The average number of integral points on the congruent number curves

TL;DR

This paper establishes an upper bound on the total number of non-torsion integral points on a family of elliptic curves associated with congruent numbers, using advanced number theoretic techniques.

Contribution

It introduces a discriminant-lowering method and applies Heath-Brown's approach to estimate the average size of the 2-Selmer group for these curves.

Findings

Total non-torsion integral points grow at most as N(log N)^{-1/4+ε}

Discriminant-lowering procedure effectively bounds integral points

Average size of 2-Selmer group estimated for the family of curves

Abstract

We show that the total number of non-torsion integral points on the elliptic curves $\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\log N)^{-1/4+\epsilon})$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the $2$-Selmer group of the curves in this family.