ABSTRACT
In the first half of the 20th century Siegel considered arithmetic subgroups of the classical groups, generalizing many aspects of the reduction theory. An arithmetic group is the group of integral points of an algebraic group defined over the rational numbers. This chapter focuses on the problem to devise algorithms that, given an algebraic group, find generators of an arithmetic subgroup. It is concerned with problem, that is, to compute an arithmetic subgroup. This problem splits into two subcases: G unipotent, and G reductive. The chapter shows how to obtain generators of the unit group of an order in a semisimple commutative matrix algebra. It describes the algorithm for tori: the first step is to embed a torus in a larger torus that is the unit group of a commutative associative algebra, the second step is to obtain generators of the arithmetic group corresponding to this larger algebraic group.
