Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T04:53:04.382Z Has data issue: false hasContentIssue false
  • English
  • Français

ON FOURIER COEFFICIENTS OF MODULAR FORMS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 November 2010

C. J. CUMMINS  and
N. S. HAGHIGHI
C. J. CUMMINS*
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada (email: [email protected])
N. S. HAGHIGHI
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recursive formulae satisfied by the Fourier coefficients of meromorphic modular forms on groups of genus zero have been investigated by several authors. Bruinier et al. [‘The arithmetic of the values of modular functions and the divisors of modular forms’, Compositio Math. 140(3) (2004), 552–566] found recurrences for SL(2,ℤ); Ahlgren [‘The theta-operator and the divisors of modular forms on genus zero subgroups’, Math. Res. Lett.10(5–6) (2003), 787–798] investigated the groups Γ0(p); Atkinson [‘Divisors of modular forms on Γ0(4)’, J. Number Theory112(1) (2005), 189–204] considered Γ0(4), and S. Y. Choi [‘The values of modular functions and modular forms’, Canad. Math. Bull.49(4) (2006), 526–535] found the corresponding formulae for the groups Γ+0(p). In this paper we generalize these results and find recursive formulae for the Fourier coefficients of any meromorphic modular form f on any genus-zero group Γ commensurable with SL(2,ℤ) , including noncongruence groups and expansions at irregular cusps. The form of the recurrence relations is well suited for the computation of the Fourier coefficients of the functions and forms on the groups which occur in monstrous and generalized moonshine. The required initial data has, in many cases, been computed by Norton (private communication).

Keywords

Fourier coefficientsmodular formsgenus zeromoonshineHauptmodul

MSC classification

Secondary: 11F30: Fourier coefficients of automorphic forms 11F03: Modular and automorphic functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 1 , February 2011 , pp. 50 - 62
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This work was supported in part by NSERC.

References

[1]Ahlgren, S., ‘The theta-operator and the divisors of modular forms on genus zero subgroups’, Math. Res. Lett. 10(5–6) (2003), 787–798.Google Scholar
[2]Alexander, D., Cummins, C., McKay, J. and Simons, C., ‘Completely replicable functions’, in: Groups, Combinatorics & Geometry (Durham, 1990), London Mathematical Society Lecture Note Series, 165 (Cambridge University Press, Cambridge, 1992), pp. 87–98.CrossRefGoogle Scholar
[3]Atkinson, J. R., ‘Divisors of modular forms on Γ0(4)’, J. Number Theory 112(1) (2005), 189–204.CrossRefGoogle Scholar
[4]Borcherds, R. E., ‘Monstrous moonshine and monstrous Lie superalgebras’, Invent. Math. 109(2) (1992), 405–444.Google Scholar
[5]Bouali, A., ‘Faber polynomials, Cayley–Hamilton equation and Newton symmetric functions’, Bull. Sci. Math. 130(1) (2006), 49–70.Google Scholar
[6]Bruinier, J. H., Kohnen, W. and Ono, K., ‘The arithmetic of the values of modular functions and the divisors of modular forms’, Compositio Math. 140(3) (2004), 552–566.CrossRefGoogle Scholar
[7]Choi, D., ‘On values of a modular form on Γ0(N)’, Acta Arith. 121(4) (2006), 299–311.CrossRefGoogle Scholar
[8]Choi, S. Y., ‘The values of modular functions and modular forms’, Canad. Math. Bull. 49(4) (2006), 526–535.CrossRefGoogle Scholar
[9]Conway, J. H. and Norton, S. P., ‘Monstrous moonshine’, Bull. Lond. Math. Soc. 11(3) (1979), 308–339.CrossRefGoogle Scholar
[10]Cummins, C. J., ‘Congruence subgroups of groups commensurable with PSL(2,ℤ) of genus 0 and 1’, Experiment. Math. 13(3) (2004), 361–382.CrossRefGoogle Scholar
[11]Cummins, C. J., ‘On conjugacy classes of congruence subgroups of PSL(2,ℝ)’, LMS J. Comput. Math. 12 (2009), 264–274.CrossRefGoogle Scholar
[12]Cummins, C. J. and Gannon, T., ‘Modular equations and the genus zero property of moonshine functions’, Invent. Math. 129(3) (1997), 413–443.Google Scholar
[13]Ford, D., McKay, J. and Norton, S. P., ‘More on replicable functions’, Comm. Algebra 22(13) (1994), 5175–5193.CrossRefGoogle Scholar
[14]Helling, H., ‘Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe’, Math. Z. 92 (1966), 269–280.CrossRefGoogle Scholar
[15]Helling, H., ‘On the commensurability class of the rational modular group’, J. Lond. Math. Soc. (2) 2 (1970), 67–72.Google Scholar
[16]Jones, G. A., ‘Congruence and noncongruence subgroups of the modular group: a survey’, in: Proceedings of Groups—St. Andrews 1985, London Mathematical Society Lecture Note Series, 121 (Cambridge University Press, Cambridge, 1986), pp. 223–234.Google Scholar
[17]Lehmer, D. H., ‘Properties of the coefficients of the modular invariant J(τ)’, Amer. J. Math. 64 (1942), 488–502.CrossRefGoogle Scholar
[18]Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd edn Oxford Mathematical Monographs (Clarendon Press, Oxford, 1995); with contributions by A. Zelevinsky.CrossRefGoogle Scholar
[19]Mahler, K., ‘On a class of nonlinear functional equations connected with modular functions’, J. Aust. Math. Soc. Ser. A 22(1) (1976), 65–118.CrossRefGoogle Scholar
[20]Newman, M., ‘Construction and application of a class of modular functions’, Proc. Lond. Math. Soc. (3) 7 (1957), 334–350.Google Scholar
[21]Newman, M., ‘Construction and application of a class of modular functions. II’, Proc. Lond. Math. Soc. (3) 9 (1959), 373–387.CrossRefGoogle Scholar
[22]Norton, S. P., ‘More on moonshine’, in: Computational Group Theory (Durham, 1982) (Academic Press, London, 1984), pp. 185–193.Google Scholar
[23]Norton, S. P., Private communication.Google Scholar
[24]Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, 11 (Iwanami Shoten, Tokyo and Princeton University Press, Princeton, NJ, 1971).Google Scholar
[25]Suetin, P. K., Series of Faber Polynomials, Analytical Methods and Special Functions, Vol. 1 (Gordon and Breach, Amsterdam, 1998); translated from the 1984 Russian original by E. V. Pankratiev.Google Scholar