2 Nevanlinna theory

We use Landau’s notation o. Thus, $o(1)$ represents a function that tends to $0$ . In addition, the subscript ‘ $exc$ ’ in inequalities and equalities between functions of a variable $r\in \mathbb {R}_{\ge 0}$ means that the claimed relationship holds for r outside a set of finite measure in $\mathbb {R}_{\ge 0}$ .

In the first half of the 1920s, Nevanlinna developed a very successful theory to study value distribution of complex meromorphic functions. In this section, we recall some basic results of this theory; we refer the reader to [Reference Vojta, Colliot-Thélène, Swinnerton-Dyer and Vojta13] for a general reference.

Let $\mathscr {M}$ be the field of (possibly transcendental) complex meromorphic functions on $\mathbb {C}$ . Given a nonconstant $h\in \mathscr {M}$ , a point $\alpha \in {\mathbb {P}}^1(\mathbb {C})=\mathbb {C}\cup \{\infty \}$ and a real number $r\ge 0$ , we define

$$ \begin{align*} n_h^{(1)}(\alpha,r)=\#\{ z_0\in \mathbb{C} : \vert z_0\vert\le r\mbox{ and }h(z_0)=\alpha\}, \end{align*} $$

where the case $h(z_0)=\infty $ is understood as the condition that h has a pole at $z_0$ . The truncated counting function $N_h^{(1)}(\alpha ,r)$ is then defined as the logarithmic average

$$ \begin{align*} N_h^{(1)}(\alpha,r) = \int_{0}^r(n_h^{(1)}(\alpha,t)-n_h^{(1)}(\alpha,0))\,\frac{dt}{t} + n_h^{(1)}(\alpha,0)\log r. \end{align*} $$

Associated to every $h\in \mathscr {M}$ , there is the Nevanlinna height (or characteristic) function

$$ \begin{align*} T_h\colon \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}, \end{align*} $$

which has the basic property that it is bounded if h is constant; otherwise $T_f(r)$ grows to infinity as $r\to \infty $ . We recall that the function $T_h(r)$ is only defined up to adding a bounded function. Intuitively, $T_h(r)$ measures the complexity of h restricted to the disk $\{z\in \mathbb {C}: \vert z\vert \le r\}$ as r grows. For our purposes, we do not need to recall the precise definition of $T_h(r)$ (which can be found, for example, in [Reference Vojta, Colliot-Thélène, Swinnerton-Dyer and Vojta13]), but instead we simply need its relationship to the truncated counting function, which is provided by the Second Main Theorem of Nevanlinna theory.

Theorem 2.1 (Second Main Theorem).

Let $h\in \mathscr {M}$ be nonconstant and let $\alpha _1,\ldots , \alpha _q$ be different points in ${\mathbb {P}}^1(\mathbb {C})$ . Then

$$ \begin{align*} \sum_{j=1}^q N^{(1)}_h(\alpha_j, r)\ge_{exc} (q-2+o(1))T_h(r). \end{align*} $$

In addition to the previous general result, we need another relationship between the Nevanlinna height and the truncated counting function in a special case.

Lemma 2.2 ([Reference Garcia-Fritz and Pasten9, Lemma 3.3]).

Let E be a complex elliptic curve and let $g\colon E\to {\mathbb {P}}^1$ be a nonconstant morphism of degree d. Let $\phi \colon \mathbb {C}\to E$ be a nonconstant holomorphic map and let $\alpha \in {\mathbb {P}}^1(\mathbb {C})$ . Consider the nonconstant meromorphic function ${h=g\circ \phi \in \mathscr {M}}$ . Then

$$ \begin{align*} N^{(1)}_h(\alpha,r)=_{exc} \bigg(\frac{\# g^{-1}(\alpha)}{d}+o(1)\bigg) T_h(r). \end{align*} $$

We remark that our proof of the previous lemma in [Reference Garcia-Fritz and Pasten9] uses the Second Main Theorem for holomorphic maps to elliptic curves rather than the case of meromorphic functions cited above.

