Structure and bases of modular spaces sequences ${\left({M}_{2k}\left({\Gamma }_{0}\left(N\right)\right)\right)}_{k\in {ℕ}^{*}}$
Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 2, pp. 181-206.

The modular discriminant $\Delta$ is known to structure the sequence of modular forms of level $1$ ${\left({M}_{2k}\left({\mathrm{SL}}_{2}\left(ℤ\right)\right)\right)}_{k\in {ℕ}^{*}}$. For any positive integer $N$, we define a strong modular unit ${\Delta }_{N}$ of level $N$ which enables us to structure the sequence ${\left({M}_{2k}\left({\Gamma }_{0}\left(N\right)\right)\right)}_{k\in {ℕ}^{*}}$ in an identical way. We then apply this novel result to the search of bases for each of the ${\left({M}_{2k}\left({\Gamma }_{0}\left(N\right)\right)\right)}_{k\in {ℕ}^{*}}$ spaces.

Le discriminant modulaire $\Delta$ est connu pour structurer la famille de formes modulaires de niveau 1, ${\left({M}_{2k}\left({\mathrm{SL}}_{2}\left(ℤ\right)\right)\right)}_{k\in {ℕ}^{*}}$. Pour tout entier $N$, nous définissons une unité modulaire forte de niveau $N$ notée ${\Delta }_{N}$, qui permet de structurer la famille ${\left({M}_{2k}\left({\Gamma }_{0}\left(N\right)\right)\right)}_{k\in {ℕ}^{*}}$ de manière identique. Nous appliquerons ce résultat à la recherche de bases pour chacun des espaces ${\left({M}_{2k}\left({\Gamma }_{0}\left(N\right)\right)\right)}_{k\in {ℕ}^{*}}$.

Published online:
DOI: 10.5802/ambp.395
Classification: 11F11,  11G16,  11F33,  33E05
Keywords: modular forms, modular units, Dedekind eta function
Jean-Christophe Feauveau 1

1 Lycée Bellevue, 135, route de Narbonne BP. 44370 31031 Toulouse Cedex 4 France
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Jean-Christophe Feauveau. Structure and bases of modular spaces sequences $(M_{2k}(\Gamma _0(N)))_{k\in \protect \mathbb{N}^*}$. Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 2, pp. 181-206. doi : 10.5802/ambp.395. https://ambp.centre-mersenne.org/articles/10.5802/ambp.395/

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