The average number of integral points on the congruent number curves

Abstract

We show that the total number of non-torsion integral points on the elliptic curves ${\mathcal{E}}_D:y^2=x^3-D^{2}x$, 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.

21 pages

References in corpus (3)

• Integral points on the congruent number curve
• A positive proportion of locally soluble quartic Thue equations are globally insoluble
• Quadratic Fields Admitting Elliptic Curves with Rational $j$-Invariant and Good Reduction Everywhere