Monatshefte für Mathematik

, Volume 173, Issue 1, pp 1–34 | Cite as

\(\ell \)-Adic properties of partition functions

  • Eva Belmont
  • Holden Lee
  • Alexandra Musat
  • Sarah Trebat-Leder


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.


Congruences Partitions Andrews’ spt-function  Modular forms Hecke operators 

Mathematics Subject Classification (1991)

11P83 11F03 11F11 11F33 11F37 



The authors would like to thank Ken Ono for hosting the Emory REU, where the research was conducted, and for his guidance and comments on this paper. We would also like to thank Zachary Kent for useful discussions, M. Boylan and J. Webb for providing a preprint of their paper which was useful for our research, and the National Science Foundation for funding the REU. Finally, we would like to thank the referee for helping to improve the bound for \(b_{\ell }(r,m)\) in Theorem 1.1.


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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  • Eva Belmont
    • 1
  • Holden Lee
    • 2
  • Alexandra Musat
    • 3
  • Sarah Trebat-Leder
    • 4
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Trinity CollegeCambridgeUK
  3. 3.Department of MathematicsStanford UniversityStanfordUSA
  4. 4.Department of MathematicsEmory UniversityAtlantaUSA

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