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

  Published:2011-04-30
 Printed: Dec 2011
  • Michael Dewar,
    Department of Mathematics and Statistics, Queen's University, Kingston, ON
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Abstract

Ramanujan famously found congruences like $p(5n+4)\equiv 0 \operatorname{mod} 5$ 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: modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crank modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crank
MSC Classifications: 11F33, 11P83 show english descriptions Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Partitions; congruences and congruential restrictions
11F33 - Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
11P83 - Partitions; congruences and congruential restrictions
 

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