Explicit Small Heights in Infinite Non-Abelian Extensions

TL;DR

This paper establishes explicit lower bounds for the heights of non-zero, non-root of unity elements in the infinite extension generated by all torsion points of an elliptic curve over rationals, depending on the curve's conductor.

Contribution

It provides the first explicit height lower bounds in the infinite non-abelian extension generated by elliptic curve torsion points, linking to the curve's conductor.

Findings

Explicit height lower bounds depending on conductor

Bound for a small supersingular prime

Results applicable to non-abelian infinite extensions

Abstract

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower bound for the height of non-zero elements in $\mathbb{Q}(E_{\text{tor}})$ that are not a root of unity, only depending on the conductor of the elliptic curve. As a side result we will give an explicit bound for a small supersingular prime for an elliptic curve.