Ramanujan type congruences for the Klingen-Eisenstein series

Abstract

In the case of Siegel modular forms of degree n, we prove that, for almost all prime ideals p in any ring of algebraic integers, mod p m cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.