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In the following I will follow the notations as in Chapter $11$, section $3$ of the book 'Cohomology of number fields' by Neukirch and others. Let $k_{\infty}$ be a $\mathbb{Z}_p$-extension of a number field $k$ of CM-type. Let $\Sigma=S_p \cup S_{\infty}$, where $S_{\infty}$ (and $S_p$) are the infinite primes (the primes above the prime $p$) of $k$. Let $\lambda_{ur},\lambda,\lambda_{cs}$ be the Lambda Iwasawa invariant of the groups $Gal(L/k_{\infty})^{ab},Gal(k_{\Sigma}(p)/k_{\infty})^{ab},Gal(L^{'}/k_{\infty})^{ab}$ respectively. Here $L,L^{'},k_{\Sigma}(p)$ are the maximal unramified, maximal unramified completely decomposed at $p$, unramified outside $p$, $p$-extension of $k_{\infty}$ respectively. Then Corollary $11.4.4$ gives

(1) $\lambda^{\pm}=\lambda_{nr}^{\mp}$.

Also proposition $11.4.5$ gives

(2) $\lambda^+=\lambda_{cs}^-+\#S_p(k_{\infty})-\#S_p(k_{\infty}^+)$,

(3) $\lambda^-=\lambda_{cs}^++\#S_p(k_{\infty}^+)-1$,

where $S_p(k_{\infty}),S_p(k_{\infty}^+)$ are (I think) the places above the prime $p$ of $k_{\infty},k_{\infty}^+$ respectively. Also if the strong Leopoldt conjecture holds for some layer $k_n^+$ of $k_{\infty}^+/k^+$ with $n \geq \lambda^+$, then proposition $11.4.7$ gives that

(4) $\lambda_{nr}^+=\lambda_{cs}^+$.

Now using $(1)$ and $(4)$ and placing it in $(3)$ I get that

$\lambda_{cs}^+=\lambda_{cs}^++\#S_p(k_{\infty}^+)-1$. It means that $\#S_p(k_{\infty}^+)=1$ (if the Leopoldt conjecture is true for $k_n^+$). So it means that there is only one prime above $p$ in $k_{\infty}^+$. I doubt this fact. Did I make any mistake? Thank you for your help.

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