On Some Unramified Families of Motivic Euler Sums

TL;DR

This paper identifies specific families of motivic Euler sums that are unramified, meaning they can be expressed as rational linear combinations of motivic multiple zeta values, and proves some identities relating them.

Contribution

It applies Glanois's criterion to find new unramified motivic Euler sums and establishes concrete identities linking these sums to motivic MZVs under certain assumptions.

Findings

Identified families of unramified motivic Euler sums

Proved identities relating motivic Euler sums to motivic MZVs

Demonstrated the unramified property using Glanois's criterion

Abstract

It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as $\Q$-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for motivic Euler sums to be unramified, namely, expressible as $\Q$-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified motivic Euler sums in two groups. In one such group we can further prove the concrete identities relating the motivic Euler sums to the motivic MZVs, determined up to rational multiple of a motivic Riemann zeta value by a result of Brown, under the assumption that the analytic version of such identities hold.