Lang's conjecture and sharp height estimates for the elliptic curves $y^{2}=x^{3}+b$

TL;DR

This paper proves Lang's conjecture for elliptic curves of the form y^2 = x^3 + b, providing sharp bounds on heights of points and showing many results are optimal.

Contribution

It establishes Lang's conjecture for a specific family of elliptic curves and derives precise bounds on height differences, many of which are optimal.

Findings

Proved lower bounds for canonical heights of non-torsion points.

Derived upper and lower bounds for the difference between canonical and logarithmic heights.

Many bounds obtained are sharp and best-possible.

Abstract

For $E_{b}: y^{2}=x^{3}+b$, we establish Lang's conjecture on a lower bound for the canonical height of non-torsion points along with upper and lower bounds for the difference between the canonical and logarithmic height. In many cases, our results are actually best-possible.