Arithmetic of Quaternion Groups

2009/02/22 by Jahangiri, Majid
#11F06 #20H10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0902.3794

Abstract

Let p be a prime and a a quadratic non-residue \bmod p. Then the set of integral solutions of the diophantine equation x₀² - ax₁² -px₂² + apx₃²=1 form a cocompact discrete subgroup Γ_p,a⊂ SL(2,ℝ) and is commensurable with the group of units of an order in a quaternion algebra over ℚ. The problem addressed in this paper is an estimate for the traces of a set of generators for Γ_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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