Unlikely Intersections in products of families of elliptic curves and the multiplicative group
Let Eλ be the Legendre family of elliptic curves of equation Y2=X(X−1)(X−λ) and let P1(λ),…,Pn(λ) be n independent points with coordinates algebraic over Q(λ). We will see that there are at most finitely many specialisations of the parameter λ such that the specialised points P1(λ0),…,Pn(λ0) satisfy two independent linear relations with integer coefficients on Eλ0. 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.