Computing $S$-unit groups of orders

TL;DR

This paper introduces an algorithm for computing presentations of $S$-unit groups of orders, leveraging the Voronoi algorithm and the strategy of Borel and Serre, and applies it to study the congruence subgroup property.

Contribution

It develops a new algorithm for $S$-unit groups of orders based on existing strategies and algorithms, enabling further investigations into their properties.

Findings

Successfully computed presentations of $S$-unit groups

Provided new insights into the congruence subgroup property

Enhanced computational methods for algebraic structures

Abstract

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of $S$-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.