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.

Ramanujan type congruences for the Klingen-Eisenstein series

“…where σ m (n) is the n-th Fourier coefficient of the Eisenstein series of weight 12 (i.e., the sum of m-th powers of the divisors of n) and τ (n) is the n-th Fourier coefficient of Ramanujan's ∆ function. In the case of degree 2 and of f = 1 for the situation (2), we already proved these congruences in [7].…”

Ramanujan type congruences for the Klingen-Eisenstein series

“…In the case of degree 2 and of f = 1 for the situation (2), we already proved these congruences in [7].…”