# Uniformity of quadratic points

Tangli Ge  Department of Mathematics
Brown University
Box 1917
Providence, RI 02912
U.S.A<EMAIL_ADDRESS>
###### Abstract.

In this paper, we extend a uniformity result of Dimitrov-Gao-Habegger [8] to
dimension two and use it to get a uniform bound on the set of all quadratic
points for non-hyperelliptic non-bielliptic curves in terms of the Mordell-
Weil rank.

Work supported in part by funds from NSF grant DMS-2100548 and DMS-1759514

##  1\. introduction

Let $C$ be a non-hyperelliptic, non-bielliptic curve (i.e. having no
$2$-to-$1$ map to an elliptic curve or $\mathbb{P}^{1}$) defined over a number
field $F$. Harris-Silverman showed in [17, Corollary 3] that the set of
quadratic points on $C$ is finite by applying Falting’s Theorem [9] to its
symmetric product. Motivated by this and the recent uniformity result of
Dimitrov-Gao-Habegger [8], we will show that

######  Theorem 1.1.

Let $g\geq 2$ and $d\geq 1$ be integers. Then there exists a constant
$c=c(g,d)\geq 1$ such that for any non-hyperelliptic non-bielliptic smooth
irreducible curve $C$ of genus $g$ defined over a number field $F$ with
$[F:\mathbb{Q}]\leq d$, we have

$\\#C(F,2)\leq c^{1+\rho}$

where $C(F,2)$ is the set of points on $C$ that are defined over some
$F^{\prime}$ with $[F^{\prime}:F]\leq 2$ and $\rho$ is the rank of
$\operatorname{Jac}(C)(F)$.

Let us call a positive dimensional variety $X$ defined over a field $K$
geometrically Mordellic or GeM, if $X_{\bar{K}}$ does not contain subvarieties
which are not of general type (see [1, Definition 1.4]). For a subvariety of
an abelian variety, Kawamata first showed that being GeM is equivalent to
containing no translates of positive dimensional abelian subvarieties, see
[19]. Let $\mathcal{A}\rightarrow S$ be a family of abelian varieties defined
over $\bar{\mathbb{Q}}$ where $S$ is a variety over $\bar{\mathbb{Q}}$. Let
$\mathcal{X}\rightarrow S$ be a family of $2$-dimensional GeM subvarieties of
$\mathcal{A}\rightarrow S$. The above theorem relies on a uniform bound of the
same quality as in [8], on the number of rational points for such a family
$\mathcal{X}\rightarrow S$. Namely we need the following (see §2 for the
notations and heights involved)

######  Theorem 1.2.

Let $\mathcal{A}\rightarrow S$ be a principally polarized abelian scheme with
level $l$ sturctures where $S$ is a variety over $\bar{\mathbb{Q}}$. Assume
the modular map $\tau:S\rightarrow\mathbf{A}_{g}$ is quasi-finite, where
$\mathbf{A}_{g}$ is the moduli space of principally polarized abelian
varieties over $\bar{\mathbb{Q}}$ with level $l$ structures. Let $h$ be a
height function on $\mathbf{A}_{g}$ associated to an ample line bundle on
$\mathbf{A}_{g}$. Let $\mathcal{X}\rightarrow S$ be a family of
$2$-dimensional irreducible GeM subvarieties in $\mathcal{A}\rightarrow S$.
Assume also $\mathcal{X}_{s}$ generates $\mathcal{A}_{s}$ for any $s\in
S(\bar{\mathbb{Q}})$. Then there is some $c>0$ such that for any $s\in
S(\bar{\mathbb{Q}})$ with $h(\tau(s))>c$, and any subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$, we have

$\\#\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq c^{1+r}.$

The idea of our proof is as follows. We first use the non-degeneracy result by
Gao [10, Theorem 10.1] and the height inequality developed by Dimitrov-Gao-
Habegger [8, Theorem 1.6] to find several families of curves in
$\mathcal{X}\rightarrow S$, over possibly different bases, with one marked
family over $S$, satisfying the following: for any point $P$ of $\mathcal{X}$
not on the marked family of curves, all of the points that are ”close” to $P$
live on a single curve among these families of curves (see Lemma 6.1 for
details). Then we prove and use the following Theorem 1.3 to bound the number
of points on those families of curves. Finally we combine it with a classical
result of Rémond for large points to get the desired bound on
$\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ for any $s\in
S(\bar{\mathbb{Q}})$ with large modular height.

######  Theorem 1.3.

Let $S$ be a variety over $\bar{\mathbb{Q}}$ and $\mathcal{A}\rightarrow S$ be
a family of abelian varieties. Let $\mathcal{C}\rightarrow S$ be a family of
GeM curves in them. Then there exists $c>0$ such that for any $s\in
S(\bar{\mathbb{Q}})$ and any subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$,

$\\#\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq c^{1+r}.$

Theorem 1.3 itself is of interest to us. Compared with [11, Theorem 1.1] or
[20, Theorem 4], it is also valid for the case when the ambient abelian
varieties are not the Jacobians of the curves. Also, the curves can be
singular or reducible.

The main new ingredient in the proof of Theorem 1.3 is the use of Hilbert
scheme to construct non-degenerate subvarieties of a family of abelian
varieties which could, for example, even be a constant family of abelian
varieties (see §3). This is motivated by the work of Hrushovski [18, Lemma
1.3.2] and Scanlon [28] on automatic uniformity, and the Betti rank formula by
Gao [10, Theorem 10.1].

From the proof of Theorem 1.3, we see that the difficulty in generalizing its
proof to higher dimensions falls in two parts. One is the need of uniform
Bogmolov conjecture for higher dimensions. The other is that we are no longer
able to compare distance between points with the base height, cf. the note
after its proof.

In a joint project [12] with Gao and Kühne, we will prove the uniform Mordell-
Lang conjecture for any subvariety. The non-degeneracy construction using
Hilbert schemes in §3 will be used. Compared to the current paper, some finer
properties of Hilbert schemes are also required. As a consequence of this
result, the constant $c$ in Theorem 1.1 can be taken independent of the degree
$d$ of the number field.

##  2\. Notation

We work over characteristic $0$. The field of definition is
$\bar{\mathbb{Q}}$. A variety is a reduced separated scheme of finite type
over $\bar{\mathbb{Q}}$. A subvariety of a variety is a closed reduced
subscheme. A variety $X$ is said to be defined over a number field $F$, if
$X=X_{0}\times_{\operatorname{Spec}F}\operatorname{Spec}\bar{\mathbb{Q}}$ for
some $X_{0}$ over $F$.

A curve is a one-dimensional variety. A family of curves is denoted by
$\mathcal{C}\rightarrow S$ for some variety $S$. A family of $2$-dimensional
varieties is denoted by $\mathcal{X}\rightarrow S$. Whenever we say a family,
the projection map is always assumed to be surjective. We do not require a
family to be flat, although we will reduce to the flat case in the proofs.

For any family $U\rightarrow H$, denote by $U^{n}=U_{H}^{n}$ the $n$-th fiber
product of $U$ over $H$. Since our fiber products for families are always over
the corresponding base, we sometimes leave out the subscript.

For a projective morphism $Y\rightarrow B$, denote by
$\operatorname{Hilb}(Y/B)$ its relative Hilbert scheme, cf. [15] and [23]. It
is a fine moduli space which represents the functor that sends any scheme
$B^{\prime}$ over $B$ to the set of closed subschemes of
$Y\times_{B}B^{\prime}$ that are flat over $B^{\prime}$.

Let $\mathcal{A}\rightarrow S$ be a family of principally polarized abelian
varieties of dimension $g$. Fix some $l\geq 3$. A level $l$ structure on
$\mathcal{A}\rightarrow S$ consists of $2g$ sections
$\sigma_{1},...,\sigma_{2g}:S\rightarrow\mathcal{A}$, such that
$\sigma_{1}(s),...,\sigma_{2g}(s)$ form a basis of the $l$-torsion subgroup of
$\mathcal{A}_{s}$. The moduli space of principally polarized abelian varieties
of dimension $g$ with level $l$ structures is representable by a quasi-
projective variety denoted by $\mathbf{A}_{g}$, see [22, Theorem 7.9 and its
proof] for the existence. Let $\mathfrak{A}_{g}\rightarrow\mathbf{A}_{g}$ be
the universal family. If $\mathcal{A}\rightarrow S$ comes with a level $l$
structure, then there exist natural modular maps
$\iota:S\rightarrow\mathbf{A}_{g}$ and
$\iota:\mathcal{A}\rightarrow\mathfrak{A}_{g}$.

Fix a relatively ample symmetric line bundle $\mathcal{L}_{0}$ for the
universal family $\mathfrak{A}_{g}\rightarrow\mathbf{A}_{g}$ which defines
fiber-wise Néron-Tate heights $\hat{h}$ on the abelian varieties. Fix an ample
line bundle $M_{0}$ on $\mathbf{A}_{g}$ which defines a height function
$h:\mathbf{A}_{g}(\bar{\mathbb{Q}})\rightarrow\mathbb{R}$.

By the $n$-th Faltings-Zhang morphism $\mathcal{D}_{n}$ for
$\mathcal{A}\rightarrow S$, we mean

$\begin{split}\mathcal{D}_{n}:\quad\mathcal{A}^{n+1}&\longrightarrow\mathcal{A}^{n}\\\
(P_{0},...,P_{n})&\longmapsto(P_{1}-P_{0},...,P_{n}-P_{0}).\end{split}$

We say a subvariety $X$ of an abelian variety $A$ generates the abelian
variety if $X-X$ is not contained in any proper abelian subvarieties of $A$.

##  3\. Non-degneracy

The notion of non-degeneracy first appeared in the work of Habegger [16]. Here
we take the definition from [8, Definition 1.5]. For an abelian scheme
$\mathcal{A}\rightarrow S$ over $\bar{\mathbb{Q}}$ and a smooth point $s\in
S(\mathbb{C})$, there is an analytic neighborhood $\Delta$ of $s$ and a so-
called Betti map

$b_{\Delta}:\mathcal{A}_{\Delta}\rightarrow\mathbb{T}^{2g}$

where $b_{\Delta}$ is real analytic and $\mathbb{T}^{2g}$ is the real torus of
dimension $2g$, see [8, §2] for details. For an irreducible subvariety
$\mathcal{X}$ of $\mathcal{A}$ dominant over $S$, restrict the Betti map to
$\mathcal{X}^{\text{sm,an}}$, the analyticification of the smooth locus of
$\mathcal{X}$. The _generic Betti rank_ is defined to be the maximal rank of
$\mathrm{d}b_{\Delta}|_{\mathcal{X}^{\text{sm,an}}\cap\mathcal{A}_{\Delta}}$,
the differential of the restricted Betti map, see [10]. We say $\mathcal{X}$
is _non-degenerate_ if the generic Betti rank of $\mathcal{X}$ is equal to
$2\dim\mathcal{X}$.

