Greenberg's fixed-point conjecture for abelian cyclotomic extensions

Let $\ell$ be a prime, let $K$ be a number field, and let

$$K_{\infty}=\bigcup_{n\in\mathbb N}K_n$$

be its cyclotomic $\mathbb Z_\ell$-extension. Let $Pl_\ell=S\sqcup T$ be an arbitrary partition of the places of $K$ above $\ell$. Write $\Lambda=\mathbb Z_\ell[[\Gamma]]$, where $\Gamma=\operatorname{Gal}(K_{\infty}/K)$, choose a topological generator $\gamma$ of $\Gamma$, and set $\omega=\gamma-1$. Let $H^T_S(K_{\infty})$ be the maximal abelian pro-$\ell$-extension of $K_{\infty}$ that is $S$-split and $T$-ramified, and put

$$\mathcal C^T_S(K_{\infty})=\operatorname{Gal}(H^T_S(K_{\infty})/K_{\infty}).$$

Greenberg's conjecture. The characteristic polynomial of the $\Lambda$-module $\mathcal C^T_S(K_{\infty})$ is not divisible by $\omega$. Equivalently, its fixed submodule is finite:

$$\mathcal C^T_S(K_{\infty})^\Gamma\sim 1.$$

This extends Greenberg's assertion in the abelian setting to every partition of the places above $\ell$. The surrounding discussion recalls that the analogous assertion for $\mu$ is known in the relevant abelian cases by the theorem of Ferrero and Washington, while the present fixed-point statement is the conjecture addressed by the paper.