.1] deduced the following estimate.

$\displaystyle\nu_{2}(m_{E})\geq\nu_{2}\left(\frac{\\#W^{\prime}}{E(\mathbb{Q})[2]}\right)$
$\displaystyle=\nu_{2}\left({\\#W^{\prime}}\right)-\nu_{2}\left({E(\mathbb{Q})[2]}\right)$
$\displaystyle=\omega(N(E))-\dim_{{\mathbb{Z}}/2{\mathbb{Z}}}(W/W^{\prime})-\mathrm{dim}_{\mathbb{Z}/2\mathbb{Z}}(E(\mathbb{Q})[2]).$

In particular, we deduced the following, that will be used for the treatment
over ${\mathcal{E}}_{2}$.

######  Corollary 2.1.

For any elliptic curve $E/\mathbb{Q}$ with
$E(\mathbb{Q})[2]={\mathbb{Z}}/2{\mathbb{Z}}$, we have the following
inequality

$\nu_{2}(m_{E})\geq\omega(N(E))-1-\dim_{{\mathbb{Z}}/2{\mathbb{Z}}}(W/W^{\prime})\geq\omega(N(E))-2.$

######  Remark 1.

Recall that for an elliptic curve $E_{A,B}$ in its minimal form as in (1), the
corresponding conductor $N(E_{A,B})$ is, up to some bounded factor of $2$ and
$3$, the product of all primes $p$ dividing the discriminant
$\Delta(E_{A,B})$. In light of Corollary 2.1, our primary objective is to
count the set of all such minimal $E_{A,B}$, for which $N(E_{A,B})$ is
divisible by many large primes. Additionally, we also need to study the rank
growth to study the $M$-Watkins elliptic curves.

###  2.2. Estimation of ranks over $\mathcal{E}_{2}$ and $\mathcal{E}_{3}$

###  2.3. Descent via 2-isogeny

Let $E_{a,b}$ be of the form $y^{2}=x^{3}+ax^{2}+bx,~{}a,b\in{\mathbb{Z}}$,
and $E^{\prime}_{a,b}$ be the dual curve, defined by
$y^{2}=x^{3}-2ax^{2}+(a^{2}-4b)x$. Now we briefly recall the method of descent
via $2$-isogeny. Consider the spaces of the Homogeneous space of the form
$C_{d_{1},d_{2},F}:Z^{2}=d_{1}U^{4}+FU^{2}V^{2}+d_{2}V^{4},$ and for any set
of prime $T$ containing $\infty$, let us denote
$Q(T):=\\{d\in\mathbb{Q}^{*}/(\mathbb{Q}^{*})^{2}\mid\nu_{p}(d)=0\pmod{2},~{}\forall
p\not\in T\\}$. Then, we have the following key bound from [1, Eqn. (5)]:

(4)
$\operatorname{rank}\left(E(\mathbb{Q})\right)\leq\dim_{{\mathbb{F}}_{2}}\operatorname{Sel}^{\phi}(E)+\dim_{{\mathbb{F}}_{2}}\operatorname{Sel}^{\hat{\phi}}(E^{\prime})-2,$

where the Selmer groups are precisely of the following forms.

(5) $\operatorname{Sel}^{\phi}(E_{a,b})=\\{d\in
Q(T_{1}):C_{d,\frac{a^{2}-4b}{d},-2a}(\mathbb{Q}_{p})\neq\phi,~{}\forall p\mid
2b(a^{2}-4b)\\},$ (6)
$\operatorname{Sel}^{\hat{\phi}}(E^{\prime}_{a,b})=\\{d\in
Q(T_{2}):C_{d,\frac{b}{d},a}(\mathbb{Q}_{p})\neq\phi,~{}\forall p\mid
2b(a^{2}-4b)\\},$

where $T_{1}$ (resp. $T_{2}$) is the set of prime factors of $a^{2}-4b$ (resp.
$b$), containing $\infty$. It follows from the proof of [1, Lem. 2.1] that,

(7) $\operatorname{Sel}^{\phi}(E_{a,b})\subset
Q(T_{1}\setminus\\{\infty\\}),~{}\mathrm{or}~{}\operatorname{Sel}^{\hat{\phi}}(E^{\prime}_{a,b})\subset
Q(T_{2}\setminus\\{\infty\\}).$

With this observation, we deduce the following.

######  Corollary 2.2.

Let $a,b$ be any two co-prime integers for which
$\operatorname{Sel}^{\phi}(E_{a,b})\subset Q(T\setminus\\{p\\})$ for some
finite prime $p$. Then, we have

(8)
$\operatorname{rank}\left(E_{a,b}(\mathbb{Q})\right)\leq\omega(N(E_{a,b}))-2.$

###### Proof.

Given that $\operatorname{Sel}^{\phi}(E_{a,b})\subset Q(T\setminus\\{p\\})$
for some finite prime $p$, it follows immediately from equations (4) and (7)
that
$\operatorname{rank}\left(E_{a,b}(\mathbb{Q})\right)\leq\omega(a^{2}-4b)+\omega(b)-1$.
Since $a,b$ are co-prime, we also have
$\omega(b(a^{2}-4b))=\omega(b)+\omega(a^{2}-4b)$. In other words,
$\operatorname{rank}\left(E_{a,b}(\mathbb{Q})\right)\leq\omega(b^{2}(a^{2}-4b))-1$.
On the other hand, $\Delta(E_{a,b})=16b^{2}(a^{2}-4b)$, which shows that
$\omega(b^{2}(a^{2}-4b))\leq\omega(N(E_{a,b}))$. This completes the proof. ∎

###  2.4. Descent via $3$-isogeny

Let $E/\mathbb{Q}$ be an elliptic curve, equipped with $\phi:E\to E^{\prime}$
a rational isogeny of degree $3$, with the dual isogeny
$\hat{\phi}:E^{\prime}\to E$ satisfying $\phi\circ\hat{\phi}=[3]$. Arguing
similarly as in [1, Secc. 2] for the $3$-isogeny $\phi$, we get

(9)
$\operatorname{rank}\left(E(\mathbb{Q})\right)\leq\dim_{{\mathbb{F}}_{3}}\operatorname{Sel}^{\phi}(E)+\dim_{{\mathbb{F}}_{3}}\operatorname{Sel}^{\hat{\phi}}(E^{\prime}).$

To understand the Selmer groups in (9), let us mention [19, Sec. 1] that the
equation of $E$ can be expressed either in the form of a Type-I curve
$E_{a}:y^{2}=x^{3}+a$, or of a Type-II curve $E_{a,b}:y^{2}=x^{3}+a(x-b)^{2}$.
In this article, we will focus on the Type-I case due to its computational
simplicity.

####  2.4.1. Treatment for Type-I

We follow the discussion from [19, Sec. 1.1]. Consider the elliptic curve
denoted by the equation $E_{a}:y^{3}=x^{2}+a$. Notably, we observe that the
discriminant of $E_{a}$ is $\Delta(E_{a})=-2^{4}\cdot 3^{2}\cdot a^{2}$, and
its $j$-invariant is $j(E_{a})=0$. Additionally, the curve possesses complex
multiplication by the field $\mathbb{Q}(\sqrt{-3})$. The set
$C=\\{O,(0,\sqrt{a}),(0,-\sqrt{a})\\}$ is a subgroup of
$E(\overline{\mathbb{Q}})$ of order $3$. Let us then consider the $3$-isogeny
$\phi:E_{a}\to E^{\prime}_{a}$, where $E^{\prime}_{a}=E_{-27a}$, and denote
$\hat{\phi}_{a}$ be the corresponding dual isogeny
$\hat{\phi}_{a}:E^{\prime}_{a}\to E_{a}$. Let us set $K_{a}$ to be the number
field $\mathbb{Q}(\sqrt{-3a})$, and denote $\mathcal{O}_{K_{a}}$ to be its
ring of integers.

######  Definition 3.

Let $S$ be any finite set of finite primes of $\mathcal{O}_{K_{a}}$. We define

$\displaystyle H(S)=\Big{\\{}x\in
K_{a}^{\times}/\left(K^{\times}_{a}\right)^{3}\Big{|}\nu_{p}(x)\equiv
0\pmod{3},~{}\text{for any}~{}p\notin S\Big{\\}}.$

From [2, Lemm. 3.4], we have

(10) $\dim_{{\mathbb{F}}_{3}}\left(H(S)\right)\leq
r_{3}(K_{a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{a}}/U_{K_{a}}^{3}\right)+\\#S,$

where $r_{3}(K)$ denotes the $3$-rank of the ideal class group of $K$, and
$U_{K}$ denotes the group of units of $K$.

For any integer $a$, consider $S_{a}$ be the set of primes defined by: $2,3\in
S_{a}$, and a prime $p\neq 2,3$ is in $S_{a}$ if and only if,
$\nu_{p}(a)=2,4~{}\text{and}~{}\left(\frac{-3}{p}\right)=1$. Combining (10)
with [3, Thm. 3.5], we have

(11) $\dim_{{\mathbb{F}}_{3}}(\operatorname{Sel}^{\phi}(E_{a}))\leq
r_{3}(K_{a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{a}}/U_{K_{a}}^{3}\right)+\\#S_{a}.$

Furthermore, arguing the same for the dual $E^{\prime}_{a}:=E_{-27a}$, we have
the bound

(12)
$\dim_{{\mathbb{F}}_{3}}(\operatorname{Sel}^{\hat{\phi}}(E^{\prime}_{a}))\leq
r_{3}(K_{-27a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{-27a}}/U_{K_{-27a}}^{3}\right)+\\#S_{-27a}.$

####  2.4.2. 3-torsion ideal classes of quadratic fields

To bound the Selmer ranks in (11) and (12), a key requirement is to establish
bounds on the $3$-torsion ranks of the class groups of quadratic fields. We
denote $h_{3}(K_{a})$ to be the size of the $3$-torsion part. Of course,
$r_{3}(a)=\log_{3}(h_{3}(K_{a}))$. It is conjectured [15, Conj. 1] that
$h_{3}(K_{a})=O_{\varepsilon}(|a|^{\varepsilon})$, or equivalently,
$r_{3}(K_{a})=O_{\varepsilon}(\log|a|)$, for any $\varepsilon>0$. For the
bound on $h_{3}$, Helfgott-Venkatesh [18, Thm. 4.2] obtained the exponent
$0.45$, and later, Ellenberg and Venkatesh improved it to
$|a|^{\frac{1}{6}+\varepsilon}$. However, the pointwise bound is not
sufficient for us, as we need to focus on $a$ for which
$r_{3}(K_{a})=O(\omega(a))$. Since $\omega(a)$ is typically of order
$\log\log|a|$, we aim for a stronger phenomenon here. It is perhaps worth
noting that Kishi and Miyake in [22] provided criteria for $r_{3}(K_{a})=1$,
though this appears challenging for quantitative analysis. One thing is to
note that, $r_{3}(K_{a})$ is just $O(1)$ over the almost squares $a$. For the
general case, while Heilbronn, Davenport [12] would suffice for our purposes,
we will instead use [5] regarding the average rank of elliptic curves of
Type-I.

