Formes modulaires modulo $2$ : L'ordre de nilpotence des op\'erateurs de Hecke (version d\'evelopp\'ee)

Let $\Delta= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $\Delta$. 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 $\Delta$ 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 $p_1,p_2,\ldots,p_g$, the relation $T_{p_1}T_{p_2}\ldots T_{p_g}(f)=0$ holds. We show how one can compute explicitly $g(f)$; if $f$ is a polynomial of degree $d\geqslant 1$ in $\Delta$, one finds that $g(f) < \frac 32 \sqrt d$.