Formes modulaires modulo 2: l'ordre de nilpotence des opérateurs de Hecke

الأصل · EN

The nilpotence order of the mod 2 Hecke operators. 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) << d(1/2).