Depth-graded motivic multiple zeta values

Francis Brown

doi:10.1112/s0010437x20007654

Abstract

We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.

Highlights

IntroductionWe begin by motivating the results of this paper from two apparently different, but equivalent, perspectives
We begin by motivating the results of this paper from two apparently different, but equivalent, perspectives.1.1 Depth filtration on multiple zeta values Multiple zeta values are defined for integers n1, . . . , nr−1 ≥ 1 and nr ≥ 2 by1 ζ(n1, . . . , nr) = 1≤k1
Even if one works modulo commutators, these two possible definitions of generators in depth 4 are related by a non-trivial isomorphism on the space of period polynomials

Summary

Introduction

We begin by motivating the results of this paper from two apparently different, but equivalent, perspectives. 1.1 Depth filtration on multiple zeta values Multiple zeta values are defined for integers n1, . Their weight is the quantity n1 + · · · + nr, and their depth is the number of indices r. Relations between multiple zeta values of depth 2 were first studied by Euler. Let ZN denote the Q-vector space spanned by multiple zeta values in weight N. That the space Z of multiple zeta values is isomorphic to the direct sum of the ZN (in other words, the weight is a grading), and secondly, that the dimension of ZN can be expressed using the generating series dimQ (ZN )sN = − 1 s2 s3.

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Conclusion