Unlikely intersections with isogeny orbits in a product of elliptic schemes

Abstract

Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the g-th fibered power $\mathcal {A}$ of the elliptic scheme, where g is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal {A}$ that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta–Rémond inequality with the Pila–Zannier strategy.

Appendix A. Generalized Vojta–Rémond inequality

Let $m \ge 2$ be an integer and let $X_1, \ldots , X_m$ be a family of irreducible positive-dimensional projective varieties, defined over $\bar{\mathbb {Q}}$. In the present work, we use a generalization [10] of Rémond’s results of [53] to the case of an algebraic point $x = (x_1,\ldots ,x_m)$ in the product $X = X_1 \times \cdots \times X_m$. Let us briefly recall the hypotheses which come into play. We follow the exposition of [10].

For an m-tuple $a = (a_1,\ldots ,a_m)$ of positive integers, we write

$$\begin{aligned} \mathcal {N}_a = \bigotimes _{i=1}^{m}{p_i^{*}\mathcal {L}_i^{\otimes a_i}},\end{aligned}$$

where $\mathcal {L}_i$ is a fixed very ample line bundle on $X_i$ and $p_i: X_1 \times \cdots \times X_m \rightarrow X_i$ is the natural projection. We relate a to a nef line bundle $\mathcal {M}$ on X which satisfies some further conditions, specified below.

By (a system of) homogeneous coordinates for a very ample line bundle we mean the set of pull-backs of the homogeneous coordinates on some $\mathbb {P}^{N'}$ under a closed embedding into $\mathbb {P}^{N'}$ that is associated to that line bundle. Let $W^{(i)} \subset \Gamma (X_i,\mathcal {L}_i)$ be a fixed system of homogeneous coordinates on $X_i$. We identify $p_i^{*}W^{(i)}$ with $W^{(i)}$.

By an injection of line bundles we mean an injective morphism between the associated invertible sheaves. We fix a non-empty open subset $U^0$ of X and suppose that there exists a very ample line bundle $\mathcal {P}$ on X, an injection $\mathcal {P} \hookrightarrow \mathcal {N}_a^{\otimes t_1}$ which induces an isomorphism on $U^0$ and a system of homogeneous coordinates $\Xi$ for $\mathcal {P}$ which are (by means of the aforementioned injection) monomials of multidegree $t_1a$ in the $W^{(i)}$.

We also suppose that there exists an injection $(\mathcal {P} \otimes \mathcal {M}^{\otimes -1}) \hookrightarrow \mathcal {N}_a^{\otimes t_2}$ which induces an isomorphism on $U^0$ and that $\mathcal {P} \otimes \mathcal {M}^{\otimes -1}$ is generated by a family Z of M global sections on X which are polynomials of multidegree $t_2a$ in the $W^{(i)}$ such that the height of the family of coefficients of all these polynomials is at most $\sum _{i} a_i \delta _i$.

The parameters $t_1, t_2, M \in \mathbb {N}$ and $\delta _1, \ldots , \delta _m \in [1,\infty )$ are fixed independently of the pair $(a,\mathcal {M})$. Using this pair, we can define the following two notions of height for a point $x \in U^0(\bar{\mathbb {Q}})$:

$$\begin{aligned} h_{\mathcal {M}}(x) = h(\Xi (x)) - h(Z(x)),\end{aligned}$$

$$\begin{aligned} h_{\mathcal {N}_a}(x) = a_1h\left( W^{(1)}(x)\right) + \cdots + a_mh\left( W^{(m)}(x)\right) .\end{aligned}$$

Let $\theta \ge 1$ and $\omega \ge -1$ be two integer parameters and set (with $\omega ' = 3 + \omega$)

$$\begin{aligned} \Lambda = \theta (2t_1u_0)^{u_0}\left( \max _{1 \le i \le m}{N_i}+1\right) \prod _{i=1}^{m}{\deg (X_i)},\end{aligned}$$

$$\begin{aligned} \psi (u) = \prod _{j=u+1}^{u_0}{(\omega 'j+1)}, \end{aligned}$$

$$\begin{aligned} c_1 = c_2 = \Lambda ^{\psi (0)},\end{aligned}$$

$$\begin{aligned} c^{(i)}_3 = \Lambda ^{2\psi (0)}(Mt_2)^{u_0}(h(X_i)+\delta _i) \quad (i=1,\ldots ,m), \end{aligned}$$

where $u_0 = \dim (X_1)+\cdots +\dim (X_m)$, $N_i+1 = \#W^{(i)}$ and the degrees and heights are computed with respect to the embeddings given by the $W^{(i)}$.

Theorem A.1

Let $x \in U^0(\bar{\mathbb {Q}})$ and let $(a,\mathcal {M})$ be a pair as defined above. Suppose that, for every subproduct of the form $Y = Y_1 \times \cdots \times Y_m$, where $Y_i \subset X_i$ is an irreducible subvariety that contains $x_i$, the following estimate holds:

$$\begin{aligned} \mathcal {M}^{\cdot \dim (Y)} \cdot Y \ge \theta ^{-1}\prod _{i=1}^{m}{(\deg (Y_i))^{-\omega }a_i^{\dim (Y_i)}}.\end{aligned}$$

If furthermore $c_2a_{i+1} \le a_i$ for every $i < m$ and $c^{(i)}_3 \le h\left( W^{(i)}(x_i)\right)$ for every $i \le m$, then we have

$$\begin{aligned} h_{\mathcal {N}_a}(x) \le c_1 h_{\mathcal {M}}(x).\end{aligned}$$

Proof

See [10] with $\mathcal {X} = X$ and $\pi = {{\,\mathrm{id}\,}}_X$. $\square$