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


Let EλE_\lambda be the Legendre elliptic curve of equation Y2=X(X1)(Xλ)Y^2=X(X-1)(X-\lambda). We recently proved that, given nn linearly independent points P1(λ),,Pn(λ)P_1(\lambda), \dots,P_n(\lambda) on EλE_\lambda with coordinates in Q(λ)ˉ\bar{\mathbb{Q}(\lambda)}, there are at most finitely many complex numbers λ0\lambda_0 such that the points P1(λ0),,Pn(λ0)P_1(\lambda_0), \dots,P_n(\lambda_0) satisfy two independent relations on Eλ0E_{\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.Comment: To appear in The Quarterly Journal of Mathematic

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