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Linear relations among algebraic points on tensor powers of the Carlitz module

Yen-Tsung Chen, Ryotaro Harada

TL;DR

The paper develops a unified framework using Anderson dual $t$-motives and Frobenius difference equations to study linear relations among algebraic points on tensor powers of the Carlitz module. It provides explicit bounds for generating sets of relation modules and applies the theory to Carlitz polylogarithms, yielding explicit sufficient conditions for their linear independence in both $\infty$-adic and $v$-adic contexts, with consequences for algebraic independence and countable rank. The results connect $t$-module theory with CPL values and extend prior work on linear relations in positive characteristic. The methods offer a versatile approach that could be adapted to broader families of $t$-modules and polylogarithmic-like functions in function field arithmetic.

Abstract

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual $t$-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence for Carlitz polylogarithms at algebraic points in both $\infty$-adic and $v$-adic settings.

Linear relations among algebraic points on tensor powers of the Carlitz module

TL;DR

The paper develops a unified framework using Anderson dual -motives and Frobenius difference equations to study linear relations among algebraic points on tensor powers of the Carlitz module. It provides explicit bounds for generating sets of relation modules and applies the theory to Carlitz polylogarithms, yielding explicit sufficient conditions for their linear independence in both -adic and -adic contexts, with consequences for algebraic independence and countable rank. The results connect -module theory with CPL values and extend prior work on linear relations in positive characteristic. The methods offer a versatile approach that could be adapted to broader families of -modules and polylogarithmic-like functions in function field arithmetic.

Abstract

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual -motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence for Carlitz polylogarithms at algebraic points in both -adic and -adic settings.
Paper Structure (14 sections, 12 theorems, 105 equations)

This paper contains 14 sections, 12 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Motivation and the main result
  3. Carlitz polylogarithms at algebraic points
  4. Organization
  5. Preliminaries
  6. Notation
  7. Anderson $t$-modules and dual $t$-motives
  8. Tensor powers of the Carlitz module
  9. Linear equations on tensor powers of the Carlitz module
  10. The main results
  11. A sufficient condition for linear independence
  12. Carlitz polylogarithms
  13. Linear relations among CPLs at algebraic points
  14. $v$-adic CPLs at algebraic points

Key Result

Theorem 1.1

Let $L\subset\overline{k}$ be a finite extension of $k$ and $n\in\mathbb{Z}_{>0}$. Let $\mathbf{C}^{\otimes n}=(\mathbb{G}_a^n,[\cdot]_n)$ be the $n$-th tensor power of the Carlitz module and $P_1,\dots,P_\ell\in \mathbf{C}^{\otimes n}(L)$ be distinct non-zero $L$-valued points. Then there exists an

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1: ABP04
  • Example 2.2
  • Definition 2.3: And86
  • Lemma 2.4: Anderson, HJ20, NP21
  • Definition 2.5
  • Lemma 2.6
  • ...and 21 more