On Zagier-Hoffman's conjectures in positive characteristic

Abstract

We study Todd-Thakur's analogues of Zagier-Hoffman's conjectures in positive characteristic. These conjectures predict the dimension and an explicit basis $\mathcal{T}_w$ of the span of characteristic $p$ multiple zeta values of fixed weight $w$ which were introduced by Thakur as analogues of classical multiple zeta values of Euler.

In the present paper we first establish the algebraic part of these conjectures which states that the span of characteristic $p$ multiple zeta values of weight $w$ is generated by the set $\mathcal{T}_w$. As a consequence, we obtain upper bounds for the dimension. This is the analogue of Brown's theorem and also those of Deligne-Goncharov and Terasoma.

We then prove two results towards the transcendental part of these conjectures. First, we establish the linear independence for a large subset of $\mathcal{T}_w$ and yield lower bounds for the dimension. Second, for small weights we prove the linear independence for the whole set $\mathcal{T}_w$ and completely solve Zagier-Hoffman's conjectures in positive characteristic. Our key tool is the Anderson-Brownawell-Papanikolas criterion for linear independence in positive characteristic.