Singular intersections in families of abelian varieties

TL;DR

This paper proves finiteness of points where a curve inside an abelian scheme over a curve intersects flat subgroup schemes tangentially, contributing to unlikely intersection problems and the Pink conjecture.

Contribution

It establishes a finiteness result for tangential intersections in families of abelian varieties, extending known results and providing an alternative proof for elliptic curves.

Findings

Finiteness of tangential intersection points in abelian schemes.

Alternative proof for elliptic curve case using Pila-Zannier method.

Potential for effective results via point-counting theorems.

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.