Kida’s theorem for a class of nonnormal extensions

Abstract:

Let $E,F$ be ${{\mathbf {Z}}_p}$-fields of CM-type such that $E/F$ is an extension of degree $p$. Let $L$, the normal closure of $E/F$, be such that $\operatorname {Gal} (L/F)$ has a normal subgroup of order $p$. Denote the fixed field of this group by $K$. We prove a Kida type formula which describes the minus part of the Iwasawa lambda invariant of $E$ in terms of the lambda invariants of $F$ and $K$.