Weakly holomorphic modular forms on $$\Gamma _{0}(4)$$ Γ 0 ( 4 ) and Borcherds products on unitary group $$\mathrm{U}(2,1)$$ U ( 2 , 1 )

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 \(k\le -1\), and we make use of the basis elements for the case \(k=-1\) to construct explicit Borcherds products on unitary group \(\mathrm{U}(2,1)\).