THE DISTRIBUTION OF SUBLATTICES OF Z(M)

Lattices Lambda, Lambda' are similar if one can be transformed into th e other by an angle-preserving linear map. Similarity classes of latti ces of rank n may be parametrized by a fundamental domain F of the act ion of GL(n)(Z) on the generalized upper half-plane Hn. Given 1 < n le ss than or equal to m and D subset of F, let N(D, T) be the number of sublattices of Z(m) which have rank n, similarity class in D, and dete rminant less than or equal to T. Our most basic result will be that N( D, T) similar to c(1)(m, n)mu(D)Tm as T --> infinity for suitable sets D, where mu is the invariant measure on Hn. The case n = 2 had been d ealt with by Roelcke and by Maass using the theory of modular forms.