# Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and assume that $p$ splits into two places $v$ and $v^{\sigma}$ in $F$ and $v$ is inert or ramified in $H$ ($\operatorname{Gal}(F/\mathbb{Q})$ is generated by $\sigma$).

Let $M_{v}$ (resp. $M_{v^{\sigma}}$) be the maximal unramified outside the primes above $v$(resp. $v^{\sigma}$) abelian pro-$p$ extension of $H$ and $H_{\infty,v}$ (resp. $H_{\infty,v^{\sigma}}$) be the fixed field by the torsion part of $\operatorname{Gal}(M_{v}/H)$ (resp. $\operatorname{Gal}(M_{v^{\sigma}}/H)$), it is known that $H_{\infty,v}$ and $H_{\infty,v^{\sigma}}$ are $\mathbb{Z}_p$-extension.

Let $H_{\infty}$ be the compositum of $H_{\infty,v}$ and $H_{\infty,v^{\sigma}}$, $L_{\infty}$ be the maximal unramified abelian $p$-extension of $H_{\infty}$ and $X_{\infty}$ be the Galois group $\operatorname{Gal}(L_{\infty}/H_{\infty})$. Now, denote by $X_{\infty}^{+}$ the quotient of $X_{\infty}$ such that the complex conjugation and $\sigma$ acts trivially, and denote by $L'_{\infty}$ the subfield of $L_{\infty}$ fixed by the kernel of the projection $X_{\infty} \rightarrow X_{\infty}^{+}$.

It is known that $\operatorname{Gal}(H_{\infty}/H)\simeq \mathbb{Z}_{p}^{2}$ acts by conjugation on $X_{\infty}^{-}$, so $X_{\infty}^{-}$ has structure of $\mathbb{Z}_p[[\operatorname{Gal}(H_{\infty}/H)]]$-module. Serre showed that there exists a non-canonical isomorphism between $\varLambda_{2}=\mathbb{Z}_p [[T_1,T_2]]$ and $\mathbb{Z}_p[[\operatorname{Gal}(H_{\infty}/H)]]$ by sending the topological generators $\tau_i$ of $\operatorname{Gal}(H_{\infty}/H)$ to $T_{i}+1$. Greenberg proved that $X_{\infty}^{-}$ is always a finitely generated, torsion $\varLambda_2$-module.}

Let $F''$ denote the maximal unramified extension of $H$ contained $H_{\infty}$ (the extension $F''/H$ is finite) and $L_0$ the fixed sub-field of $L_{\infty}'$ by the subgroup $(T_{1},..,T_{d})X_{\infty}^{-}$.

Assume that the action of the complex conjugation on $\operatorname{Gal}(H_{\infty,v}/H)$ (resp.$\operatorname{Gal}(H_{\infty,v^{\sigma}}/H)$) is the inversion (multiplication by -1).'

Is $L_{0}'$ is an abelian extension of $F''$? or equivalently, if $t_1$ and $t_2$ are an element of $\operatorname{Gal}(L_{0}'/H_{\infty})$ lifting the generators of $\operatorname{Gal}(H_{\infty}/F'')$, $t_1$ commutes with $t_2$?