ℓ-Adic properties of partition functions
Author(s)Belmont, Eva; Lee, Holden; Musat, Alexandra; Trebat-Leder, Sarah
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Folsom, Kent, and Ono used the theory of modular forms modulo ℓ to establish remarkable “self-similarity” properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers p[subscript r] of the partition function as well as Andrews’s spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for p[subscript r] and spt on arithmetic progressions of the form ℓ[superscript m]n+δℓ modulo powers of ℓ. Our work gives a conceptual explanation of the exceptional congruences of p[subscript r] observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Monatshefte für Mathematik
Belmont, Eva, Holden Lee, Alexandra Musat, and Sarah Trebat-Leder. “ℓ-Adic properties of partition functions.” Monatshefte Für Mathematik 173, no. 1 (November 6, 2013): 1–34.
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