Lattice (discrete Subgroup) - S-arithmetic Lattices

S-arithmetic Lattices

Arithmetic lattices admit an important generalization, known as the S-arithmetic lattices. The first example is given by the diagonally embedded subgroup

This is a lattice in the product of algebraic groups over different local fields, both real and p-adic. It is formed by the unimodular matrices of order 2 with entries in the localization of the ring of integers at the prime p. The set S is a finite set of places of Q which includes all archimedean places and the locally compact group is the direct product of the groups of points of a fixed linear algebraic group G defined over Q (or a more general global field) over the completions of Q at the places from S. To form the discrete subgroup, instead of matrices with integer entries, one considers matrices with entries in the localization over the primes (nonarchimedean places) in S. Under fairly general assumptions, this construction indeed produces a lattice. The class of S-arithmetic lattices is much wider than the class of arithmetic lattices, but they share many common features.