CM relations in fibered powers of elliptic families.

Let (Formula presented.) be the Legendre family of elliptic curves. Given (Formula presented.) points (Formula presented.), linearly independent over (Formula presented.), we prove that there are at most finitely many complex numbers (Formula presented.) such that (Formula presented.) has complex multiplication and (Formula presented.) are linearly dependent over End(Formula presented.). This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over (Formula presented.).