Does Ribet's construction of class fields give us eigenspaces of rank 1?

Question

Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL_2(\mathbb F_q)$ where $q = p^r$ of the specific form: $ \begin{bmatrix} 1 & *\\ 0 & \chi \end{bmatrix}$ where $\chi$ is a power of the cyclotomic character mod $p$.

In particular, if we let $K$ be the kernel of $\chi$, the matrix is of the form $ \begin{bmatrix} 1 & *\\ 0 & 1 \end{bmatrix}$ and our representation looks like $\rho: Gal(\overline{\Bbb Q}/K) \to \mathbb F_q$.

The image is a subgroup (under addition) of $\mathbb F_q$ and if $B$ is the field fixed by the kernel of this map, then Ribet shows that $B/K$ is an unramified extension with abelian Galois group of the form $(p,\dots,p)$ on which $Gal(K/\Bbb Q)$ acts by $\chi^{-1}$.

I believe it is conjectured that $B/K$ should be a cyclic extension. Morever, by the main conjecture of Iwasawa theory, we can deduce that the degree of $B/K$ is less than the p-adic valuation of a particular Bernoulli number and most of the time, this is just $1$.

This suggests that $B/K$ should always be cyclic - can we prove this directly?

Answer 1

I don't think we know how to prove this directly. Indeed, recent works by Wake and Wake–Erickson show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the effect that a suitable localization of the Hecke algebra $\mathfrak{H}_\mathfrak{p}$ is Gorenstein.

The conjuction of the two papers shows how Greenberg's conjecture that $\lambda^+=0$ implies the cyclicity you mention. This is quite remarkable, in the sense that Greenberg's conjecture is known to imply the Main Conjecture (at least for abelian fields), but so far none has been able to deduce it from the Main Conjecture. It thus seems that the Main Conjecture lies somehow shallower than Greenberg's, in turn equivalent to the cyclicity of $B/K$.

Let me finally mention Kurihara's paper where he directly studies the equivalence between the statement that Ribet's extension attached to $\omega_\mathrm{cyc}^{(1-k)}$ be cyclic and the existence of an element of precise order $p^n$ (where $n=\operatorname{ord}_p(\zeta(1-k))$ inside $\mathfrak{I}/\mathfrak{I}^2$ where $\mathfrak{I}$ is an Eisenstein Ideal.