A characteristic 2 recurrence related to $U_3$, with a Hecke algebra application

Abstract

I begin with a simple modular form motivated proof of the following: Let $C_n$ in $Z/2[[t]]$ be defined by $C_{n+4}=C_{n+3}+(t^4+t^3+t^2+t)C_n+t^n(t^2+t)$, with initial values $0$, $1$, $t$ and $t^2$ for $C_0$, $C_1$, $C_2$ and $C_3$. Then every $C_{4m}$ is a sum of $C_k$ with $k<4m$. This, combined with earlier results, yields: If $K$ consists of all mod $2$ modular forms of level ${\Gamma }_0(3)$ annihilated by $U_2$ and $U_3+I$, then $K$ has a basis adapted (in the sense of Nicolas and Serre) to the Hecke operators $T_7$ and $T_{13}$; consequently the Hecke algebra attached to $K$ is a power series ring in these two operators.

9 pages

References in corpus (3)

• Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke
• Radiation of non-relativistic particle on a conducting sphere and a string of spheres
• A Hecke algebra attached to mod 2 modular forms of level 3

Cited by in corpus (1)

• Variations on a Lemma of Nicolas and Serre