2024/11/07 by Nicolas, Jean-Louis
#14J15 #FOS: Mathematics #G.2 #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2411.12754
Let Δ= ∑m=0^∞ q(2m+1)2 ∈ \mathbbF2[[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\geqslant 1 in Δ, one finds that g(f) < \frac 32 √ d.