The Hecke system of harmonic Maass functions and applications to modular curves of higher genera

Abstract

In monstrous moonshine, the replication formula and the Hecke operator played a central role. We generalize the replication formula and the Hecke operator to higher genus modular curves, with an eye toward extending moonshine to these cases. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera and obtain uniform proofs for numerous arithmetic properties of Fourier coefficients of modular functions of arbitrary level, which have been proved only for special cases of curves of genus zero or small prime levels.