Invariants for the Weil representation and Modular Units for Orthogonal Groups of Signature (2,2)

TL;DR

This paper characterizes invariants of the Weil representation for certain discriminant groups and constructs modular units for orthogonal groups of signature (2,2) using Borcherds products.

Contribution

It demonstrates that invariants are spanned by characteristic functions of self-dual isotropic subgroups and applies this to build modular units for orthogonal groups.

Findings

Invariants are spanned by characteristic functions of self-dual isotropic subgroups.

Constructed modular units for orthogonal groups of signature (2,2).

Provided a new method for using Borcherds products in this context.

Abstract

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we construct modular units for orthogonal groups in signature (2,2) using Borcherds products.