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ON FOURIER COEFFICIENTS OF MODULAR FORMS

  • C. J. CUMMINS (a1) and N. S. HAGHIGHI (a2)

Abstract

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).

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Copyright

Corresponding author

For correspondence; e-mail: cummins@mathstat.concordia.ca

Footnotes

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This work was supported in part by NSERC.

Footnotes

References

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[1]Ahlgren, S., ‘The theta-operator and the divisors of modular forms on genus zero subgroups’, Math. Res. Lett. 10(5–6) (2003), 787798.
[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. 8798.
[3]Atkinson, J. R., ‘Divisors of modular forms on Γ0(4)’, J. Number Theory 112(1) (2005), 189204.
[4]Borcherds, R. E., ‘Monstrous moonshine and monstrous Lie superalgebras’, Invent. Math. 109(2) (1992), 405444.
[5]Bouali, A., ‘Faber polynomials, Cayley–Hamilton equation and Newton symmetric functions’, Bull. Sci. Math. 130(1) (2006), 4970.
[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), 552566.
[7]Choi, D., ‘On values of a modular form on Γ0(N)’, Acta Arith. 121(4) (2006), 299311.
[8]Choi, S. Y., ‘The values of modular functions and modular forms’, Canad. Math. Bull. 49(4) (2006), 526535.
[9]Conway, J. H. and Norton, S. P., ‘Monstrous moonshine’, Bull. Lond. Math. Soc. 11(3) (1979), 308339.
[10]Cummins, C. J., ‘Congruence subgroups of groups commensurable with PSL(2,ℤ) of genus 0 and 1’, Experiment. Math. 13(3) (2004), 361382.
[11]Cummins, C. J., ‘On conjugacy classes of congruence subgroups of PSL(2,ℝ)’, LMS J. Comput. Math. 12 (2009), 264274.
[12]Cummins, C. J. and Gannon, T., ‘Modular equations and the genus zero property of moonshine functions’, Invent. Math. 129(3) (1997), 413443.
[13]Ford, D., McKay, J. and Norton, S. P., ‘More on replicable functions’, Comm. Algebra 22(13) (1994), 51755193.
[14]Helling, H., ‘Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe’, Math. Z. 92 (1966), 269280.
[15]Helling, H., ‘On the commensurability class of the rational modular group’, J. Lond. Math. Soc. (2) 2 (1970), 6772.
[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. 223234.
[17]Lehmer, D. H., ‘Properties of the coefficients of the modular invariant J(τ)’, Amer. J. Math. 64 (1942), 488502.
[18]Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd edn Oxford Mathematical Monographs (Clarendon Press, Oxford, 1995); with contributions by A. Zelevinsky.
[19]Mahler, K., ‘On a class of nonlinear functional equations connected with modular functions’, J. Aust. Math. Soc. Ser. A 22(1) (1976), 65118.
[20]Newman, M., ‘Construction and application of a class of modular functions’, Proc. Lond. Math. Soc. (3) 7 (1957), 334350.
[21]Newman, M., ‘Construction and application of a class of modular functions. II’, Proc. Lond. Math. Soc. (3) 9 (1959), 373387.
[22]Norton, S. P., ‘More on moonshine’, in: Computational Group Theory (Durham, 1982) (Academic Press, London, 1984), pp. 185193.
[23]Norton, S. P., Private communication.
[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).
[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.
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ON FOURIER COEFFICIENTS OF MODULAR FORMS

  • C. J. CUMMINS (a1) and N. S. HAGHIGHI (a2)

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