Formes modulaires modulo $2$ : L'ordre de nilpotence des opérateurs de Hecke (version développée)

Let $\mathrm{\Delta }={\sum }_{m=0}^{\mathrm{\infty }}{q}^{(2m+1{)}^2}\in {\mathbb{F}}_2[[q]]$ be the reduction mod 2 of the $\mathrm{\Delta }$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $\mathrm{\Delta }$. 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 $\mathrm{\Delta }$ 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 $p_1,p_2,\dots ,p_g$, 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 $d⩾1$ in $\mathrm{\Delta }$, one finds that $g(f)<\frac{3}{2}\sqrt{d}$.