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?



Inverse Problem for Iwasawa Modules
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 $(1+T)^{p^n}-1$. 


