2002/06/25 by Oliver D. King · 58 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Advanced Combinatorial Mathematics #Unimodular matrix #Algorithm #Dimension (graph theory) #Annotation #Type (biology) #Computer science #Mathematics #Artificial intelligence #Combinatorics #Geology
paper · pdf · doi:10.1090/s0025-5718-02-01455-2
published in Mathematics of Computation 72(242), 839-863 (American Mathematical Society)
openalex publication_date 2002/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
We derive a mass formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 30"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>30</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n≤ 30</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 30"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>30</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n≤ 30</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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.