On the Construction and Uniqueness of the E8 Lattice

Abstract

In this thesis, we set out the basic theory of lattices, and study the E_8 lattice in particular. Lattices are finitely generated free abelian groups equipped with a symmetric bilinear form, and have many applications in mathematics and computer science. We define the most important properties of lattices, and construct the E_8 lattice based on the construction shown by Jean-Pierre Serre. We then study an article by Noam Elkies, in which he introduces the characteristic vectors and theta series of a lattice. The latter, after the necessary complex analysis, leads to a characterisation of the Z^n lattices by their shortest characteristic vectors. It turns out that E_8 and Z^8 have a common sublattice D_8, and that both are contained in the dual lattice D_8^*. After studying the lattices between D_8 and D_8^*, we conclude that E_8 is, up to isomorphism, the unique even, positive-definite, unimodular lattice in dimension 8.