Relations among Ramanujan-Type Congruences I

TL;DR

This paper explores the nature of Ramanujan-type congruences in modular forms, establishing their relation to Hecke eigenvalues, and investigates their occurrence, preservation, and obstructions across various modular form classes.

Contribution

It provides a unified framework connecting Ramanujan-type congruences with Hecke eigenvalues and classifies their occurrence in different modular form contexts, including new results for weakly holomorphic forms.

Findings

Ramanujan-type congruences are equivalent to specific Hecke eigenvalue congruences.

These congruences are preserved under the action of the shallow Hecke algebra.

Ramanujan-type congruences occur for shifts in the union of two square-classes, not just single square-classes.

Abstract

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.