Small height and infinite nonabelian extensions

Let E be an elliptic curve defined over Q without complex multiplication. The field F generated over 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(F) and use this to determine the structure of the group E(F).