Integral points on cubic twists of Mordell curves

Abstract

Fix a non-square integer $k\ne 0$. We show that the number of curves $E_B:y^2=x^3+k{B}^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N{)}^{-\frac{1}{2}+\epsilon })$. In particular, this implies that the number of positive integers $B\le N$ such that $-3k{B}^2$ is the discriminant of an elliptic curve over $\mathbb{Q}$ is $o(N)$. The proof involves a discriminant-lowering procedure on integral binary cubic forms.

References in corpus (1)

• Integral points on the congruent number curve