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

In this talk, I will talk about some congruence relations of Siegel/Hermitian modular forms
modulo a prime.
Prof. Böcherer and Prof. Nagaoka generalized theta operator, which Swinnerton-Dyer, Serre and
Katz considered in elliptic modular form case, to Siegel modular form case.
We are interested in investigating the mod p kernel of the theta operator.
In this talk, I will construct some elements of the mod p kernel of the theta operator
using Siegel-Eisenstein series of odd degrees.
I will also talk about a similar result in Hermitian modular form case if time permits.
This is a joint work with Prof. Nagaoka (Kindai university).