32-dimensional even unimodular lattices with and without roots. (32-dimensionale unimodulare Gitter mit und ohne Wurzeln.) (German) Zbl 0861.11039

Bielefeld: Univ. Bielefeld, Fak. f. Math. 46 p. (1996).
For an \(n\)-dimensional lattice \(\Lambda\) with root system \(R\), the defect of \(\Lambda\) is given by \(\text{def} (\Lambda) = n- \max_{R' \subset R} |R' |\), where \(R'\) consists of pairwise orthogonal roots. Koch and Venkov constructed 32-dimensional lattices with defect 8 by glueing the 8-dimensional lattice with no roots to certain 24-dimensional lattices with defect 0. Furthermore, they developed the idea of neighbor defect, classified all 32-dimensional lattices with neighbor defect 0 and 8, and constructed all 32-dimensional even unimodular lattices, again with defect 0 or 8. Moreover, it was shown that every 32-dimensional even unimodular lattice with root system \(kA_1\), \(k \geq 17\) has, up to isomorphism, a uniquely determined neighbor containing no roots. (For these results see H. Koch and B. B. Venkov, J. Reine Angew. Math. 398, 144-168 (1989; Zbl 0667.10020) and H. Koch and G. Nebe, Math. Nachr. 161, 309-319 (1993; Zbl 0793.11012)).
The present author uses these ideas to investigate the case of 32-dimensional even unimodular lattices with defect 12. She constructs seven even unimodular lattices of dimension 32 with root system \(20 A_1\), i.e. defect 12, and, from these, seven such lattices containing no roots having neighbor defect 12. The author concludes by remarking that the methods developed by Koch, Nebe and Venkov do not appear to be useful in the general case, as they are so complicated already in the case of defect 12 that a full classification was not possible.
Reviewer: K.Roegner (Berlin)


11H06 Lattices and convex bodies (number-theoretic aspects)
11E12 Quadratic forms over global rings and fields
11H55 Quadratic forms (reduction theory, extreme forms, etc.)