Generators of Arithmetic Quaternion Groups and a Diophantine Problem

TL;DR

This paper investigates the generators of certain arithmetic quaternion groups derived from a specific Diophantine equation, providing empirical trace estimates and exploring their complex fluctuations related to quaternion algebra units.

Contribution

It offers new empirical insights into the trace behavior of generators for quaternion groups associated with a Diophantine problem, connecting geometry and arithmetic of quaternion units.

Findings

Trace fluctuations are significant and irregular

Behavior resembles solutions of Pell's equation

Empirical data summarized in several tables

Abstract

Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ 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.