In this section, we will use Gao’s non-degeneracy criterion [10, Theorem 10.1]
to construct a new type of non-degenerate varieties.

Let us start with the following obeservation:

######  Lemma 3.1.

Let $Y\rightarrow B$ be a projective morphism and
$H\rightarrow\operatorname{Hilb}(Y/B)$ be a quasi-finite map with $H$
irreducible. Let $U\rightarrow H$ be the corresponding flat family. Assume
$U\rightarrow H$ has reduced fibers. Then for large enough n, the map
$U^{n}_{H}\rightarrow Y^{n}_{B}$ is generically finite.

###### Proof.

We may assume $B=\operatorname{Spec}k$ is a point. If $H$ is a finite set,
there’s nothing to prove. Otherwise for any $h_{1},h_{2}\in H$ whose images in
$\operatorname{Hilb}(Y)$ are distinct, there is a point $P_{1}\in
U_{h_{1}}\setminus U_{h_{2}}$ since $U_{h_{1}}$ and $U_{h_{2}}$ have different
moduli and are reduced. Let $H_{1}:=\\{h\in H:P_{1}\in U_{h}\\}$ which is a
proper closed subset of $H$. As long as $H_{1}$ is not of dimension $0$, for
the subfamily $U_{1}\rightarrow H_{1}$, we can find another point $P_{2}$ such
that $H_{2}:=\\{h\in H:P_{1},P_{2}\in U_{h}\\}$ is a proper closed subset of
$H_{1}$. Keep going and eventually we can get some $n$ and $P_{1},...,P_{n}$
such that only finitely many fibers contain them.

This implies that for the morphism $U^{n}_{H}\rightarrow Y^{n}_{B}$, the
preimage of $(P_{1},...,P_{n})$ is a finite set. In particular,
$U_{n}^{H}\rightarrow Y^{n}_{B}$ is generically finite. ∎

######  Remark 3.2.

If $H\subseteq\operatorname{Hilb}(Y/B)$ is an immersion, then the same
argument shows that $U^{n}_{H}\rightarrow Y^{n}_{B}$ is moreover birational to
its image for large $n$.

######  Corollary 3.3.

Let $Y\rightarrow B$ be as above and $U\rightarrow H$ be a flat family of
subvarieties inside $Y_{H}=Y\times_{B}H\rightarrow H$, with $H$ irreducible.
Assume that the geometric generic fiber $U_{\bar{\eta}}$ is reduced. If the
induced map $H\rightarrow\operatorname{Hilb}(Y/B)$ is generically finite, then
there exists $n>0$ such that the map $U^{n}_{H}\rightarrow Y^{n}_{B}$ is
generically finite

###### Proof.

If $H\rightarrow\operatorname{Hilb}(Y/B)$ is generically finite, then there
exists a dense open subset $H^{\prime}\subseteq H$ such that
$H^{\prime}\rightarrow\operatorname{Hilb}(Y/B)$ is finite [29, Tag 02NV] and
that all fibers over points of $H^{\prime}$ are reduced [29, Tag 0574]. By
Lemma 3.1, there exists large enough $n$ such that
$(U^{\prime})^{n}_{H^{\prime}}\rightarrow Y^{n}_{B}$ is generically finite.
Since $(U^{\prime})^{n}_{H^{\prime}}\subseteq U^{n}_{H}$ is dense open,
$U^{n}_{H}\rightarrow Y^{n}_{B}$ is also generically finite. ∎

######  Proposition 3.4.

Let $\mathfrak{A}_{g}\rightarrow\mathbf{A}_{g}$ be the universal abelian
scheme. Let $S$ be an irreducible variety with a generically finite morphism
$S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathcal{A}_{g})$ and
$U\rightarrow S$ is the induced family inside
$\mathcal{A}=\mathfrak{A}_{g}\times_{\mathbf{A}_{g}}S\rightarrow S$. For the
geometric generic fiber $U_{\bar{\eta}}$, assume the following:

  1. (1)

$U_{\bar{\eta}}$ is integral of positive dimension;

  2. (2)

$U_{\bar{\eta}}$ is not contained in any proper subgroup of
$\mathcal{A}_{\bar{\eta}}$;

  3. (3)

$U_{\bar{\eta}}$ has finite stabilizer in $\mathcal{A}_{\bar{\eta}}$.

Then for $n$ large enough, $U^{n}\subseteq\mathcal{A}^{n}$ is non-degenerate.

###### Proof.

Note that $U\rightarrow S$ satisfies the assumptions of Gao’s Betti rank
formula [10, Theorem 10.1(i)]. So for each $t\geq 0$, the Betti rank of
$U^{n}$ is at least $2(\dim\iota(U^{n})-t)$ for all $n\geq\dim S-t$. But by
Lemma 3.3, if $n$ is large enough,

$\iota:U^{n}\rightarrow\mathfrak{A}_{g}^{n}$

is generically finite which implies $\dim\iota(U^{n})=\dim U^{n}$. Therefore,
letting $t=0$ and $n\geq\dim S$ large enough, the Betti rank of $U^{n}$ is
equal to $2\dim U^{n}$, that is, $U^{n}\subseteq\mathcal{A}^{n}$ is non-
degenerate. ∎

We remark that if the induced map $S\rightarrow\mathbf{A}_{g}$ for
$\mathcal{A}\rightarrow S$ is generically finite, then
$S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ is
generically finite for free.

Sometimes we also want to consider the non-degeneracy of the image of the
Faltings-Zhang morphism. Let $U\rightarrow T$ be a flat family of subvarieties
in the abelian scheme $\mathcal{A}_{T}=\mathcal{A}\times_{S}T\rightarrow T$.
Consider the following embedding of $U\times_{T}U$:

$\begin{split}U\times_{T}U&\longrightarrow\mathcal{A}_{U}=U\times_{T}\mathcal{A}_{T}\\\
(P,Q)&\longmapsto(P,Q-P).\end{split}$

Denote its image by $U_{1}$. It can be regarded as a family over $U$ whose
fiber over any point $P\in U_{t}$ is $U_{t}-P$. We have the following

######  Lemma 3.5.

Let $\mathcal{A}\rightarrow S$ be an abelian scheme and $U\rightarrow T$ be a
flat family of subvarieties in the abelian scheme $\mathcal{A}_{T}\rightarrow
T$. Let $U_{1}\rightarrow U$ be the family whose fiber over any point $P\in
U_{t}$ is $U_{t}-P$, as defined above. If the induced map
$U\rightarrow\operatorname{Hilb}(\mathcal{A}/S)$ for $U_{1}\rightarrow U$ is
generically finite, then for $n$ large enough, the map
$\mathcal{D}_{n}(U^{n+1}_{T})\rightarrow\mathcal{A}^{n}_{S}$ is generically
finite .

###### Proof.

Again we may assume $S$ is a point and write $A$ for $\mathcal{A}$. Write
$(P,t)$ for a point of $P\in U_{t}$. Note that

$\mathcal{D}_{n}(U_{T}^{n+1})=\\{(t,P_{1}-P_{0},...,P_{n}-P_{0}):P_{0},...,P_{n}\in
U_{t}\\}.$

The map $\mathcal{D}_{n}(U_{T}^{n+1})\rightarrow A^{n}$ is given by forgetting
the $t$-coordinte.

If $U\rightarrow\operatorname{Hilb}(\mathcal{A}/S)$ is generically finite,
then by Corollary 3.3, there exists $n>0$ large enough such that the map
$(U_{1})_{U}^{n}\rightarrow A^{n}$ is generically finite. Then there exists
$(a_{1},...,a_{n})\in A^{n}$ such that its preimage is nonempty and finite.
Note that a fiber for $(U_{1})^{n}_{U}$ over $(P,t)\in U$ is $(U_{t}-P)^{n}$.
That means, there are only finitely many $(P,t)$ such that
$(a_{1},...,a_{n})\in(U_{t}-P)^{n}$. Then in particular, there are only
finitely many $t$ such that $(a_{1},...,a_{n})$ is in the fiber of
$\mathcal{D}_{n}(U_{T}^{n+1})\rightarrow T$ over $t$. So
$\mathcal{D}_{n}(U_{T}^{n+1})\rightarrow A^{n}$ is generically finite. ∎

######  Proposition 3.6.

Let $\mathfrak{A}_{g}\rightarrow\mathbf{A}_{g}$ be the universal abelian
scheme. Let $S$ be an irreducible variety and $U\rightarrow S$ is a flat
family inside $\mathcal{A}=\mathfrak{A}_{g}\times_{\mathbf{A}_{g}}S\rightarrow
S$. For the geometric generic fiber $U_{\bar{\eta}}$, assume the following:

  1. (1)

$U_{\bar{\eta}}$ is integral of positive dimension;

  2. (2)

$U_{\bar{\eta}}$ is not contained in any proper subgroup of
$\mathcal{A}_{\bar{\eta}}$;

  3. (3)

$U_{\bar{\eta}}$ has finite stabilizer in $\mathcal{A}_{\bar{\eta}}$.

Let $U_{1}\rightarrow U$ be the family whose fiber over any point $P\in U_{t}$
is $U_{t}-P$, as above. Assume that the induced map
$U\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ for
$U_{1}\rightarrow U$ is generically finite. Then for $n$ large enough,
$\mathcal{D}_{n}(U^{n+1})\subseteq\mathcal{A}^{n}$ is non-degenerate.

###### Proof.

Applying Gao’s Betti rank formula [10, Theorem 10.1(ii)] with $t=0$, the Betti
rank of $\mathcal{D}_{n}(U^{n+1})$ is at least
$2\dim\mathcal{D}_{n}(\iota(U^{n+1}))=2\dim\iota(\mathcal{D}_{n}(U^{n+1}))$
for $n\geq\dim X$. But for large enough $n$,
$\dim\iota(\mathcal{D}_{n}(U^{n+1}))=\dim\mathcal{D}_{n}(U^{n+1})$ by our
assumption and Lemma 3.5. So
$\mathcal{D}_{n}(U^{n+1})\subseteq\mathcal{A}^{n}$ is non-degenerate. ∎

######  Remark 3.7.

The assumption on the finiteness of
$U_{1}\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ is a
fairly restrictive condition. It does not allow the family to have too many
translates of a same subvariety. For example, it excludes the case when
$U\rightarrow S$ is given by the corresponding $U_{1}^{\prime}\rightarrow
U^{\prime}$ for another family $U^{\prime}\rightarrow S^{\prime}$.

