On imaginary quadratic fields with non-cyclic class groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910853853347840 |
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| author | Ouyang, Yi Song, Qimin Zhang, Chenhao |
| author_facet | Ouyang, Yi Song, Qimin Zhang, Chenhao |
| contents | For a fixed abelian group $H$, let $N_H(X)$ be the number of square-free positive integers $d\leq X$ such that H is a subgroup of $CL(\mathbb{Q}(\sqrt{-d}))$. We obtain asymptotic lower bounds for $N_H(X)$ as $X\to\infty$ in two cases: $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$, $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $2\nmid g\geq 5$. More precisely, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-ε}$ when $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{g-1}-ε}$ when $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $ g\geq 5$. The first case is an adaptation of Soundararajan's results for $H=\mathbb{Z}/g\mathbb{Z}$, and the second conditionally improves the bound $X^{\frac{1}{g}-ε}$ due to Byeon and the bound $X^{\frac{1}{g}}/(\log X)^{2}$ due to Kulkarni and Levin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On imaginary quadratic fields with non-cyclic class groups Ouyang, Yi Song, Qimin Zhang, Chenhao Number Theory For a fixed abelian group $H$, let $N_H(X)$ be the number of square-free positive integers $d\leq X$ such that H is a subgroup of $CL(\mathbb{Q}(\sqrt{-d}))$. We obtain asymptotic lower bounds for $N_H(X)$ as $X\to\infty$ in two cases: $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$, $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $2\nmid g\geq 5$. More precisely, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-ε}$ when $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{g-1}-ε}$ when $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $ g\geq 5$. The first case is an adaptation of Soundararajan's results for $H=\mathbb{Z}/g\mathbb{Z}$, and the second conditionally improves the bound $X^{\frac{1}{g}-ε}$ due to Byeon and the bound $X^{\frac{1}{g}}/(\log X)^{2}$ due to Kulkarni and Levin. |
| title | On imaginary quadratic fields with non-cyclic class groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.00787 |