Borcherds Products for U(1,1)

TL;DR

This paper explores the Borcherds lift for the U(1,1) group, providing explicit descriptions of Weyl-chambers, Weyl-vectors, and CM-point divisors for weakly holomorphic modular forms, advancing understanding of automorphic forms on this group.

Contribution

The paper offers a complete description of the Borcherds lift for U(1,1), including explicit calculations of Weyl-vectors and CM-divisors for a basis of input functions.

Findings

Explicit Weyl-chamber descriptions

Calculation of Weyl-vectors for basis functions

Determination of CM-point divisors

Abstract

The Borcherds lift for indefinite unitary groups, previously constructed by the author, is examined here in greater detail for the special case of the group U(1,1). The inputs for the lifting in this case are weakly holomorphic modular forms of weight zero, which are lifted to meromorphic modular forms on the usual complex upper half plane transforming under an arithmetic subgroup of U(1,1). In this setting, we can completely describe the Weyl-chambers involved and explicitly calculate the attached Weyl-vectors, for a family of input functions with principle part $q^{-n}$. Since these are a basis for the input space, we obtain similarly explicit results for arbitrary input functions. The Heegner divisors in this case consist of CM-points, the CM-order of which is also determined.