A statistical view on the conjecture of Lang about the canonical height on elliptic curves

Abstract

We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of nontorsion rational points on elliptic curves defined over Q \mathbb {Q} . More specifically, we prove that among the family of all elliptic curves defined over Q \mathbb {Q} and having positive rank, there is a density one subfamily of curves which satisfy a strong form of Lang’s conjecture.