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

## TL;DR

This paper investigates Lang's conjecture on the canonical height of rational points on elliptic curves, demonstrating that almost all positive rank curves satisfy a strong form of the conjecture in a statistical sense.

## Contribution

It provides a statistical proof that a density one subset of elliptic curves with positive rank adhere to a strong version of Lang's conjecture.

## Key findings

- Almost all positive rank elliptic curves satisfy the conjecture statistically.
- The paper establishes a density one subset of curves meeting the conjecture.
- It offers a probabilistic perspective on Lang's conjecture for 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 non-torsion rational points on elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that among the family of all elliptic curves defined over $\mathbb{Q}$ and having positive rank, there is a density one subfamily of curves which satisfy a strong form of Lang's conjecture.