Generators and relations for the shallow mod 2 Hecke algebra in levels Gamma₀(3) and Gamma₀(5)

Let $M(\mathit{odd})\subset Z/2[[x]]$ be the space of odd mod~2 modular forms of level $\Gamma_{0}(3)$. It is known that the formal Hecke operators $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p$ an odd prime other than $3$, stabilize $M(\mathit{odd})$ and act locally nilpotently on it. So $M(\mathit{odd})$ is an $\mathcal{O} = Z/2[[t_{5},t_{7}, t_{11}, t_{13}]]$-module with $t_{p}$ acting by $T_{p}$, $p\in \{5,7,11,13\}$. We show: (1) Each $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$, $p\ne 3$, is multiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$. (2) The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $(A^{2},AC,BC)$ where $A,B,C$ have leading forms $t_{5}+t_{7}+t_{13},\, t_{7},\, t_{11}$. We prove analogous results in level $\Gamma_{0}(5)$. Now $\mathcal{O}$ is $Z/2[[t_{3},t_{7},t_{11},t_{13}]]$, and the leading forms of $A,B,C$ are $t_{3}+t_{7}+t_{11},\, t_{7},\, t_{13}$. Let $\mathit{HE}$, "the shallow mod~2 Hecke algebra (of level $\Gamma_{0}(3)$ or $\Gamma_{0}(5)$)" be $\mathcal{O}/I$. (1) and (2) above show that $\mathit{HE}$ is a 1 variable power series ring over the 1-dimensional local ring $Z/2[[A,B,C]]/(A^{2},AC,BC)$. For another approach to all these results, based on deformation theory, see Deo and Medvedovsky, "Explicit old components of mod-2 Hecke algebras with trivial $\bar{\rho}$."