#### 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