Computations of cyclotomic lattices. (English) Zbl 0873.11026

Batut, Christian; Quebbemann, Heinz-Georg; Scharlau, Rudolf

Exp. Math. 4, No. 3, 175-179 (1995).

The authors study certain modular lattices of level \(l\) and dimension \(2(p-1), p\) a prime, using a construction involving the ideal class group of the \(p \)th cyclotomic extension of \(\mathbb Q(\sqrt{-l}).\) In particular, a family of even, hermitian-unimodular lattices related to the Craig lattices are obtained making use of \(\text{Gal} (\mathbb Q(\sqrt{-l})/\mathbb Q)\)-invariant ideal classes. In the course of their computational investigations, aided by PARI, several new (and some old) examples of extremal lattices are produced in dimensions 12, 20, 24, 32, 36 and 56, and the question of eutaxy is also decided.

Reviewer: K.Roegner (Berlin)

MSC:

11E12	Quadratic forms over global rings and fields
11H55	Quadratic forms (reduction theory, extreme forms, etc.)
11R18	Cyclotomic extensions
11-04	Software, source code, etc. for problems pertaining to number theory

Keywords:

even hermitian-unimodular lattices; Craig lattices; cyclotomic lattices; eutaxy; extremal lattices; integral quadratic forms

Software:

PARI/GP

References:

[1]	Batut C., User’s Guide to Pari-GP.
[2]	Bachoc C., Exper. Math. 1 pp 184– (1992) · Zbl 0787.11024
[3]	Bayer-Fluckiger E., J. reine angew. Math. 451 pp 51– (1994)
[4]	Conway J. H., Sphere Packings, Lattices and Groups (1988) · Zbl 0634.52002
[5]	DOI: 10.1112/plms/s3-29.4.633 · Zbl 0312.20003
[6]	Lang S., Cyclotomic Fields I and II (1990) · Zbl 0704.11038
[7]	Ozeki M., Théorie des nombres pp 772– (1989)
[8]	Plesken W., Memoirs Amer. Math. Soc. 116 pp 1– (1995)
[9]	Quebbemann H.-G., Math. Nachr. 156 pp 219– (1992) · Zbl 0774.11018
[10]	Quebbemann H.-G., J. Number Theory 54 (1995)
[11]	Scharlau R., ”Classification of integral lattices with large class number” · Zbl 0919.11031