# Families of isogenous elliptic curves ordered by height

Yu Fu

###### Abstract.

Given a family of products of elliptic curves over a rational curve defined
over a number field $K$, and assuming that there exists no isogeny between the
pair of elliptic curves in the generic fiber, we establish an upper bound for
the number of special fibers with height at most $B$ where the two factors are
isogenous. Our proof provides an upper bound that is dependent on $K$, the
family, and the bound of height $B$. Furthermore, by introducing a slight
modification to the definition of the height of the parametrizing family, we
prove a uniform bound depends solely on the degree of the family, the field
$K$, and $B$. Based on the uniformity, and the fact that the idea of using
Heath-Brown type bounds on covers and optimizing the cover to count rational
points on specific algebraic families has not been exploited much yet, we hope
that the paper serves as a good example to illustrate the strengths of the
method and will inspire further exploration and application of these
techniques in related research.

##  1\. Introduction

Let $k$ be a field. Let $X\to S$ be a family of algebraic varieties over $k$,
where $S$ is irreducible, and let $X_{\eta}$ be the generic fiber of this
family. People care about what properties of $X_{\eta}$ extend to the special
fibers and how we can measure the size of specializations such that a specific
property does not extend. For example, the Hilbert Irreducibility Theorem says
that for a Galois covering $X\to\mathbb{P}^{n}$ over a number field $K$, for
most of the rational points $t\in\mathbb{P}^{n}(K)$ the specializations over
$t$ generate a Galois extension with Galois group $G$. Moreover, the size of
the complement set, which can be considered as locus of ‘exceptional’ points,
can be bounded as in [Ser].

In [EEHK09], Ellenberg, Elsholtz, Hall, and Kowalski studied families of
Jacobians of hyperelliptic curves defined over number fields by affine
equations

$y^{2}=f(x)(x-t),\text{ }t\in\mathbf{A}^{1},$

with the assumption that the generic fiber is geometrically simple. They
proved that the number of geometrically non-simple fibers in this family, with
the height of the parametrizer $t\leq B$, is bounded above by a constant
$C(f)$ depending on $f$. Moreover, they obtained an effective bound that shows
how the number of geometrically non-simple fibers grows with $B$. Their
effective bound comes from the analytic number theory and not only depends on
the coefficients of $f(x)$ but also depends on the primes dividing the
discriminant of $f(x)$ and the genus of the family.

In this paper, we study families of pairs of elliptic curves defined over a
rational curve over a number field $K$. To be precise, let $C$ be a rational
curve over $K$ isomorphic to $\mathbb{P}^{1}$ which parametrizes a one-
dimensional family of pairs of elliptic curves and let
$(E_{t},E^{\prime}_{t})$ be the generic fiber of this family over $K(t)$, with
the assumption that there exists no isogeny between $E_{t}$ and
$E^{\prime}_{t}$. We establish an upper bound for the number of
specializations of $(E_{t},E^{\prime}_{t})$ where the two factors are
isogenous, with the assumption that the height of the parameter $t\leq B$. Our
method is from a purely arithmetic point of view and relies on constructions
and optimizations of explicit covers of the moduli space $X(1)\times X(1)$,
combined with results of the uniform bound for points of bounded height on
curves, which is completely different from [EEHK09]. Notably, by a proper
choice of definition of the height of $t$, the resulting upper bound depends
solely on $K$ and $B$ and the degree of the parameter family, as detailed in
Theorem 1.3. In comparison with the effective bound in loc. cit., the power of
the logarithmic term we got is much better. Besides this, the uniformity of
our bound adds an exciting dimension to the further exploration of our method.

To discuss the results, we first fix the general setting and terminology (see
section $\S 2$ for more details). Let $\iota$ denote the map from $C$ to
$\mathbb{P}^{3}$ which is a composition of a finite map via the $j$-invariant,
followed by the Segre embedding:

(1.1) $\iota:C\rightarrow X(1)\times
X(1)\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{3}.$

To be precise, the map $\iota$ sends $t$ to

$(E_{t},E^{\prime}_{t})\mapsto(j(E_{t}),j(E_{t}^{\prime}))\mapsto(j(E_{t})j(E_{t}^{\prime});j(E_{t});j(E_{t}^{\prime});1).$

Degrees and heights are computed with respect to this fixed embedding. We
prove the following theorem:

######  Theorem 1.1.

Let $K$ be a number field of degree $d_{K}$. let $C$ be a rational curve over
$K$ isomorphic to $\mathbb{P}^{1}$ which parametrizes a one-dimensional family
of pairs of elliptic curves $(E,E^{\prime})$. Let $(E_{t},E^{\prime}_{t})$ be
the generic fiber of this family over $K(t)$, and suppose that there exists no
$\overline{K(t)}$-isogeny between $E_{t}$ and $E^{\prime}_{t}$. Let
$d=\operatorname{deg}\iota^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)$ be the degree
of the parameter family $C$ defined with respect to $\iota$. Let $H(\iota)$ be
the height of $\iota$ defined by the projective height of the coefficients of
the defining polynomials of $j(E_{t})$ and $j(E_{t}^{\prime}).$ Let
$H:C(K)\to\mathbb{R}$ be the projective height defined over $K$. Define $S(B)$
to be the set

$S(B)=\\{t\in C(K)|H(t)\leq B,\text{ there is an
$\overline{\mathbb{Q}}$-isogeny between }E_{t}\text{ and }E^{\prime}_{t}\\}.$

There is an absolute constant $M$ such that for any $B\geq M$, we have

$|S(B)|\lesssim_{K}d^{4+\epsilon}(\log H(\iota)+\log B)^{6}.$

######  Definition 1.2.

For a point $P_{t}\in C$ parametrized by $t\in K$, define the height
$H(P_{t})$ to be the projective height of $\iota(P_{t})\in\mathbb{P}^{3}$.

Note that in Theorem 1.1, $H(t)$ is the height of $t$ as an element of $K$. If
we change the definition of the height from $H(t)$ to $H(P_{t})$ and assume
that $H(P_{t})\leq B$, then we get an uniform bound on the number of points
$t$ such that $E_{t}$ and $E^{\prime}_{t}$ are geometrically isogenous.
Moreover, this uniform bound only depends on $K$, the height $B$, and the
degree of the family.

######  Theorem 1.3.

Assume the same hypothesis as in Theorem 1.1. Let $S^{\prime}(B)$ be the set

$S^{\prime}(B)=\\{t\in C(K)|H(P_{t})\leq B,\text{ there is an
$\overline{\mathbb{Q}}$-isogeny between }E_{t}\text{ and }E^{\prime}_{t}\\}.$

Then we have

$|S^{\prime}(B)|\lesssim_{K}d^{4}(\log B)^{6}.$

### Relations with Unlikely Intersections

Although Theorem 1.1 indicates the sparsity of isogeny between elliptic curves
in a family, one should emphasize that this is not an unlikely intersection
problem on its own. There are infinitely many $t\in\overline{\mathbb{Q}}$ such
that $E_{t}$ and $E^{\prime}_{t}$ are isogenous! However, since we are working
over a fixed number field $K$, this number has to be finite. Nevertheless, one
may consider questions in more generalized settings.

Let $S$ be a GSpin Shimura variety and denote by $\\{Z_{i}\\}_{i}$ a sequence
of special divisors on $S$. Let $C\hookrightarrow S$ be a curve whose generic
fiber has maximal monodromy.

######  Definition 1.4.

Define the set $Z(C)$ to be the intersection of $C$ with the infinite union of
the special divisors

$Z(C)=C\cap\bigcup_{i}Z_{i}.$

If we take $S=X(1)\times X(1)$ as a special case of GSpin Shimura variety and
take the special divisors to be $Z_{n}=Y_{0}(n)$, which parametrizes
$n$-isogenous pairs of elliptic curves, then the Theorem 1.3 can be
reformulated as follows:

######  Theorem 1.5.

Let $K$ be a number field of degree $d_{K}$. Let $C\subset X(1)\times X(1)$ be
a rational curve defined over $K$ parametrizing a family of non-isotrivial and
generically non-isogenous elliptic curves. Let $d$ be the projective degree of
$C$ defined by $\iota$. For a positive integer $B$ define the set $Z(C;B)$ to
be the set of $K$-valued points on $C$ such that

$Z(C;B)=\\{x\in Z(C)\mid H(\iota(x))\leq B\\}.$

We have

$|Z(C;B)|\lesssim_{K}d^{4}(\log B)^{6}.$

The discussion above suggests the following question (which we do not claim to
know the answer to):

Question: Suppose $C$ is a curve defined over $\overline{\mathbb{Q}}$ and let
$Z(C;B)$ denote the set of $\overline{\mathbb{Q}}$-valued points in $Z(C)$
whose absolute height is bounded above by $B$. One may ask if $Z(C;B)$ is
finite. If this is the case, can we get an upper bound in terms of the bounded
height $B$?

### Non-simple abelian varieties in a family

Theorem 1.1 can be considered as a generalization of the results from
Ellenberg, Elsholtz, Hall, and Kowalski. In work in progress, we hope to
generalize their main results to any family of abelian varieties parametrized
by an irreducible rational curve. To be explicit, let
$X\in\mathcal{A}_{g}/\mathbb{Q}$ be a curve that parametrizes a
$1$-dimensional family of abelian varieties where we denote by $A_{x}$ the
fiber of this family over $x\in X$. Let $d$ be the projective degree of $X$.
We aim to obtain a uniform bound for the number of $x\in X(\mathbb{Q})$ such
that $A_{x}$ is geometrically non-simple, using a method similar to what we
use in this article. By contrast, all upper bounds for the number of non-
simple fibers studied in previous literature depend on $X$, and a uniform
upper bound that does not depend on $X$ would follow inexplicitly from Lang’s
conjecture via the result of Caporaso-Harris-Mazur [CHM], as explained in
[EEHK09].

Organization of the paper. In $\S 2$, we recall the notion of heights on
projective spaces, Hecke correspondence, the modular diagonal quotient
surfaces, and some results on the Heath-Brown type bounds for curves, which we
will use later. In $\S 3$, we interpret the counting problem into counting
rational points on projective curves with certain level structures and
construct ‘nice’ Galois covers that capture the information of being
isogenous. In $\S 4$, we construct certain projective embeddings with respect
to the covers in $\S 3$, such that the dimension growth conjecture applies. In
$\S 5$, we give an upper bound on the change of heights between covers so that
one can bound the height of the lifting points. Finally, we prove Theorem 1.1
and Theorem 1.3 in $\S 6$.

