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

Densuke Shiraishi

TL;DR

The paper proves the Landen formula for complex multiple polylogarithms via algebraic and geometric methods and constructs an $\,\ell$-adic Galois analogue by replacing complex KZ components with $\,\ell$-adic Galois 1-cocycles. A key innovation is a chain rule for KZ solutions under the symmetry $z\mapsto z/(z-1)$, which yields the complex Landen relation, and a parallel chain rule for $\,\ell$-adic Galois cocycles that introduces lower-weight error terms controlled by the Baker–Campbell–Hausdorff sum and Goldberg polynomials. The explicit $\,\ell$-adic corrections are encoded by coefficients $\mu_{\mathbf u,r}$ expressible as integrals against Goldberg polynomials, illuminating how non-abelian Galois actions on the pro-$\ell$ étale fundamental group yield explicit MPL relations. The results connect the algebraic structure of non-commutative formal power series with iterated integrals and Galois representations, providing concrete low-weight identities that bridge the complex and $\,\ell$-adic worlds and advancing understanding of the Galois theory of polylogarithms.

Abstract

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

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

TL;DR

The paper proves the Landen formula for complex multiple polylogarithms via algebraic and geometric methods and constructs an -adic Galois analogue by replacing complex KZ components with -adic Galois 1-cocycles. A key innovation is a chain rule for KZ solutions under the symmetry , which yields the complex Landen relation, and a parallel chain rule for -adic Galois cocycles that introduces lower-weight error terms controlled by the Baker–Campbell–Hausdorff sum and Goldberg polynomials. The explicit -adic corrections are encoded by coefficients expressible as integrals against Goldberg polynomials, illuminating how non-abelian Galois actions on the pro- étale fundamental group yield explicit MPL relations. The results connect the algebraic structure of non-commutative formal power series with iterated integrals and Galois representations, providing concrete low-weight identities that bridge the complex and -adic worlds and advancing understanding of the Galois theory of polylogarithms.

Abstract

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 of . Furthermore, by replacing complex KZ solutions with -adic Galois 1-cocycles in this proof, we obtain the Landen formula for -adic Galois multiple polylogarithms. This formula involves lower weight terms specific to the -adic Galois setting, which originate from the higher-order terms of the Baker-Campbell-Hausdorff sum . These lower weight terms are explicitly described by an integral involving Goldberg polynomials.
Paper Structure (28 sections, 17 theorems, 140 equations, 2 figures, 3 tables)

This paper contains 28 sections, 17 theorems, 140 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction and main results
  2. Key theorems on non-commutative formal power series in two variables
  3. Review of multiple polylogarithms
  4. Complex multiple polylogarithms
  5. $\ell$-adic Galois multiple polylogarithms
  6. Proofs of main results
  7. Explicit $\ell$-adic Landen formula for low weight indices
  8. Case $\mathbf{k}=(2)$ (Weight 2, Depth 1)
  9. Left-hand side
  10. Right-hand side
  11. Case $\mathbf{k}=(3)$ (Weight 3, Depth 1)
  12. Left-hand side
  13. Right-hand side
  14. Case $\mathbf{k}=(4)$ (Weight 4, Depth 1)
  15. Left-hand side
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathbf{k} \in \bigcup_{d=1}^{\infty} \mathbb{N}^d$. For the multiple polylogarithm defined by the following formula holds: Here, for $\mathbf{k}=(k_1,\dots,k_d)$, we set $\operatorname{dp}(\mathbf{k}):=d$. The notation $\mathbf{J} \preceq \mathbf{k}$ means that $\mathbf{J}$ is a refinement of $\mathbf{k}$ (see Definition saibun for the precise definition).

Figures (2)

  • Figure 1: Topological paths on ${\mathbb P}^1({\mathbb C})\backslash \{0,1,\infty\}$
  • Figure 2: Topological loops on ${\mathbb P}^1({\mathbb C})\backslash \{0,1,\infty\}$

Theorems & Definitions (53)

  • Theorem 1.1: Landen formula for multiple polylogarithms OU04
  • Theorem 1.2: The Landen formula for complex multiple polylogarithms
  • Remark 1.3
  • Theorem 1.4: The Landen formula for $\ell$-adic Galois multiple polylogarithms
  • Theorem 1.5: The Landen formula for $\ell$-adic Galois multiple polylogarithms in the case of $\mathbf{k}=n$
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1: Weight and depth of multi-indices
  • Definition 2.2: Refinement of multi-indices
  • Definition 2.3: Monomial $W_{\mathbf{k}}$ and coefficient $\operatorname{Coeff}_{\mathbf{k}}$
  • ...and 43 more