Some constructions of modular forms for the Weil representation of SL2(Z)

Abstract

Modular forms for the Weil representation of ${SL}_2\left(\mathbb{Z}\right)$ play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac–Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems $D_{12}^2$, $E_8^3$ and to the Leech lattice.