##  2\. Preliminaries

This section states and proves results and facts in arithmetic geometry,
primarily focusing on the theory of heights and the outcomes of dimension
growth conjectures for curves. Specifically, we make a survey of uniform
bounds for points of bounded height on curves, a crucial ingredient in our
proofs, which are utilized in this article. While these concepts are likely
familiar to experts, we compile them here for the reader’s convenience.

Let $C$ be a rational curve over $K$ isomorphic to $\mathbb{P}^{1}$
parametrizing a one-dimensional family of pairs of elliptic curves, and let
$(E_{t},E^{\prime}_{t})$ be the generic fiber of this family over $K(t)$, with
the assumption that there exists no isogeny between $E_{t}$ and
$E^{\prime}_{t}$. Without loss of generality, we write $E_{t}$ and
$E^{\prime}_{t}$ in the Weierstrass form:

(2.1) $E_{t}:y^{2}=x^{3}+f(t)x+g(t)$

(2.2) $E^{\prime}_{t}:y^{2}=x^{3}+f^{\prime}(t)x+g^{\prime}(t).$

Here, $f(t)$, $g(t)$, $f^{\prime}(t)$, and $g^{\prime}(t)$ are rational
functions over $K$. Consequently, the $j$-invariants of $E_{t}$ and
$E^{\prime}_{t}$, denoted by $j(E_{t})$ and $j(E^{\prime})$, are also rational
functions. 2.1. Heights of points. For a rational point $a\in K$, define the
height $H(a)$ of $a$ to be

$H(a):=\prod_{v\in M_{K}}\max\\{1,|a|_{v}\\}^{\frac{n_{v}}{[K:\mathbb{Q}]}}$

where $n_{v}=[K_{v}:\mathbb{Q}_{v}].$

The height of a $K$-rational point $P\in\mathbb{P}^{n}$ with homogeneous
coordinates $P=(x_{0},\cdots,x_{n})$ is defined to be

$H(P)=\prod_{v\in
M_{K}}\max\\{|x_{0}|_{v},\cdots,|x_{n}|_{v}\\}^{\frac{n_{v}}{[K:\mathbb{Q}]}}$

where $n_{v}=[K_{v}:\mathbb{Q}_{v}].$ Recall that we define $\iota$ to be the
map obtained via the $j$-invariant and via the Segre embedding in
$\mathbb{P}^{3}$, see (1.1).

Recall the definition of $H(P_{t})$ in Definition 1.2. The following lemma
indicates that

$H(P_{t})=H(j(E_{t}))H(j(E_{t}^{\prime})).$

######  Lemma 2.1.

Let $\sigma_{n}$ be the Segre embedding of $n$-copies of $\mathbb{P}^{1}$

$\sigma_{n}\colon\underbrace{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}}_{n\text{-times}}\hookrightarrow\mathbb{P}^{2^{n}-1}$

such that for a point
$(x_{1},\cdots,x_{n})\in\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}$

$\sigma_{n}(x_{1},\cdots,x_{n})=(\prod_{1\leq i\leq
n}x_{i},\prod_{i_{1}<i_{2}<\cdots<i_{n-1}}x_{i_{1}}\cdots
x_{i_{n-1}},\ldots,x_{1},\ldots,x_{n},1).$

Let $H(.)$ be the projective height defined above. We have

$H(\sigma_{n}(x_{1},\cdots,x_{n}))=H(x_{1})\cdots H(x_{n}).$

###### Proof.

By definition of the projective height,

$H(\sigma_{n}(x_{1},\cdots,x_{n}))=\prod_{v\in M_{K}}\max\\{|\prod_{1\leq
i\leq
n}x_{i}|_{v},\cdots,|x_{1}|_{v},\cdots,|x_{n}|_{v},1\\}^{\frac{n_{v}}{[K:\mathbb{Q}]}}$

and

$H(x_{1})\cdots H(x_{n})=\prod_{v\in
M_{K}}\\{\max\\{1,|x_{1}|_{v}\\}\cdots\max\\{1,|x_{n}|_{v}\\}\\}^{\frac{n_{v}}{[K:\mathbb{Q}]}}.$

A direct observation shows that for each $v\in M_{K}$,

$\max\\{|\prod_{1\leq i\leq
n}x_{i}|_{v},\cdots,|x_{1}|_{v},\cdots,|x_{n}|_{v},1\\}=\max\\{1,|x_{1}|_{v}\\}\cdots\max\\{1,|x_{n}|_{v}\\}.$

∎

2.2. The modular diagonal quotient surfaces. Later in this article, we will
define the modular surface $X_{\tilde{H}_{\Delta}}(m)$ (see (3.1)) that lies
in a special family, called the modular diagonal quotient surfaces, which
arise naturally as the (coarse) moduli space to the moduli problem that
classifies isomorphisms between mod $m$ Galois representations attached to
pairs of elliptic curves $E/K$.

The modular curve $X(m)$ is a Galois cover of $X(1)$ with Galois group

$G=SL_{2}(\mathbb{Z}/m\mathbb{Z})/\\{\pm 1\\}.$

Let $\epsilon$ be an element in $(\mathbb{Z}/m\mathbb{Z})^{\times}$. Let
$\alpha_{\epsilon}$ be the automorphism of $G$ defined by conjugation with
$Q_{\epsilon}=\big{(}\begin{smallmatrix}\epsilon&0\\\
0&1\end{smallmatrix}\big{)}$, i.e.
$\alpha_{\epsilon}(g)=Q_{\epsilon}gQ_{\epsilon}^{-1}$. The product surface
$X(m)\times X(m)$ carries an action of the twisted diagonal subgroup of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})\times SL_{2}(\mathbb{Z}/m\mathbb{Z})$ defined
by

$\alpha_{\epsilon}\text{ : }\Delta_{\epsilon}=\\{(g,\alpha_{\epsilon}(g)):g\in
G\\}.$

######  Definition 2.2.

The twisted diagonal quotient surface defined by $\alpha_{\epsilon}$ is the
quotient surface $X_{\epsilon}:=\Delta_{\epsilon}\backslash X(m)\times X(m)$
obtained by the action of $\Delta_{\epsilon}$.

$X_{\epsilon}$ can be viewed as the moduli space of the triple
$(E_{1},E_{2},Q)$, where

$Q:E_{1}[p]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{2}[p]$

multiplies the Weil pairing by $\epsilon$.

The modular diagonal quotient surfaces and their modular interpretation are
widely used in studying Mazur’s question[Maz78] [HK00], Frey’s
conjecture[Fre97], and so on.

2.3. Uniform bound for points of bounded height on curves.

Let $X$ be an irreducible projective curve defined over a number field $K$
with a degree $d$ embedding into $\mathbb{P}^{n}$. Denote by $N(X,B)$ the set
of $K$-rational points on $X\subseteq\mathbb{P}^{n}$ of projective heights at
most $B$. Heath-Brown proved a uniform bound for rational points on a curve
$X$ with bounded height in [Heath-Brown, Theorem 5], which addresses that

$N(X,B)\leq O_{\epsilon}(B^{2/d+\epsilon}).$

The result on irreducible projective curves with the removal of the term
$B^{\epsilon}$ without needing $\log B$ was proved by Walsh in [Walsh], using
a combination of the determinant method based on the $p$-adic approximation
introduced by Heath-Brown [Heath-Brown] with the method due to Ellenberg and
Venkatesh in [EV05]. The uniform upper bound on $N(X,B)$ with an explicit term
of polynomial growth depends on $d$ was proved by Castryck, Cluckers,
Dittmann, and Nguyen [CCDN19]:

######  Theorem 2.3.

[CCDN19, Theorem 2] Given $n\geq 1$, there exists a constant $c=c(n)$ such
that for all $d>0$ and all integral projective curves
$X\in\mathbb{P}^{n}_{\mathbb{Q}}$ of degree $d$ and all $B\geq 1$ one has

$|N(X,B)|\leq cd^{4}B^{2/d}.$

A year later, Paredes and Sasyk [PS22] extended the work of Castryck,
Cluckers, Dittmann, and Nguyen to give uniform estimates for the number of
rational points of bounded height on projective varieties defined over global
fields. More precisely, they proved the following extension of [CCDN19,
Theorem 2] to global fields.

######  Theorem 2.4.

[PS22, Theorem 1.8] Let $K$ be a global field of degree $d_{K}$. Let $H$ be
the absolute projective multiplicative height. For any integral projective
curve $C\subseteq\mathbb{P}_{K}^{N}$ of degree $d$ it holds

$|\\{\boldsymbol{x}\in C(K):H(\boldsymbol{x})\leq
B\\}|\lesssim_{K,N}\begin{cases}d^{4}B^{\frac{2d_{K}}{d}}&\text{ if }K\text{
is a number field, }\\\ d^{8}B^{\frac{2d_{K}}{d}}&\text{ if }K\text{ is a
function field. }\end{cases}$

We will discuss this result in section $\S 6$, where we apply it.

##  3\. Construct Galois Covering with Level Structures

In this section, we construct ‘nice’ Galois coverings of $X(1)\times X(1)$
with level structures. To be precise, for a large enough rational prime $m$,
these are quotients of $X(m)\times X(m)$ by certain kinds of subgroups of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})\times SL_{2}(\mathbb{Z}/m\mathbb{Z})$, with
the property that one can lift a point $t\in S(B)$ to one of the quotients. In
other words, the Galois coverings capture points that parametrize isogenous
pairs $(E_{t},E^{\prime}_{t})$ in the family. The main theorem in this section
is Lemma 3.11.

