Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the algebraic independence of logarithms of Anderson $t$-modules

Oğuz Gezmiş, Changningphaabi Namoijam

TL;DR

This work establishes algebraic independence results for tractable coordinates of logarithms of Anderson $t$-modules built from tensor/exterior constructions of a Drinfeld module with Carlitz tensor powers. The authors deploy Papanikolas's motivic Galois framework, compute explicit motivic Galois groups for the associated $t$-motives, and construct extension motives whose period data capture the quasi-logarithms. They prove that for En (and Gn) the bottom coordinates of logarithms, together with certain quasi-periods, are algebraically independent over the base field, generalizing Chang–Yu results in the Carlitz-tensor-power setting. The results illuminate the transcendence structure of logarithms and periods in function-field arithmetic and have potential applications to special values of Goss $L$-functions and related motives.

Abstract

In the present paper, we determine the algebraic relations among the tractable coordinates of logarithms of Anderson $t$-modules constructed by taking the tensor product of Drinfeld modules of rank $r$ defined over the algebraic closure of the rational function field and their $(r-1)$-st exterior powers with the Carlitz tensor powers. Our results, in the case of the tensor powers of the Carlitz module, generalize the work of Chang and Yu on the algebraic independence of polylogarithms.

On the algebraic independence of logarithms of Anderson $t$-modules

TL;DR

This work establishes algebraic independence results for tractable coordinates of logarithms of Anderson -modules built from tensor/exterior constructions of a Drinfeld module with Carlitz tensor powers. The authors deploy Papanikolas's motivic Galois framework, compute explicit motivic Galois groups for the associated -motives, and construct extension motives whose period data capture the quasi-logarithms. They prove that for En (and Gn) the bottom coordinates of logarithms, together with certain quasi-periods, are algebraically independent over the base field, generalizing Chang–Yu results in the Carlitz-tensor-power setting. The results illuminate the transcendence structure of logarithms and periods in function-field arithmetic and have potential applications to special values of Goss -functions and related motives.

Abstract

In the present paper, we determine the algebraic relations among the tractable coordinates of logarithms of Anderson -modules constructed by taking the tensor product of Drinfeld modules of rank defined over the algebraic closure of the rational function field and their -st exterior powers with the Carlitz tensor powers. Our results, in the case of the tensor powers of the Carlitz module, generalize the work of Chang and Yu on the algebraic independence of polylogarithms.
Paper Structure (20 sections, 26 theorems, 169 equations)

This paper contains 20 sections, 26 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. t-modules
  3. Tate algebras and twisting
  4. t-modules and Anderson generating functions
  5. Biderivations and quasi-periodic functions of t-modules
  6. t-motives and system of difference equations
  7. A-finite dual t-motives and t-modules
  8. The t-module En
  9. The Galois group of En
  10. Endomorphisms of En
  11. The t-module Gn
  12. The Galois group of Gn
  13. Endomorphisms of Gn
  14. The proof of the main result for En
  15. Construction of ga and ha for En
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $n \geq 0$ and each $1 \leq \ell \leq m$, let $\boldsymbol{y}_{\ell} =[y_{\ell,1}, \dots, y_{\ell,rn+r-1}]^{\mathrm{tr}} \in \mathop{\mathrm{Lie}}\nolimits(\mathcal{E}_n)(\mathbb{C}_{\infty})$ be such that $\mathop{\mathrm{Exp}}\nolimits_{\mathcal{E}_n}(\boldsymbol{y}_{\ell})\in \mathcal{E}_n(\m

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 42 more