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Duality-reflection formulas of multiple polylogarithms and their $\ell$-adic Galois analogues

Densuke Shiraishi

TL;DR

The paper derives duality-reflection formulas for both complex and $\\ell$-adic Galois multiple polylogarithms on $\\mathbb{P}^1\\setminus\\{0,1,\\infty\\}$, generalizing Oi--Ueno's reflection for polylogarithms and their specializations to multiple zeta values. It uses an algebraic chain-rule identity among associators arising from $S_3$-symmetry, namely $G_0(X,Y)(z;\\gamma)=G_0(Y,X)(1-z;\\gamma')\\cdot\\Phi(X,Y)$, and an analogous $\\ell$-adic equation with $\\mathfrak f^{z,\\gamma}_{\\sigma}(X,Y)$, to establish a unified framework for duality and reflection across the complex and $\\ell$-adic settings. The main results include a pair of functional equations (Main1 and Main2) that generalize known identities for multiple zeta values and their $\\ell$-adic Galois analogues, situating them within the KZ equation, Drinfeld associators, and Galois actions. The work points to potential $p$-adic extensions via Furusho's theory and broad arithmetic implications of symmetry-driven functional relations among polylogarithms.

Abstract

In the present paper, we derive formulas of complex and $\ell$-adic multiple polylogarithms, which have two aspects: a duality in terms of indexes and a reflection in terms of variables. We provide an algebraic proof of these formulas by using algebraic relations between associators arising from the $S_3$-symmetry of the projective line minus three points.

Duality-reflection formulas of multiple polylogarithms and their $\ell$-adic Galois analogues

TL;DR

The paper derives duality-reflection formulas for both complex and -adic Galois multiple polylogarithms on , generalizing Oi--Ueno's reflection for polylogarithms and their specializations to multiple zeta values. It uses an algebraic chain-rule identity among associators arising from -symmetry, namely , and an analogous -adic equation with , to establish a unified framework for duality and reflection across the complex and -adic settings. The main results include a pair of functional equations (Main1 and Main2) that generalize known identities for multiple zeta values and their -adic Galois analogues, situating them within the KZ equation, Drinfeld associators, and Galois actions. The work points to potential -adic extensions via Furusho's theory and broad arithmetic implications of symmetry-driven functional relations among polylogarithms.

Abstract

In the present paper, we derive formulas of complex and -adic multiple polylogarithms, which have two aspects: a duality in terms of indexes and a reflection in terms of variables. We provide an algebraic proof of these formulas by using algebraic relations between associators arising from the -symmetry of the projective line minus three points.
Paper Structure (5 sections, 2 theorems, 43 equations, 1 figure, 1 table)

This paper contains 5 sections, 2 theorems, 43 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Complex multiple polylogarithms
  4. $\ell$-adic Galois multiple polylogarithms
  5. Proof of main results

Key Result

Theorem 1.1

Given a (possibly, tangential base) point $z$ of $\mathbb{P}^1(\mathbb{C}) \backslash \{0,1,\infty\}$ and a path $\gamma \in \pi_1^{\rm top}\left(\mathbb{P}^1(\mathbb{C}) \backslash \{0,1,\infty\}; \overrightarrow{01}, z\right)$, define the path $\gamma'$ associated to $\gamma$ by where $\phi \in {\rm Aut}\left(\mathbb{P}^1(\mathbb{C}) \backslash \{0,1,\infty\}\right)$ is given by $\phi(t)=1-t$ a

Figures (1)

  • Figure 1: Topological paths on ${\mathbb P}^1({\mathbb C})\backslash \{0,1,\infty\}$

Theorems & Definitions (6)

  • Theorem 1.1: The duality-reflection formula of complex multiple polylogarithms
  • Theorem 1.2: The duality-reflection formula of $\ell$-adic Galois multiple polylogarithms
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • proof : Proof of Theorem \ref{['M1']}, Theorem \ref{['M2']}