The order of nilpotence of Hecke operators mod 2: a new proof

Abstract

Let \(\mathbb {F}_{2}[\!\Delta ]\) be the ring of modular forms of level 1, mod 2, where the coefficients of Δ are the reduction mod 2 of Ramanujan’s τ function. The Hecke operators act nilpotently on \(\mathbb {F}_{2}[\!\Delta ]\). The order of nilpotence of Δk is the smallest integer n(k) such that \(T_{p}^{n(k)}\Delta ^{k} = 0\) for all p. Nicolas and Serre have recently shown that the growth of n(k) is bounded by k1/2 but omitted the proof of a key result. We give a new, elementary proof which highlights that the Hecke action is essentially 2-adically continuous in k.