Spence-Kummer's trilogarithm functional equation and its underlying geometry

In this paper, we investigate the underlying geometry of the Spence--Kummer functional equation for the trilogarithm. Our geometry determines a certain path system on the projective line minus three points, connecting the standard tangential base point to the nine variables of the $Li_{3}$ terms in the equation, which reflects the geometry of the so-called non-Fano arrangement. Consequently, we derive a precise form of the Spence--Kummer equation together with its $\ell$-adic Galois analogue by using algebraic relations between polylogarithm generating series arising from the path system. We apply the tensor and homotopy criteria for functional equations of iterated integrals due to Zagier and Nakamura--Wojtkowiak. To compute the lower-degree terms of the functional equation in both the complex and the $\ell$-adic Galois cases, we also focus on a diagram of three geometric objects: the moduli space $M_{0,5}$, the complement to the Coxeter arrangement of type ${\rm B_3}$, and the complement to the non-Fano arrangement.