We assume that $m$ is a rational prime such that $m\geq
M^{\prime}=\operatorname{max}\\{17,M_{1}\\}$ for some absolute integer $M_{1}$
defined in Theorem 3.4. Also we want $m$ to be a prime between
$2(\log(d+1)+\log H(\iota)+\log B)^{\frac{1}{2}}$ and $4(\log(d+1)+\log
H(\iota)+\log B)^{\frac{1}{2}}$ once we fix $B$, which we will make precise in
$\S 6$. This condition implies that we need to assume $4(\log(d+1)+\log
H(\iota)+\log B)^{\frac{1}{2}}\geq M^{\prime}$ and thus we can always assume
$B$ is greater than the absolute integer $M=e^{(\frac{M^{\prime}}{4})^{2}}$ to
make everything go through. Let $X(m)\times X(m)$ be the surface parametrizing
$4$-tuples $(E,E^{\prime},\phi,\phi^{\prime})$, where $E_{t}$ and
$E^{\prime}_{t}$ are elliptic curves and $\phi$, $\phi^{\prime}$ are $m$-level
structures, i.e.

$\phi:E[m]\to(\mathbb{Z}/m\mathbb{Z})^{2},\phi^{\prime}:E^{\prime}[m]\to(\mathbb{Z}/m\mathbb{Z})^{2}$

are isomorphisms of group schemes preserving the Weil pairing. Let $H$ be a
subgroup of $SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})$ and define $X_{H}$ to be the quotient

(3.1) $X_{H}:=(X(m)\times X(m))/H.$

$p$-torsion monodromy representations and the lifting criterion. Let $E$ be an
elliptic curve over a number field $K$ and let $p$ be a prime. Denote by

$\rho_{E,p}:G_{K}\to\operatorname{Aut}(E[p])$

the $p$-torsion Galois representation associated to the $p$-torsion points
$E[p]$ of the elliptic curve $E$. It is a standard fact that the elliptic
curve $E/K$ admits an isogeny of degree $p$ defined over $K$ if and only if
the image $\rho_{E,p}\left(G_{K}\right)$ is contained in a Borel subgroup of
$\operatorname{Aut}(E[p])$. If $E/K$ and $E^{\prime}/K$ are related by an
isogeny over $K$ of degree coprime to $p$, then this isogeny induces a
$G_{K}$-module isomorphism $E[p]\simeq E^{\prime}[p]$, which identifies the
images $\rho_{E,p}\left(G_{K}\right)$ and $\rho_{E^{\prime},p}(G_{K})$ up to
change of basis. See [RV00] for an explicit description of the images of
$p$-torsion Galois representations attached to the product of two isogenous
elliptic curves with an isogeny of degree $p$.

######  Definition 3.1.

Define $H_{\Delta}$ to be the image of the diagonal map

$\Delta:SL_{2}(\mathbb{Z}/m\mathbb{Z})\to SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z}).$

We will prove, in Proposition 3.7, that there exists an isogeny
$\phi_{t}:E_{t}\to E^{\prime}_{t}$ defined over $\overline{\mathbb{Q}}$ if and
only if the monodromy of the $p$-torsion Galois representation on
$E_{t}[p]\times E^{\prime}_{t}[p]$ is contained in a group
$\tilde{H}_{\Delta}$, which contains $H_{\Delta}$ as an index $2$ subgroup for
all but finitely many $p$.

######  Remark 3.2.

It is a classical result that for elliptic curves defined over a number field
$K$, there are finitely many $j$-invariants with complex multiplication in
$K$. Denote this number by $C(K)$. We need to bound the number of points
$P_{t}$ on $C$ whose $j$-invariant lies in this set. There are at most
$C(K)^{2}$ points $P_{t}\in C$ such that
$\iota(P_{t})=(j(E_{t}),j(E^{\prime}_{t}))$ contains CM $j$-invariants.
Therefore in our setting, we can discard them and focus on pairs of elliptic
curves without complex multiplication.

######  Lemma 3.3.

Let $E_{1}$ and $E_{2}$ be two elliptic curves without complex multiplication
over a number field $K$. If there exists an isogeny $\phi:E_{1}\to E_{2}$
defined over $K$ then the $p$-torsion Galois representation of $E_{1}\times
E_{2}$

$\operatorname{Gal}(\bar{K}/K)\to GL_{2}(\mathbb{F}_{p})\times
GL_{2}(\mathbb{F}_{p})$

has image conjugate to $H_{\Delta}$ for primes $p$ not dividing the degree of
$\phi$.

###### Proof.

First, by Serre’s open image theorem, the $p$-torsionGalois representation of
each factor

$\operatorname{Gal}(\bar{K})\to GL_{2}(\mathbb{F}_{p})$

is surjective for all large enough primes $p$.

Suppose $E_{1}/K$ and $E_{2}/K$ are related by an isogeny over $K$ of degree
$d$, then for all primes $p\nmid d$ this isogeny induces a $G_{K}$-module
isomorphism from $E_{1}[p]$ to $E_{2}[p]$, which identifies the images
$\rho_{E_{1},p}\left(G_{K}\right)$ and $\rho_{E_{2},p}(G_{K})$. The lemma
follows.

∎

Note that Lemma 3.3 is a result over number fields and will serve as a main
ingredient in the proof of Proposition 3.7. However, later in the proof of
Proposition 4.9, we need to use a similar result over the function field
$K(t)$ which classifies elliptic curves up to isogeny by their $p$-torsion
Galois representations. Here we present a beautiful theorem by Bakker and
Tsimerman [BT, Theorem 1].

######  Theorem 3.4.

Let $k$ be an algebraically closed field of characteristic $0$. For any $N>0$,
there exists $M_{N}>0$ such that for any prime $p>M_{N}$ and any smooth quasi-
projective curve $U$ of gonality $n<N$, non-isotrivial elliptic curves
$\mathcal{E}$ over $U$ are classified up to isogeny by their $p$-torsion local
system $\mathcal{E}[p]$.

######  Definition 3.5.

Let $\tilde{H}_{\Delta}$ be the maximal subgroup of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})\times SL_{2}(\mathbb{Z}/m\mathbb{Z})$ that
contains $H_{\Delta}$ as an index $2$ subgroup.

The following lemma, together with the fact that if $F$ is a field with more
than $5$ elements then the only proper normal subgroup of $SL_{2}(F)$ is the
group $\\{\pm 1\\}$, proves that $\tilde{H}_{\Delta}$ is the unique proper
subgroup of $SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})$ that contains $H_{\Delta}$ as an index $2$
subgroup.

######  Lemma 3.6.

Let $A$ be a group and let $G=A\times A$. Define $\Delta=\\{(a,a)\mid a\in
A\\}$ as the diagonal subgroup of $G$. If $\Delta\leq H\leq G$ then there
exists a normal subgroup $N$ of $A$ such that $H=\\{(g,h)\in G\mid gh^{-1}\in
N\\}$.

###### Proof.

Let $N=\\{h\in A\mid(h,1)\in H\\}$ be a subgroup of $G$. We claim that $N$ is
the desired normal subgroup. Indeed, for any $a\in A$ and $(h,1)\in H$, we
have $(aha^{-1},1)=(a,a)(h,1)(a^{-1},a^{-1})\in H$, therefore $N$ is a normal
subgroup of $A$.

For any $a,a^{\prime}\in A$, we have
$(aa^{\prime-1},1)(a^{\prime},a^{\prime})=(a,a^{\prime})$. Therefore
$(a,a^{\prime})\in H$ is and only if $(aa^{\prime-1},1)\in H$, if and only if
$aa^{\prime-1}\in A$ by the definition of $A$. ∎

######  Proposition 3.7.

Let $E_{1}$ and $E_{2}$ be elliptic curves without complex multiplication over
$\mathbb{Q}$. There exists an isogeny $\phi:E_{1}\to E_{2}$ defined over
$\overline{\mathbb{Q}}$ if and only if the $p$-torsion Galois representation

$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to
GL_{2}(\mathbb{F}_{p})\times GL_{2}(\mathbb{F}_{p})$

has image contained $\tilde{H}_{\Delta}$, for all primes $p$ not dividing the
degree of $\phi$.

###### Proof.

We need the following lemma:

######  Lemma 3.8.

Let $E_{1}$, $E_{2}$ be elliptic curves without complex multiplication defined
over a number field $K$. If $E_{1}$ and $E_{2}$ are isogenous over
$\overline{\mathbb{Q}}$ then there exists a quadratic twist of $E_{2}$ that is
isogenous to $E_{1}$ over $K$.

Suppose there exists an isogeny $\varphi:E_{1}\to E_{2}$ over
$\overline{\mathbb{Q}}$ then by Lemma 3.8 and Lemma 3.3, there is a quadratic
extension $L/K$ such that $G_{L}$ has diagonal image in
$GL_{2}(\mathbb{F}_{p})\times GL_{2}(\mathbb{F}_{p})$. Therefore the image of
$G_{K}$ is contained in a subgroup of $GL_{2}(\mathbb{F}_{p})\times
GL_{2}(\mathbb{F}_{p})$ which contains $H_{\Delta}$ as an index $2$ subgroup.
For the other direction, we take the preimage of the $H_{\Delta}$, which can
be written in the form $G_{F}$ for some number field $F$, that is quadratic
over $K$. By Proposition 2.19, there is an isogeny $\varphi:E_{1}\to E_{2}$
defined over $F$ which completes the proof.

∎

###### Proof of Lemma 3.8.

Let $\varphi:E_{1}\to E_{2}$ be an isogeny over $\overline{\mathbb{Q}}$ and
$G_{\overline{Q}/K}=\operatorname{Gal}(\overline{\mathbb{Q}}/K)$. For every
$g\in G_{\overline{Q}/K}$, $\varphi^{g}=[\alpha(g)]\circ\varphi$ is another
isogeny $E_{1}\to E_{2}$ of the same degree as $\varphi$. Here we define
$\alpha:G_{\overline{Q}/K}\to\mathbb{R}$ be a character on
$G_{\overline{Q}/K}$. Since for all elliptic curves without complex
multiplication over $\overline{\mathbb{Q}}$, there exists a cyclic isogeny
$\varphi:E_{1}\to E_{2}$ up to sign and all other isogenies $\psi$ from
$E_{1}$ to $E_{2}$ can be written as $\psi=\varphi\circ[m]$ for some integer
$m$, we have $\alpha(g)=\pm 1$. Hence $\alpha(g)$ is a quadratic character and
there exists $d\in K^{*}$ such that $\alpha(g)=g(\sqrt{d})/\sqrt{d}$. Thus the
quadratic twist of $E_{2}$ by $d$ is the desired twist.

