The nilpotence order of the mod 2 Hecke operators

Let Delta = Sigma(infinity)(m=0)q((2m+1)2) is an element of F-2 parallel to q parallel to 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), ... , p(g), the relation Tp(1) Tp(2) ... Tp(g) (f) = 0 holds. We show how one can compute explicitly g(f); if f is a polynomial of degree d in Delta, one finds that g(f) << d(1/2). (C) 2012 Academie des sciences. Publie par Elsevier Masson SAS. Tous droits reserves.