Volumes and distributions for random unimodular complex and quaternion lattices

Abstract

Two themes associated with invariant measures on the matrix groups SLN(F), with F=R,C or H, and their corresponding lattices parametrised by SLN(F)/SLN(O), O being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of SLN(F) with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in SLN(O). The second is the problem of lattice reduction in the case N=2. A unified proof of the validity of the appropriate analogue of the Lagrange–Gauss algorithm for computing the shortest basi..