Un réseau fortement modulaire est dit s-extrémal, s’il maximise le minimum du réseau et son ombre simultanément. La dimension des réseaux s-extrémaux dont le minimum est pair peut être bornée par la théorie des formes modulaires. En particulier de tels réseaux sont extrémaux.
S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.
@article{JTNB_2007__19_3_683_0,
author = {Gabriele Nebe and Kristina Schindelar},
title = {S-extremal strongly modular lattices},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {683--701},
publisher = {Universit\'e Bordeaux 1},
volume = {19},
number = {3},
year = {2007},
doi = {10.5802/jtnb.608},
zbl = {1196.11097},
mrnumber = {2388794},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.608/}
}
Gabriele Nebe; Kristina Schindelar. S-extremal strongly modular lattices. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 683-701. doi : 10.5802/jtnb.608. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.608/
[1] N.G. de Bruijn, Asymptotic methods in analysis. 2nd edition, North Holland (1961). | Zbl 0109.03502
[2] J. Cannon et al., The Magma Computational Algebra System for Algebra, Number Theory and Geometry. Published electronically at .
[3] J. H. Conway, N. J. A. Sloane, A note on optimal unimodular lattices. J. Number Theory 72 (1998), no. 2, 357–362. | MR 1651697 | Zbl 0917.11027
[4] J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups. Springer, 3. edition, 1998. | Zbl 0915.52003
[5] N. D. Elkies, Lattices and codes with long shadows. Math. Res. Lett. 2 (1995), no. 5, 643–651. | MR 1359968 | Zbl 0854.11021
[6] P. Gaborit, A bound for certain s-extremal lattices and codes. Archiv der Mathematik 89 (2007), 143–151. | MR 2341725 | Zbl 1127.11046
[7] M. Kneser, Klassenzahlen definiter quadratischer Formen. Archiv der Math. 8 (1957), 241–250. | MR 90606 | Zbl 0078.03801
[8] C. L. Mallows, A. M. Odlysko, N. J. A. Sloane, Upper bounds for modular forms, lattices and codes. J. Alg. 36 (1975), 68–76. | MR 376536 | Zbl 0311.94002
[9] G. Nebe, Strongly modular lattices with long shadow. J. T. Nombres Bordeaux 16 (2004), 187–196. | Numdam | MR 2145580 | Zbl 1078.11047
[10] G. Nebe, B. Venkov, Unimodular lattices with long shadow. J. Number Theory 99 (2003), 307–317. | MR 1968455 | Zbl 1081.11049
[11] H.-G. Quebbemann, Atkin-Lehner eigenforms and strongly modular lattices. L’Ens. Math. 43 (1997), 55–65. | Zbl 0898.11014
[12] E.M. Rains, New asymptotic bounds for self-dual codes and lattices. IEEE Trans. Inform. Theory 49 (2003), no. 5, 1261–1274. | MR 1984825 | Zbl 1063.94123
[13] E.M. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices. J. Number Th. 73 (1998), 359–389. | MR 1657980 | Zbl 0917.11026
[14] R. Scharlau, R. Schulze-Pillot, Extremal lattices. In Algorithmic algebra and number theory, Herausgegeben von B. H. Matzat, G. M. Greuel, G. Hiss. Springer, 1999, 139–170. | MR 1672117 | Zbl 0944.11012
[15] K. Schindelar, Stark modulare Gitter mit langem Schatten. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen (2006).
[16] E.T. Whittaker, G.N. Watson, A course of modern analysis (4th edition) Cambridge University Press, 1963. | MR 1424469