What is the relationship between these two notions of "period"?

Question

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.

Answer 1

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 $\mathcal{P}_{KZ} \nsubseteq \mathcal{P}^+$.

Answer 2

Let me try to explain the relation, without going into technical details of inverting $2\pi i.$

There are three Tannakian categories appearing here.

1. Conjectural abelian category of mixed motives over $\mathbb{Q}.$ Periods in the sense of Kontsevich-Zagier appear as matrix coefficients of the comparison isomorphism between de Rham and Betti fiber functors. Explicitly, they will look like integrals of rational functions over domains, defined by rational inequalities. For every period one can find a smooth proper algebraic variety $X$ and a pair of devisors $D_1, D_2$ so that the period equals to $$ \int_{\gamma}\omega $$ for a cycle with boundary on $D_2$ and an algebraic form with poles on $D_1.$ Cohomology groups $$H^k(X\setminus D_1, D_2)$$ define objects in the category of mixed motives.

This category is terribly big and complicated. It also has lots of simple objects, coming from cohomology of smooth proper varieties defined over $\mathbb{Q},$ like elliptic curves.

2. Mixed Tate Motives over $\mathbb{Q}.$ This category is known to exist. It should be a subcategory of the category of mixed motives, generated by objects $$H^k(X\setminus D_1, D_2),$$ where the mixed Hodge structure on cohomology is of mixed Tate type. Typical periods here are polylogarithms at rational values, like $Li_5\left(\frac{23}{7}\right).$

This category is smaller, but still big. All its simple objects are pure Tate motives $\mathbb{Q}(n),$ it has homological dimension one and $Ext^1(\mathbb{Q}(0), \mathbb{Q}(n))$ is finite dimensional for $n\ge 2.$ Notice that $$ Ext^1(\mathbb{Q}(0), \mathbb{Q}(1))=K_1(\mathbb{Q})_\mathbb{Q}=\mathbb{Q}^{\times}_{\mathbb{Q}}, $$ is not finite dimensional. This is the reason why this category is still very big. In particular, the graded Hopf algebra of periods is not finitely generated in any degree.

3.Finally, mixed Tate motives unramified over $\mathbb{Z}.$ This is a tiny subcategory of the previous one. It has the same simple objects and the same $Ext^1(\mathbb{Q}(0), \mathbb{Q}(n))$ for $n\geq 2,$ but $$ Ext^1(\mathbb{Q}(0), \mathbb{Q}(1))=K_1(\mathbb{Z})_\mathbb{Q}=0. $$ This means that the graded Hopf algebra of its periods is finitely generated in each degree. This is (thanks to the theorem of Francis Brown) the algebra of motivic multiple zeta values.