##  4\. Known Results

###  4.1. Classical results

In this subsection, we recall some classical results developed by Rémond.

######  Proposition 4.1 (Rémond, curve case).

Let $\mathcal{C}\rightarrow S$ be a family of GeM curves in an abelian scheme
$\mathcal{A}\rightarrow S$ with $S$ quasi-projective. Let $\mathcal{L}$ be a
relatively ample line bundle for $\mathcal{A}\rightarrow S$ and $M$ be an
ample line bundle on $S$, which induce Néron-Tate heights $\hat{h}$ on abelian
varieties and height $h$ on $S$ respectively. Then there exists a constant
$c>0$ such that for any $s\in S(\bar{\mathbb{Q}})$ and
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ a subgroup of finite rank
$r$, we have

$\\#\\{P\in\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma:\hat{h}(P)>c\max\\{1,h(s)\\}\\}\leq
c^{r}.$

###### Proof.

This is a generalization of [7, Corollary 2.4] with the similar proof. We may
assume that the relatively ample line bundle $\iota^{*}\mathcal{L}$ on
$\mathcal{A}\rightarrow S$ defines an embedding
$\mathcal{A}\hookrightarrow\mathbb{P}^{n}_{S}$ for some constant $n$. By the
formulation of [7, Corollary 2.3], for any
$\mathcal{C}_{s}\subseteq\mathcal{A}_{s}$, there exists
$c_{1}=c_{1}(n,\deg\mathcal{C}_{s})>0$ such that

$\\#\\{P\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\hat{h}(P)>c_{1}\max\\{1,h(\mathcal{C}_{s}),h_{1}=h_{1}(\mathcal{A}_{s},\mathcal{L}_{s}),c_{\text{NT}}=c_{\text{NT}}(\mathcal{A}_{s},\mathcal{L}_{s})\\}\\}\leq
c_{1}^{r}.$

Since for the family $\mathcal{C}\rightarrow S$, the degree
$\deg\mathcal{C}_{s}$ is bounded, we can take $c_{1}$ uniformly for all $s\in
S(\bar{\mathbb{Q}})$.

By the proof of [8, Proposition 8.1] and [8, Lemma 6.1(ii)], we have

$h(\mathcal{C}_{s}),h_{1}(\mathcal{A}_{s},\mathcal{L}_{s}),c_{\text{NT}}(\mathcal{A}_{s},\mathcal{L}_{s})\leq
c_{2}\max\\{1,h(s)\\}$

for some $c_{2}>0$, uniformly in $s\in S(\bar{\mathbb{Q}})$. The proposition
follows by take $c:=\max\\{c_{1},c_{2}\\}$. ∎

Take a very ample symmetric line bundle $L$ on an abelian variety $A$ with the
corresponding embedding $A\hookrightarrow\mathbb{P}^{n}$. Then $L$ defines a
Néron-Tate height $\hat{h}$ on $A$ and let $|P|:=\hat{h}(P)^{1/2}$ for $P\in
A(\bar{\mathbb{Q}})$. Let us recall the following explicit Vojta and Mumford’s
inequality of Rémond for dimension two, which are special cases of [26,
Théorème 1.2] and [25, Proposition 3.4], respectively (note that increasing
the constants appeared in them will only weaken the results).

######  Lemma 4.2 (Vojta’s inequality).

For any GeM subvariety $X\subseteq A$ of dimension $2$, there exists constants
$c_{1}=c(n,\deg(X))>0$ such that if $P_{1},P_{2},P_{3}\in X(\bar{\mathbb{Q}})$
satisfy for $i=1,2$,

$\begin{split}\langle P_{i},P_{i+1}\rangle&\geq(1-1/c_{1})|P_{i}||P_{i+1}|\\\
|P_{i+1}|&\geq c_{1}|P_{i}|\end{split}$

then $|P_{1}|^{2}\leq c_{1}\max\\{1,h(X),h_{1},c_{NT}\\}$.

######  Lemma 4.3 (Mumford’s inequality).

Assume that $(P,P_{1},P_{2})\in X^{3}(\bar{\mathbb{Q}})$ is an isolated point
in the fiber of the Faltings-Zhang morphism $\mathcal{D}_{2}:X^{3}\rightarrow
A^{2}$. There exists constant $c_{2}=c_{2}(n,\deg(X))$ such that if
$P,P_{1},P_{2}$ satisfy for $i=1,2$,

$\begin{split}\langle P,P_{i}\rangle&\geq(1-1/c_{2})|P||P_{i}|\\\
\big{|}|P|-|P_{i}|\big{|}&\geq\frac{1}{c_{2}}|P|\end{split}$

then $|P|^{2}\leq c_{2}\max\\{1,h(X),h_{1},c_{NT}\\}$.

To prove the following family version of Rémond’s result for dimension $2$, we
need to invoke Theorem 1.3, the proof of which has nothing to do with the
$2$-dimensional world. It is expected to some extent, as Rémond in his
original proof of [25, Proposition 3.3], uses the most general results for
lower dimension too.

######  Proposition 4.4 (Rémond, surface case).

Let $\mathcal{X}\rightarrow S$ be a family of GeM subvarieties of dimension
$2$ in an abelian scheme $\mathcal{A}\rightarrow S$ with $S$ quasi-projective.
Let $\mathcal{L}$ be a relatively ample line bundle for
$\mathcal{A}\rightarrow S$ and $M$ be an ample line bundle on $S$, which
induce Néron-Tate heights $\hat{h}$ on abelian varieties and height $h$ on $S$
respectively. Then there exists a constant $c>0$ such that for any $s\in
S(\bar{\mathbb{Q}})$ and $\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ a
subgroup of finite rank $r$, we have

$\\#\\{P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma:\hat{h}(P)>c\max\\{1,h(s)\\}\\}\leq
c^{r}.$

###### Proof.

Consider the following subvariety
$Y\subseteq\mathcal{X}\times_{S}\mathcal{A}\xrightarrow{p_{2}}\mathcal{A}$
defined by the property that any fiber over
$a\in\mathcal{A}_{s}(\bar{\mathbb{Q}})$ for $s\in S(\bar{\mathbb{Q}})$ is
$\mathcal{X}_{s}\cap(\mathcal{X}_{s}-a)$. Since $\mathcal{X}_{s}$ is GeM, the
stabilizer of $\mathcal{X}_{s}$ is finite. So for all but finitely many (say
$N$, uniformly in $s$) $a\in\mathcal{A}_{s}(\bar{\mathbb{Q}})$, the fiber of
$Y\rightarrow\mathcal{A}$ over $a$ is at most of dimension $1$. By the upper-
semicontinuity of fiber dimension [13, Corollaire 13.1.5], there exists a
dense open subset $V\subseteq\mathcal{A}$, such that the restricted family
$Y^{\prime}:=Y\times_{\mathcal{A}}V\rightarrow V$ is a family of GeM curves.
Applying Theorem 1.3 to this family of curves, there exists $c_{1}>0$ such
that for any $s\in S(\bar{\mathbb{Q}})$, and
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ a subgroup of finite rank
$r$ and any $a\in V_{s}(\bar{\mathbb{Q}})$,
$\\#Y^{\prime}_{a}(\bar{\mathbb{Q}})\cap\Gamma\leq c_{1}^{1+r}$. Let
$N^{\prime}$ be an upper bound for $\deg(\mathcal{X}_{s})$.

######  Lemma 4.5.

For any set $\Sigma$ of points in
$\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ with cardinality
$\\#\Sigma>N+(N^{\prime})^{2}\cdot c_{1}^{1+r}$ and
$P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})$, there exist $P_{1},P_{2}$ in this set
such that $(P,P_{1},P_{2})$ is an isolated point in the fiber of the Faltings-
Zhang morphism
$\mathcal{D}_{2}:\mathcal{X}_{s}^{3}\rightarrow\mathcal{A}_{s}^{2}$.

###### Proof.

Indeed, note that $(P,P_{1},P_{2})$ is isolated in the fiber over
$(P_{1}-P,P_{2}-P)$, if and only if the local dimension of
$(\mathcal{X}_{s}-P)\cap(\mathcal{X}_{s}-P_{1})\cap(\mathcal{X}_{s}-P_{2})$ at
the identity $0$ is zero. Since the stabilizer of $\mathcal{X}_{s}-P$ has
cardinality $N$, there are at most $N$ points $P_{1}$ such that
$(\mathcal{X}_{s}-P)\cap(\mathcal{X}_{s}-P_{1})=\mathcal{X}_{s}-P$. Pick
$P_{1}\in\Sigma$ such that
$\dim((\mathcal{X}_{s}-P)\cap(\mathcal{X}_{s}-P_{1}))\leq 1$. Assume $P_{2}$
is such that the local dimension of
$(\mathcal{X}_{s}-P)\cap(\mathcal{X}_{s}-P_{1})\cap(\mathcal{X}_{s}-P_{2})$ at
the identity $0$ is not zero. Then $\mathcal{X}_{s}-P_{2}$ must contain some
irreducible component $0\in C$ of dimension one of
$(\mathcal{X}_{s}-P)\cap(\mathcal{X}_{s}-P_{1})$. This implies that

$P_{2}\in\bigcap_{a\in C}(\mathcal{X}_{s}-a)=\mathcal{X}_{s}\cap\bigcap_{a\in
C\setminus 0}(\mathcal{X}_{s}-a).$

Note that the right hand side is contained in some
$\mathcal{X}_{s}\cap(\mathcal{X}_{s}-a)$ for some $a\in C(\bar{\mathbb{Q}})$
of dimension $1$, which is in our family $Y^{\prime}\rightarrow V$. So there
are at most $c_{1}^{1+r}$ points of such $P_{2}\in\Gamma$. On the other hand,
the number of such irreducible components $C$ is at most

$\sum_{C}\deg C\leq\deg(\mathcal{X}_{s})^{2}\leq(N^{\prime})^{2}.$

So for $(P,P_{1})$, there are at most $(N^{\prime})^{2}\cdot c_{1}^{1+r}$
choices of $P_{2}$ such that $(P,P_{1},P_{2})$ is not isolated in the fiber of
the Faltings-Zhang morphism. Hence the claim. ∎

Now we are ready to use Vojta and Mumford’s inequality. Since
$\deg\mathcal{X}_{s}$ and $n$ are bounded for the family
$\mathcal{X}\rightarrow S$, we could take a uniform $c_{2}>0$, such that both
Lemma 4.2 and Lemma 4.3 are satisfied.