∎

######  Definition 3.9.

Let $H_{\text{p}}:=H_{1}\times H_{2}$, where $H_{1}$ and $H_{2}$ (possibly
$H_{1}=H_{2}$) are maximal parabolic subgroups of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$.

######  Remark 3.10.

All maximal parabolic subgroups of $SL_{2}(\mathbb{Z}/m\mathbb{Z})$ are in the
same conjugacy class. Therefore the covers constructed in this way are all
isomorphic to each other. If $t\in S(B)$ lifts to one, it lifts to all.

######  Lemma 3.11.

Any rational point $t\in S(B)$ for some $B$ admits a lifting to one of the
congruence covers: $X_{\tilde{H}_{\Delta}}(K)$ or $X_{H_{\text{p}}}(K)$ .

###### Proof.

By the argument above, when there is a $K$-isogeny of degree $m\nmid d$, it
induces a $G_{K}$-isomorphism $E_{t}[m]\simeq E^{\prime}_{t}[m]$ which implies
an isomorphism of the mod $m$ Galois images, up to a change of basis given by
conjugating by an element $(1,g)\in SL_{2}(\mathbb{Z}/m\mathbb{Z})$. Applying
Proposition 3.7, we get the conclusion that if $m$ does not divide the degree
of the isogeny, the Galois image lies in $\tilde{H}_{\Delta}$. As for the rest
of the lemma, when $d=m$, we have the Galois image of the $m$-torsion
monodromy representation of both $E_{1}$ and $E_{2}$ contained in a Borel
subgroup of $SL_{2}(\mathbb{Z}/m\mathbb{Z})$. Since any Borel subgroup is
maximal parabolic in $SL_{2}(\mathbb{Z}/m\mathbb{Z})$, this proves lemma 3.11.
∎

##  4\. Geometric Interpretation and Projective Embeddings

In this section, we explore the geometric interpretation of the question and
construct projective embeddings, which allow us to transform the question into
counting rational points with bounded height in projective spaces.

######  Definition 4.1.

Let $H$ be a subgroup of $SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})$ such that $H$ is either $\tilde{H}_{\Delta}$
or $H_{\operatorname{p}}$. Define $C_{H}$ to be the lifting of $C$ to the
modular surface $X_{H}:=X(m)\times X(m)/H$, which is given by the pullback of
the following diagram.

${C_{H}}$${X_{H}}$${C}$${X(1)\times X(1)}$

###  4.1. The lift curve is integral

In order to apply the result by Paredes and Sasyk, we need to prove that the
lifting $C_{H}$ of $C$ is integral for large enough $m$. We need the following
Goursat’s lemma:

######  Lemma 4.2.

[Hall, Theorem 5.5.1] Let $G$ and $H$ be groups, and let $K$ be a subdirect
product of $G$ and $H$; that is, $K\leq G\times H$, and
$\pi_{G}(K)=G,\pi_{H}(K)=H$, where $\pi_{G}$ and $\pi_{K}$ are the projections
onto the first and second factor, respectively from $G\times H$. Let
$N_{1}=K\cap\operatorname{ker}\left(\pi_{G}\right)$ and
$N_{2}=K\cap\operatorname{ker}\left(\pi_{H}\right)$. Then $N_{2}$ can be
identified with a normal subgroup $N_{G}$ of $G,N_{1}$ can be identified with
a normal subgroup $N_{H}$ of $H$, and the image of $K$ in $G/N_{G}\times
H/N_{H}$ is the graph of an isomorphism $G/N_{G}\cong H/N_{H}$.

###### Proof.

See [Hall, Theorem 5.5.1].

∎

Also, we prove the following proposition:

######  Proposition 4.3.

For all $m\geq 17$, the Galois image of the $p$-torsion monodromy
representation of $E_{t}[m]$ and $E^{\prime}_{t}[m]$ in the generic fiber is
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$.

###### Proof.

Suppose the Galois image of the $m$-torsion monodromy representation is some
proper subgroup $G$ of $SL_{2}(\mathbb{Z}/m\mathbb{Z})$. Then we have a
dominant map $f:C\to X(m)/G$. Since the genus of $C$ is zero, this implies
that the genus of the modular curve $X(m)/G$ is zero.

Let $\mathcal{N}(m)$ be the quantity such that

$\mathcal{N}(m):=\operatorname{min}\\{\text{genus}(X(m)/G)\mid G\subsetneq
SL_{2}(\mathbb{Z}/m\mathbb{Z}),\text{ }G\text{ maximal}\\}.$

Cojocaru and Hall proved the genus formula for $X(m)/G$ for all possible
maximal subgroup $G$ of $SL_{2}(\mathbb{Z}/m\mathbb{Z})$, which is summarized
in a table [CH05, Table 2.1]. Moreover, they proved that

$\mathcal{N}(m)=\frac{1}{12}\left[m-\left(6+3\mathrm{e}_{2}+4\mathrm{e}_{3}\right)\right]>0$

for $m\geq 17$. The proposition follows. ∎

Now we are ready to prove that $C_{H}$ is irreducible. This follows as a
consequence of Proposition 4.3:

######  Lemma 4.4.

For each choice of $H$, the curve $C_{H}$ is integral.

###### Proof.

We have the covering map $q_{\tilde{U}}:\tilde{U}\to U$ where $U$ is the
connected dense open subset of $C$ parametrizing smooth points. Later in the
proof of Proposition 4.9, we showed that the Galois image of the $m$-torsion
monodromy representation of $\pi_{1}(U)$ is the full group
$SL_{2}(\mathbb{Z}/m\mathbb{Z})\times SL_{2}(\mathbb{Z}/m\mathbb{Z})$, by
using Proposition 4.3 and Lemma 4.2. Therefore the monodromy of $\pi_{1}(U)$
acts transitively on the right cosets $SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})/H$, which implies that the cover $C_{H}$ is
connected for both $H=\tilde{H}_{\Delta}$ and $H=H_{\operatorname{p}}$. The
lemma follows by the fact that the quotient space of a connected space is
connected.

∎

###  4.2. Construct projective embeddings

In this subsection, we give an explicit construction of projective embeddings
of $C_{H}$, denoted by $\iota_{H}$. We make the following diagram commute for
each case’s choice of $N\in\mathbb{Z}$. Here the rational map
$\mathbb{P}^{N}\dashrightarrow\mathbb{P}^{3}$ is a projection of coordinates
of $\mathbb{P}^{N}$.

(4.1)
${C_{H}}$${\mathbb{P}^{N}}$${C}$${\mathbb{P}^{3}.}$$\scriptstyle{\iota_{H}}$$\scriptstyle{q}$

Case I: The modular diagonal quotient surfaces.

Recall the definition of the modular diagonal quotient surfaces in $\S 2.4$.
In our case where $\epsilon=1$, $X_{H_{\Delta}}(m)(K)$ has the moduli
interpretation that it is the set of isomorphism classes of triples
$\left(E_{1},E_{2},\psi\right)$, where $E_{1}$, $E_{2}$ are elliptic curves
over $K$ and
$\psi:E_{1}[m]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{2}[m]$ is an
isomorphism of the $m$-torsion subgroups of the elliptic curves which
preserves the Weil pairing. Let $t$ be a rational point of $C(K)$ such that
there exists a point
$(E_{t},E_{t}^{\prime},\psi:E_{t}[m]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{t}^{\prime}[m])$
which is a point of $X_{H_{\Delta}}(m)(K)$. One may notice that it is not
obvious that $C_{\tilde{H}_{\Delta}}$ is connected, and we will address this
point in Lemma 4.4.

We now define some functions on $C_{\tilde{H}_{\Delta}}$ in order to apply the
result from [CCDN19] and to bound the number of points on
$C_{\tilde{H}_{\Delta}}$.

For a fixed $t\in K$, there is a list of elliptic curves isogenous to $E_{t}$
through cyclic isogenies of degree $m$ given by the list of cyclic subgroups
of $E_{t}[m]$, say

$E_{t,1},\cdots,E_{t,m+1}.$

Similarly we have a list of $m$-cyclic subgroups of $E_{t}^{\prime}[m]$
parametrizing the $m$-cyclic isogenies of $E_{t}^{\prime}$, with the
corresponding list of elliptic curves isogenous to $E_{t}^{\prime}$:

$E_{t,1}^{\prime},\cdots,E_{t,m+1}^{\prime}.$

######  Remark 4.5.

These isogenies will most likely not be defined over $K$ unless $E_{t}[m]$ or
$E_{t}^{\prime}[m]$ has a rational cyclic subgroup. Even then, most of them
will not be defined over $K$.

For each point $(E_{t},E_{t}^{\prime},\psi)$ on $\tilde{C}$, we have $m+1$
cyclic subgroups of $E_{t}[m]$ and $m+1$ cyclic subgroups of
$E_{t}^{\prime}[m]$ which can be placed in natural bijection with each other
under $\psi$. We can re-order the lists for $E_{t}^{\prime}$ such that
$E_{t,1}$ is in correspondence with $E_{t,1}^{\prime}$ and so on.

######  Definition 4.6.

Let $F$ be a function defined on $X_{\tilde{H}_{\Delta}}$ given by

$F(E_{t},E_{t}^{\prime},\psi)=j(E_{t,1})j(E^{\prime}_{t,1})+\cdots+j(E_{t,m+1})j(E^{\prime}_{t,m+1}).$

######  Lemma 4.7.

$F$ is defined over $\mathbb{Q}$. Moreover, $F$ is an element of the function
field $\mathbb{Q}(X_{\tilde{H}_{\Delta}}).$

###### Proof.

For any $g\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$,
$t\in\overline{\mathbb{Q}}$, denote by $(t,\psi)$ one of the preimages on
$X_{\tilde{H}_{\Delta}}$.

$F((t,\psi)^{g})=\Sigma_{i=1}^{m+1}j(E_{t^{g},i})j(E^{\prime}_{t^{g},i})=\Sigma_{i=1}^{m+1}(j(E_{t,i})j(E^{\prime}_{t,i}))^{g}.$

The second equality holds since $\psi^{g}=\psi$ and for an arbitrary $1\leq
i\leq m+1$, there is a unique $k$ such that

$(j(E_{t,i}))^{g}=j(E_{t,k})$

and once we fix $i$, it is also true that

$(j(E_{t,i}^{\prime}))^{g}=j(E_{t,k}^{\prime})$

for the same $k$.

