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Algebraic Relations among Special Gamma Values and the Chowla-Selberg Phenomenon over Function Fields

Fu-Tsun Wei

TL;DR

This work addresses the problem of understanding algebraic relations among function-field gamma values and their connection to CM periods in positive characteristic. It develops a period-distribution framework built from CM dual $t$-motives, Stickelberger functions, and diamond brackets, then proves Chowla–Selberg-type formulas for quasi-periods of CM abelian and Drinfeld modules and derives a Lerch-type normalization. The results include a Lang–Rohrlich-type description of algebraic relations among $\Gamma_{geo}$, $\Gamma_{ari}$, and $\Gamma$, a Deligne–Gross-type conjecture for CM Hodge–Pink structures, and explicit constructions via shtuka functions to realize generalized CM types. Collectively, these findings deepen the link between special gamma values and abelian CM periods in the function-field setting, with universal distribution phenomena and cyclotomic specializations capturing the full period structure.

Abstract

The aim of this paper is to determine all algebraic relations among various special gamma values over function fields, and prove a Chowla-Selberg-type formula for quasi-periods of CM abelian $t$-modules. Our results are based on the intrinsic relations between gamma values in question and periods of CM dual $t$-motives, which are interpreted in terms of their "distributions". This also enables us to derive an analogue of the Deligne-Gross period conjecture for CM Hodge-Pink structures.

Algebraic Relations among Special Gamma Values and the Chowla-Selberg Phenomenon over Function Fields

TL;DR

This work addresses the problem of understanding algebraic relations among function-field gamma values and their connection to CM periods in positive characteristic. It develops a period-distribution framework built from CM dual -motives, Stickelberger functions, and diamond brackets, then proves Chowla–Selberg-type formulas for quasi-periods of CM abelian and Drinfeld modules and derives a Lerch-type normalization. The results include a Lang–Rohrlich-type description of algebraic relations among , , and , a Deligne–Gross-type conjecture for CM Hodge–Pink structures, and explicit constructions via shtuka functions to realize generalized CM types. Collectively, these findings deepen the link between special gamma values and abelian CM periods in the function-field setting, with universal distribution phenomena and cyclotomic specializations capturing the full period structure.

Abstract

The aim of this paper is to determine all algebraic relations among various special gamma values over function fields, and prove a Chowla-Selberg-type formula for quasi-periods of CM abelian -modules. Our results are based on the intrinsic relations between gamma values in question and periods of CM dual -motives, which are interpreted in terms of their "distributions". This also enables us to derive an analogue of the Deligne-Gross period conjecture for CM Hodge-Pink structures.
Paper Structure (33 sections, 55 theorems, 354 equations)

This paper contains 33 sections, 55 theorems, 354 equations.

Table of Contents

  1. Introduction
  2. Special gamma values and algebraic independence
  3. The period distribution and gamma distributions
  4. The Chowla--Selberg formula for abelian t-modules
  5. The Deligne--Gross conjecture for CM Hodge--Pink structures
  6. Preliminaries
  7. Basic settings
  8. Dual t-motives and Betti modules
  9. CM fields and CM types
  10. CM dual t-motives
  11. Period symbols and the period distribution
  12. Period symbols
  13. Stickelberger functions
  14. The period distribution
  15. The restriction to cyclotomic function fields
  16. ...and 18 more sections

Key Result

Theorem 1.1.1

Given $\mathfrak{n} \in A_+$ and $\ell \in \mathbb{N}$, we have that where $\epsilon_\mathfrak{n} = 1$ if $\deg \mathfrak{n} >0$ and $0$ otherwise. Consequently, all algebraic relations among gamma values $\Gamma_{\operatorname{geo}}(x)$, $\Gamma_{\operatorname{ari}}(y)$, $\Gamma(x,y)$ for $x \in k\setminus (-A_+\cup\{0\})$ and $y \in \mathbb{Z}_{(p)}$ are explained

Theorems & Definitions (134)

  • Theorem 1.1.1: Theorem \ref{['thm: Lang-Rohrlich-conj']} and Corollary \ref{['cor: AR-Gamma']}
  • Remark 1.1.2
  • Theorem 1.2.1: Theorem \ref{['thm: PD']}
  • Theorem 1.2.2: Theorem \ref{['thm: Gamma dist']}
  • Theorem 1.2.4: Theorem \ref{['thm: CSP']}
  • Remark 1.2.5
  • Theorem 1.3.1: Theorem \ref{['thm: ex-qp-CM']} (1)
  • Theorem 1.3.2: Theorem \ref{['thm: ex-qp-CM']} (2)
  • Theorem 1.3.3: Theorem \ref{['thm: ex-qp-CM']} (3)
  • Remark 1.3.4
  • ...and 124 more