Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917909867003904 |
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| author | Bartz, Kollin Levin, Aaron Thamminana, Aman Dhruva |
| author_facet | Bartz, Kollin Levin, Aaron Thamminana, Aman Dhruva |
| contents | We prove that there are $\gg\frac{X^{\frac{1}{3}}}{(\log X)^2}$ imaginary quadratic fields $k$ with discriminant $|d_k|\leq X$ and an ideal class group of $5$-rank at least $2$. This improves a result of Byeon, who proved the lower bound $\gg X^{\frac{1}{4}}$ in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus $2$ curve $C$ over $\mathbb{Q}$ such that $C$ has a rational Weierstrass point and the Jacobian of $C$ has a rational torsion subgroup of $5$-rank $2$. We deduce the main result from the existence of the curve $C$ and a quantitative result of Kulkarni and the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2 Bartz, Kollin Levin, Aaron Thamminana, Aman Dhruva Number Theory 11R29, 11R11 We prove that there are $\gg\frac{X^{\frac{1}{3}}}{(\log X)^2}$ imaginary quadratic fields $k$ with discriminant $|d_k|\leq X$ and an ideal class group of $5$-rank at least $2$. This improves a result of Byeon, who proved the lower bound $\gg X^{\frac{1}{4}}$ in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus $2$ curve $C$ over $\mathbb{Q}$ such that $C$ has a rational Weierstrass point and the Jacobian of $C$ has a rational torsion subgroup of $5$-rank $2$. We deduce the main result from the existence of the curve $C$ and a quantitative result of Kulkarni and the second author. |
| title | Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2 |
| topic | Number Theory 11R29, 11R11 |
| url | https://arxiv.org/abs/2502.00845 |