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Non-Existence of Ramanujan Congruences in Modular Forms of Level Four
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 6 / 01 December 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1284-1306
- Print publication:
- 01 December 2011
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- Article
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Ramanujan famously found congruences like
$p(5n\,+\,4)\,\equiv \,0$ mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on 
${{\Gamma }_{1}}(4)$ that 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.
