The fake monster algebra and singular Borcherds products

TL;DR

This paper classifies and characterizes Borcherds products related to the fake monster algebra, establishing their uniqueness, classification on prime level lattices, and their connection to twisted denominator identities via modular forms and Jacobi forms.

Contribution

It proves the uniqueness of the fake monster algebra's denominator as a Borcherds product, classifies symmetric singular weight products on prime level lattices, and links twisted identities to Fourier expansions of Borcherds products.

Findings

The fake monster algebra's denominator is the unique holomorphic Borcherds product of singular weight.

Complete classification of symmetric holomorphic Borcherds products on prime level lattices.

Twisted denominator identities correspond to Fourier expansions of Borcherds products at a cusp.

Abstract

In this paper we consider several problems in the theory of automorphic products and generalized Kac--Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type $U(N)\oplus U \oplus L$ and certain tuples of Jacobi forms of level $N$.