Graded rings of modular forms (1)

TL;DR

This paper investigates the structure of graded rings of modular forms for specific congruence subgroups with levels that are powers of 2, 3, or 5, introducing new identities and theories for rational weight modular forms.

Contribution

It provides detailed descriptions of graded rings, identities involving infinite sums and products, and introduces the theory of modular forms with rational weights for certain levels.

Findings

Structures of graded rings are characterized for levels 2, 3, and 5.

New identities involving infinite sums and products are established.

Theory of rational weight modular forms is developed.

Abstract

We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3, 1/4, 1/5 etc) weight or more formally weight modular form are introduced.