# Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

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, $$K$$)?

• 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. – Chris Wuthrich Aug 5 '18 at 22:04
• Unique unramified outside of \fp. – debanjana Aug 5 '18 at 22:12