A Hecke algebra attached to mod 2 modular forms of level 3

TL;DR

This paper investigates a specific Hecke algebra associated with mod 2 modular forms of level 3, showing it is a power series ring in two generators with a nilpotent element, extending classical results to level 3.

Contribution

It establishes that the completed Hecke algebra acting on a certain space of mod 2 modular forms at level 3 is a power series ring in two generators with a nilpotent element, generalizing level 1 results.

Findings

The Hecke algebra is a power series ring in T_7 and T_13.

The algebra includes an element of square zero.

The structure parallels level 1 results of Nicolas and Serre.

Abstract

Let $D$ in $Z/2[[x]]$ be $\sum x^{n^{2}}$, $n>0$ and prime to $6$. Let $W$ be spanned by the $D^{k}$, $k>0$ and prime to $6$. Then the formal Hecke operators $T_{p}$, $p>3$, stabilize $W$, and it can be shown that they act locally nilpotently. We show that the completion of the Hecke algebra generated by these $T_{p}$ acting on $W$, with respect to the maximal ideal generated by the $T_{p}$, is a power series ring in $T_{7}$ and $T_{13}$ with an element of square $0$ adjoined. This may be viewed as a level 3 analog of the level 1 results of Nicolas and Serre -- the Hecke stable space they study is spanned by the odd powers of the mod $2$ reduction of $\Delta$, and their resulting completed Hecke algebra is a power series ring in $T_{3}$ and $T_{5}$.