∎

######  Definition 4.8.

Define
$\iota_{\tilde{H}_{\Delta}}^{\prime}:C_{\tilde{H}_{\Delta}}\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
to be the function given by

$\iota_{\tilde{H}_{\Delta}}^{\prime}((E_{t},E_{t}^{\prime},\psi))=(F(E_{t},E_{t}^{\prime},\psi),j(E_{t}),j(E_{t}^{\prime})).$

######  Proposition 4.9.

For $m\geq 17$, $\iota^{\prime}_{\tilde{H}_{\Delta}}$ is generically an
embedding of $C_{\tilde{H}_{\Delta}}$ into
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. This is equivalent
to saying that the subfield $M\subset K(C_{\tilde{H}_{\Delta}})$ generated by
$j(E_{t})$, $j(E_{t}^{\prime})$ and $F$ is the whole function field.

The proof of Proposition 4.9 splits into two parts. The first part is to show
that $M$ is not contained in $K(C)$. The second part is to show there is no
intermediate extension between $K(C)$ and $K(C_{\tilde{H}_{\Delta}})$, so that
$K(C)\subseteq M\subseteq K(C_{\tilde{H}_{\Delta}})$ and $M\neq K(C)$ implies
$M=K(C_{\tilde{H}_{\Delta}}).$

First, we present some representation theoretical lemma which we will use
later. Let $G$ be a finite group and let $V$ and $V^{\prime}$ be permutation
representations of $G$ of the same dimension, such that $G$ acts
$2$-transitively on some finite sets by $V$ and $V^{\prime}$. It is a standard
fact that the corresponding representation $V$(resp. $V^{\prime}$) is the
direct sum of the trivial representation and an irreducible representation
whose coordinates sum to $0$. Denote the irreducible representation by
$V_{0}$(resp. $V^{\prime}_{0}$). We prove that if $v\in V$, $v^{\prime}\in
V^{\prime}$ such that $\langle v,v^{\prime}\rangle=\langle
v,(v^{\prime})^{g}\rangle$ for any $g\in G$, then either $v$ or $v^{\prime}$
is fixed by $G$. We first prove the following lemma.

######  Lemma 4.10.

If $v\in V$(resp. $v^{\prime}\in V^{\prime}$) is not in the trivial
representation generated by $[1,\cdots,1]^{T}$, the set $\\{v-v^{g}\\}$(resp.
$\\{v^{\prime}-(v^{\prime})^{g}\\}$), as $g$ ranges over $G$, span the space
of all vectors in $V_{0}$(resp. $V^{\prime}_{0}$).

###### Proof.

Let $W_{v}$(resp. $W^{\prime}_{v}$) be the subspace Span$\\{v-v^{g}\\}$(resp.
Span$\\{(v^{\prime})-(v^{\prime})^{g}\\}$) as $g$ ranges over $G$. We claim
that $W_{v}$ is a sub-representation of $V_{0}$, the irreducible
representation of the permutation representation given by the condition that
the sum of the coordinates equals $0$. Indeed, for any $v\in W_{v}$,
$\sigma,\tau\in G$, one have

$(v-v^{\sigma})^{\tau}=v^{\tau}-v^{\sigma\tau}=(v-v^{\sigma\tau})-(v-v^{\tau}).$

The lemma follows from the face that $V_{0}$(resp. $V^{\prime}_{0}$) is
irreducible. ∎

######  Lemma 4.11.

If $v\in V$, $v^{\prime}\in V^{\prime}$ such that $\langle
v,v^{\prime}\rangle=\langle v,(v^{\prime})^{g}\rangle$ for any $g\in G$, then
either $v$ or $v^{\prime}$ is fixed by $G$.

###### Proof.

If $\langle v,v^{\prime}\rangle=\langle v,(v^{\prime})^{g}\rangle$ for all
$g\in G$, then

$\langle v,(v^{\prime})-(v^{\prime})^{\gamma}\rangle=0$

for all $g\in G.$

If $W_{v^{\prime}}=0$, then $v^{\prime}=(v^{\prime})^{g}$ for any $g\in G$
thus $v^{\prime}$ is fixed by $G$. If $W_{v^{\prime}}\neq 0$, by lemma 4.10,
$v^{\prime}-(v^{\prime})^{g}$ span the space $V^{\prime}_{0}$ which is the
orthogonal complement of the trivial representation. Therefore

$v\in W_{v^{\prime}}^{\perp}=\mathbb{C}\langle[1,\cdots,1]^{T}\rangle,$

thus fixed by $G$.

∎

Now we apply Lemma 4.11 to the case where $G=SL_{2}(\mathbb{Z}/m\mathbb{Z})$
and where $V$ and $V^{\prime}$ are $m+1$ dimensional representations of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$ spanned by vectors $v_{t}$ and
$v_{t}^{\prime}$ respectively, as $t$ ranges over $C$. For $t\in C$ such that
$E_{t}$ and $E_{t}^{\prime}$ are both non-singular, define $v_{t}$ and
$v_{t}^{\prime}$ by

(4.2) $v_{t}=(j(E_{t,1}),\cdots,j(E_{t,m+1}))$

(4.3) $v_{t}^{\prime}=(j(E^{\prime}_{t,1}),\cdots,j(E^{\prime}_{t,m+1})).$

It is easy to see that $V$ and $V^{\prime}$ are permutation representations of
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$ by its transitive action on the basis.

Recall the definition of $F$ in 4.6. We may also write $F$ as an inner
product:

$F(E_{t},E_{t}^{\prime},\psi)=\langle
v_{t},v^{\prime}_{t}\rangle=j(E_{t,1})j(E^{\prime}_{t,1})+\cdots+j(E_{t,m+1})j(E^{\prime}_{t,m+1}).$

Note that $F$ is defined on $C_{\tilde{H}_{\Delta}}$ by restriction, but
$SL_{2}(\mathbb{Z}/m\mathbb{Z})\times SL_{2}(\mathbb{Z}/m\mathbb{Z})$ does not
act on $C_{\tilde{H}_{\Delta}}$. By Lemma 4.4 $C_{\tilde{H}_{\Delta}}$ is
connected, we define $C^{\prime}$ as the Galois closure of
$C_{\tilde{H}_{\Delta}}$ over $C$. This is equivalent to saying that
$C^{\prime}$ is the pullback of $C$ all the way to $X(m)\times X(m)$, see
diagram 4.4. Therefore the pullback of $F$ to $X(m)\times X(m)$ is a function
on $C^{\prime}$ with $F^{\sigma}=F$ for all $\sigma\in\tilde{H}_{\Delta}$. We
prove that $F$ is not defined on $C$.

(4.4) ${C^{\prime}}$${X(m)\times X(m)}$${C_{\tilde{H}_{\Delta}}}$${X(m)\times
X(m)/\tilde{H}_{\Delta}}$${C}$${X(1)\times X(1).}$

######  Lemma 4.12.

$F$ is not defined on $C$.

###### Proof.

For each $\gamma\in SL_{2}(\mathbb{Z}/m\mathbb{Z})$, the action of $\gamma$ on
$F$ is given by

$F(E_{t},E_{t}^{\prime},\psi)^{\gamma}=\langle
v_{t},(v^{\prime}_{t})^{\gamma}\rangle.$

If $F$ is defined on $C$, then for every $\gamma\in
SL_{2}(\mathbb{Z}/m\mathbb{Z})$ we have $F^{\gamma}=F$ therefore $\langle
v_{t},v_{t}^{\prime}\rangle=\langle v_{t},(v_{t}^{\prime})^{\gamma}\rangle$.
By Lemma 4.11, either $v_{t}$ or $v^{\prime}_{t}$ is fixed by
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$. But this implies either

$j(E_{t,1})=\cdots=j(E_{t,m+1})$

or

$j(E^{\prime}_{t,1})=\cdots=j(E^{\prime}_{t,m+1}).$

Since there exists $t\in C$ such that $E_{t}$ and $E_{t}^{\prime}$ are both
non-CM elliptic curves, this cannot happen. Therefore $F$ cannot be defined on
$C$.

∎

Now we are ready to prove the generic injectivity of
$\iota^{\prime}_{H_{\Delta}}$. We show no intermediate cover exists between
$\tilde{C}$ and $C$ by an argument using monodromy.

###### Proof of Proposition 4.9.

By lemma 4.11, we have the argument that $F$ is not defined on $C$.

Recall that $C_{\tilde{H}_{\Delta}}$ is defined to be the cover of $C$
constructed by pulling back $C\to X(1)\times X(1)$ in the previous context.
Let $U\subset C$ be the dense open locus parametrizing smooth points, i.e.,
pairs of genuine elliptic curves. The étale fundamental group $\pi_{1}(U)$ is
a quotient of the absolute Galois group of $K(t)$, which acts on the
$m$-torsion of the generic fiber, say $E_{t}[m]$ and $E^{\prime}_{t}[m]$, in
the usual Galois way. Proposition 4.3 asserts that for $m\geq 17$, the map

$\rho_{m}:\pi_{1}(U)\to SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})$

is surjective on each of the two factors. Therefore the reduction map

$\bar{\rho}_{m}:\pi_{1}(U)\to PSL_{2}(\mathbb{Z}/m\mathbb{Z})\times
PSL_{2}(\mathbb{Z}/m\mathbb{Z})$

is surjective on each factor. For $m\geq 5$, the projective special linear
group $PSL_{2}(\mathbb{Z}/m\mathbb{Z})=SL_{2}(\mathbb{Z}/m\mathbb{Z})/\\{\pm
1\\}$ is simple. By assumption, the generic fiber $E_{t}$ and $E^{\prime}_{t}$
of the family are not isogenous. Therefore by the work of Bakker and Tsimerman
[BT, Theorem 1], there exists an absolute constant $M_{1}$ such that for any
prime $m>M_{1}$, the image of $\rho_{m}$ on each of the factors is non-
isomorphic. Hence the inequivalence condition in Lemma 4.2 is satisfied. Lemma
4.2 leads to the conclusion that the Galois image of the $m$-torsion monodromy
representation of $\pi_{1}(U)$ is full in
$PSL_{2}(\mathbb{Z}/m\mathbb{Z})\times PSL_{2}(\mathbb{Z}/m\mathbb{Z})$.

