Complex and Integral Laminated Lattices

In an earlier paper we studied real laminated lattices (or Z-modules) A„, where A, is the lattice of even integers, and A„ is obtained by stacking layers of a suitable (n — l)-dimensional lattice A„_, as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing Z-module by /-module, where J may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which A„ is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the 6-dimensional integral laminated lattice over Z[ to ] of minimal norm 2. The paper includes tables of the best real integral lattices in up to 24 dimensions. Part