Presentations for Quaternionic S-Unit Groups

Abstract

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 Hamilton’s 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.

2000 AMS Subject Classification::

• 20H10
• 22E40
• 11F06
• 16H10
• 16U60
• 20F05

Keywords::

• S-unit groups
• S-arithmetic lattices
• presentations for S-arithmetic lattices
• Hurwitz order
• congruence subgroup problem