Depth-graded motivic multiple zeta values

TL;DR

This paper explores the structure of multiple zeta values through depth filtration, connecting motivic Galois groups, modular forms, and Lie algebras to conjecturally describe all their identities and relations.

Contribution

It constructs an explicit Lie algebra of solutions to double shuffle equations using period polynomials, proposing a unified conjecture linking multiple zeta values and Grothendieck-Teichmüller theory.

Findings

Constructs a Lie algebra of solutions to double shuffle equations.

Proposes a conjecture on the homology of this Lie algebra.

Links multiple zeta value identities to broader algebraic structures.

Abstract

We study the depth filtration on multiple zeta values, the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ and the Grothendieck-Teichm\"uller 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-Kreimer, Racinet, Zagier and Drinfeld on the structure of multiple zeta values and on the Grothendieck-Teichm\"uller Lie algebra.