Local Borcherds Products for Unitary Groups

TL;DR

This paper investigates the structure of Heegner divisors on modular varieties associated with unitary groups using local Borcherds products, providing criteria for torsion elements and linking obstructions to vector valued cusp forms.

Contribution

It introduces a criterion for local Heegner divisors to be torsion in the local Picard group using local Borcherds products and relates obstructions to spaces of vector valued cusp forms.

Findings

Criterion for torsion Heegner divisors established

Obstructions described via vector valued cusp forms

Connections made with theta-series and elliptic cusp forms

Abstract

For a discrete subgroup of an indefinite unitary group $U(1,n+1)$, $n\geq 1$, consider the attached modular variety. Using local Borcherds products, we study Heegner divisors in the local Picard group over a boundary component the compactified variety. We obtain a criterion for local Heegner divisors to be torsion elements in the local Picard group. As an application, we find that the obstructions for a local Heegner divisor to be a torsion element can be described through spaces of vector valued elliptic cusp forms spanned by certain theta-series.