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Landen's trilogarithm functional equation and $\ell$-adic Galois multiple polylogarithms

Hiroaki Nakamura, Densuke Shiraishi

TL;DR

The paper presents algebraic proofs of Landen's trilogarithm functional equation and its $\\ell$-adic Galois analogs by exploiting the $S_3$-symmetry of the thrice-punctured projective line and the two-variable associator formalisms. It develops dual approaches: (i) a complex-analytic route via Zagier’s tensor criterion to derive Landen’s identity from the KZ associator and, (ii) a chain-rule-based method using the $\,\\ell$-adic Galois associator to obtain $\ ilde{\chi}$-character relations and integrality properties. The results culminate in an $\\ell$-adic Landen trilogarithm formula, an $\\ell$-adic version of Oi–Ueno's relation, and a concrete bridge between classical polylogarithms and their $\\ell$-adic Galois counterparts through polylogarithmic characters. The work deepens the connection between fundamental-group symmetry, associator calculus, and functional equations for multiple polylogarithms within the Grothendieck-Teichmüller framework.

Abstract

The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in the framework of graded Lie algebras and the other is based on the chain rule for the associator power series. In the course of the second proof, we are led to investigate $\ell$-adic Galois multiple polylogarithms appearing as regular coefficients of the $\ell$-adic Galois associator. As an application, we show an $\ell$-adic Galois analog of Oi-Ueno's functional equation between $Li_{1,\dots,1,2}(1-z)$ and $Li_k(z)$'s $(k=1,2,...)$ .

Landen's trilogarithm functional equation and $\ell$-adic Galois multiple polylogarithms

TL;DR

The paper presents algebraic proofs of Landen's trilogarithm functional equation and its -adic Galois analogs by exploiting the -symmetry of the thrice-punctured projective line and the two-variable associator formalisms. It develops dual approaches: (i) a complex-analytic route via Zagier’s tensor criterion to derive Landen’s identity from the KZ associator and, (ii) a chain-rule-based method using the -adic Galois associator to obtain -character relations and integrality properties. The results culminate in an -adic Landen trilogarithm formula, an -adic version of Oi–Ueno's relation, and a concrete bridge between classical polylogarithms and their -adic Galois counterparts through polylogarithmic characters. The work deepens the connection between fundamental-group symmetry, associator calculus, and functional equations for multiple polylogarithms within the Grothendieck-Teichmüller framework.

Abstract

The Galois action on the pro- étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with -adic coefficients, called the -adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its -adic Galois analog can be derived algebraically from the -symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in the framework of graded Lie algebras and the other is based on the chain rule for the associator power series. In the course of the second proof, we are led to investigate -adic Galois multiple polylogarithms appearing as regular coefficients of the -adic Galois associator. As an application, we show an -adic Galois analog of Oi-Ueno's functional equation between and 's .
Paper Structure (7 sections, 6 theorems, 84 equations, 2 tables)

This paper contains 7 sections, 6 theorems, 84 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Set up
  3. Associators and multiple polylogarithms
  4. Polylogarithmic characters and ${\mathbb Z}_\ell$-integrality test
  5. Tensor criterion for Landen's equation for $Li_3$
  6. Complex case
  7. $\ell$-adic Galois case

Key Result

Theorem 1.1

There are suitable paths $\overrightarrow{01}{\leadsto} 1-z, \overrightarrow{01}{\leadsto}\frac{z}{z-1}$ associated to a given path $\gamma_z:\overrightarrow{01}{\leadsto} z$ such that the following functional equation holds for $\sigma\in G_K$.

Theorems & Definitions (15)

  • Theorem 1.1: $\ell$-adic Galois analog of the Landen trilogarithm functional equation
  • Theorem 1.2: $\ell$-adic Galois analog of the Oi-Ueno functional equation
  • Remark 1.3
  • Remark 3.2
  • Lemma 3.3: Key identities
  • proof
  • Remark 3.4
  • Definition 4.1: NW99: $\ell$-adic Galois polylogarithmic character
  • Proposition 4.2
  • proof
  • ...and 5 more