3 Geometric preliminaries

Let us fix some notation and assumptions for this section. Let $E_1$ and $E_2$ be complex elliptic curves. For $j=1,2$ , let $g_j\colon E_j\to {\mathbb {P}}^1$ be a nonconstant morphism of degree $d_j$ . Suppose that $g_1$ and $g_2$ have different branch loci in ${\mathbb {P}}^1$ . Let ${X\subseteq E_1\times E_2}$ be the one-dimensional algebraic set defined by the equation $g_1(P_1)=g_2(P_2)$ on ${(P_1,P_2)\in E_1\times E_2}$ : that is, X is the pre-image of the diagonal $\Delta \subseteq {\mathbb {P}}^1\times {\mathbb {P}}^1$ via the map $G=(g_1,g_2)\colon E_1\times E_2\to {\mathbb {P}}^1\times {\mathbb {P}}^1$ .

If $Z\subseteq E_1\times E_2$ is a one-dimensional algebraic set, we define its degree $\deg (Z)$ as the intersection number $Z.(V_1+V_2)$ , where Z is seen as a reduced divisor, and we define $V_1=\{0\}\times E_2$ and $V_2=E_1\times \{0\}$ , where $0$ is the neutral point of the corresponding elliptic curve. Here, we remark that the divisor $V_1+V_2$ on $E_1\times E_2$ is ample.

Proof. Let $\Delta \subseteq {\mathbb {P}}^1\times {\mathbb {P}}^1$ be the diagonal and let $L_1=\{p_1\}\times {\mathbb {P}}^1$ and $L_1={\mathbb {P}}^1\times \{p_2\}$ for a fixed choice of points $p_1,p_2\in {\mathbb {P}}^1$ . Then $\Delta $ is linearly equivalent to $L_1+L_2$ .

Note that $X=G^{-1}\Delta \le G^*\Delta $ as divisors, and by the projection formula,

$$ \begin{align*} \begin{aligned} \deg(X)&\le \deg(G^*\Delta)\\ &=(G^*(L_1+L_2))\cdot (V_1+V_2)\\ &\leq G^*G_*(G^*(L_1+L_2)\cdot (V_1+V_2))\\ &=G^*((L_1+L_2)\cdot G_*(V_1+V_2))\\ &=d_1d_2(L_1+L_2)\cdot (d_2\cdot\{g_1(0)\}\times\mathbb{P}^1+d_1\cdot\mathbb{P}^1\times\{g_2(0)\})\\ &=d_1d_2(d_1+d_2).\\[-22pt] \end{aligned} \end{align*} $$

Before we give the main result of this section, we recall the Riemann–Hurwitz formula, which is useful in our argument.

Lemma 3.2 (Riemann–Hurwitz formula).

Let $Y_1$ and $Y_2$ be smooth projective (complex) curves of genus $\mathfrak {g}_1$ and $\mathfrak {g}_2$ , respectively, and let $g\colon Y_1\to Y_2$ be a nonconstant morphism of degree d. Let $\alpha _1,\ldots , \alpha _m\in Y_2$ be all the branch values of g. Then

$$ \begin{align*} 2(\mathfrak{g}_1-1)=2d(\mathfrak{g}_2-1) + \sum_{j=1}^m (d-\#g^{-1}(\alpha_j)). \end{align*} $$

With this at hand, we can prove our geometric result.

Proof. Let $C\subseteq X$ be an irreducible component and, for the sake of contradiction, suppose that C has geometric genus $0$ or $1$ . As $C\subseteq E_1\times E_2$ , we see that, necessarily, C has geometric genus $1$ , because the projection to at least one component $E_i$ is nonconstant. Since elliptic curves can be uniformised by holomorphic functions from $\mathbb {C}$ , we obtain a nonconstant holomorphic map $\phi =(\phi _1,\phi _2)\colon \mathbb {C}\to C\subseteq E_1\times E_2$ , where at least one of $\phi _j\colon \mathbb {C}\to E_j$ is nonconstant. By the definition of X, we see that ${g_1\circ \phi _1=g_2\circ \phi _2}$ , and we conclude that both $\phi _j$ are nonconstant.

