Stabilization on ideal class groups in potential cyclic towers
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912385444347904 |
|---|---|
| author | Li, Jianing |
| author_facet | Li, Jianing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stabilization on ideal class groups in potential cyclic towers Li, Jianing Number Theory 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. |
| title | Stabilization on ideal class groups in potential cyclic towers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.15224 |