On the 3-rank of the class group of quadratic fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915692293390336 |
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| author | Chen, Shi-Chao Wu, Chuan-Chuan |
| author_facet | Chen, Shi-Chao Wu, Chuan-Chuan |
| contents | Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{f_1(D)}), \ldots, \mathbb{Q}(\sqrt{f_r(D)})$ and $\mathbb{Q}(\sqrt{-f_{r+1}(D)}), \ldots, \mathbb{Q}(\sqrt{-f_{r+s}(D)})$ is simultaneously less than $n$. Moreover, for any positive integer $k$, there exist positive integers $a, d$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{a+g_1(d)}), \ldots,\mathbb{Q}(\sqrt{a+g_k(d)})$ is simultaneously less than $n$ for polynomials $g_1(x), g_2(x), \ldots, g_k(x)$ that take integer values at the integers and have no constant terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the 3-rank of the class group of quadratic fields Chen, Shi-Chao Wu, Chuan-Chuan Number Theory Primary 11R29, Secondary 11R11 Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{f_1(D)}), \ldots, \mathbb{Q}(\sqrt{f_r(D)})$ and $\mathbb{Q}(\sqrt{-f_{r+1}(D)}), \ldots, \mathbb{Q}(\sqrt{-f_{r+s}(D)})$ is simultaneously less than $n$. Moreover, for any positive integer $k$, there exist positive integers $a, d$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{a+g_1(d)}), \ldots,\mathbb{Q}(\sqrt{a+g_k(d)})$ is simultaneously less than $n$ for polynomials $g_1(x), g_2(x), \ldots, g_k(x)$ that take integer values at the integers and have no constant terms. |
| title | On the 3-rank of the class group of quadratic fields |
| topic | Number Theory Primary 11R29, Secondary 11R11 |
| url | https://arxiv.org/abs/2512.20023 |