Certain aspects of holomorphic function theory on some genus zero arithmetic groups

TL;DR

This paper explores generalizations of fundamental holomorphic function theory results from PSL(2,Z) to other genus zero arithmetic groups generated by Atkin-Lehner involutions of square-free level N, expanding the understanding of modular forms.

Contribution

It extends classical results about modular forms and Hauptmoduln from PSL(2,Z) to a broader class of genus zero arithmetic groups with Atkin-Lehner involutions.

Findings

Generalized the structure of holomorphic modular forms to new groups

Identified generators for rings of modular forms in these groups

Established analogous relations for Hauptmodul and cusp forms

Abstract

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weight four and six; the smallest weight cusp form Delta has weight twelve and can be written as a polynomial in E4 and E6; and the Hauptmodul j can be written as a multiple of E4 cubed divided by Delta. The goal of the present article is to seek generalizations of these results to some other genus zero arithmetic groups, namely those generated by Atkin-Lehner involutions of level N with square-free level N.