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

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.