Small Height and Infinite Non-Abelian Extensions

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 Néron-Tate height has a similar gap on $E(F)$ and use this to determine the structure of the group $E(F)$.

Added new corollary on the structure of the group $E(F)$ and corrected some typos in version 2

References in corpus (1)

• On certain infinite extensions of the rationals with Northcott property

Cited by in corpus (9)

• Heights and totally $p$-adic numbers
• On the Northcott property for special values of L-functions
• Small points and free abelian groups
• Heights and totally real numbers
• Points de petite hauteur sur une variété semi-abélienne de la forme ${\mathbb{G}}_m^n\times A$
• A note on extensions of ${\mathbb{Q}}^{tr}$
• Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
• Lehmer without Bogomolov
• Torsion points with multiplicatively dependent coordinates on elliptic curves