Linear relations in families of powers of elliptic curves

Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_{\lambda }$ of equation $Y^2=X\left(X-1\right)\left(X-\lambda \right)$, we prove that, given $n$ linearly independent points $P_1\left(\lambda \right),\dots ,P_n\left(\lambda \right)$ on $E_{\lambda }$ with coordinates in $\overline{\mathbb{Q}\left(\lambda \right)}$, there are at most finitely many complex numbers ${\lambda }_0$ such that the points $P_1\left({\lambda }_0\right),\dots ,P_n\left({\lambda }_0\right)$ satisfy two independent relations on $E_{{\lambda }_0}$. This is a special case of conjectures about unlikely intersections on families of abelian varieties.