Unlikely intersections in families of abelian varieties and the polynomial Pell equation

摘要

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that, when A is a fibred product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g > 2. This, combined with the above-mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes which has applications in the study of solvability of almost-Pell equations in polynomials.