Singular intersections in families of abelian varieties

Abstract

Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove that the set of points in which $\mathcal{C}$ intersects proper flat subgroup schemes of $\mathcal{A}$ tangentially is finite. The crucial case of elliptic curves already follows from a result by Corvaja, Demeio, Masser and Zannier: in this case we provide an alternative proof using the Pila-Zannier method. Such a proof may lead to an effective result using an effective point-counting theorem. This fits in the framework of the so-called problems of unlikely intersections, and can be seen as a variation of the relative Pink conjecture for abelian varieties.