Let $h=g_1\circ \phi _1=g_2\circ \phi _2\in \mathscr {M}$ . Since $g_1$ and $g_2$ have different branch loci, we may assume, without loss of generality, that there is $\beta \in {\mathbb {P}}^1$ that is a branch value of $g_2$ but not of $g_1$ . Let $\alpha _1,\ldots ,\alpha _m\in {\mathbb {P}}^1$ be the different branch values of $g_1$ .

By the Second Main Theorem 2.1 with $q=m+1$ ,

$$ \begin{align*} N^{(1)}_h(\beta,r)+\sum_{j=1}^m N^{(1)}_h(\alpha_j,r)\ge_{exc} (m-1+o(1))T_h(r). \end{align*} $$

On the other hand, Lemma 2.2 gives

$$ \begin{align*} N^{(1)}_h(\beta,r) =_{exc} \bigg(\frac{\# g_2^{-1}(\beta)}{d_2}+o(1)\bigg)T_h(r) \end{align*} $$

and, for each $1\le j\le m$ , we similarly obtain

$$ \begin{align*} N^{(1)}_h(\alpha_j,r) =_{exc} \bigg(\frac{\# g_1^{-1}(\alpha_j)}{d_1}+o(1)\bigg)T_h(r). \end{align*} $$

Putting all of this together, we find that

$$ \begin{align*} \bigg(\frac{\# g_2^{-1}(\beta)}{d_2}+\sum_{j=1}^m\frac{\# g_1^{-1}(\alpha_j)}{d_1}+o(1)\bigg)T_h(r)\ge_{exc} (m-1+o(1))T_h(r). \end{align*} $$

Letting $r\to \infty $ , since h is nonconstant, we deduce that

$$ \begin{align*} m-1\le \frac{\# g_2^{-1}(\beta)}{d_2}+\sum_{j=1}^m\frac{\# g_1^{-1}(\alpha_j)}{d_1}. \end{align*} $$

Since $\beta $ is a branch value of $g_2$ , we have $\# g_2^{-1}(\beta )<d_2$ , and hence

(3.1) $$ \begin{align} m-1<1+\frac{1}{d_1}\sum_{j=1}^m \# g_1^{-1}(\alpha_j). \end{align} $$

On the other hand, the Riemann–Hurwitz formula (in the form of Lemma 3.2) applied to $g_1\colon E_1\to {\mathbb {P}}^1$ gives

$$ \begin{align*} 0 = -2d_1 + \sum_{j=1}^m(d_1-\# g_1^{-1}(\alpha_j)) \end{align*} $$

from which

$$ \begin{align*} \frac{1}{d_1}\sum_{j=1}^m \# g_1^{-1}(\alpha_j)=m-2. \end{align*} $$

This contradicts the bound (3.1).

4 Uniform Mordell–Lang and Manin–Mumford

The rank of an abelian group $\Gamma $ , denoted by $\mathrm {rank}\, \Gamma $ , is defined as the dimension over $\mathbb {Q}$ of the vector space $\Gamma \otimes _{\mathbb {Z}} \mathbb {Q}$ .

After the recent works [Reference Dimitrov, Gao and Habegger6, Reference Gao, Ge and Kühne8, Reference Kühne10, Reference Yuan14], the uniform Mordell–Lang conjecture is proved. Here, we recall the version obtained in [Reference Gao, Ge and Kühne8] in the case of (possibly singular) curves contained in abelian varieties.

Theorem 4.1 (Uniform Mordell–Lang for curves).

Let $n,D\ge 1$ be integers. There is a constant $c(n,D)$ , depending only on n and D, with the following property.

Let A be an abelian variety over $\mathbb {C}$ of dimension n, let $\mathscr {L}$ be an ample line sheaf on A and let $X\subseteq A$ be a one-dimensional Zariski closed subset with $\deg _{\mathscr {L}}(X)\le D$ . Let $\Gamma \le A(\mathbb {C})$ be a subgroup of finite rank and let $r=\mathrm {rank}\, \Gamma $ . If all irreducible components of X have geometric genus at least $2$ , then

$$ \begin{align*} \# (\Gamma\cap X)\le c(n,D)^{1+r}. \end{align*} $$

As usual, $\deg _{\mathscr {L}}(X)$ is defined as the intersection number of $\mathscr {L}$ with X. Theorem 4.1 follows from Theorem 1.1 in [Reference Gao, Ge and Kühne8]; note that here we do not require that X is irreducible, but this case also follows from the same theorem because $\deg _{\mathscr {L}}(X)$ is additive on X and it is a strictly positive integer as $\mathscr {L}$ is ample.

