# Borcherds Products on Unitary Group $U(2,1)$

## TL;DR

This paper constructs explicit Borcherds products on the unitary group U(2,1) by developing canonical bases for weakly holomorphic modular forms on
mma_0(4) and leveraging these bases for explicit product construction.

## Contribution

It introduces a method to explicitly construct Borcherds products on U(2,1) using canonical bases of weakly holomorphic modular forms for
mma_0(4).

## Key findings

- Canonical bases for weakly holomorphic modular forms are constructed.
- Explicit Borcherds products on U(2,1) are obtained.
- The approach links modular form bases to automorphic product construction.

## Abstract

In this note, we construct canonical bases for the spaces of weakly holomorphic modular forms with poles supported at the cusp $\infty$ for $\Gamma_{0}(4)$ of integral weight $k$ for $k\leq-1$, and we make use of the basis elements for the case $k=-1$ to construct explicit Borcherds products on unitary group $U(2,1)$.