## Rationality of the Folsom-Ono grid

HTML articles powered by AMS MathViewer

- by P. Guerzhoy PDF
- Proc. Amer. Math. Soc.
**137**(2009), 1569-1577 Request permission

## 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.## References

- Basmaji, Jacques, Em Algorithmus zur Berechnung von Hecke-Operatoren Anwendung auf modulare Kurven, Dissertation Essen (1996).
- 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**, DOI 10.1007/s00222-005-0493-5 - 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**, DOI 10.1007/s00208-006-0048-0 - Jan Hendrik Bruinier and Jens Funke,
*On two geometric theta lifts*, Duke Math. J.**125**(2004), no. 1, 45–90. MR**2097357**, DOI 10.1215/S0012-7094-04-12513-8 - Bruinier, Jan H.; Ono, Ken, Heegner divisors, $L$-functions and harmonic weak Maass forms, preprint.
- 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.
- 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) Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977, pp. 69–78 (French). MR**0472703** - 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.
- 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**, DOI 10.1112/jlms/jdm119 - Sharon Anne Garthwaite,
*Vector-valued Maass-Poincaré series*, Proc. Amer. Math. Soc.**136**(2008), no. 2, 427–436. MR**2358480**, DOI 10.1090/S0002-9939-07-08961-7 - Guerzhoy, P., On weak harmonic Maass-modular grids of even integral weights, Math. Res. Lett., to appear.
- 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** - Ono, Ken, A mock theta function for the Delta-function, Proceedings of the 2007 Integers Conference, accepted for publication.
- 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; Kanô Memorial Lectures, 1. MR**1291394** - 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** - S. P. Zwegers,
*Mock $\theta$-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**, DOI 10.1090/conm/291/04907

## Additional Information

**P. Guerzhoy**- Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273
- Email: pavel@math.hawaii.edu
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 1569-1577 - MSC (2000): Primary 11F37
- DOI: https://doi.org/10.1090/S0002-9939-08-09681-0
- MathSciNet review: 2470814