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

TL;DR

This paper links $v$-adic gamma values to $v$-adic periods of Carlitz motives, proving their algebraic independence and deriving relations from known formulas, thus deepening understanding of $v$-adic special values.

Contribution

It introduces a novel interpretation of $v$-adic gamma values via $v$-adic crystalline-de Rham periods and establishes an Ogus-type Chowla-Selberg formula, advancing the theory of $v$-adic periods.

Findings

Interpretation of $v$-adic gamma values in terms of $v$-adic periods.

Proof of algebraic independence of $v$-adic periods.

Derivation of all algebraic relations from functional equations and Thakur's formula.

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.