Geometric quadratic Chabauty and $p$-adic heights
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909128917516288 |
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| author | Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim |
| author_facet | Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim |
| contents | Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and Néron--Severi rank $ρ$. When $r < g+ ρ- 1$, the geometric quadratic Chabauty method determines a finite set of $p$-adic points containing the rational points of $X$. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of $p$-adic heights and $p$-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of $p$-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_10389 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Geometric quadratic Chabauty and $p$-adic heights Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim Number Theory Algebraic Geometry Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and Néron--Severi rank $ρ$. When $r < g+ ρ- 1$, the geometric quadratic Chabauty method determines a finite set of $p$-adic points containing the rational points of $X$. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of $p$-adic heights and $p$-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of $p$-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference. |
| title | Geometric quadratic Chabauty and $p$-adic heights |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2207.10389 |