Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918324047183872 |
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| author | Jarossay, David Lilienfeldt, David T. -B. G. Saettone, Francesco Maria Weiss, Ariel Zehavi, Sa'ar |
| author_facet | Jarossay, David Lilienfeldt, David T. -B. G. Saettone, Francesco Maria Weiss, Ariel Zehavi, Sa'ar |
| contents | Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants Jarossay, David Lilienfeldt, David T. -B. G. Saettone, Francesco Maria Weiss, Ariel Zehavi, Sa'ar Number Theory Algebraic Geometry 11D45, 11G55, 11Y50, 14G05 Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach. |
| title | Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants |
| topic | Number Theory Algebraic Geometry 11D45, 11G55, 11Y50, 14G05 |
| url | https://arxiv.org/abs/2408.07400 |