A mass formula for unimodular lattices with no roots

Abstract

We derive a mass formula fornn-dimensional unimodular lattices having any prescribed root system. We use Katsurada’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 dimensionn≤30n\leq 30. 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 dimensionn≤30n\leq 30, verifying Bacher and Venkov’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.