Integral points on cubic twists of Mordell curves

TL;DR

This paper establishes bounds on the number of integral points on certain cubic twists of Mordell curves, showing that such points are relatively rare among positive integers, using a discriminant-lowering method on binary cubic forms.

Contribution

It introduces a new bound on the count of integral points on cubic twists of Mordell curves, employing a discriminant-lowering technique on binary cubic forms.

Findings

Number of such curves with integral points is bounded by O(N(log N)^{-1/2+ε})

The count of B where -3kB^2 is a discriminant of an elliptic curve is o(N)

Method involves a discriminant-lowering procedure on binary cubic forms

Abstract

Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ 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 $B\leq N$ such that $-3kB^2$ 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.