Regard $\Gamma\otimes\mathbb{R}$ as a normed vector space of dimension $r$. By
[25, Corollaire 6.1], we could cover the vector space by at most
$(1+\sqrt{8c_{2}})^{r}$ closed cones, such that for any $x,y$ in a same cone,
we have

$\langle x,y\rangle\geq(1-1/c_{2})|x||y|.$

Inside a cone, consider all the points
$\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ whose Néron-Tate height is
greater than

$c_{2}\max\\{1,h(\mathcal{X}_{s}),h_{1}=h_{1}(s),c_{\text{NT}}=c_{\text{NT}}(s)\\}.$

We arrange them so that $|P_{0}|\leq|P_{1}|\leq...$. Then Mumford’s inequality
together with the claim tells us that for any number
$B>c_{2}\max\\{1,h(\mathcal{X}_{s}),h_{1},c_{\text{NT}}\\}$, there are at most
$N+(N^{\prime})^{2}\cdot c_{1}^{1+r}$ points $P_{j}$ with

$B\leq|P_{j}|\leq(1+1/c_{2})B$

.

Take $N_{1}$ to be an upper bound for $\log_{1+1/c_{2}}(c_{2})$ so that
$(1+1/c_{2})^{N_{1}}>c_{2}$. By above, letting

$N_{2}:=(N+(N^{\prime})^{2}\cdot c_{1}^{1+r})\cdot N_{1},$

we have $|P_{i+N_{2}}|\geq c_{2}|P_{i}|$ for any $i$. By Vojta’s inequality,
we get then there are at most $2N_{2}$ points in the cone, since otherwise we
could take the triple $(P_{0},P_{N_{2}},P_{2N_{2}})$ in Lemma 4.2 to get a
contradiction.

Hence overall, we have

$\begin{split}\\#\\{P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma:\hat{h}(P)>c_{2}\max\\{1,h(\mathcal{X}_{s}),h_{1},c_{\text{NT}}\\}\\}&\leq(1+\sqrt{8c_{2}})^{r}\cdot
2N_{2}\\\ &\leq c_{3}^{1+r}\end{split}$

for some $c_{3}>0$.

As in Proposition 4.1, we have

$h(\mathcal{X}_{s}),h_{1},c_{\text{NT}}\leq c_{4}\max\\{1,h(s)\\}.$

So take $c>0$ large enough we would have the proposition. ∎

###  4.2. Recent results

The first theorem we recall is the height inequality for a non-degenerate
subvariety, as proved in [8, Theorem 1.6 and Appendix B].

######  Theorem 4.6 (Height inequality).

Let $S$ be an irreducible quasi-projective variety and
$\pi:\mathcal{A}\rightarrow S$ be a family of abelian varieties defined over
$\bar{\mathbb{Q}}$ with Néron-Tate height
$\hat{h}=\hat{h}_{\mathcal{A},\mathcal{L}}$ on $\mathcal{A}$ associated to a
relatively ample symmetric line bundle $\mathcal{L}$ and height $h=h_{S,M}$ on
$S$ associated to an ample line bundle $M$. Let $X$ be an irreducible closed
subvariety of $\mathcal{A}$ that dominates $S$. Suppose $X$ is non-degenerate.
Then there exists constants $c_{1},c_{2}\geq 0$ and a Zariski dense open
subset $U$ of $X$ such that

$\hat{h}(P)\geq c_{1}h(s)-c_{2}\qquad\text{for all }P\in
U_{s}(\bar{\mathbb{Q}}).$

The other theorem is the Uniform Bogmolov conjecture for curves by Kühne [20,
Proposition 21]. For our purpose, we need the following version:

######  Theorem 4.7 (Uniform Bogmolov conjecture for curves).

Let $S$ be an irreducible algebraic variety over $\bar{\mathbb{Q}}$,
$\pi:\mathcal{A}\rightarrow S$ a family of principally polarized abelian
varieties with level $l$ structures, and $\mathcal{C}\rightarrow S$ a family
of irreducible GeM curves in $\mathcal{A}\rightarrow S$. Assume that for any
$s\in S(\bar{\mathbb{Q}})$, the curve $\mathcal{C}_{s}$ generates
$\mathcal{A}_{s}$. Then, there exist constants $c_{1},c_{2}>0$ such that

$\\#\\{x\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\hat{h}(x)\leq c_{1}\\}<c_{2},$

where the fiber-wise Néron-Tate heights are induced from those on the
universal family $\mathfrak{A}_{g}\rightarrow\mathbf{A}_{g}$, as defined in
§2.

###### Proof.

As our setup is slightly more general than [20, Proposition 21] and [11,
Theorem 8.3], we explain the new input to modify the proof in [11, Theorem
8.3].

As in the original case, we do induction on $\dim S$.

Note first that we may assume $\mathcal{C}\rightarrow S$ is flat by induction
hypothesis. So we have an induced morphism
$\phi:S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$. If
$\phi$ is not generically finite, then we use induction hypothesis and pull it
back. Note that pull-back works since our Néron-Tate heights are taken from
the universal family. So we can assume $\phi$ is generically finite.

To prove the results on $\mathcal{C}\rightarrow S$, we remark that the rest of
the proof in [11, Theorem 8.3] is valid except that the nondegeneracy of
$\mathcal{C}^{n}$ is provided by Proposition 3.4. ∎

##  5\. proof of theorem 1.3

###### Proof.

Induct on $\dim S$. We arrange the proof in a few steps.

Step 0: We make several reductions in this step. Note first the following
simple fact:

_If $S^{\prime}\rightarrow S$ is a dominant map, then the result on
$\mathcal{C}_{S^{\prime}}\subseteq\mathcal{A}_{S^{\prime}}$_ implies the
result on $\mathcal{C}\subseteq\mathcal{A}$.

Immediate consequences of the fact are that we may assume $S$ is _quasi-
projective_ by Chow’s lemma and the map $\mathcal{C}\rightarrow S$ is _flat_.

Without loss of generality, assume $\mathcal{C}$ and $S$ are irreducible. Then
the generic fiber $\mathcal{C}_{\eta}$ is irreducible, which may not be
geometrically irreducible. There is a finite extension of the function field
$k(S)$, which gives rise to a quasi-finite dominant map $T\rightarrow S$, such
that the generic fiber of the pullback $\mathcal{C}_{T}\rightarrow T$
decomposes into geometrically irreducible components. By the fact, we may work
on $\mathcal{C}_{T}\rightarrow T$ instead. Then we can work on each
irreducible component of $\mathcal{C}_{T}$. Let
$\mathcal{C}_{1}\subseteq\mathcal{C}_{T}$ be an irreducible component that is
dominant over $T$. Then the generic fiber of $\mathcal{C}_{1}$ is an
irreducible component of the generic fiber of $\mathcal{C}_{T}$. So the
generic fiber of $\mathcal{C}_{1}\rightarrow T$ is geometrically irreducible.
By [29, Tag 0553], there exists a dense open subset $T^{\prime}$ of $T$ such
that $\mathcal{C}_{1}\times_{T}T^{\prime}\rightarrow T^{\prime}$ has
geometrically irreducible fibers. Together with induction hypothesis, we
reduce to study such a family as
$\mathcal{C}_{1}\times_{T}T^{\prime}\rightarrow T^{\prime}$ and may assume
_every fiber is geometrically irreducible_.

We can assume _the curves pass through $0$_. Indeed, by possibly taking a
finite cover, which is justified by the fact above, we can assume there is a
point $\operatorname{Spec}k(S)\rightarrow\mathcal{C}_{\eta}$ on the generic
fiber. By spreading out the morphism, there is a dense open subset
$S^{\prime}$ of $S$ such that
$S^{\prime}\rightarrow\mathcal{C}^{\prime}:=\mathcal{C}\times_{S}S^{\prime}$
is a section. By the fact, it suffices to work on
$\mathcal{C}^{\prime}\rightarrow S^{\prime}$. Translating
$\mathcal{C}^{\prime}$ by the section, we get
$\mathcal{C}^{\prime\prime}\rightarrow S^{\prime}$. The result on
$\mathcal{C}^{\prime\prime}\rightarrow S^{\prime}$ implies the result on
$\mathcal{C}^{\prime}\rightarrow S^{\prime}$, since
$\mathcal{C}^{\prime}_{s}(\bar{\mathbb{Q}})\cap\Gamma\hookrightarrow(\mathcal{C}^{\prime}_{s}(\bar{\mathbb{Q}})-P)\cap\langle\Gamma,P\rangle$,
where $\langle\Gamma,P\rangle$ is the group generated by $\Gamma$ and $P$,
which has rank at most $1+\operatorname{rank}\Gamma$.

We may assume _the curves generate the ambient abelian varieties_. Indeed, if
the generic curve $\mathcal{C}_{\eta}$ generates $\mathcal{A}_{\eta}$, then
for some $n$ large enough, the addition map
$+:(\mathcal{C}-\mathcal{C})^{n}\rightarrow\mathcal{A}$ is surjective over the
generic point of $S$. By a spreading out result [13, Théorème 8.10.5(vi)],
there is a nonempty open subset $S^{\prime}$ of $S$ over which the addition
map is surjective. Namely, the curves over $s\in S^{\prime}$ generate the
ambient abelian varieties. By induction we are good in this case. If the
generic curve $C_{\eta}$ is contained in a proper coset $P+\mathcal{B}_{\eta}$
of $\mathcal{A}_{\eta}$, since $0\in C_{\eta}$ by our assumption in the last
paragraph, we could take $P=0$. Here $B_{\eta}$ is probably only defined over
a finite extension of $k(S)$. Then by spreading out $\mathcal{B}_{\eta}$,
there is a quasi-finite dominant map $S^{\prime}\rightarrow S$, such that
$\mathcal{C}_{s}$ generates $\mathcal{B}_{s}$ for any $s\in S^{\prime}$. By
the fact, we may assume $S^{\prime}=S$. Assume we have the result for
$\mathcal{C}\subseteq\mathcal{B}\rightarrow S$, namely there is $c>0$ such
that for any $s\in S(\bar{\mathbb{Q}})$ and any abelian subgroup $\Gamma$ of
$\mathcal{B}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$, we have
$\\#\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq c^{1+r}$. Then the result
for $\mathcal{C}\subseteq\mathcal{A}\rightarrow S$ will be an immediate
corollary because for any finite rank subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$, the intersection
$\Gamma\cap\mathcal{B}_{s}(\bar{\mathbb{Q}})$ is an abelian subgroup of
possibly lower rank, which means the same $c>0$ would work for
$\mathcal{C}\subseteq\mathcal{A}$.

