Ramanujan type congruences for modular forms of several variables

Abstract

We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the (k−1)th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight k. We will conclude by giving numerical examples for each case.