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The Galois action on the pro-$\ell$ \'etale fundamental groupoid of theprojective line minus three points with rational base points gives rise to anon-commutative formal power series in two variables with $\ell$-adiccoefficients, 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 $S_3$-symmetryof the projective line minus three points. Twofold proofs of the functionalequation will be presented, one is based on Zagier's tensor criterion devisedin the framework of graded Lie algebras and the other is based on the chainrule for the associator power series. In the course of the second proof, we areled to investigate $\ell$-adic Galois multiple polylogarithms appearing asregular 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