Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices

Abstract

We give a new existence proof for the rank 2^d even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend. Extending these ideas, we construct in dimensions 2^d, for d>>0, the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes-Wall lattices. The number Upsilon (2^d) of isometry types here is large: log_2 (Upsilon (2^d)) has dominant term at least (r/4) d 2^{2d}, for any r in [0, 1/2). The Ypsilanti lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate (log_2(mass(n)) has dominant term (1/4) log_2(n) n^2). This work continues our general uniqueness program for lattices, begun in Pieces of Eight. See also our new uniquness proof for the E_8-lattice.