Greenberg's conjectures

Let $\ell$ be a prime number and let $k$ be a finite extension of $\mathbb{Q}$. Inside the field $k(\mu_{\ell^{\infty}})$ generated by all $\ell$-power roots of unity there is a unique subextension $k_{\infty}/k$ with $\operatorname{Gal}(k_{\infty}/k)\cong\mathbb{Z}_{\ell}$ (the cyclotomic $\mathbb{Z}_{\ell}$-extension of $k$); for each $n\ge 0$ let $k_{n}$ be the unique field with $k\subseteq k_{n}\subseteq k_{\infty}$ and $[k_{n}:k]=\ell^{n}$, so that

$$k=k_{0}\subset k_{1}\subset k_{2}\subset\cdots\subset k_{n}\subset\cdots,$$

with $k_{n}/k$ cyclic of degree $\ell^{n}$. There exist integers $\lambda_{\ell}(k)$, $\mu_{\ell}(k)$, $\nu_{\ell}(k)$, depending only on $k$ and $\ell$, such that for all sufficiently large $n$ the exact power of $\ell$ dividing the class number of $k_{n}$ equals $\ell^{e_{n}}$, where

$$e_{n}=\lambda_{\ell}(k)\,n+\mu_{\ell}(k)\,\ell^{n}+\nu_{\ell}(k).$$

(1) For every prime $\ell$ and every totally real number field $k$,

$$\mu_{\ell}(k)=\lambda_{\ell}(k)=0;$$

equivalently, the power of $\ell$ dividing the class number of $k_{n}$ remains bounded as $n\to\infty$.

(2) Let $p$ be a prime and let $F$ be a totally real number field. Let $\tilde{F}$ be the compositum of all $\mathbb{Z}_{p}$-extensions of $F$, let $\tilde{L}$ be the pro-$p$ Hilbert class field of $\tilde{F}$ (the maximal abelian pro-$p$ extension of $\tilde{F}$ unramified at all places), and put

$$\tilde{X}=\operatorname{Gal}(\tilde{L}/\tilde{F}),$$

regarded as a module over the completed group ring $\tilde{\Lambda}=\mathbb{Z}_{p}[[\operatorname{Gal}(\tilde{F}/F)]]$ by lifting and conjugating. Then $\tilde{X}$ is a pseudo-null $\tilde{\Lambda}$-module, i.e. $\tilde{X}$ is finitely generated over $\tilde{\Lambda}$ and its localization $\tilde{X}_{\mathfrak{p}}$ vanishes for every prime ideal $\mathfrak{p}$ of $\tilde{\Lambda}$ of height $1$.

(3) Call a number field $K$ $p$-rational, for a prime $p$, if $\operatorname{Gal}(K_{S}/K)$ is a free pro-$p$ group, where $K_{S}$ is the maximal pro-$p$ extension of $K$ unramified outside the set $S$ consisting of the primes of $K$ above $p$ together with the archimedean places. Then for every odd prime $p$ and every integer $t\ge 1$ there exists a $p$-rational number field $K$ with

$$\operatorname{Gal}(K/\mathbb{Q})\cong(\mathbb{Z}/2\mathbb{Z})^{t}.$$