Presentations for quaternionic S-unit groups

Abstract

© 2015 Taylor and Francis Group, LLC. We give an algorithm for presenting S-unit groups of an order in a definite rational quaternion algebra B such that for every p S at which B splits, the localization of at p is maximal, and all left ideals of of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamiltons quaternions over the rational field. 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.