Height of rational points on congruent number elliptic curves

TL;DR

This paper proves that a positive proportion of congruent number elliptic curves have their lowest non-torsion rational point's height bounded below by a strong constant, advancing understanding of rational points on these curves.

Contribution

It establishes a positive density result for the height of the lowest non-torsion rational points on congruent number elliptic curves, with a new lower bound.

Findings

A positive proportion of congruent number elliptic curves have a canonical height above a certain threshold.

The result links the distribution of congruent numbers to the height of rational points.

Provides new insights into the arithmetic of elliptic curves related to congruent numbers.

Abstract

We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.