##  3\. Setting up the ground work for counting with heights

For any pair $(A,B)\in{\mathbb{Z}}^{2}$, denote $E_{A,B}$ be the elliptic
curve $y^{2}=x^{3}+Ax+B$. Recall that we are arranging them with respect to
the height function $H(E_{A,B})=\max\left\\{|A|^{3},|B|^{2}\right\\}^{1/6}$,
and studying the proportion of all such $(A,B)$, for which $E_{A,B}$ satisfies
the Watkins’s conjecture. Note that there are $\asymp X^{5}$ many elliptic
curves of height at most $X$. For each $\ell\in\\{2,3,5,7\\}$, the main aim of
this section is to discuss how many of the elliptic curves with height at most
$X$ have $\ell$-torsion. In Section 4 and Section 5, we will further
investigate how often these curves satisfy Watkins’s conjecture, or in
general, the $M$-Watkins property.

###  3.1. Counting over ${\mathcal{E}}_{2}$

We know [17, Lem. 5.1, part (a)] that an elliptic curve $E_{A,B}$ has a
rational point of order $2$ if and only if there exists $a,b\in{\mathbb{Z}}$
such that $A=a,B=b^{3}+ab$. This leads us to consider the lattice points in
the region $R_{2}(X)=\Big{\\{}(a,b)\in\mathbb{R}^{2}:|a|<X^{2}\text{ and
}|b^{3}+ab|<X^{3}\Big{\\}}$. In other words, we have
$\mathcal{E}_{2}(X)=R_{2}(X)\cap{\mathbb{Z}}^{2}$. By the principle of
Lipschitz [11], we have

(13)
$\\#{\mathcal{E}}_{2}(X)=\\#(R_{2}(X)\cap{\mathbb{Z}}^{2})\sim\mathrm{Vol}(R_{2}(1))X^{3}.$

Moreover, [17, Lem. 5.2] shows that where the numerical value of
$\mathrm{Vol}(R_{2}(1))$ is precisely given by

(14)
$2\log(\alpha_{-}/\alpha_{+})+\frac{4}{3}(\alpha_{+}+\alpha_{-}),~{}\mathrm{where~{}\alpha_{\pm}~{}is~{}the~{}unique~{}real~{}root~{}of}~{}x^{3}\pm
x-1.$

In this section, let us denote $\mathcal{E}^{\prime}_{2}$ be the set of all
elliptic curves $E_{a,b}:=y^{2}=x^{3}+ax^{2}+bx$. First of all, let us note
that any $E_{A,B}\in\mathcal{E}_{2}$ can be written in the Weierstrass form
$E_{-3b,~{}3b^{2}+a}:=y^{2}=x^{3}-3bx^{2}+(3b^{2}+a)x,~{}a,b\in{\mathbb{Z}}$.
This shows that $\mathcal{E}_{2}\subset\mathcal{E}^{\prime}_{2}$. Let us
arrange the elliptic curves in $\mathcal{E}^{\prime}_{2}$, with respect to the
natural height function defined by
$H_{2}(E_{a,b})=\max\left\\{|a|^{2},|b|\right\\}^{1/2}$. With this, we set

$R^{\prime}_{2}(X)=\Big{\\{}(a,b)\in\mathbb{R}^{2}:|a|\leq X,~{}|b|\leq
X^{2}\Big{\\}},~{}\mathcal{E}^{\prime}_{2}(X)=R^{\prime}_{2}(X)\cap{\mathbb{Z}}^{2}.$

Note that $H(E_{A,B})\leq X$ if $|a|\asymp X^{2}$ and $|b|\asymp X$, where
$A=a,~{}B=b^{3}+ab$. In other words, $H(E_{A,B})\leq X$ if and only if
$H_{2}(E_{-3b,3b^{2}+a})\asymp X$. In particular, the number of elements in
$\mathcal{E}_{2}(X)$ and $\mathcal{E}^{\prime}_{2}(X)$ are asymptotically
comparable to $X^{3}$, i.e.,
$\\#\mathcal{E}_{2}(X)\asymp\\#\mathcal{E}^{\prime}_{2}(X)\asymp X^{3}$.

Furthermore, let us consider
${\mathbb{Z}}^{2}_{1}=\\{(A,B):\mathrm{gcd}(A,B)=1\\}$, and set

$\mathcal{E}_{1,2}(X)=\mathcal{E}_{2}(X)\cap{\mathbb{Z}}^{2}_{1},~{}\mathcal{E}^{\prime}_{1,2}(X)=\mathcal{E}_{2}(X)\cap{\mathbb{Z}}^{2}_{1}.$

The requirement of considering co-prime pairs arises due to Corollary 2.2.
Since we have
$\\#\mathcal{E}_{1,2}(X)\asymp\\#\mathcal{E}^{\prime}_{1,2}(X)\asymp X^{3}$,
to conclude the proof of Theorem 1, it suffices to work with the set
$\mathcal{E}^{\prime}_{1,2}=\bigcup\limits_{X\in\mathbb{R}}\mathcal{E}_{1,2}(X)$.

###  3.2. Counting over ${\mathcal{E}}_{3}$

We know from [17, Lem. 5.1, part (b)] that $E_{A,B}$ has a rational point of
order $3$ if and only if there exists $a,b\in{\mathbb{Z}}$ such that
$A=6ab+27a^{4}$ and $B=b^{2}-27a^{6}$. In the same spirit as before, we
consider $R_{3}(X)=\\{(a,b)\in\mathbb{R}^{2}:|6ab+27a^{4}|<X^{2}\text{ and
}|b^{2}-27a^{6}|<X^{3}\\}$. Again, by the principle of Lipschitz [11], we have

(15) ${\mathcal{E}}_{3}(X)=\\#(R_{3}(X)\cap{\mathbb{Z}}^{2})\asymp X^{2}.$

To study Watkins’s conjecture for the points in $\mathcal{E}_{3}$, we will
utilize conjectures such as the Generalized Riemann Hypothesis (GRH). However,
for unconditional analysis, we hall focus on the Type-I family of elliptic
curves given by $E_{a}:y^{2}=x^{3}+a$, as described briefly in Section 2.4.1.
It turns out that such an $E_{a}$ is in ${\mathcal{E}}_{3}$ when $a$ is a
square. We organize the elliptic curves $\\{E_{a}\\}$ based on their standard
height $H(E_{a})=|a|^{1/3}$. Notably, there are $8X^{3}$ many integers
$a\in{\mathbb{Z}}$ satisfying $H(E_{a})\leq X$.

###  3.3. Universal family for
${\mathcal{E}}_{3},{\mathcal{E}}_{5},{\mathcal{E}}_{7}$

Once we study Watkins’s conjecture over $\mathcal{E}_{2}$, i.e. the points
briefly described in Section 3.1, it is sufficient to focus only on cases
where the torsion group is $\mathrm{G}=E(\mathbb{Q})_{\mathrm{tor}}$ is non-
trivial, and $2\nmid\\#\mathrm{G}$. In the sense of [17, Sec. 1.4.2], there is
a universal elliptic curve $\mathcal{E}$ over an open subset of
$\mathbb{A}^{1}$, which parametrizes the elliptic curves whose group of
rational points contains $G$. Then the universal property [17, Sec. 1.4.2]
gives an elliptic surface $\mathcal{E}\to\mathbb{A}^{1}$, capturing the
required property with $\ell=3,5$ or $7$. Let us discuss it briefly, as we
need know about the parametrizations.

Let $S$ be a scheme. An elliptic curve over $S$ is a map $\pi:E\to S$,
together with a section $\sigma:S\to E$, such that every fiber of $\pi$ is a
smooth projective curve of genus $1$. We denote $Y_{1}(N)$ to be the curve,
whose points are given by $Y_{1}(N)(S)=\\{(E/S,P/S)~{}|~{}N\cdot P=0\\},$ and
consider $X_{1}(N)$ be the Deligne-Mumford compactification of $Y_{1}(N)$. It
turns out that, $X_{1}(N)$ is a scheme for any $N>3$. Since
$X_{1}(5),X_{1}(7)$ are of genus $0$, there exists an isomorphisms
$\varepsilon_{5}:X_{1}(5)\to\mathbb{A}^{1}(\mathbb{Q}),~{}\varepsilon_{7}:X_{1}(7)\to\mathbb{A}^{1}(\mathbb{Q})$.
These isomorphisms also give rise to the polynomials
$f_{5},g_{5},f_{7},g_{7}\in{\mathbb{Z}}[t]$, parametrizing almost all the
elements of $\mathcal{E}_{5}$ and $\mathcal{E}_{7}$. Combining the data in
[17, Table 2] and [17, Prop. 3.2], we have
$\mathrm{deg}(f_{5})=4,~{}\mathrm{deg}(g_{5})=6$, and
$\mathrm{deg}(f_{7})=8,~{}\mathrm{deg}(g_{7})=12$. In particular, it turns out
from [17, Table 1] that

(16) $\mathcal{E}_{5}(X)\sim X,~{}\mathcal{E}_{7}(X)\sim X^{1/2}.$

On the other hand, even though $X_{1}(3)$ is not scheme, almost all the points
$X_{1}(3)(\mathbb{Q})$ can still be paramterized by polynomials $f_{3},g_{3}$
with $\mathrm{deg}(f_{3})=1,\mathrm{deg}(g_{3})=2$. The exceptional points in
$X_{1}(3)(\mathbb{Q})$ that can not be parametrized, are Type-I curves of the
form $y^{2}=x^{3}+a$, where $a$ is a square. For more details, the reader may
refer to [17, Sec. 3.3]

##  4\. Watkins’s conjecture over ${\mathcal{E}}_{2}$ and ${\mathcal{E}}_{3}$

Let us recall the discussions on rank bound from Section 2.3. Our objective is
to examine the lack of local solubility within a specific family of
homogeneous spaces. Throughout this section, we shall follow the terminologies
as in Section 3.1.

