GENERATORS OF ARITHMETIC QUATERNION GROUPS AND A DIOPHANTINE PROBLEM
Let p be a prime and a a quadratic non-residue ( mod p). Then the set of integral solutions of the Diophantine equation [Formula: see text] form a cocompact discrete subgroup Γp, a ⊂ SL(2, ℝ) which 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 Γp, 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.