Mixed Tate motives over $\Z$

By F. Brown. 2011

426 citations recorded by the source corpus.

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 $n_i\in \{2,3\}$.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cd3c51c8aab8d310cd92569b77144628214b1203

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Citations in the corpus

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series (2012, background): “Brown [4] proved that part of this conjectural basis, namely the multiple zeta values of the form ζ({2}a, 3, {2}b) can be expressed in terms of ordinary zeta values.”
Graphical functions and single-valued multiple polylogarithms (2013, neutral): “The dimension of Hk is conjectured (and proved in the motivic analogue [13]) to have the following generating series”
Multiple zeta values vs. multiple zeta-star values (2011, background): “It seemed to be very difficult to show the conjecture, but recently, Francis Brown [6] announced that he proved the conjecture except the uniqueness of the expression.”
On the decomposition of motivic multiple zeta values (2011, methodology): “w using the formalism of [4] that these can in turn be lifted to elements of HMT + in such a way that the motivic version of ζ(2) is non-zero. However, the tollation involves making some choices (see [1], §2 for the deﬁnitions). In summary: Theorem 3.5. There exists a sub-Hopf algebra A ⊂ AMT and a ”
Depth-graded motivic multiple zeta values (2013, methodology): “→ Rmaps the element ζm(n1,...,nr) to the real number ζ(n1,...,nr). The algebra H carries an action of the motivic Galois group of the category of mixed Tate motives over Z, and its structure is known [1, 6]. Theorem 1.1. The Hoﬀman motivic multiple zeta values (1.2) ζm(n1,...,nr) where ni = 2 or 3 a”
Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$ (2014, background): “Using the results of [6], there is a canonical isomorphism of the ring H of motivic multiple zeta values with a shuffle algebra”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, methodology): “In order to use the same matrices in both expressions, one has to fix the freedom(32) in the construction of the map φ [39] such that the only elements of weight w in table 1 whose images contain fw or (f2) w/2 are ζw and (ζ2) , respectively.”
Motivic multiple zeta values and superstring amplitudes (2012, neutral): “The latter are derived from the coaction ∆ : H → A⊗Q H [21,5]”
M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra (2010, methodology): “hat σ2j+1 is a Lie series in X, Y starting with a multiple of the term ad2j XY. The Deligne-Drinfeld conjecture states that these elements freely generate grt1. In fact, it has recently been shown in [8] that these elements generate a free Lie subalgebra of grt1. However, it is still unclear whether”
SINGLE-VALUED MOTIVIC PERIODS AND MULTIPLE ZETA VALUES (2014, methodology): “The motivic multiple zeta values ζ(w) were defined in [9], and simplified by Deligne [17].”