Formes modulaires modulo 2 : Lʼordre de nilpotence des opérateurs de Hecke

Jean-Louis Nicolas; Jean-Pierre Serre

doi:10.1016/j.crma.2012.03.013

Théorie des nombres

Formes modulaires modulo 2 : Lʼordre de nilpotence des opérateurs de Hecke

Présenté par : Jean-Pierre Serre

Jean-Louis Nicolas ¹ ; Jean-Pierre Serre ²

¹ CNRS, Université de Lyon, Institut Camille Jordan, Mathématiques, 69622 Villeurbanne cedex, France
² Collège de France, 3, rue dʼUlm, 75231 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 343-348.

Résumé
Abstract

Soit $Δ = \sum_{m = 0}^{\infty} q^{{(2 m + 1)}^{2}} \in F_{2} [[q]]$ . Une forme modulaire f mod 2 de niveau 1 est un polynôme en Δ. Si p est un nombre premier >2, lʼopérateur de Hecke $T_{p}$ transforme f en une forme modulaire $T_{p} (f)$ qui est un polynôme en Δ de degré strictement plus petit que celui de f, de sorte que $T_{p}$ est nilpotent.

Lʼordre de nilpotence de f est défini comme le plus petit entier $g = g (f)$ tel que, pour toute famille de g nombres premiers impairs $p_{1}, p_{2}, \dots, p_{g}$ , on ait $T_{p_{1}} T_{p_{2}} \dots T_{p_{g}} (f) = 0$ . Nous montrons dans ce qui suit comment on peut calculer $g (f)$ ; on a $g (f) ≪ \deg {(f)}^{1 / 2}$ .

Let $Δ = \sum_{m = 0}^{\infty} q^{{(2 m + 1)}^{2}} \in F_{2} [[q]]$ be the reduction mod 2 of the Δ series. A modular form f modulo 2 of level 1 is a polynomial in Δ. If p is an odd prime, then the Hecke operator $T_{p}$ transforms f in a modular form $T_{p} (f)$ which is a polynomial in Δ whose degree is smaller than the degree of f, so that $T_{p}$ is nilpotent.

The order of nilpotence of f is defined as the smallest integer $g = g (f)$ such that, for every family of g odd primes $p_{1}, p_{2}, \dots, p_{g}$ , the relation $T_{p_{1}} T_{p_{2}} \dots T_{p_{g}} (f) = 0$ holds. We show how one can compute explicitly $g (f)$ ; if f is a polynomial of degree d in Δ, one finds that $g (f) ≪ d^{1 / 2}$ .

Reçu le : 2012-03-15
Publié le : 2012-04-01

DOI : 10.1016/j.crma.2012.03.013

Affiliations des auteurs :

Jean-Louis Nicolas ¹ ; Jean-Pierre Serre ²

¹ CNRS, Université de Lyon, Institut Camille Jordan, Mathématiques, 69622 Villeurbanne cedex, France
² Collège de France, 3, rue dʼUlm, 75231 Paris cedex 05, France

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Jean-Louis Nicolas; Jean-Pierre Serre. Formes modulaires modulo 2 : Lʼordre de nilpotence des opérateurs de Hecke. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 343-348. doi : 10.1016/j.crma.2012.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.013/

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