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l-adic properties of partition functions

Authors
  • Belmont, Eva
  • Lee, Holden
  • Musat, Alexandra
  • Trebat-Leder, Sarah
Type
Published Article
Publication Date
Oct 05, 2015
Submission Date
Oct 05, 2015
Identifiers
arXiv ID: 1510.01202
Source
arXiv
License
Yellow
External links

Abstract

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ 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 $p_r$ 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 $p_r$ and spt on arithmetic progressions of the form $\ell^mn+\delta$ modulo powers of $\ell$. Our work gives a conceptual explanation of the exceptional congruences of $p_r$ observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

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