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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Refined configuration results for extremal Type II lattices of ranks $ 40$ and $ 80$

Author(s): Noam D. Elkies; Scott Duke Kominers
Journal: Proc. Amer. Math. Soc. 138 (2010), 105-108.
MSC (2000): Primary 11H55; Secondary 05B30, 11F11
Posted: August 27, 2009
MathSciNet review: 2550174
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Abstract | References | Similar articles | Additional information

Abstract: We show that, if $ L$ is an extremal Type II lattice of rank $ 40$ or $ 80$, then $ L$ is generated by its vectors of norm $ \operatorname{min}(L)+2$. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices $ L$ are generated by their vectors of norms $ \operatorname{min}(L)$ and $ \operatorname{min}(L)+2$.

References:

[CS99]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer-Verlag, 1999. MR 1662447 (2000b:11077)

[Elk09]
N. D. Elkies, On the quotient of an extremal Type II lattice of rank $ 40$, $ 80$, or $ 120$ by the span of its minimal vectors, in preparation, 2009.

[KA08]
S. D. Kominers and Z. Abel, Configurations of rank-$ 40r$ extremal even unimodular lattices $ (r=1,2,3$), J. Théor. Nombres Bordeaux 20 (2008), 365-371. MR 2477509

[Kom09]
S. D. Kominers, Configurations of extremal even unimodular lattices, Int. J. Number Theory 5 (2009), 457-464.

[MOS75]
C. L. Mallows, A. M. Odlyzko, and N. J. A. Sloane, Upper bounds for modular forms, lattices and codes, J. Algebra 36 (1975), 68-76. MR 0376536 (51:12711)

[Oze86a]
M. Ozeki, On even unimodular positive definite quadratic lattices of rank $ 32$, Math. Z. 191 (1986), 283-291. MR 818672 (87h:11061)

[Oze86b]
-, On the configurations of even unimodular lattices of rank $ 48$, Arch. Math. (Basel) 46 (1986), 54-61. MR 829816 (87j:11063)

[Oze89]
-, On the structure of even unimodular extremal lattices of rank $ 40$, Rocky Mountain J. Math. 19 (1989), 847-862. MR 1043254 (91d:11072)

[Ser73]
J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973. MR 0344216 (49:8956)

[Ven84]
B. B. Venkov, Even unimodular Euclidean lattices in dimension $ 32$, J. Math. Sci. (N.Y.) 26 (1984), 1860-1867. MR 0687838 (84g:10058)

[Ven01]
-, Réseaux et designs sphériques, Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, Monographie de L'Enseignement Mathematique, vol. 37, Enseignement Mathematique, Genève, 2001 (in French), pp. 10-86. MR 1878745 (2002m:11061)

[Vil68]
N. J. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, vol. 22, American Mathematical Society, 1968. MR 0229863 (37:5429)

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Additional Information:

Noam D. Elkies
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: elkies@math.harvard.edu

Scott Duke Kominers
Affiliation: Department of Mathematics and Department of Economics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Address at time of publication: 8520 Burning Tree Road, Bethesda, Maryland 20817
Email: kominers@fas.harvard.edu, skominers@gmail.com

DOI: 10.1090/S0002-9939-09-10063-1
PII: S 0002-9939(09)10063-1
Keywords: Type II lattice, extremal lattice, weighted theta function, spherical design, configuration result
Received by editor(s): May 29, 2009
Posted: August 27, 2009
Communicated by: Ken Ono
Copyright of article: Copyright 2009, Noam D. Elkies and Scott Duke Kominers




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