On fourier coefficients of modular forms

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 Theory 112(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 Γ0+(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). © Copyright Australian Mathematical Publishing Association Inc. 2010.