We prove that there is no intermediate cover between $C_{\tilde{H}_{\Delta}}$
and $C$. Suppose there is a curve $X$ such that

$C_{\tilde{H}_{\Delta}}\to X\to C$

with all maps of degrees greater than $1$. Since the connected covering space
of $C$ is in bijection with the subgroups of $\pi_{1}(C)$, $X$ corresponds to
a proper subgroup $H^{\prime}$ strictly containing $\tilde{H}_{\Delta}$.
However, $\tilde{H}_{\Delta}$ is a maximal subgroup of
$PSL_{2}(\mathbb{Z}/m\mathbb{Z})\times PSL_{2}(\mathbb{Z}/m\mathbb{Z})$, which
implies that $X$ is isomorphic to $C_{\tilde{H}_{\Delta}}$, contradiction.
Therefore $C_{\tilde{H}_{\Delta}}$ is birational to its image under
$\iota_{\tilde{H}_{\Delta}}^{\prime}$, which proves the proposition. ∎

We have constructed a generic embedding $\iota^{\prime}_{\tilde{H}_{\Delta}}$
of $C_{\tilde{H}_{\Delta}}$ into a product of projective lines from the
argument above. Composing with the Segre embedding, we get a generic embedding
of $C_{\tilde{H}_{\Delta}}$ into $\mathbb{P}^{7}$, denoted by
$\iota_{\tilde{H}_{\Delta}}$.

$\iota_{\tilde{H}_{\Delta}}:C_{\tilde{H}_{\Delta}}\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{7}.$

One notice that $\iota_{\tilde{H}_{\Delta}}$ fits into the diagram 4.1 at the
beginning of this chapter, with $N=7$.

Case II: the maximal parabolic quotient surfaces. When
$H=H_{\operatorname{p}}$ which is a product of maximal parabolic subgroups,
the quotient $X(m)\times X(m)/H$ is isomorphic to $X_{0}(m)\times X_{0}(m)$.
The corresponding projective embedding $\iota_{H_{\operatorname{p}}}$ can be
constructed in the context of Hecke correspondence of level
$SL_{2}(\mathbb{Z}/m\mathbb{Z})$ followed by the Segre embedding, as
following:

(4.5) $\iota_{H_{\operatorname{p}}}:X_{0}(m)\times
X_{0}(m)\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{15}.$

To be explicit, for a point
$\tilde{P_{t}}=(E_{t},\tilde{E_{t}},E_{t}^{\prime},\tilde{E_{t}^{\prime}})$
that lifts $P_{t}=(E_{t},E_{t}^{\prime})$, where $E_{t}$ and $\tilde{E_{t}}$
are linked by a cyclic isogeny of degree $m$ and same for $E_{t}^{\prime}$ and
$\tilde{E_{t}^{\prime}}$, one may write $\iota_{H_{\operatorname{p}}}$ as

$\displaystyle\iota_{H_{\operatorname{p}}}:(E_{t},\tilde{E_{t}},E_{t}^{\prime},\tilde{E_{t}^{\prime}})$
$\displaystyle\mapsto j(E_{t})\times j(\tilde{E_{t}})\times
j(E_{t}^{\prime})\times j(\tilde{E_{t}^{\prime}})$
$\displaystyle\mapsto(j(E_{t})j(\tilde{E_{t}})j(E_{t}^{\prime})j(\tilde{E_{t}^{\prime}});\cdots;j(E_{t});j(\tilde{E_{t}});j(E_{t}^{\prime});j(\tilde{E_{t}^{\prime}});1).$

In this case, we choose $N=15$ in diagram 4.1.

##  5\. Bound for the Change of Heights

In this section, we give an upper bound on the height of a point in $C_{H}(K)$
lying over a point $P_{t}\in C(K)$, in terms of the height $H(t)$ and the
level $m$. The main theorem of this section is Proposition 5.4.

5.1. Hecke correspondence and modular polynomials. Modular polynomials of
elliptic curves, the so-called ’elliptic modular polynomials,’ are the most
common and simplest examples of modular equations. For a positive integer $m$,
the classical modular polynomial $\Phi_{m}$ is the minimal polynomial of
$j(mz)$ over $\mathbb{C}(j)$. In other words we have $\Phi_{m}(j(mz),j(z))=0$.
The bivariate polynomial $\Phi_{m}(X,Y)$ is symmetric of degree
$\psi(m)=m\prod_{p|m}(1+p^{-1})$ in both variables, and its coefficients grow
super-exponentially in $m$. The modular curve $Y_{0}(m)$ is birational to its
image in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with $\Phi_{m}$ an equation for
this image. The graph of $\Phi_{m}$ describes the Hecke correspondence such
that there exists a cyclic isogeny of degree $m$ between projections onto each
copy of $\mathbb{P}^{1}$.

For elliptic curves $E_{1}$ and $E_{2}$ linked with a cyclic isogeny of order
$m$, we aim to find an upper bound for the height $H(j(E_{1}))$, in terms of
$H(j(E_{2}))$ and the coefficients of $\Phi_{m}$. This has been worked out by
Pazuki in [Paz19].

######  Theorem 5.1.

[Paz19, Theorem 1.1] Let $\varphi:E_{1}\rightarrow E_{2}$ be a
$\overline{\mathbb{Q}}$-isogeny between two elliptic curves defined over
$\overline{\mathbb{Q}}$. Let $j_{1}$ and $j_{2}$ be the respective
$j$-invariants. Then one has

$\left|h\left(j_{1}\right)-h\left(j_{2}\right)\right|\leq
9.204+12\log\operatorname{deg}\varphi$

where $h(.)$ denotes the absolute logarithmic Weil height.

Theorem 5.1 leads to the following corollary:

######  Corollary 5.2.

Let $\varphi:E_{1}\rightarrow E_{2}$ be a $\overline{\mathbb{Q}}$-isogeny
between two elliptic curves defined over $\overline{\mathbb{Q}}$ which is
cyclic of degree $m$. Let $j_{1}$ and $j_{2}$ be the respective
$j$-invariants. Then one has

$H(j_{1})<Am^{12}H(j_{2})$

for some absolute constant $A$. Here $H(.)$ denotes the projective height.

5.2. Bounding change of heights. We prove an upper bound on the product of
heights, which we will use later.

######  Lemma 5.3.

Let $d$ be the projective degree of $C$ under the embedding $\iota$, see
(1.1). Let $H(\iota)$ be the height of $\iota$ defined by the height of the
coefficients of the defining polynomials of $j(E_{t})$ and
$j(E_{t}^{\prime}).$ Then for every $t\in K$ with $H(t)\leq B$, we have

$H(P_{t})\leq(d+1)H(\iota)B^{d}.$

###### Proof.

By Lemma 2.1 we have

$H(P_{t})=H(j(E_{t}))H(j(E_{t}^{\prime})).$

The lemma then follows from [Sil09, VIII, Theorem 5.6] , which asserts that
when there is a map of degree $d$ between two projective spaces, say

$F:\mathbb{P}^{m}\to\mathbb{P}^{M},$

then for all points $P\in P^{m}(\overline{\mathbb{Q}})$ there are positive
constant $C_{1}$ and $C_{2}$ depending on $F$ such that

$C_{1}H(P)^{d}\leq H(F(P))\leq C_{2}H(P)^{d}.$

Write $F=[f_{0},\cdots,f_{M}]$ using homogeneous polynomials $f_{i}$ having no
common zeros. Let $H(F)$ be the height of $F$ defined by the height of the
coefficients of $f_{i}$. The constant $C_{1}$ and $C_{2}$ can be explicitly
calculated in terms of $M$, $m$ and $H(F)$. Especially, we can let
$C_{1}=\binom{m+d}{m}H(F).$ The lemma follows from the assumption that
$F=\iota$, $m=1$ and $M=3$.

∎

By Lemma 3.11, for our choice of $m\in\mathbb{Z}$, a rational point $t\in
C(K)$ with $t\in S(B)$ for some $B$ lifts to a rational point on one of the
covers $C_{\tilde{H}_{\Delta}}\subset X_{\tilde{H}_{\Delta}}$ or
$C_{H_{\operatorname{p}}}\subset X_{H_{\operatorname{p}}}$. We have the
following proposition:

######  Proposition 5.4.

Fix $m\in\mathbb{N}$. Let $t\in K$ be a rational point such that $t\in S(B)$
for some $B$. Let $P_{t}$ denote the point on $C$ parametrized by $t$, and
denote by $\tilde{P_{t}}$ a lifting of $P_{t}$ to one of the covers $C_{H}$ in
Lemma 3.11. Let $H(\iota_{H}(\tilde{P_{t}}))$ denote the projective height of
$\tilde{P_{t}}$ with respect to $\iota_{H}$.

