Unlikely intersections in products of families of elliptic curves and the multiplicative group

TL;DR

This paper investigates unlikely intersection phenomena in families of elliptic curves and semi-abelian varieties, extending previous results on finiteness of special parameters where certain algebraic relations occur among points.

Contribution

It extends prior work by analyzing unlikely intersections in products of non-isogenous elliptic curve families and semi-abelian varieties, providing new finiteness results.

Findings

Finiteness of special parameters with algebraic relations among points

Extension of unlikely intersection results to product families of elliptic curves

Analysis of semi-abelian variety families

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.