Next, we will add polarizations and level structures to the abelian varieties.
Let $(A^{\prime},L^{\prime})$ be a principally polarized abelian variety
isogenous to the generic abelian variety $A_{\eta}$. By spreading out
$(A^{\prime},L^{\prime})$ and the morphism, there is a principally polarized
abelian scheme $(\mathcal{A}^{\prime},\mathcal{L}^{\prime})$ over a dense open
subset $S^{\prime}$ of $S$ with an isogeny
$\mathcal{A}^{\prime}\rightarrow\mathcal{A}\times_{S}S^{\prime}$. Let
$\mathcal{C}^{\prime}$ be the pullback of $\mathcal{C}$ to
$\mathcal{A}^{\prime}$. Then the result on
$\mathcal{C}^{\prime}\subseteq\mathcal{A}^{\prime}$ would imply the result on
$\mathcal{C}\subseteq\mathcal{A}$ over $S^{\prime}$. In particular, we may
assume _$\mathcal{A}\rightarrow S$ is a family of principally polarized
abelian varieties_.

For level structures, note that there is a quasi-finite dominant map
$S^{\prime}\rightarrow S$ such that the $l$-torsion points of the generic
abelian variety $\mathcal{A}_{\eta}$ are all defined over $k(S^{\prime})$ and
the torsion points spread out to distinct torsion sections over $S^{\prime}$.
Then we may impose the level $l$ structures on
$\mathcal{A}\times_{S}S^{\prime}$. By the fact, we thus assume
_$\mathcal{A}\rightarrow S$ is a family of principally polarized abelian
varieties with level $l$ structures_.

Overall, we can assume the following:

  1. (1)

$S$ is an irreducible quasi-projective variety over $\bar{\mathbb{Q}}$;

  2. (2)

$\mathcal{A}\rightarrow S$ is a family of principally polarized abelian
varieties with level $l$ structures;

  3. (3)

$\mathcal{C}\rightarrow S$ is a flat family of irreducible curves passing
through $0$ in $\mathcal{A}\rightarrow S$, such that every curve
$\mathcal{C}_{s}$ generates $\mathcal{A}_{s}$.

In particular, there is an ample line bundle $M$ defining height
$h:S(\bar{\mathbb{Q}})\rightarrow\mathbb{R}$ on $S$ by (1), a natural map from
$S$ to the moduli space $\mathbf{A}_{g}$ of principally polarized abelian
varieties with level $l$ structures by (2), and a natural map from $S$ to the
Hilbert scheme $\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ by (3).
The map $S\rightarrow\mathbf{A}_{g}$ induces fiber-wise Néron-Tate heights on
$\mathcal{A}\rightarrow S$ by pulling back from the universal family.

Step 1: In this step, we will demonstrate how to apply the height inequality
4.6 to the non-degeneracy result from §3 to study the distance of geometric
points on the curves.

By Step $0$, if
$\psi:S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$
induced from the family $\mathcal{C}\rightarrow S$ is not generically finite,
then the image of $S$ has lower dimension. In this case we can simply use
induction on the image family and pull it back. So we may assume
$\psi:S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ is
generically finite.

Consider the following embedding of $\mathcal{C}\times_{S}\mathcal{C}$:

$\begin{split}\mathcal{C}\times_{S}\mathcal{C}&\longrightarrow\mathcal{A}_{\mathcal{C}}=\mathcal{C}\times_{S}\mathcal{A}\\\
(P,Q)&\longmapsto(P,Q-P).\end{split}$

Denote its image by $\mathcal{C}_{1}$. Then the projection onto the first
factor is a family of curves $\mathcal{C}_{1}\rightarrow\mathcal{C}$ inside
$\mathcal{A}_{\mathcal{C}}$. Note that $\mathcal{C}\rightarrow S$ is a
subfamily of $\mathcal{C}_{1}\rightarrow\mathcal{C}$ due to our assumption
that $0\in\mathcal{C}_{s}(\bar{\mathbb{Q}})$. By the universal property of
Hilbert schemes, there is a natural map
$\phi:\mathcal{C}\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$.
Then $\psi(S)\subseteq\phi(\mathcal{C})$. Depending on whether $\phi$ is
generically finite or not, we apply Theorem 4.6 in two ways.

Take the Néron-Tate heights as in §2. Fix an ample line bundle $M$ on $S$ and
denote a height associated to $M$ by $h$.

If $\phi$ is generically finite, we use the Faltings-Zhang morphism to study
the distance. By Proposition 3.6, for $n$ large enough,
$\mathcal{D}_{n}(\mathcal{C}^{n+1})\subseteq\mathcal{A}^{n}$ is non-
degenerate. By Theorem 4.6, there exist $c_{1},c_{2}>0$ and a dense open
subset $V$ of $\mathcal{D}_{n}(\mathcal{C}^{n+1})\subseteq\mathcal{A}^{n}$
such that

$\hat{h}(P_{1}-P_{0})+...+\hat{h}(P_{n}-P_{0})\geq n\cdot(c_{1}h(s)-c_{2})$

for any $s\in S(\bar{\mathbb{Q}})$ and $(P_{1}-P_{0},...,P_{n}-P_{0})\in
V_{s}(\bar{\mathbb{Q}})$.

If $\phi$ is not generically finite, then the scheme-theoretic images of
$\psi:S\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$ and
$\phi:\mathcal{C}\rightarrow\operatorname{Hilb}(\mathfrak{A}_{g}/\mathbf{A}_{g})$
are the same, since $\psi(S)\subseteq\phi(\mathcal{C})$ and they are both
irreducible of the same dimension. By Proposition 3.4, for $n$ large enough,
$\mathcal{C}^{n}\subseteq\mathcal{A}^{n}$ is non-degenerate. By Theorem 4.6,
there exist (by abuse of notation) $c_{1},c_{2}>0$ and a dense open subset
$V^{\prime}\subseteq\mathcal{C}^{n}$ such that

(5.0.1) $\hat{h}(P_{1})+...+\hat{h}(P_{n})\geq n\cdot(c_{1}h(s)-c_{2})$

for any $s\in S(\bar{\mathbb{Q}})$ and $(P_{1},...,P_{n})\in
V_{s}^{\prime}(\bar{\mathbb{Q}})$.

Step 2: For the rest, let us say two points
$P_{1},P_{2}\in\mathcal{A}_{s}(\bar{\mathbb{Q}})$ are _close_ if
$\hat{h}(P_{1}-P_{2})<c_{1}h(s)-c_{2}$.

Define $E_{n}:=\\{(P,Q_{1},...,Q_{n})\in\mathcal{C}^{n+1}:Q_{i}\text{ is close
to }P\text{ for any }i\\}$ and $F_{n}:=\bar{E}$. Claim: $F_{n}$ is a proper
subset of $\mathcal{C}^{n+1}$. Indeed, it is clear for the first case where
$\phi$ is generically finite. For the other case, since
$\mathcal{C}_{1}\rightarrow\mathcal{C}$ and $\mathcal{C}\rightarrow S$ are
pullbacks of the same family, $V^{\prime}$ induces an open dense subset
$V^{\prime\prime}$ of $(\mathcal{C}_{1})_{\mathcal{C}}^{n}$ with the same
height inequality as in (5.0.1). Then it is clear that $E_{n}$ is not dense in
$\mathcal{C}^{n+1}$ through the following identification

$\begin{split}\mathcal{C}^{n+1}&\longrightarrow(\mathcal{C}_{1})_{\mathcal{C}}^{n}\\\
(P,Q_{1},...,Q_{n})&\longmapsto(Q_{1}-P,...,Q_{n}-P).\end{split}$

Let $V_{n}:=\mathcal{C}^{n+1}\setminus F$. The image of $V_{n}$ in $S$, say
$S_{0}$, is open by flatness. We can use induction hypothesis on $S\setminus
S_{0}$. So without loss of generality, assume the projection $V_{n}\rightarrow
S$ is surjective.

######  Lemma 5.1.

There exists $N_{1},N_{2}>0$ such that for any $s\in S(\bar{\mathbb{Q}})$

$\\#\\{P\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\text{there are more than
}N_{1}\text{ points close to }P\\}\leq N_{2}$

###### Proof.

This is essentially what proved in [8, Proposition 7.1]. Here we give an
alternative proof using an idea from the proof of [4, Lemma 1.1].

Define $F_{i}$ inductively for $0\leq i\leq n-1$ as follows. Let $F_{i}$ be
the largest closed subset of $\mathcal{C}^{i+1}$ such that
$\pi_{i+1}^{-1}(F_{i})\subseteq F_{i+1}$ where
$\pi_{i+1}:\mathcal{C}^{i+2}\rightarrow\mathcal{C}^{i+1}$ is the projection
leaving out the last component. In other words, $F_{i}$ is the set of points
over which fibers are positive dimensional. It is closed due to upper semi-
continuity of fiber dimension [13, Corollaire 13.1.5] and properness of the
map $F_{i}\rightarrow\mathcal{C}^{i}$. Let $V_{i}:=\mathcal{C}^{i+1}\setminus
F_{i}$. Then $F_{i+1}\xrightarrow{\pi_{i+1}}\mathcal{C}^{i+1}$ is finite over
$V_{i}$. Let $d_{i}$ be an upper bound on the cardinality of the fibers. Let
$N_{1}$ be the maximum of all $d_{i}$’s.

Since $V_{n}$ surjects to $S$ by assumption above this lemma, $F_{i}$ is
proper subset of $\mathcal{C}^{i+1}$, even fiber-wise over any $s\in S$. In
particular, $F_{0}\subseteq\mathcal{C}$ maps finitely to $S$. Denote its
degree by $N_{2}$.

