Geodesic
Unimodular Hunting II
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e136

Voir la notice de l'article provenant de la source Cambridge University Press

Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete. 
Allombert, Bill; Chenevier, Gaëtan. Unimodular Hunting II. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e136. doi: 10.1017/fms.2025.10058
@article{10_1017_fms_2025_10058,
     author = {Allombert, Bill and Chenevier, Ga\"etan},
     title = {Unimodular {Hunting} {II}},
     journal = {Forum of Mathematics, Sigma},
     pages = {e136},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.10058},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10058/}
}
TY  - JOUR
AU  - Allombert, Bill
AU  - Chenevier, Gaëtan
TI  - Unimodular Hunting II
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e136
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10058/
DO  - 10.1017/fms.2025.10058
ID  - 10_1017_fms_2025_10058
ER  - 
%0 Journal Article
%A Allombert, Bill
%A Chenevier, Gaëtan
%T Unimodular Hunting II
%J Forum of Mathematics, Sigma
%D 2025
%P e136
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10058/
%R 10.1017/fms.2025.10058
%F 10_1017_fms_2025_10058

[1] Allombert, B. & Chenevier, G., ‘Database and PARI/GP source code for Unimodular Hunting II’, http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/index_uni_II.html. Also available as supplementary material to this article. Google Scholar

[2] Bacher, R., ‘Tables de réseaux entiers unimodulaires construits comme -voisins de ’, J. Théor. Nombres Bordeaux 9, 479–497 (1997). Google Scholar | DOI

[3] Bacher, R. and Venkov, B., ‘Réseaux entiers unimodulaires sans racine en dimension et ’, in Martinet, J. (ed), Réseaux euclidiens, designs sphériques et formes modulaires (Monogr. Enseign. Math) (in French) vol. 37 (L’Enseignement Mathématique, 2001), 212–267. Google Scholar

[4] Chenevier, G., ‘The characteristic masses of Niemeier lattices’, J. Théor. Nombres Bordeaux 32 (2020), 545–583. Google Scholar | DOI

[5] Chenevier, G., ‘Statistics for Kneser -neighbors’, Bull. Soc. Math. France 150(3) (2022), 473–516. With an appendix by O. Taïbi. Google Scholar

[6] Chenevier, G., ‘Unimodular hunting’, to appear in Algebraic Geometry. Google Scholar

[7] Conway, J.H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, (Grundlehren der Math. Wissenschaften) vol. 290, third edn. (Springer-Verlag, New York, 1999). Google Scholar

[8] Fincke, U. and Pohst, M., ‘Improved methods for calculating vectors of short length in a lattice, including a complexity analysis’, Math. Comp. 44 (1985), 463–471. Google Scholar | DOI

[9] The GAP group, GAP – Groups, Algorithms, and Programming, Version 4.10.2, https://www.gap-system.org (2019). Google Scholar

[10] King, O., ‘A mass formula for unimodular lattices with no roots’, Math. Comp. 72(242) (2003), 839–863. Google Scholar | DOI

[11] The PARI Group, PARI/GP version 2.13.0, Bordeaux, 2014, https://pari.math.u-bordeaux.fr/. Google Scholar

[12] Plesken, W. and Souvignier, B., ‘Computing isometries of lattices’, J. Symbolic Comput. 24, 327–334 (1997). Google Scholar | DOI

[13] Venkov, B., ‘Even unimodular Euclidean lattices in dimension ’, J. Soviet Math. 26 (1984), 1860–1867. Google Scholar | DOI

Cité par Sources :