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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.

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