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Artin twists of Drinfeld modules and Goss L-series

Jing Ye

Abstract

Twisted $L$-functions by Dirichlet characters offer deep insights into arithmetic geometry, especially in the study of elliptic curves and abelian varieties over number fields. In the function field setting, Drinfeld modules and Anderson modules serve as analogues of elliptic curves and abelian varieties, and Goss $L$-series play the role of Hasse-Weil $L$-functions. This paper introduces a motivic framework for studying twisted Goss $L$-series via Anderson motives associated to Drinfeld modules and Artin representations. For a Drinfeld module and an Artin representation on the absolute Galois group, we present a construction of Anderson motives associated to them and we show that it comes from a uniformizable abelian Anderson module. We also study their associated $L$-series, which recover the norm of the twisted Goss $L$-values. These results provide an interpretation of twisted Goss $L$-values in terms of regulators of Anderson modules with the help of Taelman's class number formula.

Artin twists of Drinfeld modules and Goss L-series

Abstract

Twisted -functions by Dirichlet characters offer deep insights into arithmetic geometry, especially in the study of elliptic curves and abelian varieties over number fields. In the function field setting, Drinfeld modules and Anderson modules serve as analogues of elliptic curves and abelian varieties, and Goss -series play the role of Hasse-Weil -functions. This paper introduces a motivic framework for studying twisted Goss -series via Anderson motives associated to Drinfeld modules and Artin representations. For a Drinfeld module and an Artin representation on the absolute Galois group, we present a construction of Anderson motives associated to them and we show that it comes from a uniformizable abelian Anderson module. We also study their associated -series, which recover the norm of the twisted Goss -values. These results provide an interpretation of twisted Goss -values in terms of regulators of Anderson modules with the help of Taelman's class number formula.
Paper Structure (35 sections, 60 theorems, 331 equations)

This paper contains 35 sections, 60 theorems, 331 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results.
  4. Preliminaries
  5. Notation
  6. Frobenius twist and twisted polynomial ring
  7. Twists of matrices
  8. $\mathbf{A}$-motives
  9. Definition of $\mathbf{A}$-motives
  10. $\mathfrak{l}$-adic cohomology of $\mathbf{A}$-motives
  11. Base change and Weil restriction of $\mathbf{A}$-motives
  12. Anderson $\mathbf{A}$-modules
  13. Tate modules and $\mathfrak{l}$-adic cohomology
  14. Taelman class number formula
  15. Fitting ideals
  16. ...and 20 more sections

Key Result

Theorem A

Let $\varphi/E$ be a Drinfeld $\mathbf{A}$-module of rank $r$ and $\rho: G_E\to \operatorname{GL}_n(\overline{\mathbb{F}}_{q})$ an $\overline{\mathbb{F}}_{q}$-representation of dimension $n$. Then, there exists an abelian $\mathbf{A}$-module $\mathcal{E}$ of dimension $N = nd$ and rank $rN$ over $E$

Theorems & Definitions (134)

  • Theorem A: Theorem \ref{['existsAnderson']}
  • Theorem B: Theorem \ref{['thmB']}
  • Corollary C: Corollary \ref{['twsit_is_uniformizable']}
  • Theorem D: Theorem \ref{['thmC']}
  • Corollary E: Corollary \ref{['corD']}
  • Corollary F: Corollary \ref{['corE']}
  • Theorem G: Theorem \ref{['thm:transcendence_criterion']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 124 more