Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916577456160768 |
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| author | Laxmi, H Saikia, Anupam |
| author_facet | Laxmi, H Saikia, Anupam |
| contents | Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we consider an infinite family of real biquadratic fields $K$. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the $2$-class groups of the $n$-th layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of $K$ and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of $K_n$ to the relative sizes of the $2$-class groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields Laxmi, H Saikia, Anupam Number Theory 11R29, 11R23, 11R11 Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we consider an infinite family of real biquadratic fields $K$. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the $2$-class groups of the $n$-th layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of $K$ and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of $K_n$ to the relative sizes of the $2$-class groups. |
| title | Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields |
| topic | Number Theory 11R29, 11R23, 11R11 |
| url | https://arxiv.org/abs/2501.12782 |