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.

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Acknowledgements

I thank my advisor Philipp Habegger for suggesting to study this problem and to apply a generalized Vojta–Rémond inequality in this context as well as for his constant support. I thank Philipp Habegger and Gaël Rémond for many useful and interesting discussions and for helpful comments on a preliminary version of this article. I thank the anonymous referee for their helpful suggestions. I thank the Institut Fourier in Grenoble for its hospitality, which I enjoyed while writing substantial portions of this article. This work was supported by the Swiss National Science Foundation as part of the project “Diophantine Problems, o-Minimality, and Heights”, no. 200021_165525.

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Appendix A. Generalized Vojta–Rémond inequality

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 \(\pi = {{\,\mathrm{id}\,}}_X\). \(\square \)

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Dill, G.A. Unlikely intersections with isogeny orbits in a product of elliptic schemes. Math. Ann. (2020). https://doi.org/10.1007/s00208-020-02024-2

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Mathematics Subject Classification

  • 11G18
  • 11G50
  • 14G40
  • 14K02