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 $Y^2=X(X-1)(X-\lambda)$, 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 e-prints
- 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
- E-Print:
- Algebra Number Theory 10 (2016) 195-214