The motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$ and its action on the fundamental group of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$

TL;DR

This paper introduces confluence relations for motivic Euler sums, showing they fully describe all linear relations and automorphisms related to the motivic Galois group acting on the fundamental group of a punctured projective line.

Contribution

It establishes confluence relations as the complete set of linear relations among motivic Euler sums and describes their role in automorphisms of the de Rham fundamental groupoid.

Findings

All linear relations among motivic Euler sums are given by confluence relations.

Explicit $Q$-linear expansions of motivic Euler sums are provided.

Coefficients in expansions are shown to be 2-adically integral.

Abstract

In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis and $2$-adic integrality of the coefficients in the expansion.