Unlikely intersections in products of families of elliptic curves and the multiplicative group
Abstract
Let $E_\lambda$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-\lambda)$. 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 non-isogenous families of elliptic curves and in a family of split semi-abelian varieties.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2016
- DOI:
- 10.48550/arXiv.1606.02063
- arXiv:
- arXiv:1606.02063
- Bibcode:
- 2016arXiv160602063B
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Algebraic Geometry;
- 11G05;
- 11U09
- E-Print:
- To appear in The Quarterly Journal of Mathematics