###  4.1. Bounding Selmer ranks

In this section, we want to eliminate sufficiently many hall divisors $d$ of
$a^{2}-4b$ for which $C_{d,\frac{a^{2}-4b}{d}}(\mathbb{Q}_{p})=\phi$ for some
prime $p\mid a^{2}-4b$.

######  Lemma 1.

Let $p>3$ be a prime dividing $b$ such that $p\nmid a$ and $d$ be a square-
free hall divisor of $a^{2}-4b$. Then
$C_{d,\frac{a^{2}-4b}{d}}(\mathbb{Q}_{p})=\phi$ if and only if the following
two conditions hold.

  * (i)

$\left(\frac{d}{p}\right)=-1$

  * (ii)

$\nu_{p}(b)=\mathrm{odd}~{}\mathrm{or}~{}\left(\frac{a}{p}\right)=1$

###### Proof.

Our goal is to ensure that none of the conditions in part (1) of [1, Lem. 3.1]
are satisfied.

Given that $p\mid b$, it is important to observe that
$\left(\frac{d}{p}\right)=-1$ if and only if
$\left(\frac{\frac{a^{2}-4b}{d}}{p}\right)=-1$. Due to the condition in $(i)$,
we see that the condition (a) in part (1) of [1, Lem. 3.1] is not satisfied,
with $d_{1}=d$ and $d_{2}=\frac{a^{2}-4b}{d}$.

On the other hand, it is worth noting that the negation of condition (b) in
part (1) of [1, Lem. 3.1] is equivalent to having $\nu_{p}(b)=\mathrm{odd}$ or
$\left(\frac{a}{p}\right)=1$, both of which are the assumptions that we have
already made. ∎

Now the idea is that we count the number of $(a,b)$ satisfying the property:
there exists at least one prime $q\mid s(a^{2}-b)$, for which there is a prime
$p$ with $\left(\frac{q}{p}\right)=-1$ and $\left(\frac{a}{p}\right)=1$. For
this, it is enough to eliminate the pairs $(a,b)$ with the property: given any
pair of signs $(\sigma_{1},\sigma_{2})$, for any prime $q\mid s(a^{2}-b)$, and
for any prime $p$ satisfying $\left(\frac{a}{p}\right)=\sigma_{2}$, we have
$\left(\frac{q}{p}\right)=\sigma_{1}$. In particular, our job will be done by
taking any pairs of signs $(\sigma_{1},\sigma_{2})\neq(-1,1)$.

######  Lemma 2.

Let $\sigma_{2}\in\\{\pm 1\\}$ be any sign, $a\in{\mathbb{Z}}$ be an integer
which is not a square, and $p$ be a prime with
$\left(\frac{a}{p}\right)=\sigma_{2}$. Then, we have the following estimate,
as we take $X\to\infty$

$N_{a}(X)=\\#\Big{\\{}b\in{\mathbb{Z}}:|b|\leq X,~{}q~{}\mathrm{prime}\mid
s(a^{2}-b)\implies\left(\frac{q}{p}\right)=\sigma_{1}\Big{\\}}=O\left((X+a^{2})^{1/2}+\frac{X+a^{2}}{(\log(X+a^{2}))^{1/2}}\right),$

where the implicit constant is absolute. In particular, we have the estimate
as $X\to\infty$

(17) $\Big{\\{}E_{a,b}:H_{2}(E_{a,b})\leq X,~{}q~{}\mathrm{prime}\mid
s(a^{2}-b)\implies\left(\frac{q}{p}\right)=\sigma_{1}\Big{\\}}=O\left(\frac{X^{3}}{\log
X}\right),$

where the implicit constant is absolute.

###### Proof.

Given any $a$ and $p$ as per the requirement, the set of primes $q$ with
$\left(\frac{q}{p}\right)=1$, has density $1/2$. In other words, we need to
demonstrate that for a rare integer $b$, all the prime factors of $s(a^{2}-b)$
belong to a set of primes with density $1/2$. For any parameter $y$, the
number of integers $0<n\leq y$ for which all the prime factors of $n$ belong
to a set of primes with density $1/2$ grows asymptotically like
$\frac{y}{\sqrt{\log y}}$. This observation shows that

$\displaystyle N_{a}(X)$
$\displaystyle\leq\sum_{n\leq(X+a^{2})^{1/2}}\frac{1}{n^{2}}\frac{X+a^{2}}{(\log(X+a^{2})-\log
n^{2})^{1/2}}$ (18)
$\displaystyle=\sum_{(X+a^{2})^{1/4}<n\leq(X+a^{2})^{1/2}}\frac{1}{n^{2}}\frac{X+a^{2}}{(\log(X+a^{2})-\log
n^{2})^{1/2}}+\sum_{n\leq(X+a^{2})^{1/4}}\frac{1}{n^{2}}\frac{X+a^{2}}{(\log(X+a^{2})-\log
n^{2})^{1/2}}.$

It is evident that the first sum in (4.1) is $O((x+a^{2})^{1/2})$. Moreover,
the second sum in (4.1) is
$O\left(\frac{X+a^{2}}{(\log(X+a^{2}))^{1/2}}\right)$, and this gives the
desired estimate for the quantity $N_{a}(X)$.

Let us now recall from Section 3.1 that $H_{2}(E_{a,b})\leq X$ implies
$|a|\leq X$ and $|b|\leq X^{2}$. The required estimate at (17) then follows
directly by estimating the sum $\sum\limits_{a\leq X}N_{a}(X^{2})$. ∎

Consequently, we obtain the following implication on the $\phi$-Selmer ranks.

######  Corollary 4.1.

For a positive proportion of elliptic curves in ${\mathcal{E}}_{2}$, we have
$\mathrm{rank}(E(\mathbb{Q}))\leq\omega(N(E))-2$.

###### Proof.

Lemma 2 implies that, for almost all pairs $(a,b)\in{\mathbb{Z}}^{2}$, there
exist primes $p,q$ (dependent on $a,b$) such that
$\nu_{q}(a^{2}-4b)=\mathrm{odd}$, and the divisor $d=q^{\nu_{q}(a^{2}-4b)}$
satisfies $\left(\frac{d}{p}\right)=-1$. Specifically, Lemma 1 implies
$C_{d,\frac{a^{2}-4b}{d}}(\mathbb{Q}_{p})=\phi$. Hence, the proof follows
combining Lemma 2 and Corollary 2.2. ∎

To prove the main result of this section, we need one more result.

######  Lemma 3.

The $2$-torsion group for $100\%$ of the all elliptic curves in
${\mathcal{E}}_{2}$ is ${\mathbb{Z}}/2{\mathbb{Z}}$.

###### Proof.

It is enough to establish that, for $100\%$ of all pairs
$(a,b)\in{\mathbb{Z}}^{2}$, the corresponding quadratic equation
$x^{2}+ax+b=0$ has no rational solutions. In particular, this means that the
discriminant $a^{2}-4b$ is not a perfect square. This assertion is evident
since, for any fixed square $S$, there are at most $O(X)$ pairs
$(a,b)\in{\mathbb{Z}}^{2}$ satisfying $|a|\leq X$, $|b|\leq X^{2}$, and
$a^{2}-4b=S$. However, since there are only $O(X)$ choices for $S$, this
completes the argument. ∎

### Proof of Theorem 1

Lemma 3 and Corollary 2.1 imply that, for almost all elliptic curves $E$ in
${\mathcal{E}}_{2}$, we have $\omega(N(E))-2\leq\nu_{2}(m_{E})$. Therefore,
the result can be deduced from Corollary 4.1. ∎

###  4.2. $M$-Watkins over $\mathcal{E}_{2}$

For any integer $M$, and for each integer $a$ which is not a perfect square,
we consider the following set of primes
$\mathbb{P}_{a}=\\{p~{}\mathrm{prime}:\left(\frac{a}{p}\right)=1\\}$. We pick
the least prime $p>3$ from $\mathbb{P}_{a}$ and denote it as $P(a)$. For any
integer $i$, we also denote $q_{i}$ to be the $i$th least prime for which
$\left(\frac{q_{i}}{p}\right)=-1$, and set $d_{i}=q_{i},~{}\forall 1\leq i\leq
M$. We then consider
$N_{a}=\Big{\\{}b\in{\mathbb{Z}}:\nu_{q_{i}}(a^{2}-4b)=1,~{}\forall 1\leq
i\leq M\Big{\\}}.$

######  Lemma 4.

For any integer $a$ that is not a square, and and for any $b\in N_{M}(a)$
which is co-prime to $a$, we have the following rank bound

$\mathrm{rank}(E_{a,b}(\mathbb{Q}))\leq\omega(N(E_{a,b}))-M.$

###### Proof.

Lemma 1 implies that the $\phi$-Selmer rank in (5) is at most
$\omega(a^{2}-4b)-M$. Furthermore, the $\hat{\phi}$-Selmer rank in (6) of
course at most $\omega(b)$. Since we assume that $a,b$ are co-prime,
$\omega(b(a^{2}-4b))\leq\omega(N(E_{a,b}))+1$. The result now follows from the
rank bound at (8). ∎

Now, we set the quantity $d_{1}d_{2}\cdots d_{M}$ to be $D_{M}(a)$, and then
we have the following estimation of $D_{M}(a)$.

######  Lemma 5.

For any integer $a\equiv 2\pmod{5}$ that is not a square, we have
$D_{M}(A)\asymp M^{M}$.

###### Proof.

For any $a\equiv 2\pmod{5}$, we have $P(a)=5$. Now we choose $q_{i}$ as the
$i$th least prime for which $q_{i}\equiv 2\pmod{5}$. This shows that,
$D_{M}(a)\asymp_{M}\prod\limits_{1\leq i\leq M}i\log i\sim M^{M}$, as desired.
∎

Now, for the counting purposes, we denote

$N_{M}(X)=\\#\\{a,b\in{\mathbb{Z}}:H_{2}(E_{a,b})\leq
X,~{}\mathrm{rank}(E_{a,b}(\mathbb{Q}))\leq\omega(a^{2}-4b)-M\\}.$

######  Lemma 6.

For any integer $M$, we have $N_{M}(X)\geq\frac{4}{5}\frac{1}{(2M)^{M}}X^{3}$,
as $X\to\infty$.

