# Inverse Problem for Iwasawa Modules

Let $$\Lambda$$ denote the Iwasawa algebra and $$M$$ a finitely generated torsion $$\Lambda$$ module. Does there exist a number field $$K$$ and a $$\mathbb{Z}_p$$-extension $$K_{\infty}/K$$ such that the $$p$$-Hilbert class field $$\Lambda$$-module $$X_{\infty}$$ is pseudo-isomorphic to $$M$$? What about the same question for $$\mathbb{Z}_p^2$$-extensions?

I guess $$X_{\infty}$$ stands for the Galois group of the maximal abelian unramified $$p$$-extension of $$K_{\infty}$$, in other words the projective limit of the $$p$$-primary part of the class groups. Since the class group of $$K$$ is finite, it is impossible that $$X_{\infty}$$ is pseudo-isomorphic to $$\mathbb{Z}_p$$ with trivial $$\Lambda$$ action. More generally the charactersitic series of $$X_{\infty}$$ must be coprime to the cyclotomic polynomials $$(1+T)^{p^n}-1$$.