Small height and infinite nonabelian extensions

Abstract

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. The field $F$ generated over $\mathbf{Q}$ by all torsion points of $E$ is an infinite, nonabelian 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 Néron–Tate height has a similar gap on $E\left(F\right)$ and use this to determine the structure of the group $E\left(F\right)$.