Theta Series and Even Unimodular Lattices

Abstract

Most of this chapter may be read independently. We first recall known properties of the Siegel theta series of even unimodular lattices in rank 16 (Witt, Igusa, Kneser) and 24 (Erokhin, Borcherds, Nebe-Venkov…). Then we give two proofs of Theorem A of the introduction (the p-neighbor problem in dimension 16): a short one relying on a construction of Ikeda, and a self-contained one based on a novel use of the triality principle. Along the way, we provide several elementary constructions of orthogonal modular forms, and simple instances of the Eichler commutation relations.