On Some Unramified Families of Motivic Euler Sums

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.