Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part I : Strong modular units

Abstract

The modular discriminant ∆ is known to structure the sequence of modular forms (M 2k (SL 2 (Z))) k∈ N * at level 1. For all positive integer N , we define a strong modular unit ∆ N at level N which enables one to structure the family (M 2k (Γ 0 (N ))) k∈ N * in an identical way. We will apply this result to the bases search for each of the spacesThis article is the first in a series of three. In the second part we will propose explicit bases of (M 2k (Γ 0 (N ))) k∈ N * for 1 N 10. Finally, in a third part, we will apply the results obtained in the first two parts to (S 2k (Γ 0 (N ))) k∈ N * .

Structure and bases of modular space sequences $(M_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(Γ_0(N)))_{k\in \mathbb{N}^*}$. Part II: a modular butterfly hunt

“…Essentially thanks to the introduction of the notion of strong modular unit, we were able to identify, in part I of this article [4], the theoretical tools allowing one to structure the family of modular spaces (M 2k (Γ 0 (N ))) k∈N * and to obtain unitary upper triangular bases of them. These results were applied to every values 1 N 10.…”

“…Although not all modularity proofs of the proposed functions are provided, this would produce a document as indigestible as it is voluminous, these have been done carefully for strong modular units in the first part of this article [4]. The modularity of other frequently appearing modular functions has also been verified, in particular thanks to the properties of the Weierstrass functions and studied in [3].…”

“…(iii) For the modularity according to the congruence group Γ 0 (N ), 1 N 10, and apart from the strong modular units represented in [4] in the form of η-products, these functions are sufficient to represent all the modular functions involved in this article.…”

“…It is a question of applying concretely the results developed in the first part of this article [4]. For a given level N , we established the existence of a function ∆ N which enables one to structure the family (M 2k (Γ 0 (N ))) k∈N * and to describe bases of each of its spaces when we know a basis of each space M 2k (Γ 0 (N )), for weights between 2 and ρ N +2, where ρ N is the weight of ∆ N .…”