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

Let $S$ be a smooth irreducible curve defined over a number field $k$ and consider an abelian scheme $cA$ over $S$ and a curve $cC$ inside $cA$, both defined over $k$. In previous works, we proved that, when $cA$ is a fibered product of elliptic schemes, if $cC$ is not contained in a proper subgroup scheme of $cA$, 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 $gge 2$. 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.