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

TL;DR

This paper explores the connection between $ ext{ell}$-adic Galois polylogarithms and triple $ ext{ell}$-th power residue symbols, revealing how functional equations imply reciprocity laws in special cases.

Contribution

It establishes a link between $ ext{ell}$-adic Galois polylogarithms and residue symbols, extending understanding of their arithmetic properties and reciprocity laws.

Findings

Functional equations of $ ext{ell}$-adic Galois polylogarithms imply reciprocity laws.

Relationship between Galois polylogarithms and triple $ ext{ell}$-th power residue symbols.

Special case analysis based on previous work of Hirano-Morishita.

Abstract

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 \'etale 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.