# Unlikely Intersections in products of families of elliptic curves and the multiplicative group

Let $E_{\lambda}$ be the Legendre family of elliptic curves of equation $Y^2=X(X-1)(X-\lambda)$ and let $P_1(\lambda), \ldots, P_n(\lambda)$ be $n$ independent points with coordinates algebraic over $\mathbb Q(\lambda)$. We will see that there are at most finitely many specialisations of the parameter $\lambda$ such that the specialised points $P_1(\lambda_0), \ldots, P_n(\lambda_0)$ satisfy two independent linear relations with integer coefficients on $E_{\lambda_0}$. This result fits in the framework of the so-called Unlikely Intersections. We will consider a generalisation of this problem, namely 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.

This is a joint work with F. Barroero.