On the Landen formula for multiple polylogarithms and its $\ell$-adic Galois analogue

Abstract: In the present paper, we provide an algebraic and geometric proof of the Landen formula for complex multiple polylogarithms originally established by Okuda and Ueno. Our approach employs a chain rule of complex KZ solutions arising from the symmetry $z \mapsto \frac{z}{z-1}$ of $\mathbb{P}^1 \backslash {0,1,\infty}$. Furthermore, by replacing complex KZ solutions with $\ell$-adic Galois 1-cocycles in this proof, we obtain the Landen formula for $\ell$-adic Galois multiple polylogarithms. This formula involves lower weight terms specific to the $\ell$-adic Galois setting, which originate from the higher-order terms of the Baker-Campbell-Hausdorff sum ${\rm log}({\rm exp}(-e_1){\rm exp}(-e_0))$. These lower weight terms are explicitly described by an integral involving Goldberg polynomials.