ℓ-Adic properties of partition functions
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 w...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Vienna
2017
|
| Online Access: | http://hdl.handle.net/1721.1/107942 https://orcid.org/0000-0003-3312-6665 |
| _version_ | 1826203192524800000 |
|---|---|
| author | Belmont, Eva Lee, Holden Musat, Alexandra Trebat-Leder, Sarah |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Belmont, Eva Lee, Holden Musat, Alexandra Trebat-Leder, Sarah |
| author_sort | Belmont, Eva |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:33:05Z |
| format | Article |
| id | mit-1721.1/107942 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:33:05Z |
| publishDate | 2017 |
| publisher | Springer Vienna |
| record_format | dspace |
| spelling | mit-1721.1/1079422022-10-01T09:44:00Z ℓ-Adic properties of partition functions Belmont, Eva Lee, Holden Musat, Alexandra Trebat-Leder, Sarah Massachusetts Institute of Technology. Department of Mathematics Belmont, Eva 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. 2017-04-07T16:55:37Z 2017-04-07T16:55:37Z 2013-11 2011-07 2016-05-23T12:08:48Z Article http://purl.org/eprint/type/JournalArticle 0026-9255 1436-5081 http://hdl.handle.net/1721.1/107942 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. https://orcid.org/0000-0003-3312-6665 en http://dx.doi.org/10.1007/s00605-013-0586-y Monatshefte für Mathematik 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. Springer-Verlag Wien application/pdf Springer Vienna Springer Vienna |
| spellingShingle | Belmont, Eva Lee, Holden Musat, Alexandra Trebat-Leder, Sarah ℓ-Adic properties of partition functions |
| title | ℓ-Adic properties of partition functions |
| title_full | ℓ-Adic properties of partition functions |
| title_fullStr | ℓ-Adic properties of partition functions |
| title_full_unstemmed | ℓ-Adic properties of partition functions |
| title_short | ℓ-Adic properties of partition functions |
| title_sort | l adic properties of partition functions |
| url | http://hdl.handle.net/1721.1/107942 https://orcid.org/0000-0003-3312-6665 |
| work_keys_str_mv | AT belmonteva ladicpropertiesofpartitionfunctions AT leeholden ladicpropertiesofpartitionfunctions AT musatalexandra ladicpropertiesofpartitionfunctions AT trebatledersarah ladicpropertiesofpartitionfunctions |