Generators and relations for the shallow mod 2 Hecke algebra in levels ${\Gamma }_0(3)$ and ${\Gamma }_0(5)$

Abstract

Let $M(odd)\subset Z/2[[x]]$ be the space of odd mod~2 modular forms of level ${\Gamma }_0(3)$. It is known that the formal Hecke operators $T_p:Z/2[[x]]\to Z/2[[x]]$, $p$ an odd prime other than $3$, stabilize $M(odd)$ and act locally nilpotently on it. So $M(odd)$ 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 $T_p:M(odd)\to M(odd)$, $p\ne 3$, is multiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$. (2) The kernel, $I$, of the action of $\mathcal{O}$ on $M(odd)$ 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 $HE$, "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 $HE$ 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 $\bar{\rho }$."

arXiv comment10 pages