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Mixed Tate motives over $\Z$

Francis Brown

TL;DR

The paper proves that mixed Tate motives over the integers are governed by the motivic fundamental group of P^1\{0,1,∞}, and establishes Hoffman’s conjecture by showing that motivic MZVs are spanned by those with entries only 2 or 3. It constructs and analyzes a level filtration on motivic MZVs, derives explicit coefficients c_w via a motivic lifting of Zagier’s formulas, and proves 2-adic invertibility of level-derivation matrices ∂_{N,ℓ} to perform induction on the level. By lifting Zagier’s identity to the motivic setting and proving the required injectivity of the ∂_{N,ℓ} maps, the authors show that the {2,3}-basis provides a full basis for motivic MZVs, implying that all periods of MT(Z) are Q[1/(2πi)]-linear combinations of ζ(n1,...,nr) with n_i∈{2,3}. The results unify motivic and classical perspectives, extend Deligne/Ihara-type insights, and yield a concrete, sparse basis for the space of MZVs, with explicit 2-adic arithmetic governing the structure.

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 $ζ(n_1,..., n_r)$ where $n_i\in \{2,3\}$.

Mixed Tate motives over $\Z$

TL;DR

The paper proves that mixed Tate motives over the integers are governed by the motivic fundamental group of P^1\{0,1,∞}, and establishes Hoffman’s conjecture by showing that motivic MZVs are spanned by those with entries only 2 or 3. It constructs and analyzes a level filtration on motivic MZVs, derives explicit coefficients c_w via a motivic lifting of Zagier’s formulas, and proves 2-adic invertibility of level-derivation matrices ∂_{N,ℓ} to perform induction on the level. By lifting Zagier’s identity to the motivic setting and proving the required injectivity of the ∂_{N,ℓ} maps, the authors show that the {2,3}-basis provides a full basis for motivic MZVs, implying that all periods of MT(Z) are Q[1/(2πi)]-linear combinations of ζ(n1,...,nr) with n_i∈{2,3}. The results unify motivic and classical perspectives, extend Deligne/Ihara-type insights, and yield a concrete, sparse basis for the space of MZVs, with explicit 2-adic arithmetic governing the structure.

Abstract

We prove that the category of mixed Tate motives over is spanned by the motivic fundamental group of minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a -linear combination of where .

Paper Structure

This paper contains 27 sections, 24 theorems, 127 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Motivic Multiple Zeta Values
  4. Preliminaries
  5. Definition of motivic MZVs
  6. The role of $\zeta^{ \mathfrak{m}}(2)$
  7. Main properties
  8. Structure of $\mathcal{H^{MT_+}}$
  9. Cogenerators of the coalgebra
  10. Infinitesimal coaction
  11. Zeta elements and kernel of $D_{<N}$
  12. Some relations between motivic multiple zeta values
  13. Arithmetic of the coefficients $c_w$
  14. Motivic version
  15. $2$-adic properties of $c_w$
  16. ...and 12 more sections

Key Result

Theorem 1.1

The set of elements are a basis of the $\mathbb Q$-vector space of motivic multiple zeta values.

Theorems & Definitions (58)

  • Conjecture 1
  • Conjecture 2
  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 48 more