CM relations in fibered powers of elliptic families

Abstract

Let $E_{\lambda }$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots ,P_n\in E_{\lambda }\left(\overline{\mathbb{Q}(\lambda )}\right)$ we prove that there are at most finitely many complex numbers ${\lambda }_0$ such that $E_{{\lambda }_0}$ has complex multiplication and $P_1({\lambda }_0),\dots ,P_n({\lambda }_0)$ are dependent over $End(E_{{\lambda }_0})$. 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 $\overline{\mathbb{Q}}$.

The formulation of Theorem 2.1 is now correct

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