ℓ-Adic properties of partition functions

Eva Belmont, Holden Lee, Alexandra Musat, Sarah Trebat-Leder*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 pr 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 pr and spt on arithmetic progressions of the form ℓmn+δ modulo powers of ℓ. Our work gives a conceptual explanation of the exceptional congruences of pr observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

Original languageEnglish (US)
Pages (from-to)1-34
Number of pages34
JournalMonatshefte fur Mathematik
Volume173
Issue number1
DOIs
StatePublished - Jan 2014

Keywords

  • Andrews' spt-function
  • Congruences
  • Hecke operators
  • Modular forms
  • Partitions

ASJC Scopus subject areas

  • Mathematics(all)

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