# What is the relationship between these two notions of “period”?

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where $\mathcal{Z}$ is the $\mathbb{Q}$-span of the set of multiple zeta values (of positive integer arguments). My picture of mixed Tate motives is not very clear, and I would like to be able to relate their periods to something I understand better.

There is a survey article of Kontsevich and Zagier which defines a period as a complex number whose real and imaginary parts are given by convergent integrals of rational functions with rational coefficients, over domains in $\mathbb{R}^n$ cut out by finitely many polynomial inequalities with rational coefficients.

What is the relationship between the set of periods of mixed Tate motives over $\mathbb{Z}$ and the set of periods in the sense of Kontsevich/Zagier? Does one of these sets contain the other?

I would be interested to see examples of periods of one kind which are not periods of the other.

-
Multiple zeta values can be defined by iterated integrals (see e.g. arxiv.org/pdf/1102.1310v2.pdf). It follows that periods of mixed Tate motives are periods in the sense of Kontsevich and Zagier (maybe you need to invert $2\pi i$). – François Brunault Apr 7 '13 at 19:26
See also math.unice.fr/~brunov/GdT/… for a more detailed explanation. – François Brunault Apr 7 '13 at 19:34
@François Brunault: But how do you invert $2\pi i$? Is this (now known to be) a period in the sense of Kontsevich--Zagier? In their paper they (if I remember well) mention specifically the (equivalent) question whether $1/\pi$ is a period or not. So, to me for the inclusion this is the point. – user9072 Apr 7 '13 at 21:13
@quid: You're right that it is only conjectured that $1/(2\pi i)$ is not a period. Anyway, it is necessary to invert $2\pi i$ to have a valid statement because the period of the Tate motive $\mathbf{Q}(n)$ is $(2\pi i)^n$. – François Brunault Apr 8 '13 at 6:45
This explains that it has become rather standard to consider the ring of extended periods $\widehat{\mathcal{P}} := \mathcal{P}[\frac{1}{2\pi i}]$. – François Brunault Apr 8 '13 at 6:47

I think the key of this issue is that recent papers, including those of Brown, routinely refer to "the algebra of periods of Kontsevich-Zagier", when they mean $\mathcal{P}[\frac{1}{2\pi i}]$. The reason is that the definition is more natural and general: it captures all periods of all mixed motives over $\mathbb{Q}$. The more classical periods of $\mathcal{P}$ defined by convergent integrals are usually called effective periods.

By analogy, the reason that you need $(2\pi i)^{-1}$ to get all periods coming from mixed motives is equivalent to having to invert $\mathbb{Z}(-1)$ to obtain an abelian category of Nori mixed motives $\mathrm{MM}(\mathbb{Q})$.

Let's denote the set of periods of mixed Tate motives over $\mathbb{Z}$ by $\mathcal{P}_\mathrm{MT}$, the effective (original) periods by $\mathcal{P}^+$, and the complete algebra of periods (i.e. $\mathcal{P}[\frac{1}{2\pi i}]$) by $\mathcal{P}_{KZ}$.

• $\mathcal{P}_\mathrm{MT} \subseteq \mathcal{P}_{KZ}$

This is an easy consequence of Brown's theorem. All the periods in $\mathcal{P}_\mathrm{MT}$ are generated by multiple zeta values and $(2\pi i)^{-1}$. The latter is in $\mathcal{P}_{KZ}$ by definition, and the former by Chen integration (I think the first one to put this on writting was Don Zagier).

• $\mathcal{P}_{KZ} \nsubseteq \mathcal{P}_\mathrm{MT}$

Special values of the L-function of a (not simple) Artin motive $M$ such that $L(M,s)$ is entire should do the trick. For an unconditional example you can use Dirichlet characters.

• $\mathcal{P}_\mathrm{MT} \nsubseteq \mathcal{P}^+$ (open)

You would need to prove that $(2\pi i)^{-1} \notin \mathcal{P}^+$, but that of course is open.

On a sad side note, we can't even prove that $\mathcal{P}_{KZ} \nsubseteq \mathcal{P}^+$.

-
Nori not Mori, I think. – Donu Arapura Oct 4 '15 at 23:21
@DonuArapura Of course. Nice catch! – Myshkin Oct 5 '15 at 0:00