A theta operator on Picard modular forms modulo an inert prime

TL;DR

This paper investigates the behavior of Picard modular forms modulo an inert prime, introducing a theta operator that affects their Fourier-Jacobi expansions and analyzing its poles.

Contribution

It constructs a new theta operator on Picard modular forms modulo an inert prime and studies its properties and effects.

Findings

The theta operator's poles are characterized.

The operator's impact on Fourier-Jacobi expansions is analyzed.

Properties of modular forms under the theta operator are established.

Abstract

We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.