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Let $K$ be an imaginary quadratic field and $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$ extensions of $K$. Here, $\widetilde{K}/K$ is a $\mathbb{Z}_p^2$ extension.

Define $M(\widetilde{K})$ to be the maximal Abelian unramified pro-$p$ extension of $\widetilde{K}$.

The generalized Greenberg's conjecture (GGC) says the Greenberg-Iwasawa module, $X_{\widetilde{K}}\simeq \textrm{Gal}(M(\widetilde{K})/\widetilde{K})$ is pseudo-null.

[There are lots of examples where this conjecture is known.]

Let $F/K$ be any finite extension of $K$, and consider the compositum $F\widetilde{K}$. This is a $\mathbb{Z}_p^2$ extension over $F$. Are there any (partial) results that prove $X_{F\widetilde{K}}$ is pseudo-null when GGC for $\widetilde{K}$ is known?

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Let $K=\mathbb{Q}(\sqrt{-q})$ where $q\equiv 7\bmod 16$ is a prime. Take $F=K(\sqrt[4]{-q},\sqrt{-1})$. Then we proved that $X_{F\tilde{K}}=0$; see the paper (https://link.springer.com/article/10.1007/s00029-021-00644-3). In particular, $X_{T\tilde{K}}=0$ for every intermediate field $T$ in the extension $F/K$.

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