The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909906455494656 |
|---|---|
| author | Ueki, Jun Yoshizaki, Hyuga |
| author_facet | Ueki, Jun Yoshizaki, Hyuga |
| contents | This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $ν$'s are arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06182 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers Ueki, Jun Yoshizaki, Hyuga Number Theory Geometric Topology Primary 57K10, 11R23, 11R29, Secondary 11G20, 57M10, 11S05 This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $ν$'s are arbitrarily large. |
| title | The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers |
| topic | Number Theory Geometric Topology Primary 57K10, 11R23, 11R29, Secondary 11G20, 57M10, 11S05 |
| url | https://arxiv.org/abs/2210.06182 |