###### Proof.

Lemma 4 implies that,

$\displaystyle N_{M}(X)$
$\displaystyle\geq\\#\Big{\\{}a,b\in{\mathbb{Z}}:(a,b)=1,~{}|a|\leq
X,~{}|b|\leq X^{2},~{}\nu_{q_{i}}(a^{2}-4b)=1,~{}\forall 1\leq i\leq
M\Big{\\}}$ $\displaystyle=\sum_{|a|\leq
X}\\#\Big{\\{}b\in{\mathbb{Z}}:(b,a)=1,~{}|b|\leq X^{2},~{}q_{i}\mid
a^{2}-4b,~{}q_{i}^{2}\nmid a^{2}-4b,~{}\forall 1\leq i\leq M\Big{\\}}$
$\displaystyle=2X^{2}\left(\sum_{|a|\leq
X}\frac{\phi(a)}{a}\prod_{i=1}^{M}\left(\frac{1}{q_{i}}-\frac{1}{q_{i}^{2}}\right)\right)+O_{M}(X)$
$\displaystyle\geq 2X^{2}\left(\sum_{|a|\leq
X}\frac{\phi(a)}{a}\prod_{i=1}^{M}\frac{1}{2q_{i}}\right)+O_{M}(X)$
$\displaystyle\geq 2X^{2}\left(\sum_{\begin{subarray}{c}|a|\leq X\\\ a\equiv
2\pmod{5}\end{subarray}}\frac{\phi(a)}{a}\frac{1}{2^{M}D_{M}(a)}\right)+O_{M}(X)$
$\displaystyle=\frac{1}{(2M)^{M}}\frac{1}{\zeta(2)}\frac{4}{5}X^{3}+O_{M}(X^{2}\log
X),$

and this completes the proof, as desired. ∎

###  4.3. Proof of Theorem 2

The proof follows combing Lemma 4 and Lemma 6.∎

###  4.4. Low $2$-Selmer rank for a specific sub-family of $\mathcal{E}_{2}$

In the introduction, we highlighted that Watkins’s conjecture is verified for
elliptic curves having a rank no greater than $1$. This serves as our
inspiration to investigate pairs $(A,B)$ where the corresponding elliptic
curve $E_{A,B}$ has a rank of either $0$ or $1$. This section of our study
heavily relies on exploring quadratic twists of a specific elliptic curve.
Given an elliptic curve $E_{A,B}/\mathbb{Q}$, we delve into the collection of
quadratic twists denoted as ${E_{A,B}^{(D)}}$. Lozano and Pasten showed in
[16] that Watkins’s conjecture is true for any $D$ with

(19) $\omega(D)\geq 6+5\omega(N(E))-\nu_{2}(m_{E}/c^{2}_{E}).$

When the elliptic curve $E_{A,B}$ has prime power conductor, Caro in [8],
using Setzer’s classification [27], established the validity of Watkins’s
conjecture for any $E_{A,B}^{(D)}$.

In this section, we specifically focus on the elliptic curve denoted as
$E_{0}:=y^{2}=x^{3}-1$. Notably, we have the values $N(E_{0})=144$,
$m_{E_{0}}=64$, and $E_{0}[2](\mathbb{Q})={\mathbb{Z}}/2{\mathbb{Z}}$.
Therefore, we are not able to apply Caro’s result for $E_{0}$. However, using
equation (19), it turns out that Watkins’s conjecture is valid for
$E_{0}^{(D)}$ when $\omega(D)\geq 10+2\nu_{2}(c_{E_{0}})$. To address the
remaining cases, we rely on the crucial results derived using techniques
outlined in [20].

######  Lemma 7.

Let $D$ be any square-free integer with one of the following properties:

  1. (i)

all prime factors are $5\pmod{12}$.

  2. (ii)

all but exactly one prime factors are $5\pmod{12}$, and the leftover prime is
$-1\pmod{4}$.

Then for any such $D$, we have $\mathrm{rank}(E_{0}^{(D)}(\mathbb{Q}))\leq 1$.

###### Proof.

If $D$ satisfies the condition at $(i)$, then according to [20, Prop. 1], we
have
$\mathrm{dim}_{\mathbb{F}_{2}}(\mathrm{Sel}^{2}(E_{0}^{(D)}))=\mathrm{dim}_{\mathbb{F}_{2}}(\mathrm{Sel}^{2}(E^{\prime(D)}))=1$,
and this shows that $\mathrm{rank}(E^{(D)}(\mathbb{Q}))=0$. Now, suppose $D$
satisfies the condition at $(ii)$. Applying case (4) of [20, Lem. 4] and case
(5) of [20, Lem. 5], we obtain
$\mathrm{dim}_{\mathbb{F}_{2}}(\mathrm{Sel}^{2}(E^{(D)}))+\mathrm{dim}_{\mathbb{F}_{2}}(\mathrm{Sel}^{2}(E^{\prime(D)}))\leq
3,$ and this shows that $\mathrm{rank}(E^{(D)}(\mathbb{Q}))\leq 1$. ∎

######  Remark 2.

We can establish that $\mathrm{rank}(E^{(p)}(\mathbb{Q}))\leq 1$ for any prime
$p$ under the conditions that $p\equiv 5\text{ or }11\pmod{12}$. Specifically,
this result is stated as [20, Thm. 2] for $p\equiv 5\pmod{12}$, while for
$p\equiv 11\pmod{12}$, it can be deduced by applying the proof of [20, Thm.
2].

As a result of the aforementioned analysis, we can derive the following
extension of [16] regarding Watkins’s conjecture.

######  Corollary 4.2.

For any squarre-free integer $D$, we have $E_{0}^{(D)}\in\mathcal{E}_{2}$.
Then, Watkins’s conjecture holds true for $E_{0}^{(D)}$, provided that $D$
satisfies one of the following:

  1. (i)

$\omega(D)\geq 10+2\nu_{2}(c_{E_{0}})$

  2. (ii)

$\omega(D)<10+2\nu_{2}(c_{E_{0}})$, and $D$ satisfies one of the conditions
specified in Lemma 7.

###### Proof.

It is evident that $E_{0}^{(D)}[2](\mathbb{Q})={\mathbb{Z}}/2{\mathbb{Z}}$ for
any $D$. Now the proof for part $(i)$ follows from [16], while the proof of
part $(ii)$ uses Lemma 7 and the fact that Watkins’s conjecture is true for
elliptic curves of rank at most $1$. ∎

### Proof of Theorem 3

#### Proof of part (iii)

The size of $\mathcal{E}_{2}(X)$ is computed in (13) in Section 3.1. For the
$M$-Watkins elliptic curves, we shall use Lemma 7. Note that
$H(E_{0}^{(D)})\leq X$, if and only if, $|D|\leq X^{1/2}$. Due to Lemma 7, it
is enough to estimate

$\Big{\\{}D\in{\mathbb{Z}}:|D|\leq
X^{1/2},~{}\omega(D)>M,~{}p~{}\mathrm{prime}\mid D\implies p\equiv
5\pmod{12}\Big{\\}}.$

Let us note that

(20) $\\#\Big{\\{}D\in{\mathbb{Z}}:|D|\leq X^{1/2},~{}\omega(D)\leq
M\Big{\\}}=O_{M}\left(\frac{X^{1/2}}{\log X}(\log\log X)^{M}\right).$

Moreover, since $\\{p~{}\mathrm{prime}:p\equiv 5\pmod{12}\\}$ has density
$1/4$, we also have

(21) $\\#\Big{\\{}D\in{\mathbb{Z}}:|D|\leq
X^{1/2},~{}~{}p~{}\mathrm{prime}\mid D\implies p\equiv
5\pmod{12}\Big{\\}}=\frac{X^{1/2}}{(\log X)^{1/4}}.$

The desired estimate now follows combining (20) and (21). ∎

To prove part $(ii)$ of Theorem 1.1, we first need to study the Watkins for
the elliptic curves of Type-I. Consequently, we will deduce part $(ii)$ along
with proving Theorem 3.

###  4.5. Watkins over $\mathcal{E}_{3}$

Let us begin by revisiting the discussion and notations outlined in Section
2.4. For any integer $a$, consider the elliptic curve $E_{a}:=y^{2}=x^{3}+a$.
In this section, we shall study Watkins’s conjecture for the family of
elliptic curves $\\{E_{a}\\}$.

### Proof of Theorem 3

###### Proof of part (i).

It is enough to show that $S=\\#\Big{\\{}a\in\mathbb{Z}:|a|\leq
X,~{}\mathrm{rank}(E_{a})>\omega(a)-M\Big{\\}}$ is
$O_{M}\left(\frac{X}{\log\log X}\right)$. To do this, let us first note that
[5, Thm. 3] implies

$\sum_{\begin{subarray}{c}a\in
S\end{subarray}}(\omega(a)-M)\leq\sum_{\begin{subarray}{c}a\in
S\end{subarray}}\mathrm{rank}(E_{a})=O(X).$

Since $S\leq 2X$, we also have that $\sum\limits_{\begin{subarray}{c}a\in
S\end{subarray}}\omega(a)=O(X)$. On the other hand, it follows from [10, Thm.
3.2.3] that $\sum\limits_{\begin{subarray}{c}a\in
S\end{subarray}}\Big{|}\omega(a)-\log\log X\Big{|}^{2}=O\left(X\log\log
X\right)$. In particular, applying triangle inequality, we have
$\\#~{}S~{}(\log\log X)^{2}=O\left(X\log\log X\right)$, and this completes the
proof. ∎

With this, we are now prepared to investigate Watkins’s conjecture over
${\mathcal{E}}_{3}$. Combining (9), (11), and (12), we have

(22) $\operatorname{rank}\left(E_{a}(\mathbb{Q})\right)\leq
r_{3}(K_{a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{a}}/U_{K_{a}}^{3}\right)+r_{3}(K_{-27a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{-27a}}/U_{K_{-27a}}^{3}\right)+\\#~{}S_{a}+\\#~{}S_{-27a}.$

Now, let us recall the set of primes $S_{a}$, defined by $2,3\in S_{a}$, and
$p\neq 2,3\in
S_{a}\Longleftrightarrow\nu_{p}(a)=2,4~{}\text{and}~{}\left(\frac{-3}{p}\right)=1$.
Our objective is to establish an upper bound on $\\#~{}S_{a}+\\#~{}S_{-27a}$
over a suitable set of integers $a$. By definition, we know that
$\\#~{}S_{a}=\\#~{}S_{-27a}$. Hence, our goal is to determine the proportion
of all $a$ for which $\\#~{}S_{a}$ is significantly smaller than
$\frac{\omega(a)}{2}$.

