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Involution on a quotient space of multiple zeta values in positive characteristic

Yoshinori Mishiba

Abstract

In this paper, we introduce multiple zeta dagger values and special values of Carlitz multiple dagger polylogarithms, and study their properties. In particular, using these values, we construct a non-trivial involution on a certain quotient space of multiple zeta values in positive characteristic.

Involution on a quotient space of multiple zeta values in positive characteristic

Abstract

In this paper, we introduce multiple zeta dagger values and special values of Carlitz multiple dagger polylogarithms, and study their properties. In particular, using these values, we construct a non-trivial involution on a certain quotient space of multiple zeta values in positive characteristic.
Paper Structure (16 sections, 11 theorems, 87 equations)

This paper contains 16 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Positive characteristic
  4. Main theorem
  5. Organization of the paper
  6. Properties of non-dagger values
  7. Algebraic setup
  8. Linear relations among non-dagger values
  9. Properties of dagger values
  10. Relations between non-dagger and dagger values
  11. Relations among the special values of CMDPLs
  12. Proof of the main theorem
  13. Relations among MZDVs
  14. $q$-shuffle product formula among single zeta dagger values
  15. Linear relations among MZDVs
  16. ...and 1 more sections

Key Result

Theorem 1.3.4

There exists a non-trivial $R$-algebra involution $\iota$ on $\mathcal{Z}_{R} / \zeta_{A}(q - 1) \mathcal{Z}_{R}$ such that for all $\mathfrak{s} \in \mathcal{I}$.

Theorems & Definitions (34)

  • Example 1.1.2
  • Definition 1.3.1
  • Remark 1.3.2
  • Example 1.3.3
  • Theorem 1.3.4
  • Remark 1.3.5
  • Example 1.3.6
  • Conjecture 1.3.7
  • Conjecture 1.3.8
  • Proposition 2.2.1
  • ...and 24 more