Arithmetic Subgroups of Algebraic Groups by Armand Borel and Harish-chandra

A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {PM[Xn, • • • , X n n ]} in n 2 indeterminates. I t is said to be defined over a field KQC if the polynomials can be chosen so as to have coefficients in K. Given a subring B of C, we denote by GB the subgroup of elements of G which have coefficients in JB, and whose determinant is a unit of B. Assume in particular G to be defined over Q. Then Gz is an "arithmetically defined discrete subgroup" of GRl or, more briefly, an arithmetic subgroup of GR. A typical example is the group of units of a nondegenerate integral quadratic form, and as a matter of fact, the main results stated below generalize facts known in this case from reduction theory. The proofs will be published elsewhere.