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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]
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Abstract

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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

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