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Arithmetic of Quaternion Groups (0902.3794v2)

Published 22 Feb 2009 in math.NT

Abstract: Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_02 - ax_12 -px_22 + apx_32=1$ 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.

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