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Densuke Shiraishi

Publications and source records attributed to Densuke Shiraishi.

5 recordsLinked to original sources
arXiv2026

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

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.

math.NT↗
arXiv2026

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 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.

math.NT↗
arXiv2025

Landen's trilogarithm functional equation and $\ell$-adic Galois multiple polylogarithms

The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in the framework of graded Lie algebras and the other is based on the chain rule for the associator power series. In the course of the second proof, we are led to investigate $\ell$-adic Galois multiple polylogarithms appearing as regular coefficients of the $\ell$-adic Galois associator. As an application, we show an $\ell$-adic Galois analog of Oi-Ueno's functional equation between $Li_{1,\dots,1,2}(1-z)$ and $Li_k(z)$'s $(k=1,2,...)$ .

math.NT↗
arXiv2021

On $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols

The $\ell$-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in $\ell$-adic numbers, which arises from Galois actions on $\ell$-adic étale paths on ${\mathbb P}^1 \backslash \{0,1,\infty\}$. In the present paper, we discuss a relationship between $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols in some special cases studied by a work of Hirano-Morishita. We show that a functional equation of $\ell$-adic Galois polylogarithms by Nakamura-Wojtkowiak implies a reciprocity law of triple $\ell$-th power residue symbols.

math.NT↗