Vol. 10, No. 1, 2016

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Linear relations in families of powers of elliptic curves

Fabrizio Barroero and Laura Capuano

Vol. 10 (2016), No. 1, 195–214
Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve ${E}_{\lambda }$ of equation ${Y}^{2}=X\left(X-1\right)\left(X-\lambda \right)$, we prove that, given $n$ linearly independent points ${P}_{1}\left(\lambda \right),\dots ,{P}_{n}\left(\lambda \right)$ on ${E}_{\lambda }$ with coordinates in $\overline{ℚ\left(\lambda \right)}$, there are at most finitely many complex numbers ${\lambda }_{0}$ such that the points ${P}_{1}\left({\lambda }_{0}\right),\dots ,{P}_{n}\left({\lambda }_{0}\right)$ satisfy two independent relations on ${E}_{{\lambda }_{0}}$. This is a special case of conjectures about unlikely intersections on families of abelian varieties.

Keywords
linear relations, elliptic curves, unlikely intersections
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11G50, 11U09, 14K05