Low-dimensional lattices V. Integral coordinates for integral lattices

Abstract

We say that an n -dimensional (classically) integral lattice ⋀ is s -integrable, for an integer s , if it can be described by vectors s -½ ( x 1 ,..., x k ), with all x i ∊ Z, in a euclidean space of dimension k ≽ n . Equivalently, ⋀ is s -integrable if and only if any quadratic form f ( x ) corresponding to ⋀ can be written as s -1 times a sum of k squares of linear forms with integral coefficients, or again, if and only if the dual lattice ⋀ * contains a eutactic star of scale s . This paper gives many techniques for s -integrating low-dimensional lattices (such as E s and the Leech lattice). A particular result is that any one-dimensional lattice can be 1-integrated with k = 4: this is Lagrange’s four-squares theorem. Let ϕ ( s ) be the smallest dimension n in which there is an integral lattice that is not s -integrable. In 1937 Ko and Mordell showed that ϕ (1) = 6. We prove that ϕ (2) = 12, ϕ (3) = 14, 21 ≼ ϕ (4) ≼ 25, 16 ≼ ϕ (5) ≼ 22, ϕ ( s ) ≼ 4 s + 2 ( s odd), ϕ ( s ) ≼ 2 π e s (1 + o (1)) ( s even) and ϕ ( s ) ≽ 2In In s /ln In In s (1 + o (1)).