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

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 $(S_{2k}(\Gamma_0(N)))_{k\in \; \mathbb{N}^*}$.