Extremal Lattices
by Rudolf Scharlau and Rainer Schulze-Pillot.
Abstract:
Extremal lattices (in the sense of integral quadratic forms) have been
introduced in the unimodular case by C.L. Mallows, A.M. Odlyzko and
N.J.A. Sloane in 1975; a finiteness result dealing with the
hypothetical theta series of such lattices was given. Recently, H.-G.
Quebbemann has extended the notion to so called modular even lattices
of levels 2, 3, 5, 7, 11 and 23, and part of the genera of levels 6,
14 and 15 containing strongly modular lattices. This kind of
``analytic extremality'' of a lattice is defined in terms of its theta
series; it roughly says that the minimal norm of the lattice is as
large as an appropriate space of modular forms allows. In the present
paper, the above mentioned finiteness result is generalized to all
genera of lattices considered by Quebbemann. For minimal norms at most
8, a detailed overview on the existence and uniqueness of extremal
lattices is given, including some information about hermitian
stuctures on such lattices. Using an obvious generalization of the
notion of extremality, similar results are obtained for other
genera of levels 6, 14 and 15, and for the levels 10 and 21 not
considered before. For the construction of lattices, a computer
implementation of Kneser's neighbor method is an important tool.
Key words and phrases:
integral lattice, integral quadratic form, modular lattice,
strongly modular lattice, extremal lattice, theta series, extremal
modular form, Atkin-Lehner involution, hermitian lattice, automorphism
group of a lattice.
1991 Mathematics Subject Classification: Primary 11H50;
Secondary 11H55, 11E12, 11F27.
Available as
To appear in: submitted
Contact: Rudolf.Scharlau@Mathematik.Uni-Dortmund.DE or
schulzep@count.uni-sb.de