A mass formula for unimodular lattices with no roots

King, Oliver

doi:10.1090/S0025-5718-02-01455-2

A mass formula for unimodular lattices with no roots
HTML articles powered by AMS MathViewer

by Oliver D. King PDF

Math. Comp. 72 (2003), 839-863

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

Roland Bacher, Unimodular lattices without nontrivial automorphisms, Internat. Math. Res. Notices 2 (1994), 91–95. MR 1264932, DOI 10.1155/S1073792894000115
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.
Etsuko Bannai, Positive definite unimodular lattices with trivial automorphism groups, Mem. Amer. Math. Soc. 85 (1990), no. 429, iv+70. MR 1004770, DOI 10.1090/memo/0429
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].
Richard E. Borcherds, Classification of positive definite lattices, Duke Math. J. 105 (2000), no. 3, 525–567. MR 1801771, DOI 10.1215/S0012-7094-00-10536-4
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. IV. The mass formula, Proc. Roy. Soc. London Ser. A 419 (1988), no. 1857, 259–286. MR 965484
J. H. Conway and N. J. A. Sloane, A note on optimal unimodular lattices, J. Number Theory 72 (1998), no. 2, 357–362. MR 1651697, DOI 10.1006/jnth.1998.2257
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to algorithms, The MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA; McGraw-Hill Book Co., New York, 1990. MR 1066870
Noam D. Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2 (1995), no. 5, 643–651. MR 1359968, DOI 10.4310/MRL.1995.v2.n5.a9
Noam D. Elkies and Benedict H. Gross, The exceptional cone and the Leech lattice, Internat. Math. Res. Notices 14 (1996), 665–698. MR 1411589, DOI 10.1155/S1073792896000426
Michael R. Garey and David S. Johnson, Computers and intractability, A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness. MR 519066
Hidenori Katsurada, An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree $3$, Nagoya Math. J. 146 (1997), 199–223. MR 1460959, DOI 10.1017/S0027763000006267
Hidenori Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999), no. 2, 415–452. MR 1680317, DOI 10.1353/ajm.1999.0013
Günter Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Math. Ann. 137 (1959), 454–476 (German). MR 121485, DOI 10.1007/BF01360845
Michel Kervaire, Unimodular lattices with a complete root system, Enseign. Math. (2) 40 (1994), no. 1-2, 59–104. MR 1279061
O. D. King, Table of masses of even unimodular 32-dimensional lattices with any given root system. Available at arXiv:math.NT/0012231
Yoshiyuki Kitaoka, A note on local densities of quadratic forms, Nagoya Math. J. 92 (1983), 145–152. MR 726146, DOI 10.1017/S0027763000020626
Yoshiyuki Kitaoka, Local densities of quadratic forms and Fourier coefficients of Eisenstein series, Nagoya Math. J. 103 (1986), 149–160. MR 858478, DOI 10.1017/S0027763000000647
Yoshiyuki Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106, Cambridge University Press, Cambridge, 1993. MR 1245266, DOI 10.1017/CBO9780511666155
Yoshiyuki Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J. 95 (1984), 73–84. MR 759464, DOI 10.1017/S0027763000020973
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Helmut Koch and Boris B. Venkov, Über ganzzahlige unimodulare euklidische Gitter, J. Reine Angew. Math. 398 (1989), 144–168 (German). MR 998477
John Leech, Notes on sphere packings, Canadian J. Math. 19 (1967), 251–267. MR 209983, DOI 10.4153/CJM-1967-017-0
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag, Berlin 1973.
G. Nebe and N. J. A. Sloane, A Catalogue of Lattices, published electronically at http://www.research.att.com/$\sim$njas/lattices/
Hans-Volker Niemeier, Definite quadratische Formen der Dimension $24$ und Diskriminante $1$, J. Number Theory 5 (1973), 142–178 (German, with English summary). MR 316384, DOI 10.1016/0022-314X(73)90068-1
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
M. Peters, On even unimodular 32-dimensional lattices, Preprint SFB 478, Mathematischen Instituts der Westfälischen Wilhelms-Universität Münster, January 2001.
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81, DOI 10.1215/S0012-7094-39-00549-1
G. L. Watson, Integral quadratic forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 51, Cambridge University Press, New York, 1960. MR 0118704