Theta operators on unitary Shimura varieties

TL;DR

This paper introduces a theta operator for p-adic vector-valued modular forms on unitary groups over quadratic imaginary fields, analyzing its effects on Fourier-Jacobi expansions and its holomorphic extension beyond the {}-ordinary locus.

Contribution

It defines a new theta operator on p-adic modular forms for unitary groups and studies its properties and extensions in the context of quadratic imaginary fields.

Findings

The theta operator extends holomorphically beyond the {}-ordinary locus.

It affects Fourier-Jacobi expansions in a controlled manner.

The operator applies to scalar-valued forms with notable properties.

Abstract

We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the {\mu}-ordinary locus, when applied to scalar-valued forms.