Then $N_{1},N_{2}$ satisfy the requirement. Indeed, we will show that if
$P\in(V_{0})_{s}(\bar{\mathbb{Q}})$, then there are at most $N_{1}$ points
close to $P$. Let $j(P)\geq 1$ be the smallest integer $j$ such that all
points $(P,Q_{1},...,Q_{j})\in\mathcal{C}^{j+1}_{s}(\bar{\mathbb{Q}})$ with
all $Q_{i}$’s close to $P$ is contained in $F_{j}$. By definition of $j(P)$,
there is a point $(P,Q_{1},...,Q_{j(P)-1})$ with all $Q_{i}$’s close to $P$ in
$V_{j(P)-1}$. But any prolongation $(P,Q_{1},...,Q_{j(P)-1},Q)$ with $Q$ close
to $P$ is in $F_{j(P)}$. Since the fiber of $F_{j(P)}$ over a point of
$V_{j(P)-1}$ is finite of cardinality bounded by $N_{2}$, the number of such
$Q$’s that are close to $P$ is bounded by $N_{1}$. ∎

Step 3: Let $\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ be a subgroup
of finite rank $r$ with $s\in S(\bar{\mathbb{Q}})$.

For $s$ such that $h(s)\geq\max\\{1,\frac{2c_{2}}{c_{1}}\\}$, we use above and
Rémond’s result to get the desired bound on
$\mathcal{A}_{s}(\bar{\mathbb{Q}})\cap\Gamma$. Note that if two points $P,Q$
in $\mathcal{X}_{s}$ satisfies $\hat{h}(P-Q)\leq\frac{c_{1}}{2}h(s)$, then
they are close. By Proposition 4.1, there exists $c_{3}>0$, such that the
number of large points in $\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ whose
Néron-Tate heights are greater than $c_{3}\max\\{1,h(s)\\}=c_{3}h(s)$ is at
most $c_{3}^{r}$. So it suffices to give a bound on the number of points
$P\in\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ with $\hat{h}(P)\leq
c_{3}h(s)$. Regard $\Gamma\otimes\mathbb{R}$ as a normed vector space with
norm given by $|\cdot|:=\hat{h}^{1/2}$. Let $R_{1}:=\sqrt{c_{3}h(s)}$ and
$R_{2}:=\sqrt{\frac{c_{1}}{2}h(s)}/2$. The closed ball of radius $R_{1}$
centered at $0$ is covered by at most
$(1+2R_{1}/R_{2})^{r}=(1+4\sqrt{2c_{3}/c_{1}})^{r}$ closed balls of radius
$R_{2}$ by [25, Lemme 6.1]. In any closed ball of radius $R_{2}$, any two
points are close. So by Lemma 5.1, there are at most $N_{1}$ in one closed
ball of radius $R_{2}$, with an overall exception of $N_{2}$ points. Thus
overall we have

$\\#\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq
c_{3}^{r}+N_{1}\cdot(1+4\sqrt{2c_{3}/c_{1}})^{r}+N_{2}\leq c^{1+r}$

for some large $c>0$.

Step 4: For $s$ such that $h(s)<\max\\{1,\frac{2c_{2}}{c_{1}}\\}$, we use
Kühne’s result and Rémond’s result to get the desired bound on
$\mathcal{A}_{s}(\bar{\mathbb{Q}})\cap\Gamma$. Write $c_{4}$ for an upper
bound of $c_{3}\max\\{1,h(s)\\}$. Rémond’s result implies that the number of
large points with Néron-Tate heights greater than $c_{4}$ is at most
$c_{3}^{r}$. To bound the small points, we use Theorem 4.7 for the family
$\mathcal{C}_{1}\rightarrow\mathcal{C}$. Then there exists $c_{5},c_{6}>0$
such that for any $P\in\mathcal{C}_{s}(\bar{\mathbb{Q}})$,

$\\#\\{Q\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\hat{h}(Q-P)\leq c_{5}\\}<c_{6}.$

This immediately tells us that a closed ball of radius $\sqrt{c_{5}}/2$
contains less than $c_{6}$ points. Now we cover the large ball of radius
$\sqrt{c_{4}}$ in $\Gamma\otimes\mathbb{R}$ by $(1+2\sqrt{c_{4}/c_{5}})^{r}$
small balls of radius $\sqrt{c_{5}}/2$. Similar argument as above then yields

$\\#\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq
c_{3}^{r}+c_{6}\cdot(1+2\sqrt{c_{4}/c_{5}})^{r}\leq c^{1+r}$

for some large $c>0$.

∎

Note that for higher dimensional analogue of Theorem 1.3, there would be an
intermediate case to deal with, namely the image of $\phi$ could have
dimension $\dim S+1$, which may cause trouble.

######  Remark 5.2.

A different proof to Theorem 1.3 discovered by Gao is sketched in the
Appendix. The idea is to deduce the so-called New Gap Principle for curves in
arbitrary abelian varieties from the New Gap Principle for smooth curves in
their Jacobians, cf. [11, Theorem 4.1]. The proof requires a careful
comparison of the involved heights. Also we need a stronger version of
Rémond’s result than Proposition 4.1 with $h(s)$ replaced by the moduli height
of the ambient abelian variety.

##  6\. Proof of Theorem 1.2

###### Proof.

Induct on $\dim S$.

By [10, Theorem 1.3], for $m$ large enough,
$\mathcal{D}_{m}(\mathcal{X}^{m+1})\subseteq\mathcal{A}^{m}$ is nondegenerate.
By Theorem 4.6, there is a dense open subset $U$ of
$\mathcal{D}_{m}(\mathcal{X}^{m+1})$ with constants $c_{1},c_{2}>0$, such that
for any $(Q_{1}-P,...,Q_{m}-P)$ in the dense open $U(\bar{\mathbb{Q}})$, we
have

(6.0.1) $\sum_{i}\hat{h}(Q_{i}-P)\geq m\cdot(c_{1}h(\tau(s))-c_{2}).$

Again, we say two points $P,Q$ in a fiber of $\mathcal{X}\rightarrow S$ are
close if $\hat{h}(P-Q)<c_{1}h(s)-c_{2}$. By induction hypothesis, we may
moreover assume $U$ surjects into $S$.

######  Lemma 6.1.

There exist $m$ $S$-schemes $S_{1}=S,S_{2},...,S_{m}$ (not necessarily
irreducible) and $m$ families of curves
$\mathcal{C}_{i}\subseteq\mathcal{X}_{S_{i}}\rightarrow S_{i}$ such that for
any $P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})$, either
$P\in(\mathcal{C}_{1})_{s}(\bar{\mathbb{Q}})$, or the points that are close to
$P$ are on a curve $C=(\mathcal{C}_{i})_{s^{\prime}}$ for some $2\leq i\leq m$
and $s^{\prime}\in S_{i}(\bar{\mathbb{Q}})$ over $s\in S(\bar{\mathbb{Q}})$.

###### Proof.

The strategy is similar to our proof of Lemma 5.1. By (6.0.1), there is a
proper closed subset $F_{m}$ of $\mathcal{X}^{m+1}$ such that if
$(P,Q_{1},...,Q_{m})\in\mathcal{X}^{m+1}(\bar{\mathbb{Q}})$ is such that all
$Q_{1},...,Q_{n}$’s are close to $P$, then $(P,Q_{1},...,Q_{m})\in F_{m}$.

Define $F_{i}$ inductively for $0\leq i\leq m-1$ as follows. Let $F_{i}$ be
the largest closed subset of $\mathcal{X}^{i+1}$ such that
$\pi_{i+1}^{-1}(F_{i})\subseteq F_{i+1}$ where
$\pi_{i+1}:\mathcal{X}^{i+2}\rightarrow\mathcal{X}^{i+1}$ is the projection
leaving out the last component. In other words, $F_{i}$ is the set of points
over which fibers are $2$ dimensional, which is closed due to upper semi-
continuity of fiber dimension [13, Corollaire 13.1.5]. Let
$V_{i}:=\mathcal{X}^{i+1}\setminus F_{i}$. Then by definition
$F_{i+1}\xrightarrow{\pi_{i+1}}\mathcal{X}^{i+1}$ is at most a curve when
restricted to $V_{i}$.

Since $V_{m}$ surjects to $S$ by assumption above this lemma, $F_{i}$ is a
proper subset of $\mathcal{C}^{i+1}$, even fiber-wise over any $s\in S$. In
particular, $F_{0}\subseteq\mathcal{X}$ is a family of (at most) curves over
$S$.

Let $\mathcal{C}_{1}\rightarrow S_{1}$ be $F_{0}\rightarrow S$. Let
$\mathcal{C}_{i}\rightarrow S_{i}$ be $F_{i}|_{V_{i}}\rightarrow V_{i-1}$ for
$i\geq 2$.

If $P\notin\mathcal{C}_{0}$, then let $j(P)\geq 1$ be the smallest integer $j$
such that all points
$(P,Q_{1},...,Q_{j})\in\mathcal{X}^{j+1}_{s}(\bar{\mathbb{Q}})$ with $Q_{i}$’s
close to $P$ is contained in $F_{j}$. By definition of $j(P)$, there is a
point $(P,Q_{1},...,Q_{j(P)-1})$ with all $Q_{i}$’s close to $P$ in
$V_{j(P)-1}$. Any prolongation $(P,Q_{1},...,Q_{j(P)-1},Q)$ with $Q$ close to
$P$ is in $F_{j(P)}$. The fiber of $F_{j(P)}$ over a point of $V_{j(P)-1}$ is
a curve in the family $\mathcal{C}_{j(P)}\rightarrow S_{j(P)}$. So all points
that are close to $P$ must live in this curve. ∎

Since all $\mathcal{X}_{s}$ are GeM, curves appeared in the families are GeM.
By applying Theorem 1.3 to these families of curves, there exists $c_{3}>0$
such that for any $s\in S(\bar{\mathbb{Q}})$ and a subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$, and for
any $s^{\prime}\in S_{i}(\bar{\mathbb{Q}})$ over $s$, we have
$\Gamma\cap(\mathcal{C}_{i})_{s^{\prime}}(\bar{\mathbb{Q}})\leq c_{3}^{1+r}$.

Now if we take $s\in S(\bar{\mathbb{Q}})$ such that
$h(s)>\max\\{1,\frac{2c_{2}}{c_{1}}\\}$, then
$\frac{c_{1}}{2}h(s)<c_{1}h(s)-c_{2}$. This means, if two points $P,Q$ in
$\mathcal{X}_{s}$ satisfies $\hat{h}(P-Q)\leq\frac{c_{1}}{2}h(s)$, then they
are close. Consider a subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$. By
Proposition 4.4, there exists a constant $c_{4}>0$ such that the number of
large points $P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ with
$\hat{h}(P)>c_{4}\max\\{1,h(s)\\}=c_{4}h(s)$ is at most $c_{4}^{n}$. So it
suffices to give a bound on the number of points
$P\in\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma$ with $\hat{h}(P)\leq
c_{4}h(s)$.

