Borcherds Products on Unitary Groups

TL;DR

This paper constructs a Borcherds lift for unitary groups U(1,m), transforming weakly holomorphic modular forms into automorphic forms with infinite product expansions and analyzing their properties.

Contribution

It extends Borcherds' theory to unitary groups by developing a new lifting method via embeddings into orthogonal groups, and explores applications to modularity of generating series.

Findings

Constructed the Borcherds lift for U(1,m)

Derived a modularity result for generating series with Heegner divisors

Analyzed the behavior of Borcherds products at cusps

Abstract

We provide a construction of the multiplicative Borcherds lift for unitary groups U(1,m), which takes weakly holomorphic elliptic modular forms and lifts them to meromorphic automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. The result is obtained through the transfer of Borcherds' theory to unitary groups via a suitable embedding of U(1,m) into O(2,2m). Further, we examine the values taken by the Borcherds products around the cusps of the symmetric domain of the unitary group. Finally, as an application of the lifting, we derive a modularity result for generating series with Heegner divisors as coefficients, along the lines of Borcherds' generalization of the Gross-Zagier-Kohnen theorem.