×

zbMATH — the first resource for mathematics

Tables of integral unimodular lattices constructed as \(k\)-neighbors in \(\mathbb{Z}^n\). (Tables de réseaux entiers unimodulaires construits comme \(k\)-voisins de \(\mathbb{Z}^n\).) (French) Zbl 0906.11033
Two lattices \(L\) and \(M\) on euclidean \({\mathbb R}^n\) are said to be \(k\)-neighbors for some positive integer \(k\) if both \(L/(L\cap M)\) and \(M/(L\cap M)\) are cyclic groups of order \(k\). The integral \(k\)-neighbors of the standard lattice \({\mathbb Z}^n\) represent all classes of self-dual lattices (the proof is sketched). In the tables published here one can find such a description for any class up to \(n=24\), together with some information related to the enumeration of these lattices in [J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups. 2nd ed. (1993; Zbl 0785.11036)].

MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
11E12 Quadratic forms over global rings and fields
11-04 Software, source code, etc. for problems pertaining to number theory
Citations:
Zbl 0785.11036
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML EMIS
References:
[1] Bacher, R., Réseaux unimodulaires sans automorphismes, Thèse No 2597, Université de Genève, 1993.
[2] Conway, J.H. et Sloane, N.J.A., Sphere Packings, Lattices and Groups, Springer, 1988. · Zbl 0634.52002
[3] Kneser, M., Klassenzahlen definiter quadratischer Formen, Arch. der Math. vol.VIII (1957), 241-250. · Zbl 0078.03801
[4] Mimura, Y., Explicit examples of unimodular lattices with the trivial automorphism group, Proceeding of KAIST Mathematics Workshop, vol.5, Algebra and topology, Korea Adv.Inst.Sci.Tech., Taejon, 91-95. · Zbl 0735.11026
[5] Schulze-Pillot, R., An algorithm for computing genera of ternary and quaternary quadratic forms, Proceedings of the International Symposium on Symbolic and Algebraic Computation, Bonn (1991), 134-143. · Zbl 0925.11045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.