On distribution formulas for complex and $\ell$-adic polylogarithms

TL;DR

This paper develops an $-adic Galois framework for polylogarithm distribution formulas, introducing universal measures that interpolate these relations across all degrees, with a focus on path and arithmetic dependencies.

Contribution

It introduces universal Kummer-Heisenberg measures that generalize $$-adic polylogarithmic distribution relations for all degrees, advancing the understanding of their arithmetic and path-dependent properties.

Findings

Defined universal Kummer-Heisenberg measures

Interpolated $$-adic polylogarithmic distribution relations

Analyzed path dependency and arithmetic behaviors

Abstract

We study an $\ell$-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer-Heisenberg measures that enable interpolating the $\ell$-adic polylogarithmic distribution relations for all degrees.