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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.17458 |
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| _version_ | 1866914985279488000 |
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| author | Avila, Josue |
| author_facet | Avila, Josue |
| contents | For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\infty}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{D})$ and show how to use it to calculate the order of $X_{\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields Avila, Josue Number Theory For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\infty}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{D})$ and show how to use it to calculate the order of $X_{\infty}$. |
| title | Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.17458 |