Formes modulaires modulo 2 : Lʼordre de nilpotence des opérateurs de Hecke

Abstract

Let Δ=∑m=0∞q(2m+1)2∈F2[[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)≪d1/2.