Abstract
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.
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References
Ahlgren, S., Bringmann, K., Lovejoy, J.: \(\ell \)-adic properties of smallest parts functions. Adv. Math. 228, 1 (2011)
Andrews, G.: The number of smallest parts in the partitions of \(n\). J. Reine Angew. Math. 624, 133–142 (2008)
Andrews, G., Garvan, F.G., Liang, J.: Self-conjugate vector partitions and the parity of the spt-function. Acta Arith. 158(3), 199–218 (2013)
Atkin, A.: Ramanujan congruences for \(p_{-k}(n)\). Can. J. Math. 20, 67–78 (1968)
Boylan, M.: Exceptional congruences for powers of the partition function. Acta Arith. 111, 187–203 (2004)
Boylan, M., Webb, J.: The partition function modulo prime powers. Trans. Am. Math. Soc. (2013, accepted)
Bringmann, K.: On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144, 195–233
Folsom, A., Kent, Z., Ono, K.: \(\ell \)-adic properties of the partition function. Adv. Math. 229, 1586–1609 (2012)
Folsom, A., Ono, K.: The \(spt\)-function of Andrews. Proc. Natl. Acad. Sci. USA 105(51), 20152–20156 (2008)
Garvan, F.G.: Congruences for Andrews’ spt-function modulo powers of 5, 7 and 13. Trans. Am. Math. Soc. 364(9), 4847–4873 (2012)
Kiming, I., Olsson, J.: Congruences like Ramanujan’s for powers of the partition function. Arch. Math. 59, 348–360 (1992)
Ono, K.: Distribution of the partition function modulo \(m\). Ann. Math. 151, 293–307 (2000)
Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and \(q\)-series. AMS, Providence (2003)
Ono, K.: Unearthing the visions of a master: harmonic maass forms and number theory. Current developments in mathematics, pp. 347–454 (2008)
Ono, K.: Congruences for the Andrews’ spt function. Proc. Natl. Acad. Sci. USA 108(2), 473–476 (2011)
Serre, J.-P.: Formes modulaires et fonctions zêta \(p\)-adiques. Springer Lect. Notes Math. 350, 191–268 (1973)
Swinnerton-Dyer, H.P.F.: On \(\ell \)-adic representations and congruences for coefficients of modular forms. Springer Lect. Notes Math. 350, 1–55 (1973)
Acknowledgments
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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Communicated by C. Krattenthaler.
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Belmont, E., Lee, H., Musat, A. et al. \(\ell \)-Adic properties of partition functions. Monatsh Math 173, 1–34 (2014). https://doi.org/10.1007/s00605-013-0586-y
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DOI: https://doi.org/10.1007/s00605-013-0586-y