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A mass formula for unimodular lattices with no roots

Author: Oliver D. King
Journal: Math. Comp. 72 (2003), 839-863
MSC (2000): Primary 11H55; Secondary 11E41
Published electronically: June 25, 2002
MathSciNet review: 1954971
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Abstract: 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.

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Additional Information

Oliver D. King
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 250 Longwood Avenue, SGMB-322, Boston, Massachusetts 02115

Received by editor(s): March 29, 2001
Received by editor(s) in revised form: May 8, 2001
Published electronically: June 25, 2002
Additional Notes: This work was partially supported by grants from the NSF and the Royal Society
Article copyright: © Copyright 2002 Oliver D. King

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