Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke

Abstract

The nilpotence order of the mod 2 Hecke operators. Let $Δ=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ 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 Tp transforms f in a modular form Tp(f) which is a polynomial in $Δ$ 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 $Δ$, one finds that g(f) << d^(1/2).