Linear relations in families of powers of elliptic curves


Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve EλE_\lambda of equation Y2=X(X1)(Xλ)Y^2=X(X-1)(X-\lambda), we prove that, given nn linearly independent points P1(λ),...,Pn(λ)P_1(\lambda), ...,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), ...,P_n(\lambda_0) satisfy two independent relations on Eλ0E_{\lambda_0}. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties

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