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

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.