Borcherds products with prescribed divisor

TL;DR

This paper proves the existence of Borcherds products with divisors supported on a given set of special divisors and shows that any meromorphic Borcherds product can be expressed as a quotient of holomorphic ones, using Eisenstein series properties.

Contribution

It establishes the existence of Borcherds products with prescribed divisors and characterizes meromorphic Borcherds products as quotients of holomorphic ones, advancing understanding of their structure.

Findings

Existence of Borcherds products with prescribed divisors.

Meromorphic Borcherds products are quotients of holomorphic products.

Proofs rely on properties of vector valued Eisenstein series.

Abstract

Given an infinite set of special divisors satisfying a mild regularity condition, we prove the existence of a Borcherds product of non-zero weight whose divisor is supported on these special divisors. We also show that every meromorphic Borcherds product is the quotient of two holomorphic ones. The proofs of both results rely on the properties of vector valued Eisenstein series for the Weil representation.