Rationality of the Folsom-Ono grid

Guerzhoy, P.

doi:10.1090/S0002-9939-08-09681-0

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