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Rationality of the Folsom-Ono grid


Author: P. Guerzhoy
Journal: Proc. Amer. Math. Soc. 137 (2009), 1569-1577
MSC (2000): Primary 11F37
Published electronically: December 11, 2008
MathSciNet review: 2470814
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Abstract: In a recent paper Folsom and Ono constructed a grid of Poincaré series of weights $ 3/2$ and $ 1/2$. They conjectured that the coefficients of the holomorphic parts of these series are rational integers. We prove that these coefficients are indeed rational numbers with bounded denominators.


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  • 1. Basmaji, Jacques, Em Algorithmus zur Berechnung von Hecke-Operatoren Anwendung auf modulare Kurven, Dissertation Essen (1996).
  • 2. Kathrin Bringmann and Ken Ono, The 𝑓(𝑞) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266. MR 2231957, 10.1007/s00222-005-0493-5
  • 3. Kathrin Bringmann and Ken Ono, Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann. 337 (2007), no. 3, 591–612. MR 2274544, 10.1007/s00208-006-0048-0
  • 4. Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, 10.1215/S0012-7094-04-12513-8
  • 5. Bruinier, Jan H.; Ono, Ken, Heegner divisors, $ L$-functions and harmonic weak Maass forms, preprint.
  • 6. Bruinier, Jan H.; Ono, Ken; Rhoades, Robert C., Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), no. 3, 673-693.
  • 7. H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 69–78. Lecture Notes in Math., Vol. 627 (French). MR 0472703
  • 8. Duke, W.; Jenkins, Paul, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (2008), no. 4, part 1, 1327-1340.
  • 9. Amanda Folsom and Ken Ono, Duality involving the mock theta function 𝑓(𝑞), J. Lond. Math. Soc. (2) 77 (2008), no. 2, 320–334. MR 2400394, 10.1112/jlms/jdm119
  • 10. Sharon Anne Garthwaite, Vector-valued Maass-Poincaré series, Proc. Amer. Math. Soc. 136 (2008), no. 2, 427–436 (electronic). MR 2358480, 10.1090/S0002-9939-07-08961-7
  • 11. Guerzhoy, P., On weak harmonic Maass-modular grids of even integral weights, Math. Res. Lett., to appear.
  • 12. 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
  • 13. Ono, Ken, A mock theta function for the Delta-function, Proceedings of the 2007 Integers Conference, accepted for publication.
  • 14. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kan\cflex o Memorial Lectures, 1. MR 1291394
  • 15. Don Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 211–244. MR 1977587
  • 16. S. P. Zwegers, Mock 𝜃-functions and real analytic modular forms, 𝑞-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR 1874536, 10.1090/conm/291/04907

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Additional Information

P. Guerzhoy
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273
Email: pavel@math.hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09681-0
Received by editor(s): June 23, 2008
Received by editor(s) in revised form: June 28, 2008
Published electronically: December 11, 2008
Additional Notes: The author was supported by NSF grant DMS-0700933
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.