Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866908955186298880 |
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| author | Betts, L. Alexander Kumpitsch, Theresa Lüdtke, Martin |
| author_facet | Betts, L. Alexander Kumpitsch, Theresa Lüdtke, Martin |
| contents | Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a point of $X$ in fact globally comes from a point of $X$. We show that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and give the appropriate generalisation to $S$-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over $\mathbb{Z}[1/2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09462 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections Betts, L. Alexander Kumpitsch, Theresa Lüdtke, Martin Number Theory Algebraic Geometry Primary: 14H30. Secondary: 11G30, 14H25 Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a point of $X$ in fact globally comes from a point of $X$. We show that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and give the appropriate generalisation to $S$-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over $\mathbb{Z}[1/2]$. |
| title | Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections |
| topic | Number Theory Algebraic Geometry Primary: 14H30. Secondary: 11G30, 14H25 |
| url | https://arxiv.org/abs/2305.09462 |