Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

   
 
 

 

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
DOI: https://doi.org/10.1090/S0025-5718-02-01455-2
Published electronically: June 25, 2002
MathSciNet review: 1954971
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. R. Bacher, Unimodular lattices without nontrivial automorphisms, Internat. Math. Res. Notes 2 (1994) 91-95. MR 95b:11067
  • 2. R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racines en dimension 27 et 28, Réseaux euclidiens, designs sphériques et formes modulaires, 212-267, Monogr. Enseign. Math., 37, Enseignement Math., Geneva, 2001.
  • 3. E. Bannai, Positive definitive unimodular lattices with trivial automorphism group,Mem. Amer. Math. Soc. 429 (1990) 1-70. MR 90j:11030
  • 4. R. E. Borcherds, The Leech lattice and other lattices, Ph.D. Dissertation, University of Cambridge, 1984. Available at arXiv:math.NT/9911195 Much of this material also appears in [5].
  • 5. R. E. Borcherds, Classification of positive definite lattices, Duke Math. J. 105 (2000), no. 3, 525-567. Available at arXiv:math.NT/9912236 MR 2001k:11057
  • 6. R. E. Borcherds, E. Freitag and R. Weissauer, A Siegel cusp form of degree 12 and weight 12, J. Reine Angew. Math. 494 (1998) 141-153. MR 99d:11047 Available at arXiv:math.AG/9805132
  • 7. J. H. Conway and N. J. A. Sloane, Low-dimensional lattices IV: The mass formula, Proc. R. Soc. Lond. A 419 (1988), 259-286. MR 90a:11074
  • 8. J. H. Conway and N. J. A. Sloane, A note on optimal unimodular lattices, J. Number Theory 72 (1998), 357-362. MR 99k:11104
  • 9. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd edition, 1998. MR 2000b:11077
  • 10. T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, Cambridge, MA, 1990. MR 91i:68001
  • 11. N. D. Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2 (1995) 643-651. MR 96h:11065
  • 12. N. D. Elkies and B. H. Gross, The exceptional cone and the Leech lattice, Internat. Math. Res. Notices 14 (1996), 665-698. MR 97g:11070
  • 13. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, SF, 1979. MR 80g:68056
  • 14. H. Katsurada, An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree 3, Nagoya Math J. 146 (1997), 199-223. MR 98g:11051
  • 15. H. Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999), 415-452. MR 2000a:11068
  • 16. G. Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2: Grades, Math. Ann. 137 (1959), 454-476. MR 22:12223
  • 17. M. Kervaire, Unimodular lattices with a complete root system, L'Enseign. Math. 40 (1994), 59-140. MR 95g:11063
  • 18. O. D. King, Table of masses of even unimodular 32-dimensional lattices with any given root system. Available at arXiv:math.NT/0012231
  • 19. Y. Kitaoka, A note on local densities of quadratic forms, Nagoya Math J. 92 (1983), 145-152. MR 85e:11029
  • 20. Y. Kitaoka, Local densities of quadratic forms and Fourier coefficients of Eisenstein series, Nagoya Math J. 103 (1986), 149-160. MR 87m:11041
  • 21. Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Math., vol. 106, Cambridge Univ. Press, Cambridge, 1993. MR 95c:11044
  • 22. Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math J. 95 (1984), 73-84. MR 86b:11038
  • 23. M. Kneser, Klassenzahlen definiter quadratischer Formen, Arch. Math. 8 (1957), 241-250. MR 19:838c
  • 24. H. Koch and B. B. Venkov, Über ganzzahlige unimodulare euklidische Gitter, J. Reine Angew. Math. 398 (1989), 144-168. MR 90g:11082
  • 25. J. Leech, Notes on sphere packings, Canadian J. Math. 19 (1967), 251-267. MR 35:878
  • 26. H. Maaß, Die Fourierkoeffizienten der Eisensteinreihen zweiten Grades, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (1973), 1-13. MR 58:22129
  • 27. J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag, Berlin 1973.
  • 28. G. Nebe and N. J. A. Sloane, A Catalogue of Lattices, published electronically at http://www.research.att.com/$\sim$njas/lattices/
  • 29. H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number Theory 5 (1973), 142-178. MR 47:4931
  • 30. R. E. O'Connor and G. Pall, The construction of integral quadratic forms of determinant 1, Duke Math. J. 11 (1944), 319-331. MR 5:254e
  • 31. M. Peters, On even unimodular 32-dimensional lattices, Preprint SFB 478, Mathematischen Instituts der Westfälischen Wilhelms-Universität Münster, January 2001.
  • 32. J.-P. Serre, A Course in Arithmetic, Springer-Verlag, NY 1973. MR 49:8956
  • 33. B. B. Venkov, The classification of integral even unimodular 24-dimensional quadratic forms. Trudy Matematicheskogo Instituta imeni V. A. Steklova 148 (1978), 65-76. Also Chapter 18 of [9]; also Proc. Steklov Inst. Math. 1980, no. 4 (148), 63-74. MR 81d:11024
  • 34. G. L. Watson, Integral Quadratic Forms, Cambridge Univ. Press, Cambridge, 1960. MR 22:9475

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11H55, 11E41

Retrieve articles in all journals with MSC (2000): 11H55, 11E41


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
Email: ok@csua.berkeley.edu

DOI: https://doi.org/10.1090/S0025-5718-02-01455-2
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

American Mathematical Society