To do this, we use a classical argument by regarding $\Gamma\otimes\mathbb{R}$
as a normed vector space of rank $r$ with norm given by
$|\cdot|=\hat{h}^{1/2}$. Let $R_{1}:=\sqrt{c_{4}h(s)}$ and
$R_{2}:=\sqrt{\frac{c_{1}}{2}h(s)}/2$. The closed ball of radius $R_{1}$
centered at $0$ is covered by at most
$(1+2R_{1}/R_{2})^{r}=(1+4\sqrt{2c_{4}/c_{1}})^{r}$ closed balls of radius
$R_{2}$ by [25, Lemme 6.1]. In any closed ball of radius $R_{2}$, we have two
possibilities. One is that it contains only points in
$(\mathcal{C}_{1})_{s}(\bar{\mathbb{Q}})\cap\Gamma$, which is fine since we
know its number is bounded by $c_{3}^{1+r}$. The other is the opposite: it
contains some point $P$ outside
$(\mathcal{C}_{1})_{s}(\bar{\mathbb{Q}})\cap\Gamma$. But by our choice of
$R_{2}$, all other points in the ball are close to $P$. So by Lemma 6.1, they
must belong to a curve $(C_{i})_{s^{\prime}}$ for some $2\leq i\leq m$ and
$s^{\prime}\in S_{i}(\bar{\mathbb{Q}})$ over $s\in S(\bar{\mathbb{Q}})$. Since
$\\#(C_{i})_{s^{\prime}}(\bar{\mathbb{Q}})\cap\Gamma\leq c_{3}^{1+r}$, the
number of points in the closed ball of radius $R_{2}$ is at most
$c_{3}^{1+r}$. In any case, a closed ball of radius $R_{2}$ contains at most
$c_{3}^{1+r}$ points. So the ball of radius $R_{1}$ contains at most

$c_{3}^{1+r}\cdot(1+4\sqrt{2c_{4}/c_{1}})^{r}$

points. Overall, we have

$\\#\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq
c_{4}^{r}+c_{3}^{1+r}\cdot(1+4\sqrt{2c_{4}/c_{1}})^{r}\leq c^{1+r}$

for some large $c>0$. ∎

######  Remark 6.2.

Lemma 5.1 and Lemma 6.1 have a similar flavor and can be generalized to higher
dimension.

##  7\. Application

Consider a smooth irreducible curve $C$ of genus $g\geq 3$ over a number field
$F$. Let $\operatorname{Jac}(C)$ be its Jacobian variety. Suppose there is a
quadratic point $P\in C(F^{\prime})$ for some quadratic extension
$[F^{\prime}:F]=2$ and let $P^{\prime}$ be its Galois conjugate. The symmetric
product $C^{(2)}:=\frac{C\times C}{S_{2}}$ corresponds to divisors of degree
$2$ on $C$. It is well-known that the map

$\begin{split}\mathcal{C}^{(2)}&\longrightarrow\operatorname{Jac}(C)\\\
(Q_{1},Q_{2})&\longmapsto[Q_{1}+Q_{2}-P-P^{\prime}]\end{split}$

is an embedding if and only if $C$ is not hyperelliptic. Let $W_{2}(C)$ be its
image. It is shown in [17, Theorem 2] that $W_{2}(C)$ contains no curves of
genus $1$ if $C$ is neither hyperelliptic nor bielliptic. Therefore Faltings’
Theorem [9] tells us that there are only finitely many rational points on
$W_{2}(C)$ which implies the same for $C^{(2)}$. Note that there is an
bijection between the set of rational points of $C^{(2)}$ and the set of
quadratic points $C(F,2)$ of $C$ modulo Galois action, see [17, Corollary 3]
for details. Therefore $\\#C(F,2)$ is finite in this case.

Now consider the moduli space $\mathcal{M}_{g}$ of smooth irreducible curves
of genus $g$ over $\bar{\mathbb{Q}}$ with level $l$ structures. It is
representable by an irreducible quasi-projective variety over
$\bar{\mathbb{Q}}$, see for example [2, XVI.2]. It is well-known that the
locus of hyperelliptic curves of genus $g$ forms a $(2g-1)$-dimensional
irreducible closed subvariety $\mathcal{H}_{g}$ in $\mathcal{M}_{g}$, see [2,
XIII.8]. The locus of bielliptic curves of genus $g$ forms a
$(2g-2)$-dimensional irreducible closed subvariety
$\mathcal{M}_{g}^{\mathrm{be}}$ in $\mathcal{M}_{g}$, see [3]. We are
interested in the complement of these two loci and define
$\mathcal{M}_{g}^{\circ}:=\mathcal{M}_{g}\setminus(\mathcal{H}_{g}\cup\mathcal{M}_{g})$,
which is an open subset of dimension $3g-3$ in $\mathcal{M}_{g}$.

Let $\mathcal{C}_{g}^{\circ}\rightarrow\mathcal{M}_{g}^{\circ}$ be the
universal non-hyperelliptic non-bielliptic curve with level $l$ structures.
Let
$\mathcal{J}_{g}^{\circ}:=\operatorname{Jac}(\mathcal{C}_{g}^{\circ}/\mathcal{M}_{g}^{\circ})$
be the relative Jacobian. It is a principally polarized abelian scheme with an
induced level $l$ sturcutre, see [2, XVI.2]. To embed the curves in their
Jacobians, we need a quasi-section. By [14, Corollaire 17.6.2 and 17.6.3],
there exists an étale surjective map $S\rightarrow\mathcal{M}_{g}^{\circ}$ for
some affine variety $S$, such that the base change
$\mathcal{C}:=\mathcal{C}_{g}^{\circ}\times_{\mathcal{M}_{g}^{\circ}}S\rightarrow
S$ has a section

$i:S\rightarrow\mathcal{C}.$

Write $\mathcal{J}\rightarrow S$ for the base change of
$\mathcal{J}_{g}^{\circ}\rightarrow\mathcal{M}_{g}^{\circ}$. Then we embed
$\mathcal{C}\rightarrow S$ into $\mathcal{J}\rightarrow S$ using the section
$i$, by sending $\mathcal{C}_{s}$ to
$\mathcal{C}_{s}-i(s)\subseteq\mathcal{J}_{s}$ for any $s\in S$.

Let $\mathcal{X}:=W_{2}(\mathcal{C})$ be the image of the symmetric product
$\mathcal{C}^{(2)}:=\frac{\mathcal{C}\times_{S}\mathcal{C}}{S_{2}}$ of
$\mathcal{C}\rightarrow S$ inside $\mathcal{J}$. Let
$\tau:S\rightarrow\mathbf{A}_{g}$ be the modular map induced by
$\mathcal{J}\rightarrow S$. Then $\tau$ is quasi-finite since the Torelli
morphism with level structures $\mathcal{M}_{g}\rightarrow\mathbf{A}_{g}$ is
quasi-finite.

The argument of Theorem 1.1 is similar to [8, Proof of Theorem 1.1].

###### Proof of Theorem 1.1.

Let us fix $l=3$ for this proof and consider level $3$ structures only.

By applying Theorem 1.2 to the family $\mathcal{X}\rightarrow S$ inside
$\mathcal{J}\rightarrow S$, there exist constants $c>0$ such that for any
$s\in S(\bar{\mathbb{Q}})$ with $h(\tau(s))>c$, and any subgroup
$\Gamma\subseteq\mathcal{J}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$, we have
$\mathcal{X}_{s}(\bar{\mathbb{Q}})\cap\Gamma\leq c^{1+r}$.

On the other hand, by Northcott’s property, there are only finitely many
points defined over a number field of degree at most $2d$ in $\mathbf{A}_{g}$
with height at most $c$. Since the morphism $\tau:S\rightarrow\mathbf{A}_{g}$
is quasi-finite, there are also only finitely many points $s_{1},...,s_{N}\in
S(\bar{\mathbb{Q}})$ such that $h(\tau(s))\leq c$ and $\tau(s_{i})$ is defined
over a number field of degree at most $2d$. For all such $s_{i}$, we apply
Rémond’s estimate [5, page 643] to say that there is a constant $c>0$,
$\mathcal{X}_{s_{i}}(\bar{\mathbb{Q}})\cap\Gamma\leq c^{1+r}$ for any subgroup
$\Gamma\subseteq\mathcal{J}_{s_{i}}(\bar{\mathbb{Q}})$ of rank $r$.

We remark that if we translate $\mathcal{X}_{s}$ by any point
$a\in\mathcal{J}_{s}(\bar{\mathbb{Q}})$, then
$(\mathcal{X}_{s}(\bar{\mathbb{Q}})-a)\cap\Gamma\leq c^{2+r}$, since the group
generated by $\Gamma$ and $a$ is of rank at most $r+1$.

Now take a non-hyperelliptic non-bielliptic smooth irreducible curve $C$ of
genus $g$ defined over some $F$ with $[F:\mathbb{Q}]\leq d$. The curve
corresponds to an $F$-rational point of $\mathcal{M}_{g,1}$, the coarse moduli
space of smooth irreducible curves of genus $g$ without level structures. The
fine moduli space of smooth irreducible curves of genus $g$ with level $3$
structures $\mathcal{M}_{g}$ is a finite cover of $\mathcal{M}_{g,1}$. We
recall that $\mathcal{M}_{g}$ is defined over $\mathbb{Q}(\zeta_{3})$ where
$\zeta_{3}\in\bar{\mathbb{Q}}$ is a third root of unity. Let
$K:=F(\zeta_{3})$. Then we may give
$C_{K}:=C\times_{\operatorname{Spec}F}\operatorname{Spec}K$ a level $3$
sturcture. So $C_{K}$ corresponds to a $K$-point
$s^{\prime}\in\mathcal{M}_{g}(\bar{\mathbb{Q}})$. Note that
$[K:\mathbb{Q}]\leq 2d.$ Let $s\in S(\bar{\mathbb{Q}})$ be a point over
$s^{\prime}$. Then $\tau(s)$ is defined over $K$, which is a number field of
degree at most $2d$. So either $h(\tau(s))>c$ or $s=s_{i}$ for some $i$.