####  4.5.1. Bounding $r_{3}$ part

For any integer $a$, let us recall that $K_{a}=\mathbb{Q}(\sqrt{-3a})$, and
$r_{3}(K_{a})$ denotes order of the $3$-torsion part of the class group of
$K_{a}$. Note that $r_{3}(K_{a})=O(1)$, when $a$ is a square, up to a constant
factor. Based on this observation, we can derive the following result.

######  Lemma 8.

For any integers $m,k_{1},k_{2},\cdots k_{m}$, consider the set of integers
$N_{\vec{k},m}=\bigcup\limits_{i=1}^{m}\\{k_{i}n^{2}:n\in{\mathbb{Z}}\\}$,
where $\vec{k}=(k_{1},k_{2},\cdots,k_{m})$. For any $a\in N_{\vec{k},m}$, we
have

(23)
$r_{3}(K_{a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{a}}/U_{K_{a}}^{3}\right)+r_{3}(K_{-27a})+\dim_{{\mathbb{F}}_{3}}\left(U_{K_{-27a}}/U_{K_{-27a}}^{3}\right)=O_{\vec{k}}(1).$

Let us now note that, the discussions in Section 3.1 and Section 3.2 implies
that

(24) $\displaystyle E_{a}\in\begin{cases}\mathcal{E}_{2}&\text{if $a$ is a
cube,}\\\ \mathcal{E}_{3},&\text{if $a$ is a square.}\\\ \end{cases}$

#### Proof of part (ii)

First of all for any $a\in N_{\vec{k},m}$, we have

(25) $\mathrm{rank}(E_{a}(\mathbb{Q}))\leq 2~{}\\#S_{a}+O_{\mathcal{S}}(1).$

For any integer $M>0$, it is now enough to show that
$\\#S_{a}\leq\frac{\omega(a)}{2}-M$, for at least
$\gg_{\vec{k},m}\left(\frac{X}{\log X}\right)^{1/2}$ many elements in
$N_{\vec{k},m}(X)$. Since $N_{\vec{k},m}(X)$ is contained in
$\bigcup\limits_{i=1}^{m}\\{k_{i}n^{2}\\}$, it is enough to only consider $a$
of the form $kn^{2}$, for a given integer $k$.

We have $\\#\Big{\\{}a\in{\mathbb{Z}}:|a|\leq y^{1/2}:~{}p\mid
a\implies\left(\frac{-3}{p}\right)=-1\Big{\\}}\gg\left(\frac{y}{\log
y}\right)^{1/2}$, as the limit $y\to\infty$. On the other hand for any integer
$M$, we also have $\\#\Big{\\{}a\in{\mathbb{Z}}:a\leq y,\omega(a)\leq
M\Big{\\}}=O_{M}\left(\frac{y^{1/2}}{\log y}(\log\log y)^{M}\right)$.

Since we have, $\left(\frac{y}{\log y}\right)^{1/2}\gg\frac{y^{1/2}}{\log
y}(\log\log y)^{M}$, the proof is complete taking
$y=\min\left\\{\frac{X^{1/2}}{k_{i}}\right\\}_{1\leq i\leq m}$. ∎

### Proof of Corollary 1.1

First of all, the size of $\mathcal{E}_{3}(X)$ follows from (15) in Section
3.2. The lower bound on the number of $M$-Watkins elliptic curves in
$\mathcal{E}_{3}$, taking $m=1$ and $k_{1}=1$ in part $(i)$ of Theorem 3. ∎

######  Remark 3.

We expect that $\\#S_{a}\leq\frac{\omega(a)}{2}-M$ will hold for $50\%$ of the
times over $N_{\vec{k},m}$. This is because for any set of primes $S$ of
density $1/2$ if we denote $\omega_{S}(n)$ be the number of prime factors of
$n$ from $S$, then even though $\omega_{S}(n)-\frac{\omega(n)}{2}$ typically
is small, it is negative if and only if
$\omega_{S^{\prime}}(n)-\frac{\omega(n)}{2}$ is positive. Here $S^{\prime}$ is
the complement of $S$, which also has a density of $1/2$. Since $S$ and
$S^{\prime}$ both are of density $1/2$, we see no reason to believe that signs
in each of the sets
$\\{\omega_{S}(n)-\frac{\omega(n)}{2}\\}_{n\in{\mathbb{Z}}}$ and
$\\{\omega_{S^{\prime}}(n)-\frac{\omega(n)}{2}\\}_{n\in{\mathbb{Z}}}$ are not
equidistributed.

##  5\. $M$-Watkins over ${\mathcal{E}}_{3},{\mathcal{E}}_{5}$ and
${\mathcal{E}}_{7}$

The key tool that we will be using is the main counting lemma due to Harron,
Snowden [17], designed to count the twists of a given $1$-parameter family of
elliptic curves.

######  Lemma 9.

Let us consider $f_{\ell}$ and $g_{\ell}$, for $\ell\in\\{3,5,7\\}$. Denote
$\mathcal{E}_{\ell}(X)$ be the set of pairs $(A,B)\in{\mathbb{Z}}^{2}$
satisfying the following conditions:

  * (i)

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

  * (ii)

$\gcd(A^{3},B^{2})$ is not divisible by any 12th power.

  * (iii)

$|A|<X^{2}$ and $|B|<X^{3}$.

  * (iv)

There exist $u,r\in\mathbb{Q}$ such that $A=u^{4}f(r)$ and $B=u^{6}g(r)$.

Denote $\mathcal{E}^{\prime}_{\ell}(X)$ to be the set of all
$(u,r)\in\mathbb{Q}^{2}$ appearing in the last point above. Then, we have

(26)
$\Big{\\{}r:(u,r)\in\mathcal{E}^{\prime}_{\ell}(X)\Big{\\}}\subseteq\Big{\\{}a/b:|a|=O_{f_{\ell},~{}g_{\ell}}\left(|X|^{m_{\ell}}\right),~{}|b|=O_{f_{\ell},~{}g_{\ell}}\left(X^{n_{\ell}}\right)\Big{\\}},$

where the exponents $m_{\ell},n_{\ell}$ are explicitly given by

$\displaystyle(m_{\ell},~{}n_{\ell})=\begin{cases}(3/2,~{}1/2),&\text{if
$f=f_{5},~{}\mathrm{and}~{}g=g_{5}$},\\\ (1/4,~{}1/4),&\text{if
$f=f_{5},~{}\mathrm{and}~{}g=g_{5}$},\\\ (1/2,~{}1/2),&\text{if
$f=f_{7},\mathrm{and}~{}g=g_{7}$.}\\\ \end{cases}$

###### Proof.

It follows from [17, Table 2] that the relevant $m$ is $1$ for both $f_{5}$
and $f_{7}$. In particular, [17, Lem. 2.5] implies the result for $\ell=5$ and
$7$. Now for $\ell=3$, [17, Table 2] and [17, Lem. 2.4] indicates that $r$ can
be written as $a/b^{2}$ such that $\mathrm{gcd}(a,b^{2})$ is square-free.
Consequently, the result follows from [17, Lem. 2.5]. ∎

Consequently, we shall deduce a result on the average rank for the fibers of
an elliptic surface over $\mathbb{Q}(t)$, which will play a key role in this
section. As mentioned in the introduction, it is recently conjectured by Park,
Poonen, Voight, and Wood in [25] that there are only finitely many elliptic
curves of greater than rank $21$. For this section, we only need to know that
only a small proportion of the elliptic curves have a large rank. In this
regard, we prove the following.

######  Lemma 10.

Consider $f_{\ell}$ and $g_{\ell}$, for $\ell\in\\{3,5,7\\}$, and assume that
the conjectures $B,C$ and $D$ in [28] hold. Then for any integer $M\geq 1$,
the proportion of elliptic curves $E\in\mathcal{E}_{f,g}$ with a rank greater
equals to $M$ is at most $O_{f,g}\left(\frac{1}{M}\right)$.

Additionally, we now need to examine the conductors of the elliptic curves in
$\mathcal{E}_{\ell}(X)$. Proposition 5.2 is central to this analysis. For the
preparations of the two tasks as described above, we require the following
facts for polynomials $f_{3}$, $f_{5}$, $f_{7}$ (and their counterparts
$g_{3}$, $g_{5}$, $g_{7}$):

######  Lemma 11.

Let us consider the polynomials $\Delta_{\ell}=4f_{\ell}^{3}+27g_{\ell}^{2}$,
for $\ell\in\\{3,5,7\\}$. It follows that any of these polynomials, namely
$\Delta_{3}$, $\Delta_{5}$, and $\Delta_{7}$, have at least one non-trivial
irreducible factor in $\mathbb{Z}[t]$ that divides with multiplicity one.

To prove both Lemma 11 and Lemma 10, we explicitly write down the equations of
$\Delta_{\ell}$ for $\ell=3,5$ and $7$, which we briefly do in the next
section.

###  5.1. Structures of $f_{3},f_{5}$ and $f_{7}$

#### Equation for $\Delta_{3}$

The authors in [17, Lem. 3.4] demonstrate that the elements in
$\mathcal{E}_{3}$ can be represented by the equation $y^{2}+axy+by=x^{3}$,
where $P=(0,0)$ is a point on any such curve with order $3$. Notably, when
$a=0$, this corresponds to a family that covers only $0\%$ of the elements in
$\mathcal{E}_{3}$. This aspect is crucial because the curve $y^{2}+by=x^{3}$
is isomorphic to $y^{2}=x^{3}+16b^{2}$.

