Geometry of numbers, class group statistics and free path lengths

Abstract: This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies.In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants.In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures.In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.