Class groups of imaginary quadratic points on $X_1(16)$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912468667727872 |
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| author | Derickx, Maarten |
| author_facet | Derickx, Maarten |
| contents | The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Class groups of imaginary quadratic points on $X_1(16)$ Derickx, Maarten Number Theory The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$. |
| title | Class groups of imaginary quadratic points on $X_1(16)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.04604 |