Relations among Ramanujan-Type Congruences II

TL;DR

This paper explores the structure of Ramanujan-type congruences, showing their preservation under Hecke algebra actions, and investigates their origins, especially in relation to partition congruences and algebraic parts of twisted L-values.

Contribution

It introduces new structural results for Ramanujan-type congruences, including a dichotomy based on their origin, and applies these findings to partition congruences and experimental methods.

Findings

Ramanujan-type congruences are preserved by shallow Hecke algebra actions.

A dichotomy exists between congruences from Hecke eigenvalues and those on arithmetic progressions.

Partition congruences can be analyzed through the lens of algebraic parts of twisted L-values.

Abstract

We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The scarcity of the latter was investigated recently. We explain that they provide congruences among algebraic parts of twisted central $\mathrm{L}$-values. We specialize our results to partition congruences, for which we investigate the proofs of partition congruences by Atkin and by Ono, and develop a heuristic that suggests that their approach by Hecke operators acting diagonally modulo $\ell$ on modular forms is optimal. In an extended example, we showcase how to employ our conclusions to benefit experimental work.