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 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 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.
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机译:我们导出具有任何规定根系统的n维单模晶格的质量公式。我们将Katsurada公式用于Siegel Eisenstein级数的傅立叶系数,以计算尺寸为 n italic> <30的偶数单模32维格和奇数单模格的所有根系的质量。 n italic> <30的偶数单模32维晶格的质量和无根的奇数单模晶格的质量,证明了Bacher和Venkov枚举在维数27和28上的正确性。与Minkowski-Siegel质量常数提供的尺寸相比,尺寸26至30中的不等价单模晶格的数量的下界更小。
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