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In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field.

If $K$ is an imaginary quadratic field and $F/K$ is any finite extension in which a prime number $p$ splits completely, then for any prime $\beta$ of $F$ dividing $p$, one can consider the (unique) $\mathbb{Z}_p$-extension of $F$ which is ramified only at $\beta$. Fixing a choice of $\beta|p$, we call this the split prime $\mathbb{Z}_p$-extension and it is known to have many properties similar to the cyclotomic extension. (eg. By an adaptation of the work of Sinnott, Schnepps and Gillard could show that $\mu=0$ conjecture holds for $p\neq 2,3$, when $K$).

Is there a known proof of the weak Leopoldt conjecture for the split prime $\mathbb{Z}_p$ extension (of a number field $F$ containing an imaginary quadratic field, $K$)?

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    $\begingroup$ What is the "(unique) split prime $\mathbb{Z}_p$-extension"? If $p$ splits in $\mathfrak p \mathfrak p^*$ in $K$, then there are two $\mathbb{Z}_p$-extensions one in which only $\mathfrak p$ is ramified in the other only the other prime is ramified. Otherwise there is the unique anti-cyclotomic $\mathbb{Z}_p$-extension. $\endgroup$ – Chris Wuthrich Aug 5 '18 at 22:04
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    $\begingroup$ Unique unramified outside of \fp. $\endgroup$ – debanjana Aug 5 '18 at 22:12

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