2012/04/04 by Jean-Louis Nicolas, Jean-Pierre Serre, Nicolas, Jean-Louis +1
Mathematics · #11F25 #11F33 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1204.1036
openalex publication_date 2012/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).