A note on rank zero quadratic twists of a Mordell curve

TL;DR

The paper investigates rank zero quadratic twists of the Mordell curve $y^2=x^3+2$ by focusing on the twists $E_{-D}$ and $E_{3D}$ with $D\equiv 2\pmod{3}$ square-free. It combines Scholz reflection, the Ankeny–Artin–Chowla relation, and descent techniques in $\mathbb{Q}(\sqrt{-2D})$ to show that, under $3\nmid h(-2D)$, both twists have trivial Mordell–Weil groups, i.e., $r(E_{-D})=r(E_{3D})=0$; conditional on Cohen–Lenstra heuristics, infinitely many such $D$ are expected. The paper also provides numerical evidence for $D\le 10000$ validating rank zero in observed cases. These results contribute to understanding the distribution of ranks in quadratic twist families of Mordell curves and highlight connections between elliptic curves, class numbers, and unit groups of related quadratic fields.

Abstract

We produce two families of rank zero quadratic twists of the Mordell curve $y^2=x^3+2$. At the end, we give numerical examples supporting the result.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem A
• Theorem B
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.1