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arXiv · 2407.15024

$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited

Chieh-Yu Chang·Fu-Tsun Wei·Jing Yu
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Abstract

Let $v$ be a finite place of $\mathbb{F}_q(\theta)$. 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 $\mathbb{F}_q(\theta)$ can be derived from standard functional equations together with Thakur's analogue of the Gross-Koblitz formula.

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Chieh-Yu Chang, Fu-Tsun Wei, Jing Yu. 2024-07-21. $v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited. https://arxiv.org/abs/2407.15024

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