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

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 ${\mathbb{P}}^1\mathrm{\backslash }\{0,1,\mathrm{\infty }\}$. Furthermore, by replacing complex KZ solutions with $\mathrm{\ell }$-adic Galois 1-cocycles in this proof, we obtain the Landen formula for $\mathrm{\ell }$-adic Galois multiple polylogarithms. This formula involves lower weight terms specific to the $\mathrm{\ell }$-adic Galois setting, which originate from the higher-order terms of the Baker-Campbell-Hausdorff sum $\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{e}\mathrm{x}\mathrm{p}(-e_1)\mathrm{e}\mathrm{x}\mathrm{p}(-e_0))$. These lower weight terms are explicitly described by an integral involving Goldberg polynomials.