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

LetSbe a smooth irreducible curve defined over a number fieldkand consider an abelianschemeAoverSand a curveCinsideA,bothdefinedoverk. In previous works, we provedthat, whenAis a fibred product of elliptic schemes, ifCis not contained in a proper subgroupscheme ofA, then it contains at most finitely many points that belong to a flat subgroupscheme of codimension at least 2. In this article, we continue our investigation and settle thecrucial case of powers of simple abelian schemes of relative dimensiong2. This, combinedwith the above-mentioned result and work by Habegger and Pila, gives the statement for generalabelian schemes which has applications in the study of solvability of almost-Pell equations inpolynomials.