2.1 Mixed Hodge–Tate structures and their period matrices

We call  $H_{\text{B}}$ and  $H_{\text{dR}}$ the Betti realization and the de Rham realization of the mixed Hodge–Tate structure, and  $\unicode[STIX]{x1D6FC}$ the comparison isomorphism. The filtration  $W$ on  $H_{\text{B}}$ is called the weight filtration. The grading on  $H_{\text{dR}}$ is called the weight grading, and the corresponding filtration  $W_{2n}H_{\text{dR}}:=\bigoplus _{k\leqslant n}(H_{\text{dR}})_{2k}$ the weight filtration.

It is convenient to view the comparison isomorphism  $\unicode[STIX]{x1D6FC}:H_{\text{dR}}\otimes _{\mathbb{Q}}\mathbb{C}\stackrel{\simeq }{\longrightarrow }H_{\text{B}}\otimes _{\mathbb{Q}}\mathbb{C}$ as a pairing

where  $(\cdot )^{\vee }$ denotes the linear dual. The weight filtration on  $H_{\text{B}}^{\vee }$ is defined by

so that we have

The pairing (8) is compatible with the weight filtrations in that we have  $\langle \unicode[STIX]{x1D711},v\rangle =0$ for  $\unicode[STIX]{x1D711}\in W_{-2m}H_{\text{B}}^{\vee }$ ,  $v\in W_{2n}H_{\text{dR}}$ and  $m<n$ .

If we choose bases for the  $\mathbb{Q}$ -vector spaces  $H_{\text{dR}}$ and  $H_{\text{B}}$ , then the matrix of  $\unicode[STIX]{x1D6FC}$ in these bases, or equivalently the matrix of the pairing (8), is called a period matrix of the mixed Hodge–Tate structure. We will always make the following assumptions on the choice of bases.

1. – The basis of  $H_{\text{B}}$ is compatible with the weight filtration.

2. – The basis of  $H_{\text{dR}}$ is compatible with the weight grading.

3. – For every  $n$ , the matrix of the comparison isomorphism  $\unicode[STIX]{x1D6FC}_{n}$ in the corresponding basis is  $(2\unicode[STIX]{x1D70B}i)^{n}$ times the identity.

This implies that any period matrix is block upper-triangular with successive blocks of  $(2\unicode[STIX]{x1D70B}i)^{n}\,\text{Id}$ on the diagonal. Conversely, any block upper-triangular matrix with successive blocks of  $(2\unicode[STIX]{x1D70B}i)^{n}\,\text{Id}$ on the diagonal is a period matrix of a mixed Hodge–Tate structure.

2.2 The category of mixed Hodge–Tate structures

We denote by  $\mathsf{MHTS}$ the category of mixed Hodge–Tate structures. It is a neutral tannakian category over  $\mathbb{Q}$ , which means in particular that it is an abelian  $\mathbb{Q}$ -linear category equipped with a  $\mathbb{Q}$ -linear tensor product  $\otimes$ . We note that an object  $H\in \mathsf{MHTS}$ is endowed with a canonical weight filtration  $W$ by subobjects such that the morphisms in  $\mathsf{MHTS}$ are strictly compatible with  $W$ . We have two natural fiber functors

from  $\mathsf{MHTS}$ to the category of finite-dimensional vector spaces over  $\mathbb{Q}$ , which only remember the Betti realization  $H_{\text{B}}$ and the de Rham realization  $H_{\text{dR}}$ respectively. We note that the de Rham realization functor  $\unicode[STIX]{x1D714}_{\text{dR}}$ factors through the category of finite-dimensional graded vector spaces. The comparison isomorphisms  $\unicode[STIX]{x1D6FC}$ gives an isomorphism between the complexifications of the two fiber functors:

For an integer  $n$ , we denote by  $\mathbb{Q}(-n)$ the mixed Hodge–Tate structure whose period matrix is the  $1\times 1$ matrix  $((2\unicode[STIX]{x1D70B}i)^{n})$ . Its weight grading and filtration are concentrated in weight  $2n$ , hence we call it the pure Tate structure of weight  $2n$ . For  $H$ a mixed Hodge–Tate structure, the tensor product  $H\otimes \mathbb{Q}(-n)$ is simply denoted by  $H(-n)$ and called the  $n$ th Tate twist of  $H$ . A period matrix of  $H(-n)$ is obtained by multiplying a period matrix of  $H$ by  $(2\unicode[STIX]{x1D70B}i)^{n}$ . The weight grading and filtration of  $H(-n)$ are those of  $H$ , shifted by  $2n$ .

