Location of small points on an elliptic curve by an equidistribution argument

Abstract

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. If $\Gamma \subset E(\overline{K})$ is a subgroup of finite rank, a very special case of a conjecture of R\'emond predicts that points of small height in $E(K(\Gamma))$ lie in the division group of $\Gamma$. Using an equidistribution argument, we will show that this conjecture is true for groups of rank arbitrarily large.