An explicit structure of the graded ring of modular forms of small level

TL;DR

This paper explicitly determines the structure of the graded ring of elliptic modular forms for various small levels by expressing it as a quotient of a polynomial ring, using relations between Fourier expansions.

Contribution

It provides an explicit algebraic description of the graded ring of modular forms for small levels, extending prior knowledge with concrete generators and relations.

Findings

Explicit generators for the graded rings identified.

Relations between Fourier expansions used to determine the structure.

The rings are described as quotients of polynomial rings.

Abstract

In this paper, we study the explicit structure of the graded ring of elliptic modular forms for the congruence subgroup $\Gamma_0(N)$ with N=1,2,3,4,5,6,7,8,9,10,12,16,18,25. More precisely, making use of some relations between Fourier expansions, we determine the ring by a quotient of the polynomial ring in several variables.