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.
Categories: math.NT
