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 $E_λ$ of equation $Y^2=X(X-1)(X-λ)$, we prove that, given $n$ linearly independent points $P_1(λ), ...,P_n(λ)$ on $E_λ$ with coordinates in $\bar{\mathbb{Q}(λ)}$, there are at most finitely many complex numbers $λ_0$ such that the points $P_1(λ_0), ...,P_n(λ_0)$ satisfy two independent relations on $E_{λ_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.