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

Abstract

In this paper, we investigate the underlying geometry of the Spence--Kummer functional equation for the trilogarithm. Our geometry naturally 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, reflecting the symmetry of the 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 complex and $\ell$-adic iterated integrals developed by 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.