Non-existence of Ramanujan congruences in modular forms of level four

TL;DR

This paper develops a method to identify simple congruences in the coefficients of certain modular forms and applies it to show the non-existence of Ramanujan-like congruences in various partition-related generating functions.

Contribution

It introduces a new approach to find all simple congruences in modular form coefficients and proves the non-existence of Ramanujan-type congruences in specific partition functions.

Findings

No Ramanujan congruences in overpartition generating functions

No Ramanujan congruences in crank difference generating functions

No Ramanujan congruences in 2-colored F-partition functions

Abstract

Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.