ABSTRACT
Often, classical number-theoretic problems can be re-phrased in terms of groups; the groups which arise in this context are called arithmetic groups. For instance, the classical Dirichlet unit theorem can be interpreted and generalized in this framework. A prime example of an arithmetic group is the modular group S L(2,Z). It is a discrete subgroup of S L(2,R) such that the quotient space S L(2,Z)\S L(2,R) is non-compact, but has a finite S L(2,R)-invariant measure. One defines a discrete subgroup Γ of a Lie group G to be an arithmetic group if there exists an algebraic group H defined over Q such that the group H(R) of real points is isomorphic to G × K for a compact group K and Γ is ‘commensurable’ with the image of H(Z) in G. Here, commensurability of two groups means that their intersection is of finite index in both. Some examples of arithmetic groups are:
(i) any finite group,
(ii) a free abelian group of finite rank,
(iii) the group of units in the ring of integers of an algebraic number field,
(iv) GL(n,Z), S L(n,Z), S p2n(Z), S Ln(Z[i]), SO(Q)(Z) := {g ∈ S Ln(Z) : tgQg = Q} where Q is a quadratic form in n variables over Q.
