On the ${\zeta }_3$-Pell equation

Abstract

Let $K=\mathbb{Q}({\zeta }_3)$, where ${\zeta }_3$ is a primitive root of unity. In this paper we study the distribution of integers $\alpha \in {\mathcal{O}}_K$ for which the norm equation $N_{K(\sqrt[3]{\alpha })/K}(\mathbf{x})={\zeta }_3$ is solvable for integers $\mathbf{x}\in {\mathcal{O}}_{K(\sqrt[3]{\alpha })}$. The analogous question for ${\zeta }_2=-1$ is the well-known negative Pell equation. We also address the natural generalization of Stevenhagen's conjecture on the negative Pell equation in this setting.

Removed the dependence on GRH; lower bounds and upper bounds improved