Formes modulaires modulo l'ordre de nilpotence des

Jean-Louis Nicolas

Description

ar X iv :1 20 4. 10 36 v1 [ ma th. NT ] 4 A pr 20 12 Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke Jean-Louis NICOLAS a, Jean-Pierre SERRE b aCNRS, Université de Lyon, Institut Camille Jordan, Mathématiques, F-69622 Villeurbanne Cedex, France. bCollège de France, 3 rue d'Ulm, F-75231 Paris Cedex 05, France. Abstract The nilpotence order of the mod 2 Hecke operators. Let ∆ = ∑∞m=0 q(2m+1) 2 ? F2[[q]] be the reduction mod 2 of the ∆ series. A modular form f modulo 2 of level 1 is a polynomial in ∆. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in ∆ whose degree is smaller than the degree of f , so that Tp 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 p1, p2, . . . , pg, the relation Tp1Tp2 . . . Tpg(f) = 0 holds. We show how one can compute explicitly g(f); if f is a polynomial of degree d in ∆, one finds that g(f) << d1/2.

• f2-espaces vectoriels


• pr modulo


• ordre de nilpotence


• exposant dominant


• extension finie de f2


• opérateur de hecke tp

Sujets

• Nicolas
• Jordan
• Université de Lyon

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 19
• Langue: Français

Extrait

Formes modulaires modulo2: l’ordre de nilpotence des opérateurs de Hecke a b JeanLouis NICOLAS , JeanPierre SERRE a CNRS, Université de Lyon, Institut Camille Jordan, Mathématiques, F69622 Villeurbanne Cedex, France. b Collège de France, 3 rue d’Ulm, F75231 Paris Cedex 05, France.

Abstract

The nilpotence order of the mod 2 Hecke operators. P 2 ∞ (2m+1) LetΔ =q∈F 2 [[q]]be the reduction mod 2 of theΔseries. A modular formfmodulo2of level m=0 1 is a polynomial inΔ. Ifpis an odd prime, then the Hecke operatorT p transformsfin a modular formT p (f) which is a polynomial inΔwhose degree is smaller than the degree off, so thatT p is nilpotent. The order of nilpotence offis defined as the smallest integerg=g(f)such that, for every family ofgodd primesp 1 , p 2 , . . . , p g , the relationT p 1 T p 2 . . . T p g (f) = 0holds. We show how one can compute explicitlyg(f); iff 1/2 is a polynomial of degreedinΔ, one finds thatg(f)<< d.

Keywords:modular forms modulo2, Hecke operators, order of nilpotence.

Mathematics Subject Classification 2000:11F33, 11F25.

1. Introduction

∞ ∞ Y X n24n SoitΔ(q) =q(1−q) =τ(n)qoùτest la fonction de Ramanujan. Soitkun entier>0. n=1n=1 ∞ X k n On écritΔ (q) =τ k (n)q .Les congruences connues surτ(n2)) (mod (cf. [5]), montrent queΔ(q)≡ n=k ∞ X 2 (2m+1) q(mod 2), ce qui entraîne m=0

Email addresses:[e-mail], http://math.univlyon1.fr/∼nicolas/(JeanLouis NICOLAS), [e-mail](JeanPierre SERRE).

Cet article va paraître aux C.R. Acad. Sci. Paris, Ser. I 350 (2012),http://dx.doi.org/10.1016/j.crma.2012.03.013