Structure and bases of modular space sequences $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$. Part I : Strong modular units

TL;DR

This paper introduces strong modular units at level N to structure spaces of modular forms, facilitating basis construction and extending classical results from level 1 to higher levels.

Contribution

It defines strong modular units at level N, enabling the structuring of modular form spaces similar to the level 1 case, and applies this to basis search for levels 1 to 10.

Findings

Defined strong modular units at level N

Structured modular form spaces using these units

Facilitated basis search for specific levels

Abstract

The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $\Delta_N$ at level $N$ which enables one to structure the sequence $(M_{2k}(\Gamma_0(N)))_{k\in \; \mathbb{N}^*}$ in an identical way. We will apply this result to the bases search for each of the spaces $(M_{2k}(\Gamma_0(N)))_{k\in \; \mathbb{N}^*}$.\\ This article is the first in a series of three. In the second part we will propose explicit bases of $(M_{2k}(\Gamma_0(N)))_{k\in \; \mathbb{N}^*}$ for $1\leq N \leq 10$. Finally, in a third part, we will apply the results obtained in the first two parts to $(S_{2k}(\Gamma_0(N)))_{k\in \; \mathbb{N}^*}$.