How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$?

Question

EDIT There appears to be a numerical zeta function $\zeta(2)$ as well as at least two different "motivic" zeta function realizations (Betti and de Rham) $\zeta^{\mathfrak{m}}(2)$. The "period map" of equally mysterious properties relates the two objects and proves the Hoffman conjecture.

Implicit in the comment is that $\zeta(2)$ is a period, which got lifted to a motive... Possibly $\zeta(2)$ has become a homological object? The comments did not explain anything.

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I tried to read a paper of Francis Brown Mixed Tate Motives over $\mathbb{Z}$ and I was confused about the status of the conjecture he states:

Conjecture Every multiple zeta value $\zeta(n_1, \dots, n_r)$ is a $\mathbb{Q}$-linear combination of multiple zeta values at $n \in \{ 2, 3\}$: $$ \big\{ \zeta(n_1, \dots, n_r): \text{ where } n_i \in \{ 2, 3\} \big\} $$

He proves a verion of this using motivic zeta functions, which do not seem to be numerical at all. Brown proves:

Theorem The set of elements $ \{ \zeta^{\mathfrak{m}}(n_1, \dots, n_r): \text{ where } n_i \in \{ 2, 3\} \} $ are a basis in the space of motivic multiple zeta values.

In fact, I haven't any idea what a motive is. Despite it's elementary appearance, it seems to be related to work of Goncharov and Deligne [1].

Wikipedia's example for the affine line and projectiv line could potentially make sense:

$$ Z(\mathbb{A}^n, t) = \frac{1}{1 - \mathbb{L}^n t} \hspace{0.25in}\text{ and }\hspace{0.25in} Z(\mathbb{P}^n, t) = \prod_{i=0}^n \frac{1}{1 - \mathbb{L}^i t}$$

Wikipedia offers another exampls with the Hilbert scheme of points. None of that seems relevant.

Confusingly Brown, notes that in his conjecture $\zeta^{\mathfrak{m}}(2) \neq 0$ unlike in Goncharov's work.

How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$ ? Does Brown's Theorem prove the Conjecture (of Hoffman)?

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Right now Brown's discussion does not mean very much to me because it rests on some rather difficult concepts:

• Tannakian Categories (and it's Galois Group)
• Mixed Motives
• Motivic Iterated Integrals
• Betti and de Rham realizations of motives
• Why are Lyndon words relevant?
• Do the motivic relations he finds descend to relations of plain zeta values?
• Which "period map" is being used?

My impression is that the Shuffle Relations (which one might find in Kaneko-Ihara-Zagier) get lifted to something very complicated (which could be the Tannakian Category). He has two notes on Feynman Amplitudes [1, 2] which are broader than multiple zeta functions.

Answer 1

I am not sure what exactly your question is, but the question in the title ``How does $\zeta^{\mathfrak{m}}(2)$ relate to $\zeta(2)$?'' can be answered without really knowing what a motivic multiple zeta values is.

First, it follows essentially from the representation of multiple zeta values as iterated integrals on $\mathbb P^1 \setminus \{0,1,\infty\}$ that the $\mathbb{Q}$-algebra $\mathcal{Z}$ of multiple zeta values is a quotient

$$ \mathbb{Q}\langle e_0,e_1\rangle/I $$

of the shuffle $\mathbb{Q}$-algebra $\mathbb{Q}\langle e_0,e_1\rangle$ by some ideal $I$. Here, the multiple zeta value $\zeta(n_1,\ldots,n_r)$ corresponds to the class modulo $I$ of the word $e_1e_0^{n_1-1}\ldots e_1e_0^{n_r-1}$. On the other hand, in Mixed Tate motives over $\mathbb Z$, Brown defines the $\mathbb Q$-algebra $\mathcal{Z}^{\mathfrak m}$ of motivic multiple zeta values to be the quotient

$$ \mathbb{Q}\langle e_0,e_1\rangle/J, $$

where $J \subset I$ is now the ideal of motivic relations (whatever those are), and the motivic multiple zeta value $\zeta^{\mathfrak m}(n_1,\ldots,n_r)$ is defined to be the class of $e_1e_0^{n_1-1}\ldots e_1e_0^{n_r-1}$. Now since $J \subset I$, basic algebra implies that the association $\zeta^{\mathfrak m}(n_1,\ldots,n_r) \mapsto \zeta(n_1,\ldots,n_r)$ induces a well-defined surjection of $\mathbb Q$-algebras

$$ \mathcal{Z}^{\mathfrak m} \cong \mathbb{Q}\langle e_0,e_1\rangle/J \rightarrow \mathbb{Q}\langle e_0,e_1\rangle/I \cong \mathcal{Z}. $$

As already mentioned in the comments, this is precisely what Brown calls the period map (it should be noted that there is also another definition of motivic multiple zeta values and the period map which adapts better to other situations, see for example Brown's ICM 2014 talk).

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As you can see, there is no difficulty at all in relating motivic multiple zeta values and multiple zeta values; the hard part is the definition of motivic multiple zeta values themselves, and more precisely the definition of the ideal $J \subset I$ of motivic relations (this is exactly where Tannakian categories, Betti and de Rham realization functors, etc. come into play). Besides Brown's articles on the subject, which you seem to be well aware of, there is also the excellent book Multiple zeta values: From numbers to motives, by Burgos and Fresán.

Finally, it should be said that one conjectures that $J=I$, i.e. that every relation between multiple zeta values is motivic. In particular, the algebras of motivic and ordinary multiple zeta values would be isomorphic, $\mathcal{Z}^{\mathfrak{m}} \cong \mathcal{Z}$. In order to appreciate the extraordinary depth of this conjecture, note that the odd motivic zeta values $\zeta^{\mathfrak{m}}(2k+1)$ are all known to be transcendental, while the analogous statement is not known for a single odd zeta value $\zeta(2k+1)$. Even worse, the only odd zeta value we know is irrational is $\zeta(3)$, thanks to Apéry.