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A mass formula for unimodular lattices with no roots. (English) Zbl 1099.11035
Summary: We derive a mass formula for \(n\)-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 dimension \(n\leq 30\). 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 dimension \(n\leq 30\), 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.

MSC:
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E41 Class numbers of quadratic and Hermitian forms
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