Stabilization on ideal class groups in potential cyclic towers
Jianing Li
TL;DR
The paper addresses stabilization of $p$-class groups in towers that are not necessarily cyclic, by introducing potential cyclic $p$-towers built from a Galois extension with $G=H\rtimes\Delta$ where $H\cong \mathbb{Z}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$. It generalizes Fukuda's stabilization results to these towers under RamHyp and establishes an Iwasawa-like class number formula in this setting, showing $e_n=\mu p^n+\lambda n+\nu$ for large $n$ with $|A_{F_n}|=p^{e_n}$. The method hinges on a descent framework for general Galois extensions, expressing $A_F$ in terms of $\mathbb{Z}_p[G]$-modules and explicit presentations $A_{F_n}\cong X/(\omega_n C+D)$ with $\omega_n=(h^{p^n}-1)/(h-1)$. The results yield concrete stabilization criteria: if $A_{F_1}/p^kA_{F_1}\cong A_{F_0}/p^kA_{F_0}$ (or $A_{F_1}\cong A_{F_0}$), then the corresponding groups stabilize along the entire potential cyclic tower, with several explicit radical-tower examples illustrating the theory and connections to prior work such as Lei and Caputo–Nuccio.
Abstract
Let $p$ be a prime and let $F$ be a number field. Consider a Galois extension $K/F$ with Galois group $H\rtimes Δ$ where $H\cong \mathbb{Z}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$, and $Δ$ is an arbitrary Galois group. The subfields fixed by $H^{p^n} \rtimes Δ$ $(n=0,1,\cdots)$ 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.