Diophantine Equation $x_1^{3}-x_2^{2}x_1+1=0$ over number fields

Abstract

Let $\mathbb{K}=\mathbb{Q}(\sqrt{-d})\text{ or }\mathbb{Q}(\sqrt{d})$, where $d$ is a positive square-free integer, and denote by $\mathcal{O}_\mathbb{K}$ the ring of integers of $\mathbb{K}$. I investigate the solution of the equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\in \mathbb{K}$ and $x_2\in\mathcal{O}_\mathbb{K}$. The case $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ for $d\equiv 2,3\pmod{4}$ faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings $\mathbb{Z}[\sqrt{d}]]$, together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\text{ and }~(1,\sqrt{2})$ for $\mathbb{K}=\mathbb{Q}(\sqrt{2})$ and one solution $(-1,0)$ for $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\in\mathcal{O}_\mathbb{K},$ and deduce that, for every $m\neq0,\sqrt{2}$ the Mordell--Weil group $C_m(\mathbb{K})$ contains no rational point of order two.