no. 3
On the number of perfect lattices
Bacher, Roland1
1 Institut Fourier Univ. Grenoble Alpes, CNRS 38000 Grenoble, France
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 917-945.

We show that the number p d of non-similar perfect d-dimensional lattices satisfies eventually the inequalities e d 1-ϵ <p d <e d 3+ϵ for arbitrary small strictly positive ϵ.

Le nombre p d de classes de similitude de réseaux parfaits en dimension d vérifie asymptotiquement les inégalités e d 1-ϵ <p d <e d 3+ϵ pour ϵ>0 arbitrairement petit.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1057
Classification: 11H55, 11T06, 20K01, 05B30, 05E30
Keywords: Perfect lattice
Bacher, Roland 1

1 Institut Fourier Univ. Grenoble Alpes, CNRS 38000 Grenoble, France 
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Bacher, Roland. On the number of perfect lattices. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 917-945. doi : 10.5802/jtnb.1057. http://archive.numdam.org/articles/10.5802/jtnb.1057/

[1] Bacher, Roland Constructions of some perfect integral lattices with minimum 4, J. Théor. Nombres Bordx, Volume 27 (2015) no. 3, pp. 655-687 | DOI | Numdam | MR | Zbl

[2] Böttcher, Albrecht; Eisenbarth, Simon; Fukshansk, Lenny; Garcia, Stephan Ramon; Maharaj, Hiren Spherical 2-designs and lattices from Abelian groups, Discrete Comput. Geom. (2017) | DOI

[3] Böttcher, Albrecht; Fukshansky, Lenny; Garcia, Stephan Ramon; Maharaj, Hiren On lattices generated by finite abelian groups, SIAM J. Discrete Math., Volume 29 (2015) no. 1, pp. 382-404 | DOI | MR | Zbl

[4] Conway, John Horton; Sloane, Neil James Alexander Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290, Springer, 1999 | MR | Zbl

[5] Dutour Sikirić, Mathieu; Schürmann, Achill; Vallentin, Frank Classification of eight-dimensional perfect forms, Electron. Res. Announc. Am. Math. Soc., Volume 13 (2007), pp. 21-32 | MR | Zbl

[6] Gruber, B. Alternative formulae for the number of sublattices, Acta Crystallogr., Volume 53 (1997) no. 6, pp. 807-808 | DOI | MR | Zbl

[7] Keller, Wolfgang; Martinet, Jacques; Schürmann, Achill On classifying Minkowskian sublattices. With an Appendix by Mathieu Dutour Sikirić, Math. Comput., Volume 81 (2012) no. 278, pp. 1063-1092 | Zbl

[8] Kitaoka, Yoshiyuki Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, 106, Cambridge University Press, 1993 | MR | Zbl

[9] Martinet, Jacques Les réseaux parfaits des espaces euclidiens, Masson, 1996 (an English translation has been published by Springer) | Zbl

[10] Minkowski Geometrie der Zahlen, Teubner, 1896 | Zbl

[11] Nebe, Gabriele Boris Venkov’s theory of lattices and spherical designs, Diophantine methods, lattices, and arithmetic theory of quadratic forms (BIRS, 2011) (Contemporary Mathematics), Volume 587, American Mathematical Society, 2013, pp. 1-19 | DOI | MR | Zbl

[12] Soulé, Christophe Perfect forms and the Vandiver conjecture, J. Reine Angew. Math., Volume 517 (1999), pp. 209-221 | DOI | MR | Zbl

[13] Zou, Yi Ming Gaussian binomials and the number of sublattices, Acta Crystallogr., Volume 62 (2006) no. 5, pp. 409-410 | DOI | MR | Zbl

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