Computing in arithmetic groups with Voronoi's algorithm

TL;DR

This paper introduces a versatile algorithm combining Voronoi's method and Bass-Serre theory to compute presentations of unit groups in orders of semisimple algebras over Q, demonstrated through comprehensive examples.

Contribution

It presents a novel, general algorithm for computing unit group presentations in semisimple algebra orders, extending beyond prior quaternion-specific methods.

Findings

Successfully computes unit group presentations in various examples.

Demonstrates the algorithm's generality and effectiveness.

Provides complete computational examples.

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.