Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915153262411776 |
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| author | Lüdtke, Martin |
| author_facet | Lüdtke, Martin |
| contents | The Chabauty--Kim method and its refined variant by Betts and Dogra aim to cut out the $S$-integral points $X(\mathbb{Z}_S)$ on a curve inside the $p$-adic points $X(\mathbb{Z}_p)$ by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line when $S$ contains two primes. We describe an algorithm for computing refined Chabauty--Kim loci and verify Kim's conjecture over $\mathbb{Z}[1/6]$ for all choices of auxiliary prime $p < 10{,}000$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$ Lüdtke, Martin Number Theory 14G05 (Primary) 11G55, 11Y50, 11D88 (Secondary) The Chabauty--Kim method and its refined variant by Betts and Dogra aim to cut out the $S$-integral points $X(\mathbb{Z}_S)$ on a curve inside the $p$-adic points $X(\mathbb{Z}_p)$ by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line when $S$ contains two primes. We describe an algorithm for computing refined Chabauty--Kim loci and verify Kim's conjecture over $\mathbb{Z}[1/6]$ for all choices of auxiliary prime $p < 10{,}000$. |
| title | Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$ |
| topic | Number Theory 14G05 (Primary) 11G55, 11Y50, 11D88 (Secondary) |
| url | https://arxiv.org/abs/2402.03573 |