As a consequence, [17, Prop. 3.5] establishes that nearly all elements in
$\mathcal{E}_{3}$ can be expressed through integral polynomials as follows:

$f_{3}(t)=3^{4}\left(2t-\frac{1}{3}\right),~{}~{}~{}g_{3}(t)=3^{6}\left(t^{2}+\frac{2}{3}t+\frac{2}{27}\right).$

In this case, we have

(27)
$\Delta_{3}(t)=4f_{3}(t)^{3}-27g_{3}(t)^{2}=-14348907t^{4}-2125764t^{3}-17006112t^{2}-157464,$

which is irreducible in ${\mathbb{Z}}[t]$.

Let us recall that the Tate normal form of an elliptic curve with $P=(0,0)$ is
given by $E=E(b,c):Y^{2}+(1-c)XY-bY=X^{3}-bX^{2}.$ The discriminant of
$E(b,c)$ is $\Delta(b,c)=b^{3}(16b^{2}-8bc^{2}-20bc+b+c(c-1)^{3})$. With this,
one has the following:

(28) $5P=\infty\iff b=c,~{}7P=\infty\iff b=t^{3}-t^{2},~{}c=t^{2}-t.$

#### Equation for $\Delta_{5}$

The corresponding family of elliptic curves is $\left\\{Y^{2}+(1-t)XY-
tY=X^{3}-tX^{2}\right\\}_{t\in\mathbb{Q}}$. A simple computation of SageMath,
immediately gives us the discriminant

(29) $\Delta_{5}(t)=(t^{2}-11t-1)t^{5}.$

#### Equation for $\Delta_{7}$

The corresponding family is
$\left\\{Y^{2}+(1-t^{2}+t)XY-(t^{3}-t^{2})Y=X^{3}-(t^{3}-t^{2})X^{2}\right\\}_{t\in\mathbb{Q}}$.
Again, a SageMath computation gives us

(30) $\Delta_{7}(t)=(t^{3}-8t^{2}+5t+1)(t-1)^{7}t^{7}.$

### Proof of Lemma 11

The proof is based on the explicit equations derived in the earlier sections
for $\Delta_{3}$, $\Delta_{5}$, and $\Delta_{7}$, respectively. More
specifically, the proof follows from the equations (27), (29) and (30). ∎

###  5.2. Aerage rank of elliptic curves and the conclusions

Let $\mathcal{E}/\mathbb{Q}(t)$ be an elliptic surface, i.e., a $1$-parameter
family of elliptic curves, given by

$\mathcal{E}_{f,g}:y^{2}=x^{3}+f(t)x+g(t),~{}f(t),g(t)\in{\mathbb{Z}}[t].$

For each rational $r\in\mathbb{Q}$, let us consider the curve
$E_{f(r),g(r)}:=y^{2}=x^{3}+f(r)x+g(r)$. The main goal of this section is to
study the ranks of the elliptic curves $E_{f(r),g(r)}$, as we vary $r$ over
all the rationals. We follow the approach in [28], and obtain the following
estimate on average rank.

######  Proposition 5.1.

Let $f,g\in{\mathbb{Z}}[t]$ be two arbitrary polynomials, and assume that the
conjectures $B,C$, and $D$ in [28] hold. For any two parameters
$X,Y\to\infty$, we have the following estimate

$R_{f,g}(X,Y)=\sum\limits_{\begin{subarray}{c}r=a/b\in\mathbb{Q}\\\ |a|\leq
X,~{}|b|\leq Y\end{subarray}}\mathrm{rank}(E_{f(r),g(r)})=O_{f,g}(XY),$

provided that $\mathcal{E}_{f,g}$ is a non-split elliptic surface.

Before proving Proposition 5.1, let us first consider the average of Frobenius
trace over the fibers of any elliptic surface
$\mathcal{E}_{f,g}/\mathbb{Q}(t),~{}f,g\in{\mathbb{Z}}[t]$, and any prime $p$,
as follows

$A_{p}(\mathcal{E})=\frac{1}{p}\sum_{\bar{r}\in\mathbb{F}_{p}}a_{p}(\mathcal{E}_{\bar{r}}),$

where we denote $a_{p}(E)$ to be the the quantity $p+1-\\#E(\mathbb{F}_{p})$,
for any elliptic curve $E/\mathbb{Q}$.

######  Lemma 12.

For any polynomials $f,g\in{\mathbb{Z}}[t]$, and for any prime $p$, we have
the estimate

$A_{p}(\mathcal{E})\leq
3\deg(\Delta(\mathcal{E}_{f,g}))+\deg(c_{4}(\mathcal{E}_{f,g}))-2.$

In particular, $A_{p}(\mathcal{E})$ is bounded independent of $p$ and
$\mathcal{E}$, as the quantity on the right above is uniformly bounded.

###### Proof.

It follows from [24, Prop. 1.2] that

$\sum\limits_{\begin{subarray}{c}\bar{r}\in\mathbb{F}_{p}\\\
\Delta(\bar{r})\neq 0\end{subarray}}a_{p}(\mathcal{E}_{\bar{r}})\leq
p(2\deg(\Delta(\mathcal{E}_{f,g}))+\deg(c_{4}(\mathcal{E}_{f,g}))-2).$

The proof follows immediately, as the number of $\bar{r}\in{\mathbb{F}}_{p}$
with $\Delta(\bar{r})=0$ is at most $\deg(\Delta)$. ∎

### Proof of Proposition 5.1

Let us denote $d=\max\left\\{\mathrm{deg}(f),\mathrm{deg}(g)\right\\}$. For
any rational $r=a/b$, note that $E_{f(r),g(r)}$ is isomorphic to
$E_{b^{4d}f(r),b^{6d}g(r)}$ over $\mathbb{Q}$. Additionally, both $b^{4d}f(r)$
and $b^{6d}g(r)$ are integers. For each $b$, observe that both of
$f_{b}(t):=b^{4d}f(t/b)$, and $g_{b}(t):=b^{6d}g(t/b)$, are polynomials with
integer coefficients.

For each $b\in{\mathbb{Z}}$, let us consider the elliptic surface

$\mathcal{E}_{f_{b},g_{b}}:y^{2}=x^{3}+f_{b}(t)x+g_{b}(t),~{}f_{b}(t),~{}g_{b}(t)\in{\mathbb{Z}}[t].$

It is clear that
$\mathrm{rank}(\mathcal{E}_{f,g}(\mathbb{Q}(t))=\mathrm{rank}(\mathcal{E}_{f_{b},g_{b}}(\mathbb{Q}(t))$,
for any $b\in{\mathbb{Z}}$. Additionally, $\Delta(E_{f,g})(\alpha)=0$ if and
only if, $\Delta_{E_{f_{b},g_{b}}}(b\alpha)=0$. In particular,
$\mathrm{deg}(\Delta_{\mathcal{E}_{f,g}})=\mathrm{deg}(\Delta_{\mathcal{E}_{f_{b},g_{b}}})$.
Similarly, we also have
$\mathrm{deg}(c_{4}(\mathcal{E}_{f,g}))=\mathrm{deg}(c_{4}(\mathcal{E}_{f_{b},g_{b}}))$
and
$\mathrm{deg}(c_{6}(\mathcal{E}_{f,g}))=\mathrm{deg}(c_{6}(\mathcal{E}_{f_{b},g_{b}}))$.
Summarizing the discussion, we have the following two crucial ingredients

(31)
$\mathrm{rank}(\mathcal{E}_{f,g}(\mathbb{Q}(t))=\mathrm{rank}(\mathcal{E}_{f_{b},g_{b}}(\mathbb{Q}(t)),~{}\mathrm{deg}(N(\mathcal{E}_{f,g}))=\mathrm{deg}(N(\mathcal{E}_{f_{b},g_{b}})),~{}\forall
b\in{\mathbb{Z}}.$

Furthermore, we can write

(32) $\displaystyle
R_{f,g}(X,Y)=\sum\limits_{\begin{subarray}{c}r=a/b\in\mathbb{Q}\\\ |a|\leq
X,~{}|b|\leq Y\end{subarray}}\mathrm{rank}(E_{f(r),g(r)})$
$\displaystyle=\sum_{\begin{subarray}{c}b\in{\mathbb{Z}}\\\ |b|\leq
Y\end{subarray}}~{}\sum_{\begin{subarray}{c}a\in{\mathbb{Z}}\\\ |a|\leq
X\end{subarray}}\mathrm{rank}(E_{f_{b}(a),~{}g_{b}(a)}).$

For any $\lambda>0$, let us now consider the triangle function
$F_{\lambda}(x)=\max\\{0,1-|x/\lambda|\\}$. Then, we have

$\lambda\mathrm{rank}(E_{f(r),g(r)})\leq\log~{}N(E_{f(r),g(r)})-2\sum\limits_{p^{m}\leq\lambda}F_{\lambda}(m\log
p)\frac{a_{p^{m}}(E_{f(r),g(r)})\log p}{p^{m}}+O(1),~{}\forall
r\in\mathbb{Q},$

where the constant term $O(1)$ is absolute, i.e., independent to
$r\in\mathbb{Q}$. In particular, we have

$\displaystyle\lambda R_{f,g}(X,Y)$
$\displaystyle\leq\sum_{\begin{subarray}{c}b\in{\mathbb{Z}}\\\ |b|\leq
Y\end{subarray}}~{}\sum_{\begin{subarray}{c}a\in{\mathbb{Z}}\\\ |a|\leq
X\end{subarray}}\log~{}N(E_{f_{b}(a),g_{b}(a)})-2\sum_{m\geq
1}S_{m}(X,Y)+O(XY),$
$\displaystyle\leq\sum_{\begin{subarray}{c}b\in{\mathbb{Z}}\\\ |b|\leq
Y\end{subarray}}~{}\big{(}\sum_{\begin{subarray}{c}a\in{\mathbb{Z}}\\\ |a|\leq
X\end{subarray}}\deg(N(\mathcal{E}_{f_{b},g_{b}}))\log|a|+O_{f,g}(1)\big{)}-2\sum_{m\geq
1}S_{m}(X,Y)+O(XY)$ (33)
$\displaystyle\leq\mathrm{deg}(N(\mathcal{E}_{f,~{}g}))XY\log X-2\sum_{m\geq
1}S_{m}(X,Y)+O(XY).$

where the implicit constant is absolute, and
$S_{m}(X,Y)=\sum\limits_{\begin{subarray}{c}r=a/b\in\mathbb{Q}\\\ |a|\leq
X,~{}|b|\leq Y\end{subarray}}\sum\limits_{p^{m}\leq
e^{\lambda}}F_{\lambda}(m\log p)\frac{a_{p^{m}}(E_{f(r),g(r)})\log p}{p^{m}}.$

