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?

up vote 2 down vote accepted

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 $(1+T)^{p^n}-1$.

  • There will be other restriction that this one. I wonder what the motivation of the question is. – Chris Wuthrich Nov 4 at 13:26
  • 1
    Thank you, I believe so. The result is known for Iwasawa modules of finite cardinality thanks to Ozaki "Construction of Zp extensions with prescribed Iwasawa Modules" in Japan Math Society and therefore it seems like a natural question to ask what is known for torsion Iwasawa modules. – Anwesh Ray Nov 4 at 14:25

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