On the proportion of locally soluble superelliptic curves
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909790667538432 |
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| author | Beneish, Lea Keyes, Christopher |
| author_facet | Beneish, Lea Keyes, Christopher |
| contents | We investigate the proportion of superelliptic curves that have a $\mathbb{Q}_p$ point for every place $p$ of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in $p$ for the proportion of such curves over $\mathbb{Z}_p$ having a $\mathbb{Q}_p$-point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04697 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the proportion of locally soluble superelliptic curves Beneish, Lea Keyes, Christopher Number Theory 11G20, 11G30 We investigate the proportion of superelliptic curves that have a $\mathbb{Q}_p$ point for every place $p$ of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in $p$ for the proportion of such curves over $\mathbb{Z}_p$ having a $\mathbb{Q}_p$-point. |
| title | On the proportion of locally soluble superelliptic curves |
| topic | Number Theory 11G20, 11G30 |
| url | https://arxiv.org/abs/2111.04697 |