Unlikely intersections in products of families of elliptic curves and the multiplicative group

Abstract

Let $E_{\lambda }$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-\lambda )$. We recently proved that, given $n$ linearly independent points $P_1(\lambda ),\dots ,P_n(\lambda )$ on $E_{\lambda }$ with coordinates in $\overline{\mathbb{Q}(\lambda )}$, there are at most finitely many complex numbers ${\lambda }_0$ such that the points $P_1({\lambda }_0),\dots ,P_n({\lambda }_0)$ satisfy two independent relations on $E_{{\lambda }_0}$. In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two non-isogenous families of elliptic curves and in a family of split semi-abelian varieties.

arXiv commentTo appear in The Quarterly Journal of Mathematics