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On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules

Oğuz Gezmiş, Changningphaabi Namoijam

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(θ)$. For a Drinfeld module $φ$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_φ$ of $φ$. Moreover, when $φ$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_φ$ at any positive integer is transcendental over $K$.

On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules

Abstract

Let be the finite field with elements and consider the rational function field . For a Drinfeld module defined over , we study the transcendence of special values of the Goss -function attached to the abelian -motive of . Moreover, when is a Drinfeld module of rank defined over which has everywhere good reduction, we prove that the value of the Goss -function attached to the -st exterior power of at any positive integer is transcendental over .

Paper Structure

This paper contains 20 sections, 8 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Special values of Goss L-functions
  4. Outline of the paper
  5. t-modules and Taelman t-motives
  6. t-modules
  7. Effective t-motives and abelian t-modules
  8. Taelman t-motives
  9. An interpretation of Theorem \ref{['T:mainThm']} in terms of Taelman L-values
  10. L-function of abelian t-motives
  11. Explicitly constructed abelian t-motives
  12. Construction of $M_n'$ and $M_{\mathcal{E}_{n,A}'}$
  13. Construction of $\textbf{C}_n^{(\mathfrak{b}')}$
  14. Construction of $\widetilde{M}_n$
  15. Local factors at places with good reduction
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $n$ be a positive integer and $\phi$ be a Drinfeld $\mathbf{A}$-module of rank $r\geq 2$ defined over $K$. Moreover, for any $\mathfrak{b}\in K\setminus\{0\}$, let $C^{(\mathfrak{b})}$ be the Drinfeld $\mathbf{A}$-module of rank one given by $C^{(\mathfrak{b})}_t:=\theta+\mathfrak{b}\tau$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 14 more