Stabilization on ideal class groups in potential cyclic towers

Abstract

Let $p$ be a prime and let $F$ be a number field. Consider a Galois extension $K/F$ with Galois group $H⋊\hat{I}”$ where $H\cong {\mathbb{Z}}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$, and $\hat{I}”$ is an arbitrary Galois group. The subfields fixed by $H^{p^n}⋊\hat{I}”$ $(n=0,1,\cdots )$ form a tower which we call it a potential cyclic $p$-tower in this paper. A radical $p$-tower is a typical example, say $\mathbb{Z}\subset \mathbb{Z}(\sqrt[p]{a})\subset \mathbb{Z}(\sqrt[p^2]{a})\subset \cdots$ where $a\in \mathbb{Z}$. We extend the stabilization result of Fukuda in Iwasawa theory on $p$-class groups in cyclic $p$-towers to potential cyclic $p$-towers. We also extend Iwasawa's class number formula in ${\mathbb{Z}}_p$-extensions to potential ${\mathbb{Z}}_p$-extensions.

6 pages