Stabilization on ideal class groups in potential cyclic towers

TL;DR

This paper extends Iwasawa theory results on class group stabilization and class number formulas from cyclic p-towers to more general potential cyclic p-towers in number fields, broadening the scope of classical Iwasawa theory.

Contribution

It generalizes Fukuda's stabilization results and Iwasawa's class number formula to potential cyclic p-towers, which include more complex Galois extensions beyond classical cyclic p-towers.

Findings

Stabilization of p-class groups in potential cyclic p-towers is established.

Iwasawa's class number formula is extended to potential zp extensions.

Results broaden the applicability of Iwasawa theory to more general Galois extensions.

Abstract

Let $p$ be a prime and let $F$ be a number field. Consider a Galois extension $K/F$ with Galois group $H\rtimes \Delta$ where $H\cong \mathbb{Z}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$, and $\Delta$ is an arbitrary Galois group. The subfields fixed by $H^{p^n} \rtimes \Delta$ $(n=0,1,\cdots)$ 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.