Asymptotic class numbers of lattices

Abstract

We study the frequency of two- and three-dimensional integral lattices with a given automorphism group G. We estimate the number of isomorphism classes H(D) of lattices with determinant at most D.

A general approach is given by counting the possible reduced Gram matrices of lattices. If the underlying reduction theory admits only one reduced lattice per isomorphism class, this gives us H(D). We use the fact that D implies bounds on the entries of the Gram matrix. We have to evaluate multiple sums with linear constraints on the coefficients of the Gram matrices. In certain cases the asymptotic behaviour of H(D) can be derived directly. An additional method exploits a theorem of Delange on Dirichlet series. We first calculate a corresponding Dirichlet series F(s) with coefficients given by the class number of a fixed determinant, and then use the analytic properties of F(s) (position of poles and their orders).

As a general result, we have: For D tending towards infinity, the function H(D) grows like some multiple of a power of D. The more symmetries a lattice has (that is, the higher the order of its automorphism groups is) the smaller is the rate of growth given by the exponent.