Computing in arithmetic groups with Voronoi's algorithm

Abstract

We describe an algorithm, meant to be very general, to compute a presentation of the group of units of an order in a (semi)simple algebra over Q. Our method is based on a generalisation of Vorono\"i's algorithm for computing perfect forms, combined with Bass-Serre theory. It differs essentially from previously known methods to deal with such questions, e.g. for units in quaternion algebras. We illustrate this new algorithm by a series of examples where the computations are carried out completely.