To prove the theorem, we may assume there is a quadratic point $P\in
C(F^{\prime})$ for some $[F^{\prime}:F]=2$ and $P^{\prime}$ be its Galois
conjugate. Embed $C^{(2)}$ in $\operatorname{Jac}(C)$ by sending
$(Q_{1},Q_{2})$ to $[Q_{1}+Q_{2}-P-P^{\prime}]$. This is an $F$-morphism.
Denote the image by $W_{2}(C)$. Note that
$\operatorname{Jac}(C)=\mathcal{J}_{s}$. We regard
$\Gamma=\operatorname{Jac}(C)(F)$ as a subgroup of
$\mathcal{J}_{s}(\bar{\mathbb{Q}})$. By Mordell-Weil Theorem, $\Gamma$ is of
finite rank. Note also $W_{2}(C)$ is a translate of $\mathcal{X}_{s}$. By our
discussion, then

$\\#W_{2}(C)(F)=W_{2}(C)(\bar{\mathbb{Q}})\cap\Gamma\leq c^{2+\rho}.$

Therefore by the correspondence between the Galois orbits of $C^{(2)}(F)$ and
the quadratic points of $C$, we have

$\\#C(F,2)\leq 2\cdot\\#W_{2}(C)(F)\leq 2c^{2+\rho}\leq(2c)^{1+\rho}.$

∎

## Appendix

In this appendix, we sketch the proof to reduce Theorem 1.3 to the case of
smooth curves inside their Jacobians proved by Dimitrov-Gao-Habegger [8] and
Kühne [20]. The main idea of this proof is communicated to us by Gao. We will
use the formulation of the New Gap Principle, introduced by Gao in [11].

As in the previous proof of the theorem, we may use induction on $\dim S$ and
reduce to assume everything in Step $0$ of that proof. In particular, each
fiber $\mathcal{C}_{s}$ is geometrically irreducible and generates
$\mathcal{A}_{s}$.

Consider the normalization
$\tilde{\mathcal{C}}_{\eta}\rightarrow\mathcal{C}_{\eta}$. By a spreading out
argument (see for example [24, Theorem 3.2.1]), after possibly replacing $S$
by an open dense subset, we may assume there is a family of smooth irreducible
curves $\tilde{\mathcal{C}}\rightarrow S$ of genus $g$ with a surjective
morphism $\tilde{\mathcal{C}}\rightarrow\mathcal{C}$ over $S$, such that any
fiber $\tilde{\mathcal{C}}_{s}$ is the normalization of $\mathcal{C}_{s}$.

Let $\mathcal{J}$ be the Jacobian of $\tilde{\mathcal{C}}\rightarrow S$. By
the New Gap principle [11, Theorem 4.1], there exists $c_{1},c_{2}>0$
depending only on $g$ such that for each $s\in S(\bar{\mathbb{Q}})$ and
$\tilde{P}\in\tilde{\mathcal{C}}_{s}(\bar{\mathbb{Q}})$

$\\#\\{\tilde{Q}\in\tilde{\mathcal{C}_{s}}(\bar{\mathbb{Q}}):\hat{h}(\tilde{Q}-\tilde{P})\leq
c_{1}\max\\{1,h_{\mathrm{Fal}}(\mathcal{J}_{s})\\}\\}\leq c_{2}.$

There is a map $\phi:\mathcal{J}\rightarrow\mathcal{A}$ induced by
$\tilde{\mathcal{C}}\rightarrow\mathcal{C}$. Note that $\phi$ restricted to
$\tilde{\mathcal{C}}-\tilde{\mathcal{C}}$ is a finite map onto
$\mathcal{C}-\mathcal{C}$. This implies the pullback of an $S$-relatively
ample line bundle on $\mathcal{C}-\mathcal{C}$ is $S$-relatively ample on
$\tilde{\mathcal{C}}-\tilde{\mathcal{C}}$ as well, see [21, Proposition 1.2.13
and Theorem 1.7.8]. In particular, the Néron-Tate heights on
$\tilde{\mathcal{C}}-\tilde{\mathcal{C}}$ are equivalent to the pullback
Néron-Tate heights from $\mathcal{C}-\mathcal{C}$. Thus we have for any point
$P\in\mathcal{C}_{s}(\bar{\mathbb{Q}})$,

$\\#\\{Q\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\hat{h}(Q-P)\leq
c_{1}\max\\{1,h_{\mathrm{Fal}}(\mathcal{J}_{s})\\}\\}\leq c_{2}.$

On the other hand, note that
$\phi_{s}:\mathcal{J}_{s}\rightarrow\mathcal{A}_{s}$ is surjective. So
$\mathcal{A}_{s}$ is a quotient of $\mathcal{J}_{s}$. By a property of the
Faltings heights, we have $h_{\mathrm{Fal}}(\mathcal{A}_{s})\leq
h_{\mathrm{Fal}}(\mathcal{J}_{s})$, see for example [27, Corollaire 1.2].
Therefore in particular we have

$\\#\\{Q\in\mathcal{C}_{s}(\bar{\mathbb{Q}}):\hat{h}(Q-P)\leq
c_{1}\max\\{1,h_{\mathrm{Fal}}(\mathcal{A}_{s})\\}\\}\leq c_{2}.$

Let us remark that $c_{\mathrm{NT}}(s),h_{1}(s)$ are bounded by
$c_{3}\max\\{1,h_{\mathrm{Fal}}(\mathcal{A}_{s})\\}$. Then for any subgroup
$\Gamma\subseteq\mathcal{A}_{s}(\bar{\mathbb{Q}})$ of finite rank $r$ and any
$\eta>0$ we are able to bound the number of points of the following set

$\\{P\in\mathcal{C}_{s}(\bar{\mathbb{Q}})\cap\Gamma:\hat{h}(P)\leq\eta\max\\{1,c_{\mathrm{NT}}(s),h_{1}(s)\\}\\}$

by using the standard ball-packing argument.

By a stronger version of Rémond’s result, taking into account not only the
small points and large points but also the middle points, see for example [6,
Théorème 6.8], this would imply Theorem 1.3.

## Acknowledgement

I am extremely grateful to my advisor Dan Abramovich for introducing me to
this interesting topic and the arithmetic world. His guidance and insights,
both mathematically and mentally, help me survive the hard quarantine days. I
am deeply indebted to Ziyang Gao for the enlightening discussions during the
preparation of this paper and the insightful comments to the first draft. I
would also like to thank Joseph Silverman for his wonderful lectures which
triggered my interest in arithmetic geometry.

## References

  * [1] Dan Abramovich and José Felipe Voloch.  Lang’s conjectures, fibered powers, and uniformity.  New York J. Math., 2:20–34, electronic, 1996.
  * [2] Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths.  Geometry of algebraic curves. Volume II, volume 268 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer, Heidelberg, 2011.  With a contribution by Joseph Daniel Harris.
  * [3] F. Bardelli and A. Del Centina.  Bielliptic curves of genus three: canonical models and moduli space.  Indag. Math. (N.S.), 10(2):183–190, 1999.
  * [4] Lucia Caporaso, Joe Harris, and Barry Mazur.  Uniformity of rational points.  J. Amer. Math. Soc., 10(1):1–35, 1997.
  * [5] Sinnou David and Patrice Philippon.  Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II.  Comment. Math. Helv., 77(4):639–700, 2002.
  * [6] Sinnou David and Patrice Philippon.  Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques.  Int. Math. Res. Pap. IMRP, (3):Art. ID rpm006, 113, 2007.
  * [7] Vesselin Dimitrov, Ziyang Gao, and Philipp Habegger.  Uniform bound for the number of rational points on a pencil of curves.  Int. Math. Res. Not. IMRN, (2):1138–1159, 2021.
  * [8] Vesselin Dimitrov, Ziyang Gao, and Philipp Habegger.  Uniformity in mordell-lang for curves.  2021\.
  * [9] Gerd Faltings.  Diophantine approximation on abelian varieties.  Ann. of Math. (2), 133(3):549–576, 1991.
  * [10] Ziyang Gao.  Generic rank of Betti map and unlikely intersections.  Compos. Math., 156(12):2469–2509, 2020.
  * [11] Ziyang Gao.  Recent developments of the uniform mordell-lang conjecture, 2021.
  * [12] Ziyang Gao, Tangli Ge, and Lars Kühne.  The uniform mordell-lang conjecture.  available on arXiv, 2021.
  * [13] A. Grothendieck.  Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III.  Inst. Hautes Études Sci. Publ. Math., (28):255, 1966.
  * [14] A. Grothendieck.  Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV.  Inst. Hautes Études Sci. Publ. Math., (32):361, 1967.
  * [15] Alexander Grothendieck.  Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert.  In Séminaire Bourbaki, Vol. 6, pages Exp. No. 221, 249–276. Soc. Math. France, Paris, 1995.
  * [16] Philipp Habegger.  Special points on fibered powers of elliptic surfaces.  J. Reine Angew. Math., 685:143–179, 2013.
  * [17] Joe Harris and Joe Silverman.  Bielliptic curves and symmetric products.  Proc. Amer. Math. Soc., 112(2):347–356, 1991.
  * [18] Ehud Hrushovski.  The Manin-Mumford conjecture and the model theory of difference fields.  Ann. Pure Appl. Logic, 112(1):43–115, 2001.
  * [19] Yujiro Kawamata.  Characterization of abelian varieties.  Compositio Math., 43(2):253–276, 1981.
  * [20] Lars Kühne.  Equidistribution in families of abelian varieties and uniformity, 2021\.
  * [21] Robert Lazarsfeld.  Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, 2004.  Classical setting: line bundles and linear series.
  * [22] D. Mumford, J. Fogarty, and F. Kirwan.  Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)].  Springer-Verlag, Berlin, third edition, 1994.
  * [23] Nitin Nitsure.  Construction of Hilbert and Quot schemes.  In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr., pages 105–137. Amer. Math. Soc., Providence, RI, 2005.
  * [24] Bjorn Poonen.  Rational points on varieties, volume 186 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2017.
  * [25] Gaël Rémond.  Décompte dans une conjecture de Lang.  Invent. Math., 142(3):513–545, 2000.
  * [26] Gaël Rémond.  Inégalité de Vojta en dimension supérieure.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29(1):101–151, 2000.
  * [27] Gaël Rémond.  Propriétés de la hauteur de Faltings.  https://www-fourier.ujf-grenoble.fr/~remond/2476.pdf, 2020.
  * [28] Thomas Scanlon.  Automatic uniformity.  Int. Math. Res. Not., (62):3317–3326, 2004.
  * [29] The Stacks project authors.  The stacks project.  https://stacks.math.columbia.edu, 2021.