Heights of Rational Points on Mordell Curves

TL;DR

This paper proposes a conjecture on the minimal canonical height of rational points on Mordell curves and provides partial results towards proving it, focusing on a large subset of sextic twists.

Contribution

It introduces a new conjecture on height bounds for rational points on Mordell curves and establishes partial lower bounds for a significant subset of these curves.

Findings

Conjectured a lower bound for minimal canonical heights on Mordell curves.

Proved a partial lower bound for a density 1 subset of sextic twists.

Advances understanding of rational points' distribution on Mordell curves.

Abstract

We conjecture a lower bound for the minimal canonical height of non-torsion rational points on a natural density 1 subset of the sextic twist family of Mordell curves. We then establish a lower bound that yields a partial result towards this conjecture.