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

TL;DR

This paper explores algebraic derivations of Landen's trilogarithm functional equation and its $ ext{ell}$-adic Galois analog, revealing new insights into $ ext{ell}$-adic Galois multiple polylogarithms and their symmetries.

Contribution

It provides two algebraic proofs of Landen's trilogarithm functional equation using $S_3$-symmetry and introduces $ ext{ell}$-adic Galois multiple polylogarithms as coefficients of the associator.

Findings

Two proofs of Landen's functional equation are presented.

An $ ext{ell}$-adic Galois analog of a known functional equation is established.

Investigation of $ ext{ell}$-adic Galois multiple polylogarithms as associator coefficients.

Abstract

The Galois action on the pro-$\ell$ \'etale 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 $(k=1,2,...)$ .