Lang's Conjecture and Sharp Height Estimates for the elliptic curves $y^{2}=x^{3}+ax$

TL;DR

This paper proves optimal bounds related to Lang's conjecture for elliptic curves of the form y^2 = x^3 + ax, providing precise estimates for heights of points.

Contribution

It establishes the best-possible lower bounds for the canonical height and tight bounds for the difference between canonical and logarithmic heights on these elliptic curves.

Findings

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

Precise upper and lower bounds for height differences.

Validation of conjectured bounds in specific elliptic curve cases.

Abstract

For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.