Six unlikely intersection problems in search of effectivity

TL;DR

This paper explores six unlikely intersection problems involving elliptic curves in Legendre form, providing effective solutions and explicit cases for properties like complex multiplication and roots of unity.

Contribution

It introduces and solves six new intersection problems related to elliptic curves, with effective methods and explicit solutions in some cases.

Findings

All six intersection problems are solved effectively.

Explicit solutions are provided for certain cases.

The work advances understanding of unlikely intersections in elliptic curves.

Abstract

We investigate four properties related to an elliptic curve $E_t$ in Legendre form with parameter $t$: the curve $E_{t}$ has complex multiplication, $E_{-t}$ has complex multiplication, a point on $E_t$ with abscissa $2$ is of finite order, and $t$ is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. We solve these problems effectively and in certain cases also explicitly.