Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910047304417280 |
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| author | Deng, Li-Tong Li, Yong-Xiong |
| author_facet | Deng, Li-Tong Li, Yong-Xiong |
| contents | Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic divisibility of Dirichlet $L$-series at negative integers, we derive an asymptotic formula that determines the order of the $2$-primary part of even $K$-groups of rings of integers of $F_n$ for sufficiently large $n$. As a corollary, we determine their $λ$ and $μ$ invariants. We also establish a lower bound for $n$ beyond which this asymptotic formula holds. Our results have two main applications: (1) For $K=\mathbb{Q}$, $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{2p})$ with $p\equiv\pm3\mod 8$, we determine the structure of the $2$-primary tame kernels $K_2\mathcal{O}_{K_n}(2)$; (2) We explicitly determine the three Iwasawa invariants $λ,μ,ν$ for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09176 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields Deng, Li-Tong Li, Yong-Xiong Number Theory 11R70 (primary), 11R42, 11R23 (secondary) Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic divisibility of Dirichlet $L$-series at negative integers, we derive an asymptotic formula that determines the order of the $2$-primary part of even $K$-groups of rings of integers of $F_n$ for sufficiently large $n$. As a corollary, we determine their $λ$ and $μ$ invariants. We also establish a lower bound for $n$ beyond which this asymptotic formula holds. Our results have two main applications: (1) For $K=\mathbb{Q}$, $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{2p})$ with $p\equiv\pm3\mod 8$, we determine the structure of the $2$-primary tame kernels $K_2\mathcal{O}_{K_n}(2)$; (2) We explicitly determine the three Iwasawa invariants $λ,μ,ν$ for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors. |
| title | Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields |
| topic | Number Theory 11R70 (primary), 11R42, 11R23 (secondary) |
| url | https://arxiv.org/abs/2603.09176 |