Upper bounds on class numbers of real quadratic fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912451560210432 |
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| author | Bernardini, Riccardo |
| author_facet | Bernardini, Riccardo |
| contents | We prove that, for any $\varepsilon>0$, the number of real quadratic fields $\mathbb{Q}(\sqrt{d})$ of discriminant $d<x$ whose class number is $\ll \sqrt{d}(\log{d})^{-2}(\log\log{d})^{-1}$ is at least $x^{1/2-\varepsilon}$ for $x$ large enough. This improves by a factor $\log\log{d}$ a result from 1971 by Yamamoto. We also establish a similar estimate for $m$-tuples of discriminants for any $m\geq 1$. Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of $\mathbb{Q}(\sqrt{d})$, generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper bounds on class numbers of real quadratic fields Bernardini, Riccardo Number Theory Primary 11R29, Secondary 11R11, 11R27, 11A55 We prove that, for any $\varepsilon>0$, the number of real quadratic fields $\mathbb{Q}(\sqrt{d})$ of discriminant $d<x$ whose class number is $\ll \sqrt{d}(\log{d})^{-2}(\log\log{d})^{-1}$ is at least $x^{1/2-\varepsilon}$ for $x$ large enough. This improves by a factor $\log\log{d}$ a result from 1971 by Yamamoto. We also establish a similar estimate for $m$-tuples of discriminants for any $m\geq 1$. Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of $\mathbb{Q}(\sqrt{d})$, generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch. |
| title | Upper bounds on class numbers of real quadratic fields |
| topic | Number Theory Primary 11R29, Secondary 11R11, 11R27, 11A55 |
| url | https://arxiv.org/abs/2506.21301 |