Unimodular Lattices In Dimension Research Articles

Siegel theta series of degree 2 of extremal lattices

It is shown that there is a unique Siegel theta series of degree 2 for extremal even unimodular lattices in dimension N for N = 32, 48, 56. For N = 40 a question of Ozeki ( J. Number Theory 28 (1988), 119–131) is answered.

Voronoi parallelohedra for certain unimodular lattices

An effective description of (n-1) -dimensional faces is given for a Voronoi parallelohedron formed in an even unimodular lattice of dimension 32 without roots.

The Unimodular Lattices of Dimension up to 23 and the Minkowski-Siegel Mass Constants

In an earlier paper we enumerated the integral lattices of determinant one and dimension not exceeding 20. The present paper extends this enumeration to dimension 23, finding 40 lattices of dimension 21, 68 of dimension 22, and 117 of dimension 23. We also give explicit formulae for the Minkowski-Siegel mass constants for unimodular lattices (apparently not stated correctly elsewhere in the literature) and an exact table of the mass constants up to 32 dimensions, which provided a valuable check on our enumeration.

A characterisation of Leech's lattice

[-5, 6] promises to be the subject of many investigations. We give here a short proof that this lattice is characterised by some of its simplest properties. Although we must quote two theorems to open and close the proof, the reader can take these on trust if he wishes, and he will find that the proof is otherwise completely self-contained. In particular the Leech lattice will itself be defined in the course of its characterisation, so that no acquaintance with it is presupposed, although for the convenience of certain readers we have adhered to the notation of [1, 2]. The methods of this paper immediately give the order of the group which is the theme of our two papers [1, 2], and can be used to give other information about the group and the lattice. But the main interest of this paper probably lies in the possibility that the argument can be extended to yield constructions for other interesting lattices. We conjecture, however, that any such extended argument must be considerably more complicated, since it is very doubtful that the extreme "tightness" which the reader will observe so often in our proof can ever occur again. We proceed at once to the proof. We shall show that Leech's is the only lattice in fewer than 32 dimensions which has one point per unit volume and in which the square of every non-zero distance is an even integer greater than 2. A lattice with one point per unit volume is called unimodular, and then also even if every squared distance is an even integer. In an even unimodular lattice of dimension d we use u, for the number of vectors of squared length n. Our theorem then asserts that Leech's is the only even unimodular lattice with d < 32 and u 2 = 0.