If $H=\tilde{H}_{\Delta}$, then

$H(\iota_{H}(\tilde{P_{t}}))\leq(m+1)m^{24(m+1)}A^{2(m+1)}((d+1)H(\iota))^{m+2}B^{d(m+2)}.$

If $H=H_{\operatorname{p}}$, then

$H(\iota_{H}(\tilde{P_{t}}))\leq m^{24}(d+1)^{2}H(\iota)^{2}B^{2d}.$

###### Proof.

Case 1: $P_{t}$ lifts to $\tilde{C}_{\tilde{H}_{\Delta}}(K)$

As noted above, passing to the quotient $C_{\tilde{H}_{\Delta}}$ we have
$\iota_{\tilde{H}_{\Delta}}$ embeds $C_{\tilde{H}_{\Delta}}$ into
$\mathbb{P}^{7}$ by composing $\iota^{\prime}_{\tilde{H}_{\Delta}}$ with the
Segre embedding. A point
$\tilde{P_{t}}=(E_{t},E_{t}^{\prime},\psi:E_{t}[m]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{t}^{\prime}[m])$
that is a lift of $P_{t}=(E_{t},E_{t}^{\prime})$ embedded into
$\mathbb{P}^{7}$ as following:

$(E_{t},E_{t}^{\prime},\psi)\to F\times j(E_{t})\times
j(E_{t}^{\prime})\hookrightarrow[Fj(E_{t})j(E_{t}^{\prime}),Fj(E_{t}),Fj(E_{t}^{\prime}),\cdots,1].$

Since our ultimate goal is to count rational points parametrized by $t\in K$,
we need a formula relating the height of $\iota_{H}(\tilde{P_{t}})$ with
heights of $F$, $j(E_{t})$ and $j(E_{t}^{\prime})$. By Lemma 2.1, we have

(5.1) $H(\iota_{H}(\tilde{P_{t}}))=H(F)H(j(E_{t}))H(j(E_{t}^{\prime})).$

Let $i$ be an integer between $1$ and $m+1$ such that

$H(j(E_{t,i})j(E_{t,i}^{\prime}))=\max_{1\leq k\leq
m+1}H(j(E_{t,k})j(E_{t,k}^{\prime})).$

By Definition 4.6 and Corollary 5.2 and Lemma 5.3, together with the fact that
for any $\alpha,\beta,\alpha_{1}\cdots\alpha_{r}\in\overline{\mathbb{Q}}$,

$H(\alpha\beta)\leq H(\alpha)H(\beta)$

and

$H(\alpha_{1}+\cdots\alpha_{r})\leq rH(\alpha_{1})\cdots H(\alpha_{r}),$

we have

(5.2) $\displaystyle H(F)$
$\displaystyle=H(j(E_{t,1})j(E^{\prime}_{t,1})+\cdots+j(E_{t,m+1})j(E^{\prime}_{t,m+1}))$
(5.3) $\displaystyle\leq(m+1)H(j(E_{t,i}))^{m+1}H(j(E_{t,i}^{\prime}))^{m+1}$
(5.4)
$\displaystyle\leq(m+1)m^{24(m+1)}A^{2(m+1)}H(j(E_{t}))^{m+1}H(j(E_{t}^{\prime}))^{m+1}$
(5.5)
$\displaystyle\leq(m+1)m^{24(m+1)}A^{2(m+1)}((d+1)H(\iota))^{m+1}B^{d(m+1)}.$

The constant $A$ comes from Corollary 5.2, and the first part of the lemma
follows from $(\ref{3})$.

Case 2: $P_{t}$ lifts to one of the maximal parabolic quotient surfaces

Recall the definition of $H_{\operatorname{p}}$ (3.9) and
$\iota_{H_{\operatorname{p}}}$ (4.5). As in the previous case, Lemma 2.1
implies that

$H(\iota_{H_{\operatorname{p}}}(\tilde{P_{t}}))=H(j(E_{t}))H(j(\tilde{E_{t}}))H(j(E_{t}^{\prime}))H(j(\tilde{E_{t}^{\prime}}))$

By Corollary 5.2 and Lemma 5.3, where $\tilde{E}_{t}$ is $m$-isogenous to
$E_{t}$ and $\tilde{E}^{\prime}_{t}$ is $m$-isogenous to $E_{t}^{\prime}$, we
have

$H(\iota_{H_{\operatorname{p}}}(\tilde{P_{t}}))\ll
m^{24}(d+1)^{2}H(\iota)^{2}B^{2d}.$

∎

##  6\. Proof of the Main Theorems

###  6.1. Proof of Theorem 1.1

The previous sections show that for a rational point $P_{t}$ on $C$, we have
two types of possible liftings to some modular surfaces with $m$-level
structures. Accordingly, we divide the proof of Theorem 1.1 into two parts and
analyze each part’s contribution to $|S(B)|$. We make optimization of $m$ in
terms of the height $B$ of $t$ as.

Case 1: Contributions from modular diagonal quotient surfaces.

Recall that we have the following commutative diagram:

${C_{\tilde{H}_{\Delta}}}$${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{7}}$${C}$${\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{3}}$$\scriptstyle{\iota_{\tilde{H}_{\Delta}}^{\prime}}$$\scriptstyle{q}$Segre

Let $\iota_{\tilde{H}_{\Delta}}$ be the composition of
$\iota^{\prime}_{\tilde{H}_{\Delta}}$ with the Segre embedding. In order to
apply Theorem 2.4, we bound the degree of $\iota_{\tilde{H}_{\Delta}}$, which
depends on $m$ and the projective degree of $C$.

Let $\deg_{C_{\tilde{H}_{\Delta}}}(F)$ be the degree of $F$ as a function on
$C_{\tilde{H}_{\Delta}}$. The degree of the function
$\iota_{\tilde{H}_{\Delta}}^{\prime}$ on $C_{\tilde{H}_{\Delta}}$ can be
viewed as a tridegree, which we denote by
$(\deg_{C_{\tilde{H}_{\Delta}}}(F),e,e^{\prime})$. When we pass to
$\mathbb{P}^{7}$ by compose with the Segre embedding, we have

$\iota_{\tilde{H}_{\Delta}}=\deg(F)+e+e^{\prime}.$

Here $e$ denotes the degree of the function $j(E_{t})$ on
$C_{\tilde{H}_{\Delta}}$ and $e^{\prime}$ is the degree of $j(E^{\prime}_{t})$
on $C_{\tilde{H}_{\Delta}}$. Let $\alpha$ be the degree of the cover $q$ and
let $d_{E}$ be the degree of the $j$-invariant map $C\to\mathbb{P}^{1}$.
Therefore $e=\alpha d_{E}$. Similarly, we have $e^{\prime}=\alpha
d_{E^{\prime}}.$ Therefore

$\deg(\iota_{\tilde{H}_{\Delta}})>e+e^{\prime}=\alpha(d_{E}+d_{E^{\prime}})=\alpha
d.$

The degree of $q$ is equal to the index of $\tilde{H}_{\Delta}$ inside the
Galois group $G=SL_{2}(\mathbb{Z}/m\mathbb{Z})\times
SL_{2}(\mathbb{Z}/m\mathbb{Z})$. We have

$\alpha=[G:\tilde{H}_{\Delta}]=m^{3}(1-\frac{1}{m^{2}})=m(m+1)(m-1).$

Therefore

$\operatorname{deg}(\iota_{\tilde{H}_{\Delta}})\geq m(m+1)(m-1)d.$

In order to get an upper bound of $\deg(\iota_{\tilde{H}_{\Delta}})$, we need
an upper bound for the degree of $F$ over $C_{\tilde{H}_{\Delta}}$. Recall
from diagram 4.4, $C^{\prime}$ is the Galois closure of
$C_{\tilde{H}_{\Delta}}$ over $C$ and the function $F$ is defined to be

$F=j(E_{t,1})j(E^{\prime}_{t,1})+\cdots+j(E_{t,m+1})j(E^{\prime}_{t,m+1}).$

The individual terms $j(E_{t,i})$ and $j(E^{\prime}_{t,i})$ are defined on
$C^{\prime}$, instead of on $C_{\tilde{H}_{\Delta}}$. A point on $C^{\prime}$
is a product of triples
$(E_{t},P,G)\times(E^{\prime}_{t},P^{\prime},G^{\prime})$, where $P$(resp.
$P^{\prime}$) is a point of $E_{t}[m]$(resp. $E_{t}^{\prime}[m]$) and
$G$(resp. $G^{\prime}$) is a cyclic subgroup of $E_{t}[m]$(resp.
$E_{t}^{\prime}[m]$). Fix a $j$-invariant $x$, the degree of
$j(E_{t,i})$(resp. $j(E^{\prime}_{t,i})$) is the number of points on
$C^{\prime}$ such that $j(E_{t}/G)=x$. As long as $E_{t}$ and
$E_{t}^{\prime})$ is not CM, there are $m+1$ $j$-invariants which are
$m$-isogenous to $x$ and there are $d_{E}$ points on $C$ mapping to each of
those $(m+1)$ $j$-invariants. Hence the degree $\deg_{C}(j(E_{t,i}))$(resp.
$\deg_{C}(j(E^{\prime}_{t,i}))$) for each $1\leq i\leq m+1$ is
$(m+1)d_{E}$(resp. $(m+1)d_{E^{\prime}}$).

The argument above, together with the fact that if $f$ and $g$ are functions
on a curve $X$ then

$\deg(f+g)\leq\deg(f)+\deg(g)$

and

$\deg(fg)\leq\deg(f)+\deg(g),$

yields that

$\deg_{C_{\tilde{H}_{\Delta}}}(F)\leq\deg_{C^{\prime}}(F)\leq(m+1)^{2}d.$

Hence we get an upper bound on the degree of $\iota_{\tilde{H}_{\Delta}}$
which is

$\operatorname{deg}(\iota_{\tilde{H}_{\Delta}})\leq(m(m+1)(m-1)+(m+1)^{2})d.$

We can write the argument in the paragraph above in a lemma:

######  Lemma 6.1.

$m(m+1)(m-1)d\leq\deg(\iota_{\tilde{H}_{\Delta}})\leq(m(m+1)(m-1)+(m+1)^{2})d.$

Let $S_{B,m,H_{\Delta}}$ be the set of rational points on
$C_{\tilde{H}_{\Delta}}(K)$ which are liftings of $P_{t}$ for some $t\in
S(B)$. Recall that we prove an upper bound for the heights of points in
$S_{B,m,H_{\Delta}}$ in Proposition 5.4. Theorem 2.4 ([PS22, Theorem 1.8])
then applies, along with Lemma 4.4 and Lemma 6.1, yielding

(6.1) $\displaystyle S_{B,m,H_{\Delta}}$
$\displaystyle\lesssim_{K}((\alpha+(m+1)^{2})d)^{4}((m+1)m^{24(m+1)}A^{2(m+1)}((d+1)H(\iota))^{m+2}B^{d(m+2)})^{\frac{2d_{K}}{\alpha
d}}$ (6.2)
$\displaystyle\lesssim_{K}((m^{2}+1)(m+1)d)^{4}((m+1)m^{24(m+1)}A^{2(m+1)}((d+1)H(\iota))^{m+2}B^{d(m+2)})^{\frac{2d_{K}}{m(m-1)(m+1)d}}$
(6.3)
$\displaystyle\lesssim_{K}(m^{3}d)^{4}(m+1)^{\frac{2d_{K}}{m(m-1)(m+1)d}}A^{\frac{4d_{K}}{m(m-1)d}}m^{\frac{48d_{K}}{m(m-1)d}}((d+1)H(\iota)B)^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)}}$

The terms in (6.3) other than $(m^{3}d)^{4}$ are bounded above by an absolute
constant. The argument requires optimizations on the choice of $m$, which we
will prove in the following lemma.

