Lattices with many Borcherds products

TL;DR

This paper classifies a finite set of even lattices of signature (2,n) with trivial cusp form spaces, showing each has a Borcherds product divisor, and extends results to finite quadratic modules.

Contribution

It proves finiteness and provides a classification of lattices with Borcherds products, extending to finite quadratic modules.

Findings

Finite classification of lattices with trivial cusp form spaces

Explicit list of such lattices

All Heegner divisors are realized as Borcherds product divisors

Abstract

We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.