Iwasawa invariants on non-cyclotomic ${\mathbf {Z}_{p}}$-extensions of CM fields

Abstract

Let $p$ be an odd prime which splits completely into distinct primes in a CM field $K$. By considering ray class field of $K$ with respect to prime ideals lying above $p$, one can define a certain special non-cyclotomic $\mathbf{Z}_{p}$-extension over $K$. We will give some examples of such non-cyclotomic $\mathbf{Z}_{p}$-extensions whose Iwasawa $λ$- and $µ$-invariants both vanish, using a variant of a criterion due to Greenberg.