Low-dimensional lattices V. Integral coordinates for integral lattices

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(x1,...,xk), with all xi ∊ 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 Es 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) ≼ 4s + 2 (s odd), ϕ(s) ≼ 2πes(1 + o(1)) (s even) and ϕ(s) ≽ 2In In s/ln In In s(1 + o(1)).