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
MathSciNet review: 3009655
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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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  • 1. Scott Ahlgren, Distribution of the partition function modulo composite integers 𝑀, Math. Ann. 318 (2000), no. 4, 795–803. MR 1802511, 10.1007/s002080000142
  • 2. Scott Ahlgren and Ken Ono, Congruence properties for the partition function, Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 12882–12884 (electronic). MR 1862931, 10.1073/pnas.191488598
  • 3. George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067
  • 4. George E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171. MR 929094, 10.1090/S0273-0979-1988-15637-6
  • 5. A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14–32. MR 0205958
  • 6. A. O. L. Atkin, Multiplicative congruence properties and density problems for 𝑝(𝑛), Proc. London Math. Soc. (3) 18 (1968), 563–576. MR 0227105
  • 7. A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(𝑚), Math. Ann. 185 (1970), 134–160. MR 0268123
  • 8. A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. MR 0060535
  • 9. E. Belmont, H. Lee, A. Musat, S. Trebat-Leder, $ \ell $-adic properties of partition functions, preprint.
  • 10. F. Calegari, A remark on a theorem of Folsom, Kent, and Ono, 2011, preprint.
  • 11. Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
  • 12. A. Folsom, Z. Kent, K. Ono. $ \ell $-adic properties of the partition function, preprint.
  • 13. Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964
  • 14. Nicholas M. Katz, 𝑝-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 69–190. Lecture Notes in Mathematics, Vol. 350. MR 0447119
  • 15. Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • 16. Ken Ono, Distribution of the partition function modulo 𝑚, Ann. of Math. (2) 151 (2000), no. 1, 293–307. MR 1745012, 10.2307/121118
  • 17. Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • 18. Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 347–454. MR 2555930
  • 19. S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), no. 1-2, 147–153. MR 1544457, 10.1007/BF01378341
  • 20. Nick Ramsey, Geometric and 𝑝-adic modular forms of half-integral weight, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 599–624 (English, with English and French summaries). MR 2244225
  • 21. Nick Ramsey, The half-integral weight eigencurve, Algebra Number Theory 2 (2008), no. 7, 755–808. With an appendix by Brian Conrad. MR 2460694, 10.2140/ant.2008.2.755
  • 22. Jean-Pierre Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, pp. 319–338. Lecture Notes in Math., Vol. 317 (French). MR 0466020
  • 23. Jean-Pierre Serre, Formes modulaires et fonctions zêta 𝑝-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 191–268. Lecture Notes in Math., Vol. 350 (French). MR 0404145
  • 24. H. P. F. Swinnerton-Dyer, On 𝑙-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 1–55. Lecture Notes in Math., Vol. 350. MR 0406931
  • 25. G.N. Watson, Ramanujan's Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. 179 (1938), 97-128.
  • 26. Y. Yang, Congruences of the partition function, Int. Math. Res. Not. (2010) doi: 10.1093/imrn/rnq194

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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
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.