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

Let Eλ be the Legendre elliptic curve of equation Y2=X(X−1)(X−λ). We recently proved that, given n linearly independent points P1(λ),…,Pn(λ) on Eλ with coordinates in Q(λ)¯, there are at most finitely many complex numbers λ0 such that the points P1(λ0),…,Pn(λ0) satisfy two independent relations on Eλ0. 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.