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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.

Submission history

From: Densuke Shiraishi [view email]
[v1] Tue, 18 Jul 2023 16:32:49 UTC (36 KB)
[v2] Tue, 1 Aug 2023 16:30:41 UTC (36 KB)
[v3] Wed, 26 Nov 2025 02:29:02 UTC (37 KB)
[v4] Tue, 23 Jun 2026 06:36:12 UTC (51 KB)