On extremal lattices in jump dimensions

Let (L,Q) be an even unimodular lattice, so L is a free Z-module of rank n, and Q : L → Z a positive definite regular integral quadratic form. Then L can be embedded into Euclidean n-space (R, (, )) with bilinear form defined by (x, y) := Q(x + y) − Q(x) − Q(y) and L defines a lattice sphere packing, whose density measures its error correcting properties. One of the main goals in lattice theory is to find dense lattices. This is a very difficult problem, the densest lattices are known only in dimension n ≤ 8 and in dimension 24 [3], for n = 8 and n = 24 the densest lattices are even unimodular lattices. The density of a unimodular lattice is proportional to its minimum, min(L) := min{Q(`) | 0 6= ` ∈ L}. For even unimodular lattices the theory of modular forms allows to bound this minimum min(L) ≤ 1 + b n 24c and extremal lattices are those even unimodular lattices L that achieve equality. The link is the theta series of L,