ℓ-Adic properties of partition functions

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

doi:10.1007/s00605-013-0586-y

ℓ-Adic properties of partition functions

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

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

4 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 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 ℓ^mn+δ modulo powers of ℓ. 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.

Original language	English (US)
Pages (from-to)	1-34
Number of pages	34
Journal	Monatshefte fur Mathematik
Volume	173
Issue number	1
DOIs	https://doi.org/10.1007/s00605-013-0586-y
State	Published - Jan 2014

Keywords

Andrews' spt-function
Congruences
Hecke operators
Modular forms
Partitions

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00605-013-0586-y

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title = "ℓ-Adic properties of partition functions",

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.",

keywords = "Andrews' spt-function, Congruences, Hecke operators, Modular forms, Partitions",

author = "Eva Belmont and Holden Lee and Alexandra Musat and Sarah Trebat-Leder",

note = "Funding Information: 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 in Theorem 1.1.",

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doi = "10.1007/s00605-013-0586-y",

language = "English (US)",

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AU - Belmont, Eva

AU - Lee, Holden

AU - Musat, Alexandra

AU - Trebat-Leder, Sarah

N1 - Funding Information: 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 in Theorem 1.1.

PY - 2014/1

Y1 - 2014/1

N2 - 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.

AB - 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.

KW - Andrews' spt-function

KW - Congruences

KW - Hecke operators

KW - Modular forms

KW - Partitions

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ℓ-Adic properties of partition functions

Abstract

Keywords

ASJC Scopus subject areas

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