Linear relations in families of powers of elliptic curves
Abstract
Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_\lambda$ of equation $Y^2=X(X1)(X\lambda)$, we prove that, given $n$ linearly independent points $P_1(\lambda), ...,P_n(\lambda)$ on $E_\lambda$ with coordinates in $\bar{\mathbb{Q}(\lambda)}$, there are at most finitely many complex numbers $\lambda_0$ such that the points $P_1(\lambda_0), ...,P_n(\lambda_0)$ satisfy two independent relations on $E_{\lambda_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 DOI:
 10.48550/arXiv.1412.3252
 arXiv:
 arXiv:1412.3252
 Bibcode:
 2014arXiv1412.3252B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G05;
 11G50;
 11U09;
 14K05
 EPrint:
 Algebra Number Theory 10 (2016) 195214