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The nilpotence order of the mod 2 Hecke operators.  Let $\Delta=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod2 of the $\Delta$ series. A modular form f modulo 2 of level 1 is a polynomialin $\Delta$. If p is an odd prime, then the Hecke operator Tp transforms f in amodular form Tp(f) which is a polynomial in $\Delta$ whose degree is smallerthan the degree of f, so that Tp is nilpotent. The order of nilpotence of f isdefined as the smallest integer g = g(f) such that, for every family of g oddprimes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds. We show howone can compute explicitly g(f); if f is a polynomial of degree d in $\Delta$,one finds that g(f) << d^(1/2).

Formes modulaires modulo 2 : l'ordre de nilpotence des op\'erateurs de Hecke