$\zeta(n)$ as a mixed Tate motive

Question

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 (*)?

Answer 1

Partially inspired by an unpublished work of Wojtkoviak for $\zeta (3)$, all of this follows from this result by Deligne:

Theorem. $\pi_1(\mathbb{P}^1-\{0,1,\infty\})$ is a smooth mixed Tate motive over $\mathrm{Spec}\mathbb{Z}$.

Now, over $\mathbb{P}^1_\mathbb{Z}-\{0,1,\infty\})$ he can define motivic (polylogarithm) local system $P$, each fiber of which is again a mixed Tate motive $M$.

In general, the realizations of those motives $M$ give rise to a polylogarithm $\mathrm{Li}_n(x)$.

But if you take the fiber $P_\vec{\rho}$, with base point $\vec{\rho}$ the tangent vector $\partial /\partial t$ of $\mathbb{P}^1$ at $0$, you get for example a Betti realization:

$$M_{\vec{\rho},B}=-(n-1)!\zeta (n)+(2\pi i)^n\mathbb{Z}$$

That's essentially the idea.

The original paper by Deligne is:

• Le groupe fondamental de la droite projective moins trois points (1989)

Also, research by Richard Hain is relevant here, for example his paper with Makoto Matsumoto, "Tannakian Fundamental Groups Associated to Galois Groups" for the "Galois Groups and Fundamental Groups" volume, or his paper on classical polylogarithms.

Answer 2

The motive $H_2(\overline{M}_{0,5}\setminus A, B\setminus A\cap B)$ seems to be the right one. The relative singular cohomology is 2-dimensional as one would expect from an extension in $Ext^1(\mathbb{Q}(0),\mathbb{Q}(n))$. The computation can be found as Theorem 5.3 in

• Q. Wang. Moduli spaces and multiple polylogarithm motives. Adv. Math. 206 (2006), 329-357.

However, it should be pointed out that just looking at the dimension of the Betti realization is not the right thing anyway. For example, there are also motives of the form $H_2(\overline{M}_{0,5}\setminus A, B\setminus A\cap B)$ (for other divisors $A$ and $B$) which realize dilogarithms $Li_2(x)$ for $x\neq 0,1$. The corresponding relative singular cohomology groups are 3-dimensional, again by Theorem 5.3 of the paper of Wang mentioned above. This would be an example where the Betti realization of the motive is too big for an extension.

The extensions in $Ext^1(\mathbb{Q}(0),\mathbb{Q}(n))$ are realized by framed motives, so the point in the paper of Goncharov and Manin is not just the construction of a motive but also of an appropriate framing. The framing of $M=H^n(\overline{M}_{0,n+3}\setminus A, B\setminus A\cap B)$ consists of elements $[\Omega_A]\in Gr^W_{2n}H^n(\overline{M}_{0,n+3}\setminus A)$ (given more concretely by a differential form) and $[\Delta_B]\in (Gr^W_0 H^n(\overline{M}_{0,n+3},B))^\vee$ (given as a relative cycle). The integral of the differential form over the cycle gives the zeta-value, and this is the sense in which the framed motive realizes the zeta-value. If the dimension of cohomology is bigger than 2, framing the motive corresponds to picking a $(2\times 2)$-submatrix of the period matrix which contains $\zeta(n)$. The framing described above corresponds to supplying maps $\mathbb{Q}(-n)\to M$ and $M\to\mathbb{Q}(0)$ which in the case $\zeta(2)$ arise from the relative motive exact sequence $$ H^1(B\setminus A\cap B)\to H^2(\overline{M}_{0,5}\setminus A,B\setminus A\cap B)\to H^2(\overline{M}_{0,5}\setminus A) $$

I think the short answer to the question is that framed motives can be used to describe extensions in $\operatorname{Ext}^1_{\operatorname{MT}(k)}(\mathbb{Q}(0),\mathbb{Q}(n))$, but for this is not necessary for the underlying motive of the framed motive to have the ``right dimension''. In general, only a subquotient of it will give the extension (provided the vanishing of a coproduct). The essential idea (going back to Beilinson-MacPherson-Schechtman's notes on motivic cohomology, it seems) is that there is an equivalence relation on framed motives such that each equivalence class contains a minimal representative corresponding to an actual extension (and hence equivalence classes of framed motives describe Ext-groups of motives). General discussion of framed motives can be found in

• A.A. Beilinson, A.B. Goncharov, V.V. Schechtman and A.N. Varchenko: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. Grothendieck Festschrift I, 1990, pp. 135-172.

• A.B. Goncharov. Periods and mixed motives. http://arxiv.org/abs/math/0202154

Answer 3

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.