Structure and bases of modular space sequences $(M_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ and $(S_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$. Part I : Strong modular units

Abstract

The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(S{L}_2(\mathbb{Z})){)}_{k\in {\mathbb{N}}^{\ast }}$ 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 $(M_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ in an identical way. We will apply this result to the bases search for each of the spaces $(M_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$.\\ This article is the first in a series of three. In the second part we will propose explicit bases of $(M_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$ for $1\le N\le 10$. Finally, in a third part, we will apply the results obtained in the first two parts to $(S_{2k}({\Gamma }_0(N)){)}_{k\in {\mathbb{N}}^{\ast }}$.