Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo $\ell$
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Abstract:
Let $\ell$ be a prime and $\lambda ,j\geq 0$ be an integer. Suppose that $f(z)=\sum _{n}a(n)q^n$ is a weakly holomorphic modular form of weight $\lambda +\frac {1}{2}$ and that $a(0)\not \equiv 0 \pmod {\ell }$. We prove that if the coefficients of $f(z)$ are not “well-distributed” modulo $\ell ^j$, then \[ \lambda =0 \text { or } 1 \pmod {\frac {\ell -1}{2}}.\] This implies that, under the additional restriction $a(0)\not \equiv 0 \pmod {\ell }$, the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight $\lambda +\frac {1}{2}$ are almost (but not all) divisible by $\ell$, then either $\lambda \equiv 0\pmod {\frac {\ell -1}{2}}$ or $\lambda \equiv 1 \pmod {\frac {\ell -1}{2}}$. We also prove that if $\lambda \not \equiv 0 \text { and } 1 \pmod {\frac {\ell -1}{2}},$ then there does not exist an integer $\beta$, $0\leq \beta <\ell$, such that $a(\ell n+ \beta )\equiv 0 \pmod {\ell }$ for every nonnegative integer $n$. As an application, we study congruences for the values of the overpartition function.References
- Scott Ahlgren and Matthew Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487–502. MR 2000466, DOI 10.1007/s00222-003-0295-6
- Scott Ahlgren and Matthew Boylan, Coefficients of half-integral weight modular forms modulo $l^j$, Math. Ann. 331 (2005), no. 1, 219–239. MR 2107445, DOI 10.1007/s00208-004-0555-9
- Scott Ahlgren and Matthew Boylan, Addendum: “Coefficients of half-integral weight modular forms modulo $l^ j$” [Math. Ann. 331 (2005), no. 1, 219–239], Math. Ann. 331 (2005), no. 1, 241–242.
- Scott Ahlgren and Matthew Boylan, Central critical values of modular $L$-functions and coefficients of half-integral weight modular forms modulo $l$, Amer. J. Math. 129 (2007), no. 2, 429–454. MR 2306041, DOI 10.1353/ajm.2007.0006
- Antal Balog, Henri Darmon, and Ken Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of $L$-functions, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 105–128. MR 1399333
- Jan Hendrik Bruinier, Nonvanishing modulo $l$ of Fourier coefficients of half-integral weight modular forms, Duke Math. J. 98 (1999), no. 3, 595–611. MR 1695803, DOI 10.1215/S0012-7094-99-09819-8
- Jan H. Bruinier and Ken Ono, Coefficients of half-integral weight modular forms, J. Number Theory 99 (2003), no. 1, 164–179. MR 1957250, DOI 10.1016/S0022-314X(02)00061-6
- Dohoon Choi and Timothy Kilbourn, The weight of half-integral weight modular forms with few non-vanishing coefficients mod $l$, Acta Arith. 127 (2007), no. 2, 193–197. MR 2289984, DOI 10.4064/aa127-2-8
- Sylvie Corteel and Jeremy Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623–1635. MR 2034322, DOI 10.1090/S0002-9947-03-03328-2
- Basil Gordon and Kim Hughes, Multiplicative properties of $\eta$-products. II, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 415–430. MR 1210529, DOI 10.1090/conm/143/01008
- Benedict H. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$), Duke Math. J. 61 (1990), no. 2, 445–517. MR 1074305, DOI 10.1215/S0012-7094-90-06119-8
- Ian Kiming and Jørn B. Olsson, Congruences like Ramanujan’s for powers of the partition function, Arch. Math. (Basel) 59 (1992), no. 4, 348–360. MR 1179461, DOI 10.1007/BF01197051
- 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
- Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $l$, Ann. of Math. (2) 147 (1998), no. 2, 453–470. MR 1626761, DOI 10.2307/121015
- Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $l$, Ann. of Math. (2) 147 (1998), no. 2, 453–470. MR 1626761, DOI 10.2307/121015
- Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
- 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, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
- Stephanie Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. (3) 93 (2006), no. 2, 304–324. MR 2251155, DOI 10.1112/S0024611506015814
- M.-F. Vignéras, Facteurs gamma et équations fonctionnelles, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977, pp. 79–103 (French). MR 0485739
Additional Information
- D. Choi
- Affiliation: School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi, 412-791, Korea
- MR Author ID: 784974
- Email: choija@postech.ac.kr
- Received by editor(s): May 1, 2007
- Received by editor(s) in revised form: August 15, 2007
- Published electronically: March 4, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3817-3828
- MSC (2000): Primary 11F11, 11F33
- DOI: https://doi.org/10.1090/S0002-9947-09-04708-4
- MathSciNet review: 2491901