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

TL;DR

This paper proves a special case of Rémond's conjecture for elliptic curves over number fields, showing that points of small height in certain extensions are contained in the division group of a subgroup, using equidistribution methods.

Contribution

The paper demonstrates the validity of Rémond's conjecture for elliptic curves with subgroups of arbitrary large rank through equidistribution techniques.

Findings

Points of small height lie in the division group of the subgroup.

The conjecture holds for groups of arbitrarily large rank.

Equidistribution methods are effective in this context.

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.