Integral points on the congruent number curve

Abstract

We study integral points on the quadratic twists ${\mathcal{E}}_D:y^2=x^3-D^{2}x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2{\mathcal{E}}_D(\mathbb{Q})$ in ${\mathcal{E}}_D(\mathbb{Q})$ and show that their total is $\ll (3.8{)}^{\mathrm{rank}{\mathcal{E}}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $a{X}^2-b{Y}^2=d$, $b{Y}^2-c{Z}^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6{)}^{\omega (abcd)}$ integer solutions.

24 pages

References in corpus (1)

• The average number of integral points on elliptic curves is bounded

Cited by in corpus (2)

• Integral points on cubic twists of Mordell curves
• The average number of integral points on the congruent number curves