2021/05/27 by Raum, Martin
#11F30 #11F33 #11F67 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2105.13170
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 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 ℓ on modular forms is optimal. In an extended example, we showcase how to employ our conclusions to benefit experimental work.