Computing in arithmetic groups with Voronoï's algorithm

Oliver Braun (1) , Renaud Coulangeon (2) , Gabriele Nebe (1) , Sebastian Schönnenbeck (1)

1 Lehrstuhl II für Mathematik
2 IMB - Institut de Mathématiques de Bordeaux

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 QQ. Our method is based on a generalisation of Voronoï'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.

Mots clés: Unit groups of orders; Generators; Presentation; Word problem; Lattices; Voronoï's algorithm

Domaines: Théorie des nombres [math.NT]; Théorie des groupes [math.GR]

Oliver Braun, Renaud Coulangeon, Gabriele Nebe, Sebastian Schönnenbeck. Computing in arithmetic groups with Voronoï's algorithm. Journal of Algebra, 2015, 435, pp.263-285. ⟨10.1016/j.jalgebra.2015.01.022⟩. ⟨hal-01208022⟩