Mixed Tate motives over $\Z$

TL;DR

This paper establishes that the category of mixed Tate motives over integers is generated by the motivic fundamental group of the projective line minus three points, and confirms Hoffman's conjecture on expressing multiple zeta values.

Contribution

It proves the category of mixed Tate motives over Z is spanned by a specific motivic fundamental group and verifies Hoffman's conjecture on multiple zeta values.

Findings

The category of mixed Tate motives over Z is spanned by the motivic fundamental group of P^1 minus three points.

Every multiple zeta value can be expressed as a Q-linear combination of zeta values with arguments 2 or 3.

Confirmed Hoffman's conjecture on the structure of multiple zeta values.

Abstract

We prove that the category of mixed Tate motives over $\Z$ is spanned by the motivic fundamental group of $\Pro^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\Q$-linear combination of $\zeta(n_1,..., n_r)$ where $n_i\in \{2,3\}$.