######  Lemma 6.2.

Recall that $m$ is a prime between $2(\log(d+1)+\log H(\iota)+\log
B)^{\frac{1}{2}}$ and $4(\log(d+1)+\log H(\iota)+\log B)^{\frac{1}{2}}$. There
is an absolute constant $A_{0}$ such that

$(m+1)^{\frac{2d_{K}}{m(m-1)(m+1)d}}A^{\frac{4d_{K}}{m(m-1)d}}m^{\frac{48d_{K}}{m(m-1)d}}((d+1)H(\iota)B)^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)}}\leq
A_{0}.$

###### Proof.

Once we write

$(m+1)^{\frac{2d_{K}}{m(m-1)(m+1)d}}$

as

$e^{(\frac{2d_{K}}{m(m-1)(m+1)d})\log(m+1)},$

it is easy to see that for $m\geq 2$ we have

$(m+1)^{\frac{2d_{K}}{m(m-1)(m+1)d}}\ll e^{\frac{2d_{K}}{d}}\leq e^{2d_{K}}.$

This is because $\frac{\log(m+1)}{m(m-1)(m+1)}$ is bounded above by $1$. A
similar argument shows that

$m^{\frac{48d_{K}}{m(m-1)d}}\ll e^{\frac{48d_{K}}{d}}\leq e^{48d_{K}}$

which also contributes as a constant independent of $m$ and $d$.

It is left to consider $((d+1)H(\iota)B)^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)}}$
which plays an important role in the optimization process. We make the
optimization by choosing suitable $m$ in terms of $B$, $d$, and $H(\iota)$.
The following inequalities, together with Proposition 4.3, allows one to take
$m$ to be any prime between $2(\log(d+1)+\log H(\iota)+\log B)^{\frac{1}{2}}$
and $4(\log(d+1)+\log H(\iota)+\log B)^{\frac{1}{2}}$, so that

$\displaystyle((d+1)H(\iota)B)^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)}}$
$\displaystyle=e^{{\frac{2d_{K}(m+2)}{m(m-1)(m+1)}}}(\log(d+1)+\log
H(\iota)+\log B)$ $\displaystyle=e^{\frac{2d_{K}(\log(d+1)+\log H(\iota)+\log
B)}{(m-1)(m+1)}}\cdot e^{\frac{4d_{K}(\log(d+1)+\log H(\iota)+\log
B)}{m(m-1)(m+1)}}$ $\displaystyle\leq e^{N_{1}d_{K}}\cdot e^{N_{2}d_{K}}$

where $\frac{2(\log(d+1)+\log H(\iota)+\log B)}{(m-1)(m+1)d}$ and
$\frac{4(\log(d+1)+\log H(\iota)+\log B)}{m(m-1)(m+1)d}$ are bounded above by
some absolute constant $N_{1}$ and $N_{2}$.

∎

We have the following proposition as a conclusion of the case.

######  Proposition 6.3.

The number of points in $|S(B)|$ that comes from the modular diagonal quotient
surface is bounded up by

$\displaystyle S_{B,m,H_{\Delta}}$
$\displaystyle\lesssim_{K}d^{4}((\log(d+1)+\log H(\iota)+\log B))^{6}$
$\displaystyle\lesssim_{K}d^{4+\epsilon}(\log H(\iota)+\log B)^{6}.$

###### Proof.

The proposition follows from the inequality (6.3) and Lemma 6.2. ∎

Case 2: Contributions from maximal parabolic quotient surfaces.  Recall that
in this case, we have the following commutative diagram.

${C_{H_{\operatorname{p}}}}$${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{15}}$${C}$${\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{3}}$$\scriptstyle{\iota^{\prime}_{{H}_{\operatorname{p}}}}$$\scriptstyle{q}$Segre

The degree of the covering space $q$, denoted by $\beta$, is equal to the
index of $H_{\operatorname{p}}$ inside the Galois group $G$, similar to the
previous case. The index of $H_{\operatorname{p}}$ as a product of maximal
parabolic subgroups is equal to

$\beta=[G:H_{\operatorname{p}}]=\frac{1}{4}(m+1)^{2}.$

One notice that the degree of $\iota_{H_{\operatorname{p}}}$ satisfies
$\operatorname{deg}(\iota_{{H}_{\operatorname{p}}})=\beta d$. Let
$S_{B,m,H_{\operatorname{p}}}$ be the number of rational points in
$C_{H_{\operatorname{p}}}(K)$ that lift $P_{t}$ for some $t\in S(B)$. Applying
Proposition 5.4 and Theorem 2.3, we get the following inequality

$S_{B,m,H_{\operatorname{p}}}\lesssim_{K}((m+1)^{2}d)^{4}(m^{24}(d+1)^{2}H(\iota)^{2}B^{2d})^{\frac{8d_{K}}{(m+1)^{2}d}}$

which allows us to make the same optimization as in case 1. By taking

$m\sim(\log B+\log(d+1)+\log H(\iota))^{\frac{1}{2}},$

we get an upper bound of the total contribution to $|S(B)|$ from the maximal
parabolic surfaces:

######  Proposition 6.4.

$\displaystyle S_{B,m,H_{\operatorname{p}}}$
$\displaystyle\lesssim_{K}d^{4}(\log B+\log(d+1)+\log H(\iota))^{4}$
$\displaystyle\lesssim_{K}d^{4+\epsilon}(\log B+\log H(\iota))^{4}.$

###### Proof of Theorem 1.1.

It is easy to see that the contribution from case $1$ dominates that of case
2. Theorem 1.1 then follows from Proposition 6.3 and 6.4.

∎

###  6.2. Proof of Theorem 1.3

Recall that in Theorem 1.1, $H(t)$ is defined as the height of $t$ as an
element of $K$. Instead, if we calculate the height of $t$ as the height of a
point on $X(1)\times X(1)$, which we have been calling $H(P_{t})$. Assume that
$H(P_{t})\leq B$. In this section, we prove an uniform bound on the number of
points $t$ such that $E_{t}$ and $E^{\prime}_{t}$ are geometrically isogenous,
which only depends on $K$, $B$, and the degree of the parametrize family.

We need a slightly modified version of Proposition 5.4, which we state as a
corollary.

######  Corollary 6.5.

Fix $m\in\mathbb{N}$. Let $t\in K$ be a rational point such that $t\in S(B)$
for some $B$. Let $P_{t}$ denote the point on $C$ parametrized by $t$, and
denote by $\tilde{P_{t}}$ a lifting of $P_{t}$ to one of the covers $C_{H}$ in
Lemma 3.11. Let $H(\iota_{H}(\tilde{P_{t}}))$ denote the projective height of
$\tilde{P_{t}}$ with respect to $\iota_{H}$.

If $H=\tilde{H}_{\Delta}$, then

$H(\iota_{H}(\tilde{P_{t}}))\leq(m+1)m^{24(m+1)}A^{2(m+1)}B^{(m+2)}.$

If $H=H_{\operatorname{p}}$, then

$H(\iota_{H}(\tilde{P_{t}}))\leq m^{24}B^{2}.$

###### Proof.

The proof is the same as Proposition 5.4, except that we have

$H(j(E_{t}))H(j(E_{t}^{\prime}))\leq B$

instead of Lemma 5.3. ∎

###### Proof of Theorem 1.3.

Let $S^{\prime}_{B,m,H}$ be the set of rational points on $C_{H}(K)$ which are
preimages of $P_{t}$ for some $t\in S^{\prime}(B).$ A similar argument to the
proof of Theorem 1.1 yields

(6.4) $\displaystyle S^{\prime}_{B,m,\tilde{H}_{\Delta}}$
$\displaystyle\lesssim_{K}((\alpha+(m+1)^{2})d)^{4}((m+1)m^{24(m+1)}A^{2(m+1)}B)^{(m+2)\frac{2d_{K}}{\alpha
d}}$ (6.5)
$\displaystyle\lesssim_{K}((m^{2}+1)(m+1)d)^{4}((m+1)m^{24(m+1)}A^{2(m+1)}B^{(m+2)})^{\frac{2d_{K}}{m(m-1)(m+1)d}}$
(6.6)
$\displaystyle\lesssim_{K}(m^{3}d)^{4}(m+1)^{\frac{2d_{K}}{m(m-1)(m+1)d}}A^{\frac{4d_{K}}{m(m-1)d}}m^{\frac{48d_{K}}{m(m-1)d}}B^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)d}}$

when $H=\tilde{H}_{\Delta},$ and

(6.7)
$S^{\prime}_{B,m,H_{\operatorname{p}}}\lesssim_{K}((m+1)^{2}d)^{4}(m^{24}B^{2})^{\frac{8d_{K}}{(m+1)^{2}d}}.$

We choose $m$ to be a prime between $2(\log B)^{\frac{1}{2}}$ and $4(\log
B)^{\frac{1}{2}}$ as to control the growth of the $B$-power factors in both
6.6 and 6.7, such that

$\displaystyle B^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)d}}$
$\displaystyle=e^{\frac{2d_{K}(m+2)}{m(m-1)(m+1)d}\log B}$
$\displaystyle=e^{\frac{2d_{K}(\log B)}{(m-1)(m+1)}}\cdot e^{\frac{4d_{K}(\log
B)}{m(m-1)(m+1)}}$ $\displaystyle\leq e^{N^{\prime}_{1}d_{K}}\cdot
e^{N^{\prime}_{2}d_{K}}$

and

$B^{\frac{16d_{K}}{(m+1)^{2}d}}=e^{\frac{16d_{K}}{(m+1)^{2}d}\log B}\leq
e^{N^{\prime}_{3}d_{K}}$

for some absolute constant $N^{\prime}_{1}$, $N^{\prime}_{2}$ and
$N^{\prime}_{3}$. ∎

## Acknowledgement

The author wishes to thank Jordan Ellenberg for many helpful suggestions.
Also, The author wants to show her gratitude to Asvin G. and Ananth Shankar
for the useful conversations, and Chunhui Liu for pointing out the reference
[PS22] to her which helps sharpen the bound.

## Compliance with ethical standards

### Conflicts of interest

The author states that there are no conflicts of interest.

### Data availability

Data sharing not applicable to this article as no data sets were generated or
analyzed.

## References