A new perspective on the rank of Mazur's Eisenstein Hecke algebra

Abstract

Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$.