  In Monstrous moonshine, genus 0 property and the notion of replicability are
strongly connected. With regards to recent developments of moonshine, we
investigate a higher genus generalization of replicability for a general
automorphic form. Specifically, we extend the definitions of replicates and a
Hecke operator to harmonic Maass functions on modular curves of higher genera
to obtain number theoretic generalizations of important results in Monstrous
moonshine. Furthermore, we show the utility of the extended notions in yielding
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.
