A mass formula for unimodular lattices with no roots

Abstract

We derive a mass formula for n-dimensional unimodular lattices having any prescribed root system. We use Katsurada&#39;s formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension n < 31. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension n < 31, verifying Bacher and Venkov&#39;s enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants. The ASCII text file table.txt (1.3 MB) at arXiv:math.NT/0012231 includes the complete table of masses of 32-dimensional even unimodular lattices with any given root system, and may be dowloaded using the source link.