On automorphisms of some semidirect product groups and ranks of Iwasawa modules
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912359937736704 |
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| author | Fujii, Satoshi |
| author_facet | Fujii, Satoshi |
| contents | Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ does not split. Based on their heuristic, Kundu and Washington posed a question which asks whether $λ$- and $μ$-invariant of the anti-cyclotomic ${\Bbb Z}_p$-extension $k_{\infty}^a$ of $k$ are always trivial. Also, if $k_{\infty}^a/k$ is totally ramified, for $n\geq 1$, they showed that the $p$-part of the ideal class group of the $n$th layer of the anti-cyclotomic ${\Bbb Z}_p$-extension of $k$ is not cyclic. In this article, inspired by their paper, we study anti-cyclotomic like ${\Bbb Z}_p$-extensions, extending both the above question and Kundu-Washington's result. We show that the values of $λ$ of certain anti-cyclotomic like ${\Bbb Z}_p$-extensions are always even. We also show the $p$-part of the ideal class groups of certain anti-cyclotomic like ${\Bbb Z}_p$-extensions of CM-fields are always not cyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On automorphisms of some semidirect product groups and ranks of Iwasawa modules Fujii, Satoshi Number Theory Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ does not split. Based on their heuristic, Kundu and Washington posed a question which asks whether $λ$- and $μ$-invariant of the anti-cyclotomic ${\Bbb Z}_p$-extension $k_{\infty}^a$ of $k$ are always trivial. Also, if $k_{\infty}^a/k$ is totally ramified, for $n\geq 1$, they showed that the $p$-part of the ideal class group of the $n$th layer of the anti-cyclotomic ${\Bbb Z}_p$-extension of $k$ is not cyclic. In this article, inspired by their paper, we study anti-cyclotomic like ${\Bbb Z}_p$-extensions, extending both the above question and Kundu-Washington's result. We show that the values of $λ$ of certain anti-cyclotomic like ${\Bbb Z}_p$-extensions are always even. We also show the $p$-part of the ideal class groups of certain anti-cyclotomic like ${\Bbb Z}_p$-extensions of CM-fields are always not cyclic. |
| title | On automorphisms of some semidirect product groups and ranks of Iwasawa modules |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.14284 |