Arithmetic of Quaternion Groups

TL;DR

This paper investigates the traces of generators for quaternion groups derived from specific diophantine equations, revealing irregular fluctuations similar to Pell's equation solutions, with insights from quaternion algebra geometry and arithmetic.

Contribution

It provides empirical estimates and analysis of generator traces in quaternion groups related to diophantine equations, highlighting their complex fluctuation patterns.

Findings

Trace fluctuations are significant and irregular.

Behavior resembles solutions to Pell's equation.

Quaternion algebra geometry influences trace properties.

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.