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

Chieh-Yu Chang, Fu-Tsun Wei, Jing Yu

Abstract

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.

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

Abstract

Let be a finite place of . In this paper, we interpret -adic arithmetic gamma values in terms of the -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 -adic periods by employing the technique of switching " and ", and determining the dimension of relevant motivic Galois groups on the "-adic" side through an adaptation and refinement of existing methods. As a consequence, all algebraic relations among -adic arithmetic gamma values over can be derived from standard functional equations together with Thakur's analogue of the Gross-Koblitz formula.
Paper Structure (30 sections, 33 theorems, 292 equations)

This paper contains 30 sections, 33 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. v-adic periods of Carlitz motives and gamma values
  4. Algebraic independence of the v-adic crystalline--de Rham periods
  5. Algebraic relations among v-adic arithmetic gamma values
  6. Organization of this paper
  7. v-adic arithmetic gamma values
  8. Notation
  9. Definition of v-adic arithmetic gamma function
  10. Product expansions of v-adic arithmetic gamma values
  11. Carlitz t-motive and the crystalline--de Rham comparison
  12. Carlitz t-motive
  13. Notation
  14. Crystalline module of Rtau(Cl)
  15. de Rham module of Rtau(Cl)
  16. ...and 15 more sections

Key Result

Theorem 1.2.1

(See Theorem thm: CSF-1.) Choose the 'global' basis $\{\omega_0,...,\omega_{\ell-1}\}$ of the de Rham module $H_{\textrm{dR}}(\matheur{R}_\tau(\matheur{C}_{\ell}),k)$ as in eqn: global-omega. Let $d:=\deg v$. For each integer $s$ with $0\leq s<\ell$, we write $s+d = s_0 + n_s'\ell$, where $s_0 \in \ and $\langle z \rangle_{\mathop{\mathrm{ari}}\nolimits}$ is the fractional part of $z$ for every $z

Theorems & Definitions (81)

  • Theorem 1.2.1
  • Remark 1.2.2
  • Theorem 1.2.3
  • Theorem 1.3.1
  • Remark 1.3.3
  • Theorem 1.4.1
  • Corollary 1.4.2
  • Remark 1.4.3
  • Remark 1.4.4
  • Remark 1.4.5
  • ...and 71 more