Class groups and Iwasawa Theory

Question

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 $K/ k$.

Answer 1

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.

[FG] L.-J. Federer, B.-H. Gross: Regulators and Iwasawa modules, Invent. Math, 62 (1981), 443-457