# Class groups and Iwasawa Theory

In a paper by Iwasawa (On Cohomology Groups of Units in $$\mathbb{Z}_p$$-Extensions, 1983) the following two facts have been mentioned without any explanation (or examples). I do not see how to justify these statements or construct examples.

1. Let $$k$$ be a field of CM type and $$K/k$$ be the cyclotomic $$\mathbb{Z}_p$$-extension. Let $$\Gamma = Gal(K/k) \simeq \mathbb{Z}_p$$ and $$C(p)$$ be the $$p$$-primary part of the ideal class group of $$K$$. Claim: Then, it is possible that $$C(p)^{\Gamma}$$ is infinite.
2. Fix the ground field $$k$$ and find a $$\mathbb{Z}_p$$-extension $$K/k$$ such that the number of primes of $$k$$ that ramify in $$K$$ is minimum in the family of all $$\mathbb{Z}_p$$-extensions over $$k$$. Claim: Then, $$C(p)^{\Gamma}$$ is finite for this $$K/ k$$.

I keep all your notations but I'll introduce also the $$\Lambda$$-modules $$X$$ (rep. $$X'$$) = Galois group over $$K$$ of the maximal abelian pro-$$p$$-extension of $$K$$ which is uramified (resp. is unramified and splits totally all the $$p$$-places of $$K$$). It is known that the Pontryagin dual of $$C(p)$$ is pseudo-isomorphic to $$X^*$$ = the module $$X$$ with inverted action of $$\Gamma = \bar{<\gamma>}$$. The same property holds for $$X'$$. It follows that the finiteness of $$C(p)^{\Gamma}$$ is equivalent to that of $$X^*_{\Gamma}$$, i.e. $$T^*=\gamma^{-1} -1$$ does not divide the characteristic series of $$X^*$$, or equivalently, $${X^*}^{\Gamma}$$ is finite, or equivalently, $${X}^{\Gamma}$$ is finite. So we are brought back to construct an example of an infinite $${X}^{\Gamma}$$.
Suppose moreover that $$k$$ is CM and $$p$$ is odd. Let $$r$$ be the number of $$p$$-places of $$k^+$$ which split completely in $$k$$. It is known that the kernel of the canonical surjection $$X^- \to (X')^-$$ is isomorphic to a direct sum $$r$$ factors of the form $$\Lambda/\omega_{n_i}\Lambda$$, where $$\omega_n = {\gamma}^{p^n}-1$$ (a more precise statement is in the appendix of [FG]). This implies that $$(X^-)^{\Gamma}$$ is infinite, and we are done.