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 Rémond 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.

arXiv commentAmoroso pointed out that [14, Lemma 5.8] was false (a counterexample has been provided). Hence, the previous version has been modified in order to make it independent of this lemma. Consequently, the descent argument presented in Section 3 (and used in Section 4) is different. Moreover, the main theorem suffered a slight degradation. Final version