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Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

Published online by Cambridge University Press:  20 November 2018

Michael Dewar
Michael Dewar*
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, ON email: mdewar@mast.queensu.ca
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Abstract

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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.

Keywords

11F3311P83modular formRamanujan congruencegeneralized Frobenius partitionoverpartitioncrank
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 6 , 01 December 2011 , pp. 1284 - 1306
Copyright
Copyright © Canadian Mathematical Society 2011

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