Unramified Euler sums and Hoffman $\star$ basis

TL;DR

This paper introduces a new basis for motivic multiple zeta values using Euler sums and provides evidence for its completeness, advancing the understanding of the algebraic structure of these special numbers.

Contribution

The paper presents a new motivic Euler sum basis for multiple zeta values and proves its basis property under certain conjectures, using a Galois descent approach.

Findings

Motivic Hoffman star basis is a basis of the space of motivic multiple zeta values.

Explicit criteria for embedding multiple zeta values into Euler sums via coaction.

Examples of unramified Euler sums with alternating even and odd patterns.

Abstract

When looking at how periods of $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace )$, i.e. multiple zeta values, embeds into periods of $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, \pm 1, \infty \rbrace )$, i.e. Euler sums, an explicit criteria via the coaction $\Delta$ acting on their motivic versions comes out. In this paper, adopting this Galois descent approach, we present a new basis for the space $\mathcal{H}^{1}$ of motivic multiple zeta values via motivic Euler sums. Up to an analytic conjecture, we also prove that the motivic Hoffman star basis $\zeta^{\star, \mathfrak{m}} (2^{a_{1}},3,\cdots,3, 2^{a_{p}}, 3, 2^{b})$ is a basis of $\mathcal{H}^{1}$. Under a general motivic identity that we conjecture, these bases are identical. Other examples of unramified ES with alternating patterns of even and odds are also provided.