Volumes and distributions for random unimodular complex and quaternion lattices

Abstract

Two themes associated with invariant measures on the matrix groups ${\rm SL}_N(\mathbb F)$, with $\mathbb F = \mathbb R, \mathbb C$ or $\mathbb H$, and their corresponding lattices parametrised by ${\rm SL}_N(\mathbb F)/{\rm SL}_N(\mathbb O)$, $\mathbb O$ being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of ${\rm SL}_N(\mathbb F)$ 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 ${\rm SL}_N(\mathcal O)$. 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 basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With $\mathbb F = \mathbb C$ this allows for the exact computation of the PDF of the first minimum (for $\mathcal O = \mathbb Z[i]$ and $\mathbb Z[(1+\sqrt{-3})/2]$), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for $\mathcal O = \mathbb Z[i]$). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for $\mathbb F = \mathbb H$ and $\mathcal O$ the Hurwitz integers, for which direct integration was not possible.