Cubic rings and the exceptional Jordan algebra

Abstract

In a previous paper [EG] we described an integral structure (J, E) on the exceptional Jordan algebra of Hermitian 3×3 matrices over the Cayley octonions. Here we use modular forms and Niemeier's classification of even unimodular lattices of rank 24 to further investigate J and the integral, even lattice J₀=(ZE)^⊥ in J. Specifically, we study ring embeddings of totally real cubic rings A into J which send the identity of A to E, and we give a new proof of R. Borcherds's result that J₀ is characterized as a Euclidean lattice by its rank, type, discriminant, and minimal norm.