ℓ-Adic properties of partition functions
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Author(s)
Belmont, Eva
Lee, Holden
Musat, Alexandra
Trebat-Leder, Sarah
Date Issued
November 2013
Journal
Monatshefte für Mathematik
Publisher
Springer Vienna
Citation
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.
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00605-013-0586-y
