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Uniqueness of $v$-adic Gamma Functions in the Gross-Koblitz-Thakur Formulas

Ting-Wei Chang, Hung-Chun Tsui

TL;DR

The paper addresses the problem of uniqueness for Gross-Koblitz-Thakur type formulas in the function-field setting by classifying all continuous non-vanishing functions that preserve the GK identities for the three $v$-adic gamma functions. Building on Adolphson's approach, the authors establish a unified framework: a continuous non-vanishing function $H$ satisfies the GK formula if and only if $H(\mathbf{x}) = \Gamma^\bullet_v(\mathbf{x}) \cdot \frac{G(\mathbf{x})}{G(-\varphi(-\mathbf{x}))}$ for some continuous non-vanishing $G$, with explicit corollaries for the arithmetic, geometric, and two-variable cases. The proof proceeds by reformulating the GK identities, reducing to a coboundary problem for $F=H/\Gamma$, and employing a sequence of carefully constructed maps to derive the required $G$ via limit arguments, following the strategy of Adolphson. The results illuminate the structure of GK-type interpolations in positive characteristic and parallel the classical p-adic theory, providing a complete characterization of all such $H$ and extending Greenberg-type uniqueness to a broad function-field context.

Abstract

In this paper, we determine all continuous non-vanishing functions satisfying Gross-Koblitz-Thakur formulas in positive characteristic.

Uniqueness of $v$-adic Gamma Functions in the Gross-Koblitz-Thakur Formulas

TL;DR

The paper addresses the problem of uniqueness for Gross-Koblitz-Thakur type formulas in the function-field setting by classifying all continuous non-vanishing functions that preserve the GK identities for the three -adic gamma functions. Building on Adolphson's approach, the authors establish a unified framework: a continuous non-vanishing function satisfies the GK formula if and only if for some continuous non-vanishing , with explicit corollaries for the arithmetic, geometric, and two-variable cases. The proof proceeds by reformulating the GK identities, reducing to a coboundary problem for , and employing a sequence of carefully constructed maps to derive the required via limit arguments, following the strategy of Adolphson. The results illuminate the structure of GK-type interpolations in positive characteristic and parallel the classical p-adic theory, providing a complete characterization of all such and extending Greenberg-type uniqueness to a broad function-field context.

Abstract

In this paper, we determine all continuous non-vanishing functions satisfying Gross-Koblitz-Thakur formulas in positive characteristic.

Paper Structure

This paper contains 11 sections, 10 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. v-adic Gamma Functions
  4. Gross-Koblitz-Thakur Formulas for v-adic Gamma Functions
  5. Main Theorem
  6. Strategy of Proof
  7. Proof of the Theorem \ref{['main result']}
  8. The "if" Direction
  9. The "only if" Direction
  10. Proof of Lemma \ref{['all limits']}
  11. Generalization

Key Result

Theorem 1.1

For $x \in (q-1)^{-1}\mathbb{Z}$ with $0\leq x < 1$, one has Here, $\langle y \rangle := y-[y]$ is the fractional part of a non-negative real number $y$.

Theorems & Definitions (22)

  • Theorem 1.1: Gross-Koblitz formula
  • Theorem 1.2: Adolphson
  • Definition 1.3
  • Theorem 1.4: Gross-Koblitz-Thakur formula, arithmetic case
  • Theorem 1.5: Gross-Koblitz-Thakur formulas, geometric and two-variable cases
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 2.1
  • Theorem 2.2
  • ...and 12 more