Effective results on linear dependence for elliptic curves

TL;DR

This paper establishes bounds on the canonical height of rational points on elliptic curves that are not in a given subgroup but reduce to it modulo many primes, advancing understanding of linear dependence over elliptic curves.

Contribution

It provides new upper and lower bounds on the canonical height for points outside a subgroup but with specific reduction properties, under certain rank conditions.

Findings

Bounds on canonical heights are derived for points outside a subgroup.

Results depend on the rank difference between the subgroup and the full group.

The bounds hold for large enough x and primes of good reduction.

Abstract

Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined over ${\mathbf Q}$ of rank $r \ge 1$ and any sufficiently large $x \ge 2$, assuming that the rank of $\Gamma$ is less than $r$, we give upper and lower bounds on the canonical height of a rational point $Q$ which is not in the group $\Gamma$ but belongs to the reduction of $\Gamma$ modulo every prime $p \le x$ of good reduction for $E$.