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

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