TITLE: $\zeta(n)$ as a mixed Tate motive
QUESTION [29 upvotes]: I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that 
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ 
and $\zeta(n)$, the Riemann zeta function at $n$, is a period of $M$. 
For the easiest case $n=2$, Goncharov and Manin explain that the integral representation
$$
\zeta(2)=\int_{0 \leq x \leq y \leq 1} \frac{dx}{1-x} \frac{dy}{y}
$$ 
allows to consider $\zeta(2)$ as a period of the relative homology
$$
H_2(\overline{M_{0, 5}}-A, B-A \cap B)) \quad (*)
$$
where $A$ and $B$ are certain unions of irreducible components of the boundary of $M_{0, 5}$. 
Then one can make sense of this object in the category $MT(\mathbb{Q})$, but it does not seem to be the required $M$ (I do not think it has dimension 2...). So how to get $M$? Does one need to consider some subobject of (*)?

REPLY [6 votes]: As Myshkin wrote, the existence of a motive $$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$$ having $\zeta(n)$ as a period follows from work of Deligne (for compatible systems of realizations) and by Deligne and Goncharov (for motives in the sense of Voevodsky).
For compatible systems of realizations, Deligne constructs this extension very explicitly in two ways: one using a rather axiomatic description (section 3 of his famous ``La droite projective...'') and a geometric construction using the unipotent fundamental group of $\mathbb P^1 \setminus \{0,1,\infty\}$ (section 16 of loc.cit.).
On the other hand, an explicit geometric construction of $M$ as a motive in the sense of Voevodsky is probably very hard, and, rather intriguingly, linked with Apéry-Beukers type irrationality proofs for $\zeta$-values (see recent work of Brown: arxiv1412.6508 and of Dupont arxiv1601.00950).
I'm not an expert, but it seems to me that in a nutshell, Brown constructs ``Apéry motives'' for $n=2$ and $n=3$, which reply to your question (Corollary 11.3), while Dupont constructs a large motive (Theorem 1.3), whose period matrix contains all $\zeta(n)$ for $n\geq 2$ and also a smaller motive (Theorem 1.4), which contains only the odd $\zeta(n)$ (apart from the $(2\pi i)^n$, of course). The construction of Apéry motives for $n \geq 4$ seems to remain open.