Similarly as in [28, p. 232], we have $\sum\limits_{m>2}S_{m}(X,Y)=O(XY)$,
where the implicit constant is absolute. Now, we need to estimate $S_{1}(X,Y)$
and $S_{2}(X,Y)$. Note that,

(34) $\displaystyle
S_{2}(X,Y)=\sum\limits_{\begin{subarray}{c}r=a/b\in\mathbb{Q}\\\ |a|\leq
X,~{}|b|\leq Y\end{subarray}}\sum\limits_{p\leq
e^{\lambda/2}}F_{\lambda}(2\log p)\frac{a_{p^{2}}(E_{f(r),g(r)})\log
p}{p^{2}}=-2XY\lambda+o(XY),$

where the last estimate follows from [28, Lem. 4.2]. Now for the estimation of
$S_{1}$, note that

$\displaystyle S_{1}(X,Y)$ $\displaystyle=\sum\limits_{|b|\leq
Y}\sum\limits_{|a|\leq X}\sum\limits_{p\leq e^{\lambda}}F_{\lambda}(\log
p)\frac{a_{p}(E_{f_{b}(a),g_{b}(a)})\log p}{p}$ (35)
$\displaystyle=\sum\limits_{|b|\leq Y}\sum\limits_{p\leq
e^{\lambda}}F_{\lambda}(\log
p)\frac{2X}{p}A_{p}(\mathcal{E}_{b})+\sum\limits_{|b|\leq Y}\sum\limits_{p\leq
e^{\lambda}}\sum_{\bar{r}\in I_{p}\subset\mathbb{F}_{p}}F_{\lambda}(\log
p)\frac{\log p}{p}a_{p}(E_{f_{b}(\bar{r}),g_{b}(\bar{r})}),$

for some subset $I_{p}$ of $\mathbb{F}_{p}$. Lemma 12 and [28, Cor. 3.2]
implies that the first sum above is

(36) $-\left(X\lambda+o(X\lambda)\right)\sum\limits_{|b|\leq
Y}\mathrm{rank}(\mathcal{E}_{b}(\mathbb{Q}(t)))=-XY\mathrm{rank}(\mathcal{E}(\mathbb{Q}(t)))(2\lambda+o(1)).$

Now, the inner sum in the second sum in (5) is crudely bounded by
$\sum\limits_{p\leq e^{\lambda}}p^{1/2}\log p=e^{3\lambda/2}$. In particular,
now combining (5), (34), and (5), we have

$R_{f,g}(X,Y)\leq\frac{\mathrm{deg}(N(\mathcal{E}_{f,~{}g}))}{\lambda}XY\log
X-XY\mathrm{rank}(\mathcal{E}(\mathbb{Q}(t)))(2+o(1/\lambda))+Ye^{3\lambda/2}.$

Now the proof follows taking $\lambda=\frac{\log X}{3}$. ∎

As an immediate consequence, we deduce that the most elliptic curves in the
$1$-parameter family as in Proposition 5.1 have a small rank. More
specifically, we have the following.

### Proof of Lemma 10

Take $E\in\mathcal{E}_{\ell}(X)$, and take the corresponding
$r\in\mathcal{E}^{\prime}_{\ell}(X)$. We have
$\mathrm{rank}(E)=\mathrm{rank}(E_{f_{\ell}(r),g_{\ell}(r)})$. With this
observation, it follows from Lemma 9 that

(37) $\displaystyle\sum_{E\in\mathcal{E}_{\ell}(X)}\mathrm{rank}(E)$
$\displaystyle\leq\sum_{\begin{subarray}{c}r=a/b~{}\in~{}\mathbb{Q}\\\ |a|\leq
CX^{m_{\ell}},~{}|b|\leq
CX^{n_{\ell}}\end{subarray}}\mathrm{rank}(E_{f_{\ell}(r),~{}g_{\ell}(r)})$
(38)
$\displaystyle=R_{f_{\ell},g_{\ell}}(CX^{m_{\ell}},X^{n_{\ell}})=O(X^{m_{\ell}+n_{\ell}})$

where $C$ is the maximum of the implicit constants in (26), and the implicit
constant is absolute. The result now follows immediately, as
$\\#\mathcal{E}_{\ell}(X)\asymp X^{m_{\ell}+n_{\ell}}$. ∎

###  5.3. Normal number of prime factors of square-free parts of polynomial
values

Let $f(t)\in{\mathbb{Z}}[t]$ be any polynomial of degree $d$. At any rational
$r\in\mathbb{Q}$, let us denote $s(f(r))$ to be the square-free part of
$f(r)$. Poonen [26], explored the prevalence of square-free values in the
polynomials. Over the integers $n$, Poonen investigated scenarios where $f(n)$
is not divisible by any fixed square $m^{2}\neq 1$, utilizing the ABC
conjecture. However, in our context, our focus is on understanding how
frequently the square-free parts of $f(r)$ possess a substantial number of
prime factors. First of all, for any rational $r=\frac{a}{b}$, let us denote
the height $h(r)=\max\\{|a|,|b|\\}$. Then we arrange all the rationals with
respect to height $h$ and prove the following.

######  Proposition 5.2.

Let $M\geq 1$ be any integer, and $f(t)\in{\mathbb{Z}}[t]$ be any irreducible
non-constant polynomial. Then, for almost all the rationals $r\in\mathbb{Q}$,
we have $\omega(s(f(r)))\geq M$. More precisely, we have the following
estimate:

$\\#\Big{\\{}r\in\mathbb{Q}:h(r)\leq
X,~{}\omega(s(f(r)))<M\Big{\\}}=O_{M}\left(\frac{X^{2}}{\log\log X}\right).$

Proposition 5.2 will be one of our key tools to prove the main results of this
article. To prove the same, we need the following two results.

######  Lemma 13.

Let $f(t)\in\mathbb{Z}[t]$ be a non-constant irreducible polynomial, and $p$
be any given prime. Then the number of solutions to the equation
$f(\bar{r})=0\pmod{p^{2}}$ over $\bar{r}\in\mathbb{Z}/p^{2}\mathbb{Z}$ is
bounded by $2\mathrm{deg}(f)$, for any prime
$p\nmid\mathrm{gcd}(c(f),c(f^{\prime}))$. Here $f^{\prime}$ is the derivative
of $f$, and $c(f)$ (resp. $c(f^{\prime})$) denotes the $\mathrm{gcd}$ of the
coefficients of $f$ (resp. $\mathrm{gcd}$ of the coefficients of
$f^{\prime}$).

###### Proof.

We can express any solution $\bar{r}\in\mathbb{Z}/p^{2}\mathbb{Z}$ as
$\bar{r}_{1}+p\bar{r}_{2}$, where $0\leq\bar{r}_{1},\bar{r}_{2}\leq p-1$.
Using this representation, we have
$f(\bar{r})=f(\bar{r}_{1})+p\bar{r}_{2}f^{\prime}(\bar{r}_{1})\pmod{p^{2}}$.
Since $f(\bar{r})=0\pmod{p^{2}}$, we have $f(\bar{r}_{1})=0\pmod{p}$,
indicating that there are at most $\mathrm{deg}(f)$ possibilities for
$0\leq\bar{r}_{1}\leq p-1$.

Now to count the number of possibilities of $\bar{r}_{2}$, without loss of
generality let us assume that $\bar{r}_{2}\neq 0$. If $p$ divides
$f^{\prime}(\bar{r}_{1})$, then $p$ divides
$\mathrm{gcd}(f(\bar{r}_{1}),f^{\prime}(\bar{r}_{1}))$. But since
$f(t)\in\mathbb{Q}[t]$ is irreducible, we know that the $\mathrm{gcd}$ of $f$
and $f^{\prime}$ over $\mathbb{Q}[t]$ is $1$, and hence, the gcd of $f$ and
$f^{\prime}$ over ${\mathbb{Z}}[t]$ is $\mathrm{gcd}(c(f),c(f^{\prime}))$. In
particular, $p\mid\mathrm{gcd}(c(f),c(f^{\prime}))$. This is not possible, as
per the assumption on $p$. Therefore, we may assume that $p$ does not divide
$f^{\prime}(\bar{r}_{1})$. Now for each possible $\bar{r}_{1}$, setting
$f(\bar{r}_{1})=p\bar{t}_{1}$, we can write
$\bar{r}_{2}=-\bar{t}_{1}f^{\prime}(\bar{r}_{1})^{-1}$. In particular,
$\bar{r}_{2}$ is uniquely determined by $\bar{r}_{1}$, as long as
$\bar{r}_{2}\neq 0$. This shows that the number of possible pairs
$(\bar{r}_{1},\bar{r}_{2})$ is only $2\mathrm{deg}(f)$, for any prime $p$ not
dividing $\mathrm{gcd}(c(f),c(f^{\prime}))$. ∎

######  Lemma 14.

Consider $S$ to be the set of all prime factors of the leading coefficient of
$f$. Then, we have the following estimate:

$\sum_{h(r)\leq X}\Big{|}\omega_{S}(s(f(r)))-\log\log
X\Big{|}^{2}=O_{S}\left(X^{2}\log\log X\right).$

###### Proof.

Let us start by noting that

$\displaystyle\sum_{h(r)\leq X}\omega_{S}(f(r))$ $\displaystyle=\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))=\mathrm{odd}\end{subarray}}1+~{}\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))\neq 0\\\ \nu_{p}(f(r))=\mathrm{even}\end{subarray}}1$ (39)
$\displaystyle=\sum_{h(r)\leq X}\omega_{S}(s(f(r))~{}+\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))\neq 0\\\ \nu_{p}(f(r))=\mathrm{even}\end{subarray}}1.$

For any rational number $r$, let us denote $r_{p}=r\pmod{p}$. Then we have,

(40) $\nu_{p}(f(r))>0\implies
r_{p}\in\mathbb{F}_{p},~{}\mathrm{and}~{}p\not\in
S\implies\nu_{p}(f(r))<0,~{}\mathrm{if~{}and~{}only~{}if},~{}r_{p}=\infty.$

This shows that

$\displaystyle\sum_{h(r)\leq X}\omega_{S}(f(r))$ $\displaystyle=\sum_{h(r)\leq
X}~{}~{}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))>0\end{subarray}}1+\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))<0\end{subarray}}1$ (41) $\displaystyle=\sum_{h(r)\leq
X}~{}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
r_{p}\in\mathbb{F}_{p}\\\ \nu_{p}(f(r))>0\end{subarray}}1+\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
r_{p}=\infty\end{subarray}}1.$

It is now enough to estimate the second sum in (5.3), and both the sums in
(5.3).

