Isogeny relations in products of families of elliptic curves

Abstract

Let $E_{\lambda}$ be the Legendre family of elliptic curves with equation $Y^2=X(X-1)(X-\lambda)$. Given a curve $\mathcal{C}$, satisfying a condition on the degrees of some of its coordinates and parametrizing $m$ points $P_1, \ldots, P_m \in E_{\lambda}$ and $n$ points $Q_1, \ldots, Q_n \in E_{\mu}$ and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points $\mathbf{c}_0$ on $\mathcal{C}$, such that there exists an isogeny $\phi: E_{\mu(\mathbf{c}_0)} \rightarrow E_{\lambda(\mathbf{c}_0)}$ and the $m+n$ points $P_1(\mathbf{c}_0), \ldots, P_m(\mathbf{c}_0), \phi(Q_1(\mathbf{c}_0)), \ldots, \phi(Q_n(\mathbf{c}_0)) \in E_{\lambda(\mathbf{c}_0)}$ are linearly dependent over $\mathrm{End}(E_{\lambda(\mathbf{c}_0)})$.