A converse theorem for Borcherds products in signature $(2,2)$

TL;DR

This paper characterizes when a modular unit on two upper half-planes is a Borcherds product by linking it to special boundary divisors, and introduces a subspace of invariant vectors related to this correspondence.

Contribution

It provides a necessary and sufficient condition for modular units to be Borcherds products in signature (2,2) and constructs a surjective map from invariant vectors to these products.

Findings

Characterization of Borcherds products via boundary divisors

Construction of a surjective map from invariant vectors to modular units

Derivation of new eta product identities

Abstract

We show that a modular unit on two copies of the upper half-plane is a Borcherds product if and only if its boundary divisor is a special boundary divisor. Therefore, we define a subspace of the space of invariant vectors for the Weil representation which maps surjectively onto the space of modular units that are Borcherds products. Moreover, we show that every boundary divisor of a Borcherds product can be obtained in this way. As a byproduct we obtain new identities of eta products.