Relations among Ramanujan-Type Congruences II
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 -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 on modular forms is optimal. In an extended example, we showcase how to employ our conclusions to benefit experimental work.
Cite
@article{arxiv.2105.13170,
title = {Relations among Ramanujan-Type Congruences II},
author = {Martin Raum},
journal= {arXiv preprint arXiv:2105.13170},
year = {2023}
}