Unlikely intersections in products of families of elliptic curves and the multiplicative group
Abstract
Let $E_\lambda$ be the Legendre elliptic curve of equation $Y^2=X(X1)(X\lambda)$. We recently proved that, given $n$ linearly independent points $P_1(\lambda), \dots,P_n(\lambda)$ on $E_\lambda$ with coordinates in $\bar{\mathbb{Q}(\lambda)}$, there are at most finitely many complex numbers $\lambda_0$ such that the points $P_1(\lambda_0), \dots,P_n(\lambda_0)$ satisfy two independent relations on $E_{\lambda_0}$. In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two nonisogenous families of elliptic curves and in a family of split semiabelian varieties.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 DOI:
 10.48550/arXiv.1606.02063
 arXiv:
 arXiv:1606.02063
 Bibcode:
 2016arXiv160602063B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G05;
 11U09
 EPrint:
 To appear in The Quarterly Journal of Mathematics