A refinement of B\'ezout's Lemma, and order 3 elements in some quaternion algebras over $\mathbb{Q}$

TL;DR

This paper refines Be9zout's Lemma by showing that the solutions can be chosen as Loeschian numbers using quaternion algebra techniques, and applies this to count conjugacy classes of order 3 elements.

Contribution

It introduces a method to select Be9zout solutions as Loeschian numbers via quaternion algebra, enabling enumeration of order 3 elements in certain orders.

Findings

Solutions to Be9zout's Lemma can be chosen as Loeschian numbers.

The method counts conjugacy classes of order 3 elements in quaternion orders.

Uses Atkin-Lehner elements in quaternion algebras for the analysis.

Abstract

Given coprime positive integers $d',d''$, B\'ezout's Lemma tells us that there are integers $u,v$ so that $d'u-d''v=1$. We show that, interchanging $d'$ and $d''$ if necessary, we may choose $u$ and $v$ to be Loeschian numbers, i.e., of the form $|\alpha|^2$, where $\alpha\in\mathbb{Z}[j]$, the ring of integers of the number field $\mathbb{Q}(j)$, where $j^2+j+1=0$. We do this by using Atkin-Lehner elements in some quaternion algebras $\mathcal{H}$. We use this fact to count the number of conjugacy classes of elements of order 3 in an order $\mathcal{O}\subset\mathcal{H}$.