Structure and bases of modular spaces sequences ${\left(M_{2k}\left({\Gamma }_0\left(N\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$

Feauveau, Jean-Christophe

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 2, pp. 181-206.

Le discriminant modulaire $\Delta$ est connu pour structurer la famille de formes modulaires de niveau 1, ${\left(M_{2k}\left({\mathrm{SL}}_2\left(\mathbb{Z}\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$. Pour tout entier $N$, nous définissons une unité modulaire forte de niveau $N$ notée ${\Delta }_N$, qui permet de structurer la famille ${\left(M_{2k}\left({\Gamma }_0\left(N\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$ de manière identique. Nous appliquerons ce résultat à la recherche de bases pour chacun des espaces ${\left(M_{2k}\left({\Gamma }_0\left(N\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$.

The modular discriminant $\Delta$ is known to structure the sequence of modular forms of level $1$ ${\left(M_{2k}\left({\mathrm{SL}}_2\left(\mathbb{Z}\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$. For any positive integer $N$, we define a strong modular unit ${\Delta }_N$ of level $N$ which enables us to structure the sequence ${\left(M_{2k}\left({\Gamma }_0\left(N\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$ in an identical way. We then apply this novel result to the search of bases for each of the ${\left(M_{2k}\left({\Gamma }_0\left(N\right)\right)\right)}_{k\in {\mathbb{N}}^{*}}$ spaces.

Classification : 11F11, 11G16, 11F33, 33E05
Keywords: modular forms, modular units, Dedekind eta function
Mot clés : formes modulaires, unités modulaires, fonction de Dedekind

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