# Generalized Greenberg's conjecture for imaginary quadratic fields

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?

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 $$F/K$$.