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Multiple polylogarithms and mixed Tate motives

A. B. Goncharov

TL;DR

This paper develops a comprehensive framework for multiple polylogarithms by integrating analytic, Hodge-theoretic, and motivic viewpoints. It constructs the associated Hopf algebras of framed mixed Tate structures and mixed Tate motives, and introduces the motivic torsor of paths on G_m−μ_N, linking these objects to cyclotomic Hopf algebras and modular geometry. The work demonstrates that multiple polylogarithms arise as periods of mixed Tate motives and outlines conjectures on motivic Galois groups and universality, with explicit regularization and shuffle-relations structures. It also connects to the broader program of understanding the arithmetic of special values and the geometry of modular varieties via motivic and Hodge-theoretic frameworks.

Abstract

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.

Multiple polylogarithms and mixed Tate motives

TL;DR

This paper develops a comprehensive framework for multiple polylogarithms by integrating analytic, Hodge-theoretic, and motivic viewpoints. It constructs the associated Hopf algebras of framed mixed Tate structures and mixed Tate motives, and introduces the motivic torsor of paths on G_m−μ_N, linking these objects to cyclotomic Hopf algebras and modular geometry. The work demonstrates that multiple polylogarithms arise as periods of mixed Tate motives and outlines conjectures on motivic Galois groups and universality, with explicit regularization and shuffle-relations structures. It also connects to the broader program of understanding the arithmetic of special values and the geometry of modular varieties via motivic and Hodge-theoretic frameworks.

Abstract

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.

Paper Structure

This paper contains 2 sections, 28 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Analytic properties

Key Result

Theorem 1.3

The Hodge-Tate structure ${\cal P}^{\cal H}({\Bbb C} P^1 - \{0, 1, \infty\}; v_0, v_1)(-n)$ with an arbitrary framing given by $\gamma(\varepsilon_1, ...,\varepsilon_n)$ and $\omega(\delta_1, ..., \delta_m)$ is equivalent to a product of the multiple zeta structures (3.6.01.12). In particular all pe

Theorems & Definitions (33)

  • Conjecture 1.1
  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • Theorem 2.1
  • ...and 23 more