The rank–order conjecture for Mazur's Eisenstein ideal

Let $N$ and $p$ be as in the paper, let $\mathbb{T}^0$ denote the relevant cuspidal Hecke algebra, and let

$$$$

be the Stickelberger element whose order $\operatorname{ord}_1(\zeta)$ is measured in the augmentation ideal. Assume that

$$\operatorname{rank}_{\mathbb{Z}_p}\mathbb{T}^0 \ge 2.$$

Rank–order conjecture. The following are equivalent:

1. $\operatorname{rank}_{\mathbb{Z}_p}\mathbb{T}^0=2$.
2. $\operatorname{ord}_1(\zeta)=2$.

The conjecture refines the observed relationship between the rank of the Eisenstein ideal and the order of vanishing of the associated Stickelberger element; the paper reports computational evidence for $N<10000$, while the analogous unrestricted equality is noted to fail in some examples.