Unramified Motivic Multiple Mixed Values

Abstract

The multiple mixed values (MMVs) are level two variants of multiple zeta values produced by restricting the summation indices to fixed parity patterns. One particularly interesting problem is to determine exactly when such values are actually in level one, namely, expressible in terms of multiple zeta values. To solve this completely is beyond our current knowledge since it calls for new ideas from transcendental number theory. However, much progress has been made on the motivic level. Previously, using the descent theory developed by Brown et al. we have tackled this problem for a few special classes of MMVs with regular parity patterns among the summation indices, including Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and our own multiple $S$-values. In this paper, we turn to the general case and determine completely all the unramified motivic MMVs of depth less than four as well as a few families of unbounded depths. At the end of the paper, we will present some general conjectures to describe the unramified MMVs at all depths greater than three.