In the special case where $\Gamma $ is the full torsion subgroup of A, one has $r=0$ and the previous result specialises to the uniform Manin–Mumford conjecture.

Theorem 4.2 (Uniform Manin–Mumford for curves).

Let $n,D\ge 1$ be integers. There is a constant $c(n,D)$ , depending only on n and D, with the following property.

Let A be an abelian variety over $\mathbb {C}$ of dimension n, let $\mathscr {L}$ be an ample line sheaf on A and let $X\subseteq A$ be a one-dimensional Zariski closed subset with $\deg _{\mathscr {L}}(X)\le D$ . Let $A[\infty ]$ be the subgroup of all torsion points of $A(\mathbb {C})$ . If all irreducible components of X have geometric genus at least $2$ , then

$$ \begin{align*} \# (A[\infty]\cap X)\le c(n,D). \end{align*} $$

We point out that we use these results only when the abelian variety A is the product of two elliptic curves, so one may refer to [Reference Kühne11].

5 Torsion and ranks

In this section, we prove Theorems 1.3 and 1.4. For this, let us fix some common notation. Let k be $\mathbb {C}$ in the case of Theorem 1.3 or a number field in the case of Theorem 1.4. Let $E_1$ and $E_2$ be elliptic curves and let $g_j\colon E_j\to {\mathbb {P}}^1$ be morphisms of degrees $d_j\le d$ for $j=1,2$ , all defined over k. We assume that the branch loci of $g_1$ and $g_2$ in ${\mathbb {P}}^1$ are different.

Let $G=(g_1,g_2)\colon E_1\times E_2\to {\mathbb {P}}^1\times {\mathbb {P}}^1$ , let $\Delta \subseteq {\mathbb {P}}^1\times {\mathbb {P}}^1$ be the diagonal and let ${X=G^{-1}\Delta \subseteq E_1\times E_2}$ . We note that X is the locus of geometric points $(P_1,P_2)$ in $E_1\times E_2$ satisfying $g_1(P_1)=g_2(P_2)$ .

By Lemma 3.1, $\deg (X)\le d_1d_2(d_1+d_2)\le 2d^3$ , where $\deg (X)$ is the degree with respect to the ample divisor $V_1+V_2$ , as defined in Section 3. Furthermore, by Theorem 3.3, the (geometric) irreducible components of X have geometric genus at least $2$ .

Proof of Theorem 1.3.

Let $\Gamma =E_1[\infty ]\times E_2[\infty ]$ ; this is the group of torsion points of the abelian surface $E_1\times E_2$ . By Theorem 4.2,

$$ \begin{align*} \#(\Gamma\cap X)\le c(2,2d^3) \end{align*} $$

with $c(n,D)$ as in Theorem 4.2. We note that

$$ \begin{align*} g_1(E_1[\infty])\cap g_2(E_2[\infty]) = G(\Gamma)\cap \Delta = G(\Gamma\cap X) \end{align*} $$

and we obtain the result with $c_0(d)=c(2,2d^3)$ .

Proof of Theorem 1.4.

The proof is very similar. Let $\Gamma =E_1(k)\times E_2(k)$ ; by the Mordell–Weil theorem, this group is finitely generated and its rank is ${r=\mathrm {rank}\, E_1(k) + \mathrm {rank}\, E_2(k)}$ . By Theorem 4.1,

$$ \begin{align*} \#(\Gamma\cap X)\le c(2,2d^3)^{1+r} \end{align*} $$

with $c(n,D)$ as in Theorem 4.1. We note that

$$ \begin{align*} g_1(E_1(k))\cap g_2(E_2(k)) = G(\Gamma)\cap \Delta = G(\Gamma\cap X) \end{align*} $$

and we obtain the result with $\kappa (d)=1/\log c(2,2d^3)$ .