Generators of Arithmetic Quaternion Groups and a Diophantine Problem

Abstract

Let $p$ be a prime and $a$ a quadratic non-residue $\mathrm{mod}p$. Then the set of integral solutions of the diophantine equation $x_0^2-a{x}_1^2-p{x}_2^2+ap{x}_3^2=1$ form a cocompact discrete subgroup ${\Gamma }_{p,a}\subset SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an estimate for the traces of a set of generators for ${\Gamma }_{p,a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a generator for the solutions of Pell's equation. The geometry and arithmetic of the group of units of an order in a quaternion algebra play a key role in the development of the code for the purpose of this paper.

15 pages; 5 figures; experimental results