An Analogue of Greenberg's Conjecture for CM Fields

Let $K$ be a CM field and $K^+$ be the maximal totally real subfield of $K$. Assume that the primes above $p$ in $K^+$ split in $K$. Let $S$ be a set containing exactly half of the prime ideals in $K$ above $p$. We show, assuming Leopoldt's conjecture is true for $K$ and $p$, that there is a unique $\mathbb{Z}_p$-extension of $K$ unramified outside of $S$ (the $S$-ramified $\mathbb{Z}_p$-extension of $K$). Such $\mathbb{Z}_p$-extensions for CM fields have similar properties to the cyclotomic $\mathbb{Z}_p$-extensions of a totally real field. For example, Greenberg proved some criterion for the Iwasawa invariants $\mu=\lambda=0$ of the cyclotomic $\mathbb{Z}_p$-extension of a totally real field, and we will prove analogous results for the $S$-ramified $\mathbb{Z}_p$-extension of a CM field. We also give a numerical criterion for the Iwasawa invariants $\mu=\lambda=0$ for an imaginary biquadratic field, which is analogous to the one given by Fukuda and Komatsu for real quadratic fields.