On a conjecture of Furusho over function fields

Abstract

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers \({\mathfrak {s}}=(s_1, \ldots , s_r)\), we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the \(\infty \)-adic MZV and v-adic MZV at \({\mathfrak {s}}\), completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of \(r=1\). Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any \(\infty \)-adic linear relation on MZV’s with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV’s.