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

Densuke Shiraishi

TL;DR

The paper clarifies the geometric origin of Spence–Kummer's trilogarithm functional equation by embedding the problem in a path-system framework on $\mathbb{P}^1\setminus\{0,1,\infty\}$ tied to the non-Fano arrangement and Goncharov's moduli space. It then establishes explicit 9-term equations for both the complex trilogarithm and its $\ell$-adic Galois analogue, deriving lower-weight terms through a careful analysis of the fundamental groups of $V_{ m non\text{-}Fano}$ and its B3-cover, via the moduli space $M_{0,5}$. The method leverages Zagier and Nakamura–Wojtkowiak tensor-homotopy criteria to obtain the dilogarithm relations and, in the $\ell$-adic setting, reveals important error terms that connect to generalized Soulé characters and integrality properties. Overall, the work links deep geometric configurations with concrete functional equations for polylogarithms, providing both complex-analytic and $\ell$-adic perspectives and highlighting the role of moduli and arrangement theory in polylogarithm identities.

Abstract

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.

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

TL;DR

The paper clarifies the geometric origin of Spence–Kummer's trilogarithm functional equation by embedding the problem in a path-system framework on tied to the non-Fano arrangement and Goncharov's moduli space. It then establishes explicit 9-term equations for both the complex trilogarithm and its -adic Galois analogue, deriving lower-weight terms through a careful analysis of the fundamental groups of and its B3-cover, via the moduli space . The method leverages Zagier and Nakamura–Wojtkowiak tensor-homotopy criteria to obtain the dilogarithm relations and, in the -adic setting, reveals important error terms that connect to generalized Soulé characters and integrality properties. Overall, the work links deep geometric configurations with concrete functional equations for polylogarithms, providing both complex-analytic and -adic perspectives and highlighting the role of moduli and arrangement theory in polylogarithm identities.

Abstract

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 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 -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 -adic Galois cases, we also focus on a diagram of three geometric objects: the moduli space , the complement to the Coxeter arrangement of type , and the complement to the non-Fano arrangement.
Paper Structure (11 sections, 5 theorems, 155 equations, 8 figures, 12 tables)

This paper contains 11 sections, 5 theorems, 155 equations, 8 figures, 12 tables.

Table of Contents

  1. Notation
  2. Spaces
  3. Base points
  4. Fundamental groups
  5. Fundamental groups of $V_{\rm B_3}^{\mathrm{an}}$ and $V_{\rm non\text{-}Fano}^{\mathrm{an}}$
  6. Review of Polylogarithms
  7. Complex polylogarithms
  8. $\ell$-adic Galois polylogarithms
  9. Proof of the main results
  10. Complex case
  11. $\ell$-adic Galois case

Key Result

Theorem 1

Given a ${\mathbb C}$-rational point $(x,y) \in V_{\rm non\text{-}Fano}({\mathbb C})$ and a path $\gamma_0 \in \pi_1^{\rm top}\left(V_{\rm non\text{-}Fano}^{\mathrm{an}};{\vec{v}},(x,y)\right)$, define the path system $\{\gamma_i\}_{i=1,\ldots,9}$ associated with $\gamma_0$ by where $\delta_i \in \pi_1^{\rm top}\left({{\mathbb P}}^{1}({\mathbb C}) \backslash \{0,1,\infty\};\overrightarrow{01},f_i

Figures (8)

  • Figure 1: The non-Fano arrangement
  • Figure 2: Topological paths on ${{\mathbb P}}^{1}({\mathbb C}) \backslash \{0,1,\infty\}$
  • Figure 3: Key diagram to capture $\pi_1\left(V_{\rm non\text{-}Fano}\right)$
  • Figure 4: Goncharov's $7$-point configuration on $\mathbb{P}^2$
  • Figure 5: (Pseudo-)line arrangements appearing in the present paper
  • ...and 3 more figures

Theorems & Definitions (18)

  • Theorem 1: The $9$-term functional equation for the complex trilogarithm
  • Theorem 2: The $9$-term functional equation for the $\ell$-adic Galois trilogarithm
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 1.1
  • Remark 2.1
  • Theorem 4.1: Functional equations for complex dilogarithms
  • proof
  • ...and 8 more