Small Height and Infinite Non-Abelian Extensions

TL;DR

This paper investigates the properties of infinite non-abelian extensions generated by torsion points of elliptic curves over rationals, establishing height bounds and analyzing the structure of the resulting Mordell-Weil group.

Contribution

It proves a height gap for elements in the field generated by torsion points and characterizes the structure of the Mordell-Weil group over this extension.

Findings

Height of elements in F is either zero or bounded away from zero.

The Néron-Tate height exhibits a similar gap on E(F).

The structure of E(F) is determined using these height bounds.

Abstract

Let $E$ be an elliptic curve defined over the rationals without complex multiplication. The field $F$ generated by all torsion points of $E$ is an infinite, non-abelian Galois extension of the rationals which has unbounded, wild ramification above all primes. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the N\'eron-Tate height has a similar gap on $E(F)$ and use this to determine the structure of the group $E(F)$.