Local Borcherds Products for Unitary Groups

Abstract

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature (1, n + 1), with n ≥ 1, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector valued elliptic cusp forms, transforming under a Weil-representation.