Hecke System of Harmonic Maass Functions and Applications to Modular Curves of Higher Genera

TL;DR

This paper generalizes the concepts of replicability and Hecke operators to harmonic Maass functions on higher genus modular curves, providing new tools for understanding their Fourier coefficients and extending moonshine-related results.

Contribution

It introduces a higher genus generalization of replicability and Hecke operators for harmonic Maass functions, enabling broader applications in number theory and moonshine.

Findings

Unified proofs for arithmetic properties of Fourier coefficients.

Extension of moonshine concepts to higher genus modular curves.

New number theoretic insights into automorphic forms.

Abstract

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.