Unramified Euler sums and Hoffman $\star$ basis

Abstract

When looking at how periods of ${\pi }_1^{\mathfrak{m}}({\mathbb{P}}^1╲\{0,1,\infty \})$, i.e. multiple zeta values, embeds into periods of ${\pi }_1^{\mathfrak{m}}({\mathbb{P}}^1╲\{0,\pm 1,\infty \})$, 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.