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

Abstract

Given coprime positive integers $d^{\prime },d^{\prime \prime }$, Bézout's Lemma tells us that there are integers $u,v$ so that $d^{\prime }u-d^{\prime \prime }v=1$. We show that, interchanging $d^{\prime }$ and $d^{\prime \prime }$ 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 $\mathcal{O}\subset \mathcal{H}$.

20 pages, comments welcome

Cited by in corpus (1)

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