Small Heights in Large Non-Abelian Extensions

TL;DR

This paper establishes an effective lower bound for the height of non-zero elements in a specific infinite Galois extension of the rationals, related to elliptic curves and torsion points, extending understanding of height bounds in number theory.

Contribution

It provides the first effective lower bound for heights in the infinite extension generated by torsion points of an elliptic curve over the rationals.

Findings

Effective height lower bound proved for elements in L(E_tor)

Extension L(E_tor) includes all torsion point coordinates of E

Results contribute to the understanding of heights in infinite Galois extensions

Abstract

Let E be an elliptic curve over the rationals. Let L be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension L(E_tor) of the rationals where we adjoin all coordinates of torsion points of E. In this paper we will prove an effective lower bound for the height of non-zero elements in L(E_tor) that are not a root of unity.