Extremal Lattices

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. 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.