Let be an elliptic curve defined over without complex multiplication. The field generated over by all torsion points of 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 is either zero or bounded from below by a positive constant depending only on . We also show that the Néron–Tate height has a similar gap on and use this to determine the structure of the group .
P. Habegger. "Small height and infinite nonabelian extensions." Duke Math. J. 162 (11) 2027 - 2076, 15 August 2013. https://doi.org/10.1215/00127094-2331342