On higher congruences between cusp forms and Eisenstein series

TL;DR

This paper investigates congruences between cusp forms and Eisenstein series of higher weights, proposing a conjecture about prime ideals and validating it through numerical examples.

Contribution

It introduces a new conjecture relating prime ideals to weights in congruences between cusp forms and Eisenstein series, supported by computational evidence.

Findings

Conjecture links prime ideals to weights in higher weight congruences.

Numerical checks support the conjecture's validity.

Provides new insights into the structure of congruences in modular forms.

Abstract

In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their relation to the weights. We check the validity of the conjecture on several numerical examples.