On the number of perfect lattices
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 917-945.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1057
Classification : 11H55, 11T06, 20K01, 05B30, 05E30
Mots clés : Perfect lattice
Roland Bacher 1

1 Institut Fourier Univ. Grenoble Alpes, CNRS 38000 Grenoble, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{JTNB_2018__30_3_917_0,
     author = {Roland Bacher},
     title = {On the number of perfect lattices},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {917--945},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1057},
     mrnumber = {3938634},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1057/}
}
TY  - JOUR
AU  - Roland Bacher
TI  - On the number of perfect lattices
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 917
EP  - 945
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1057/
DO  - 10.5802/jtnb.1057
LA  - en
ID  - JTNB_2018__30_3_917_0
ER  - 
%0 Journal Article
%A Roland Bacher
%T On the number of perfect lattices
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 917-945
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1057/
%R 10.5802/jtnb.1057
%G en
%F JTNB_2018__30_3_917_0
Roland Bacher. On the number of perfect lattices. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 917-945. doi : 10.5802/jtnb.1057. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1057/

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

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

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

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

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

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

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

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

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

Cité par Sources :