Presentations for Quaternionic S-Unit Groups

In this paper, we give an algorithm for presenting S-unit groups of an order O in a definite rational quaternion algebra B such that, for every p ∈ S at which B splits, the localization of O at p is maximal and all left ideals of O of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field Q. To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. We also include some discussion and experimentation related to the congruence subgroup problem, which is open for S-units of the Hurwitz order when S contains at least two odd primes. We now introduce the objects studied in this paper, assuming some familiarity with the theory of quaternion algebras over number fields, e.g., from [10]. Let B denote a definite quaternion algebra over Q and O ⊂ B an order, that is, a Z-order of full rank. Tensoring over the real numbers, we have B ⊗Q R ∼= H, where H denotes Hamilton’s quaternions. For each prime p, let Bp = B ⊗Q Qp be the completion of B at p. When p splits B,