Ramanujan type congruences for modular forms of several variables

TL;DR

This paper establishes Ramanujan-type congruences between Eisenstein series and cusp forms for Siegel and Hermitian modular forms, highlighting a link with generalized Bernoulli numbers and providing numerical examples.

Contribution

It introduces new congruences for modular forms of several variables and explores their relation to Bernoulli numbers, with explicit examples.

Findings

Congruences between Eisenstein series and cusp forms for Siegel and Hermitian cases

Connection between prime divisors of Bernoulli numbers and existence of cusp forms

Numerical examples illustrating the theoretical results

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.