Nonexistence of Extremal Lattices in Certain Genera of Modular Lattices

✍ Scribed by Gabriele Nebe; Boris B. Venkov

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 353 KB
• Volume: 60
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0125

✦ Synopsis

In this note we consider integral lattices 4 in euclidean space (R n , ,), i.e. 4 R n is the Z-span of an R-basis of R n with ,(4, 4) Z. The minimum of 4 is min[,(4, 4) | 0{* # 4]. It is interesting to find lattices of given determinant or of given genus with large minimum.

We prove the following Theorem. Let l=11 or l=13. There are no even l-elementary lattices of dimension 12 and determinant l 6 with minimum 8.

Note that (n, l)= (12, 11) is the first open case in the existence problem for extremal _ l -modular lattices of dimension n for some similarity _ l of norm l with l+1 | 24 (cf. [Que 95] and [ScH 95]). A lattice of determinant 11 6 with minimum 8 would have a higher density than the Coxeter Todd lattice, which is the densest lattice known in dimension 12 (cf. [CoS 93]).

Recall that an integral lattice 4 in euclidean space (R n , ,) is called even, if ,(*, *) # 2Z for all * # 4 and it is called l-elementary, if l4* 4, where article no.