Relations among Ramanujan-Type Congruences I

Martin Raum

Published 2020-10-13Version 1

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific Hecke congruences.More generally, for weakly holomorphic modular forms of integral weight, we show 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. 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.