On the mod $p$ kernel of the theta operator and Eisenstein series

TL;DR

This paper constructs Siegel and Hermitian modular forms in the mod p kernel of the theta operator using Eisenstein series, revealing new structures in odd-degree cases.

Contribution

It introduces new methods to construct modular forms in the mod p kernel of the theta operator via Eisenstein series for specific cases.

Findings

Construction of Siegel modular forms in the mod p kernel for odd degrees

Extension of results to Hermitian modular forms

Identification of Eisenstein series as key tools

Abstract

Siegel modular forms in the space of the mod $p$ kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.