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 $d\equiv 2,3(\mathrm{mod}4)$ 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 $m\ne 0,\sqrt{2}$ the Mordell--Weil group $C_m(\mathbb{K})$ contains no rational point of order two.