#### Case 1: estimate of the first sum in (5.3)

For this, let us note that for any rational $r-\frac{a}{b}$, we have

(42)
$f\left(\frac{a+p}{b}\right)=f\left(\frac{a}{b}\right)\pmod{p},~{}f\left(\frac{a}{b+p}\right)=f\left(\frac{a}{b}\right)\pmod{p}.$

Again, since we are assuming that $r_{p}\in\mathbb{F}_{p}$, we can write the
following combining with (42):

$\displaystyle\sum_{h(r)\leq X}\omega_{S}(f(r))=\sum_{\begin{subarray}{c}p\leq
X\\\
p~{}\not\in~{}S\end{subarray}}\left(X^{2}\frac{\rho_{f}(p)}{p}+O(\rho_{f}(p))\right)=X^{2}\log\log
X+O(X^{2}),$

where
$\rho_{f}(p)=\\{\bar{r}\in{\mathbb{Z}}/p{\mathbb{Z}}:f(\bar{r})=0\pmod{p}\\}$,
and the last equality follows from [10, Cor. 3.2.2]. Furthermore, we can run
the summation above for $p$ up to at most $X$ because $h(f(r))=X^{O(1)}$, and
there are only $O(1)$ many primes $p$ for which $\nu_{p}(f(r))\neq 0$, and
$O(1)\leq\deg(f)$.

#### Case 2: estimate of the second sum in (5.3)

For a rational $r$, denote $d(r)$ to be the denominator of $r$ in the lowest
term. Then, we have

(43) $\sum\limits_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
r_{p}=\infty\end{subarray}}1=\sum_{h(r)\leq X}\omega_{S}(d(r))=X^{2}\log\log
X+O(X^{2}),$

where the last equality follows by the proof of Corollary 3.2.2 in [10].

#### Case 3: estimate of the second sum in (5.3)

Let us write

$\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))\neq 0\\\
\nu_{p}(f(r))=\mathrm{even}\end{subarray}}1=\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))>0\\\
\nu_{p}(f(r))=\mathrm{even}\end{subarray}}1+~{}\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))<0\\\ \nu_{p}(f(r))=\mathrm{even}\end{subarray}}1.$

The second sum above is, of course, bounded by (43). To bound the first sum,
we use Lemma 13, and get

(44) $\sum_{h(r)\leq
X}~{}\sum_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\not\in~{}S\\\
\nu_{p}(f(r))>0\\\ \nu_{p}(f(r))=\mathrm{even}\end{subarray}}1\leq\sum_{p\leq
X}\left(X^{2}\frac{\rho_{f}(p^{2})}{p^{2}}+O(\rho_{f}(p^{2}))\right)=O(X^{2}),$

where
$\rho_{f}(p^{2})=\\{\bar{r}\in{\mathbb{Z}}/p^{2}{\mathbb{Z}}:f(\bar{r})=0\pmod{p^{2}}\\}$,
and this is at most $\mathrm{deg}(f)^{2}$, due to Lemma 13. Therefore,
combining (5.3), (5.3), (43), and (44), we get $\sum\limits_{h(r)\leq
X}\omega_{S}(s(f(r)))=X^{2}\log\log X+O(X^{2})$. To complete the proof of the
lemma, it is now enough to show that

(45) $\sum\limits_{h(r)\leq X}\omega_{S}(s(f(r)))^{2}\leq(X\log\log
X)^{2}+O(X^{2})$

Since $\omega(f(r))\leq\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}\\\
\nu_{p}(f(r))>0\end{subarray}}1+\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}\\\
r_{p}=\infty\end{subarray}}1$, we get, $\sum\limits_{h(r)\leq
X}\omega_{S}(s(f(r)))^{2}\leq\sum\limits_{h(r)\leq X}\omega(f(r))^{2}\leq
2(S_{1}+S_{2})$, where

(46) $S_{1}=\sum\limits_{h(r)\leq
X}\Big{(}\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}\\\
\nu_{p}(f(r))>0\end{subarray}}1\Big{)}^{2},~{}\mathrm{and}~{}S_{2}=\sum\limits_{h(r)\leq
X}\Big{(}\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}\\\
r_{p}=\infty\end{subarray}}1\Big{)}^{2}.$

For any suitable parameter $Y\leq X$, applying Chinese remainder theorem, we
have

$\displaystyle S_{1}(Y)=\sum\limits_{h(r)\leq
X}\Big{(}\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\leq Y\\\
\nu_{p}(f(r))>0\end{subarray}}1\Big{)}^{2}=X\Big{(}\sum_{\begin{subarray}{c}p\neq
q~{}\mathrm{primes}\\\ p,q\leq
Y\end{subarray}}X\frac{\rho_{f}(p)\rho_{f}(q)}{pq}+O(1)\Big{)}=(X\log\log
Y)^{2}+O(XY^{2}),$ $\displaystyle S_{2}(Y)=\sum\limits_{h(r)\leq
X}\Big{(}\sum\limits_{\begin{subarray}{c}p~{}\mathrm{prime}~{}\leq Y\\\
r_{p}=\infty\end{subarray}}1\Big{)}^{2}=X\Big{(}\sum_{\begin{subarray}{c}p\neq
q~{}\mathrm{primes}\\\ p,q\leq
Y\end{subarray}}X\frac{1}{pq}+O(1)\Big{)}=(X\log\log Y)^{2}+O(XY^{2}).$

The proof of equation (45), and consequently the proof of the lemma is now
complete taking $Y=X^{1/2}$, as we have
$|S_{1}(Y)-S_{1}|,~{}|S_{2}(Y)-S_{2}|=O(X^{2}).$ ∎

#### Proof of Proposition 5.2

Denote $S_{M}(X)$ to be the number of $r\in\mathbb{Q}$ such that
$\omega(s(f(r)))\leq M$, and $h(r)\leq X$. It follows from Lemma 14 that
$S_{M}(X)=O_{M}\left(\frac{X^{2}}{\log\log X}\right)$. This completes the
proof of the proposition. ∎

######  Remark 4.

Note that Proposition 5.2 addresses the number of prime factors in the square-
free parts of the values of $f$. In contrast, [26] focuses on square-free
values. When we restrict the evaluation of $f$ to integers, [26, Thm. 3.2]
shows that the proportion of square-free values of $f$ is precisely
$\prod\limits_{p,~{}\mathrm{prime}}\left(1-\frac{c_{p}}{p^{2}}\right)$, where
the quantity $c_{p}$ is studied in Lemma 13. In particular, the quantity
$\prod\limits_{p,~{}\mathrm{prime}}\left(1-\frac{c_{p}}{p^{2}}\right)$, is
positive. In other words, if $f(t)\in{\mathbb{Z}}[t]$ is irreducible, then
$f(n)$ is square-free for a positive proportion of the times.

Finally, let us address the last preparatory counting tool, that is needed to
prove the main result of this section.

######  Lemma 15.

For any $\ell\in\\{3,5,7\\}$, and any integer $M$, we have the following
estimate:

$\\#\Big{\\{}(A,B)\in S_{f_{\ell},g_{\ell}}(X):\omega(N(E_{A,B}))\leq
M\Big{\\}}=O_{M}\left(\frac{\\#\mathcal{E}_{\ell}(X)}{\log\log X}\right).$

###### Proof.

Consider an elliptic curve $E_{A,B}$ in the minimal form. Then the conductor
$E_{A,B}$ is (up to some bounded factor of $2$ and $3$) the product over all
primes $p$ dividing the discriminant $|\Delta(E_{A,B})|=|4A^{3}+27B^{2}|$.
Therefore, it suffices to estimate the quantity, $\\{(A,B)\in
S_{f,g}(X):\omega(4A^{3}+27B^{2})\leq M+2\\}$. Now Lemma 9 implies that it is
enough to estimate for each $\ell\in\\{3,5,7\\}$,

$\Big{\\{}r=a/b:|a|\leq X^{m_{\ell}},~{}|b|\leq
X^{n_{\ell}}~{}\omega(4s(f(r)^{3}+27g(r)^{2}))\leq M+2\Big{\\}}.$

Denoting $\Delta(r)=4f(r)^{3}+27g(r)^{2}$, it follows from Lemma 11 that there
exists an irreducible factor $F(t)$ of $\Delta(t)$ that divides $\Delta(t)$
with a valuation of exactly $1$. We then have:

$\displaystyle\\#\Big{\\{}r=a/b:|a|\leq X^{m_{\ell}},|b|\leq X^{n_{\ell}},~{}$
$\displaystyle\omega(s(\Delta(r)))\leq M+2\Big{\\}}$
$\displaystyle\leq\\#\Big{\\{}r=a/b:|a|\leq X^{m_{\ell}},|b|\leq
X^{n_{\ell}},~{}\omega(F(r))\leq M+2\Big{\\}}$
$\displaystyle=O_{M}\left(\frac{X^{\max\\{m_{\ell},n_{\ell}\\}}}{\log\log
X}\right)=O_{M}\left(\frac{\\#\mathcal{E}_{\ell}(X)}{\log\log X}\right),$

where the last estimate follows from Proposition 5.2, and the fact that
$\\#\mathcal{E}_{\ell}(X)\asymp X^{m_{\ell}+n_{\ell}}$. ∎

With this groundwork laid, we are now prepared to prove the main result of
this section.

### Proof of Theorem 4

The proof follows combining Lemma 15 and Lemma 10. ∎

######  Remark 5.

Additionally, Lemma 15 can be utilized to investigate the validity of
$M$-Watkins over the fibers of certain elliptic surfaces over $\mathbb{Q}(t)$.
This extends the work in [4, Thm. 8], where the authors examined the frequency
of occurrences of even modular degrees over any such family.

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