Integral points on the congruent number curve

Abstract

We study integral points on the quadratic twists E D : y 2 = x 3 − D 2 x \mathcal {E}_D:y^2=x^3-D^2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2 E D ( Q ) 2\mathcal {E}_D(\mathbb {Q}) in E D ( Q ) \mathcal {E}_D(\mathbb {Q}) and show that their total is ≪ ( 3.8 ) rank ⁡ E D ( Q ) \ll (3.8)^{\operatorname {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 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 aX^2-bY^2=d,\ bY^2-cZ^2=d for pairwise coprime positive integers a , b , c , d a,b,c,d , has at most ≪ ( 3.6 ) ω ( a b c d ) \ll (3.6)^{\omega (abcd)} integer solutions.