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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The partition function modulo prime powers


Authors: Matthew Boylan and John J. Webb
Journal: Trans. Amer. Math. Soc. 365 (2013), 2169-2206
MSC (2010): Primary 11F03, 11F11, 11F33, 11P83
Published electronically: October 25, 2012
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Abstract: Let $ \ell \geq 5$ be prime, let $ m\geq 1$ be an integer, and let $ p(n)$ denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer $ b_{\ell }(m)$ of size roughly $ m^2$ such that the module formed from the $ \mathbb{Z}/\ell ^m\mathbb{Z}$-span of generating functions for $ p\left (\frac {\ell ^bn + 1}{24}\right )$ with odd $ b\geq b_{\ell }(m)$ has finite rank. The same result holds with ``odd'' $ b$ replaced by ``even'' $ b$. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of $ m$; it is $ \left \lfloor \frac {\ell + 12}{24}\right \rfloor $.

In this paper, we prove, with a mild condition on $ \ell $, that $ b_{\ell }(m)\leq 2m - 1$. Our bound is sharp in all computed cases with $ \ell \geq 29$. To deduce it, we prove structure theorems for the relevant $ \mathbb{Z}/\ell ^m\mathbb{Z}$-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.


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Additional Information

Matthew Boylan
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: boylan@math.sc.edu

John J. Webb
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
Email: webbjj3@email.sc.edu, webbjj@wfu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05702-3
PII: S 0002-9947(2012)05702-3
Received by editor(s): May 20, 2011
Received by editor(s) in revised form: September 7, 2011
Published electronically: October 25, 2012
Additional Notes: The first author thanks the National Science Foundation for its support through grant DMS-0901068.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.