On higher congruences between cusp forms and Eisenstein series. II

TL;DR

This paper investigates congruences between cusp forms and Eisenstein series at square-free levels, extending previous prime-level results and providing bounds and existence conditions for such congruences.

Contribution

It generalizes earlier prime-level results to square-free levels, establishing bounds on congruence exponents and discussing conditions for their existence.

Findings

Established upper bounds on congruence exponents for square-free levels.

Extended previous prime-level results to more general square-free levels.

Discussed the existence of rational coefficient congruences at weight 2.

Abstract

We study congruences between cuspidal modular forms and Eisenstein series at levels which are square-free integers and for equal even weights. This generalizes our previous results from Naskr\k{e}cki [17] for prime levels and provides further evidence for the sharp bounds obtained under restrictive ramification conditions. We prove an upper bound on the exponent in the general square-free situation and also discuss the existence of the congruences when the coefficients belong to the rational numbers and weight equals 2.