Borcherds Forms and Generalizations of Singular Moduli

TL;DR

This paper explores the factorization of Borcherds forms over CM points, revealing prime occurrence patterns and demonstrating the finiteness of regularized theta lifts of weakly holomorphic modular forms.

Contribution

It provides a new factorization formula for Borcherds forms over CM points and establishes a theorem on prime occurrence, extending classical results.

Findings

Factorization of Borcherds forms over CM points

Prime occurrence patterns in the factorization

Finiteness of regularized theta lifts

Abstract

We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature (n,2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can occur in this factorization. One remarkable phenomenon we observe is that the regularized theta lift of a weakly holomorphic modular form is always finite.