$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913700954243072 |
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| author | Chang, Chieh-Yu Wei, Fu-Tsun Yu, Jing |
| author_facet | Chang, Chieh-Yu Wei, Fu-Tsun Yu, Jing |
| contents | Let $v$ be a finite place of $\mathbb{F}_q(θ)$. In this paper, we interpret $v$-adic arithmetic gamma values in terms of the $v$-adic crystalline-de Rham periods of Carlitz motives with Complex Multiplication, and establish an Ogus-type Chowla-Selberg formula. Furthermore, we prove the algebraic independence of these $v$-adic periods by employing the technique of switching "$v$ and $\infty$", and determining the dimension of relevant motivic Galois groups on the "$\infty$-adic" side through an adaptation and refinement of existing methods. As a consequence, all algebraic relations among $v$-adic arithmetic gamma values over $\mathbb{F}_q(θ)$ can be derived from standard functional equations together with Thakur's analogue of the Gross-Koblitz formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited Chang, Chieh-Yu Wei, Fu-Tsun Yu, Jing Number Theory 11R58, 11J93 Let $v$ be a finite place of $\mathbb{F}_q(θ)$. In this paper, we interpret $v$-adic arithmetic gamma values in terms of the $v$-adic crystalline-de Rham periods of Carlitz motives with Complex Multiplication, and establish an Ogus-type Chowla-Selberg formula. Furthermore, we prove the algebraic independence of these $v$-adic periods by employing the technique of switching "$v$ and $\infty$", and determining the dimension of relevant motivic Galois groups on the "$\infty$-adic" side through an adaptation and refinement of existing methods. As a consequence, all algebraic relations among $v$-adic arithmetic gamma values over $\mathbb{F}_q(θ)$ can be derived from standard functional equations together with Thakur's analogue of the Gross-Koblitz formula. |
| title | $v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited |
| topic | Number Theory 11R58, 11J93 |
| url | https://arxiv.org/abs/2407.15024 |