2.3 Extensions between pure Tate structures

The pure Tate structures  $\mathbb{Q}(-n)$ are the only simple objects of the category  $\mathsf{MHTS}$ . The extensions between them are easily described. Up to a Tate twist, it is enough to describe the extensions of  $\mathbb{Q}(-n)$ by  $\mathbb{Q}(0)$ for some integer  $n$ . The corresponding extension group is given by

More concretely, the extension corresponding to a number  $z\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}\,$ has a period matrix

We note that the higher extension groups vanish:  $\text{Ext}_{\mathsf{MHTS}}^{r}(H,H^{\prime })=0$ for  $r\geqslant 2$ and  $H$ ,  $H^{\prime }$ two mixed Hodge–Tate structures.

2.4 Mixed Tate motives over  $\mathbb{Q}$

Let  $\mathsf{DM}(\mathbb{Q})$ denote Voevodsky’s triangulated category of motives over  $\mathbb{Q}$  [Reference VoevodskyVoe00]. It is a  $\mathbb{Q}$ -linear triangulated tensor category whose objects can be described in terms of complexes of varieties and whose morphisms can be described in terms of algebraic cycles (in particular, in terms of Bloch’s higher Chow groups). There are invertible objects  $\mathbb{Q}(-n)\in \mathsf{DM}(\mathbb{Q})$ , where  $\mathbb{Q}(-1)$ is the reduced motive of the multiplicative group  $\mathbb{G}_{m}$ , shifted by  $-1$ (we work with cohomological conventions). The triangulated subcategory of  $\mathsf{DM}(\mathbb{Q})$ generated by these objects is denoted by  $\mathsf{DMT}(\mathbb{Q})$ . By using the relation between higher Chow groups and rational  $K$ -theory [Reference BlochBlo86] and Borel’s computation of the rational  $K$ -theory of number fields [Reference BorelBor77], Levine defined a natural  $t$ -structure on  $\mathsf{DMT}(\mathbb{Q})$  [Reference LevineLev93]. The heart of this  $t$ -structure is denoted by  $\mathsf{MT}(\mathbb{Q})$ and called the category of mixed Tate motives over  $\mathbb{Q}$ . It is a (neutral) tannakian  $\mathbb{Q}$ -linear category which contains the objects  $\mathbb{Q}(-n)$ .

There is a faithful and exact functor

from  $\mathsf{MT}(\mathbb{Q})$ to the category  $\mathsf{MHTS}$ of mixed Hodge–Tate structures, which is called the Hodge realization functor ([Reference Deligne and GoncharovDG05, § 2.13], see also [Reference HuberHub00, Reference HuberHub04]). It sends the object  $\mathbb{Q}(-n)\in \mathsf{MT}(\mathbb{Q})$ to the object  $\mathbb{Q}(-n)\in \mathsf{MHTS}$ . Composing (11) with the fiber functors (9) gives the Betti and de Rham realization functors, still denoted by

and we still have a comparison isomorphism (10). We note that any object in  $\mathsf{MT}(\mathbb{Q})$ is endowed with a canonical weight filtration  $W$ by subobjects such that the morphisms in  $\mathsf{MT}(\mathbb{Q})$ are strictly compatible with  $W$ . The realization morphisms are compatible with the weight filtrations.

The extension groups between the objects  $\mathbb{Q}(-n)$ are computed by the rational  $K$ -theory of  $\mathbb{Q}$  [Reference LevineLev93, § 4] and hence given, after Borel [Reference BorelBor77], by

As in the category  $\mathsf{MHTS}$ , the higher extension groups vanish in  $\mathsf{MT}(\mathbb{Q})$ . The morphisms

induced by (11) are easy to describe. For  $n=1$ , the image of the direct summand indexed by a prime  $p$ is the line spanned by the class of the Kummer extension of parameter  $p$ , i.e., by  $\log (p)\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)\mathbb{Q}$ . For  $n\geqslant 3$ odd, the image is the line spanned by  $\unicode[STIX]{x1D701}(n)\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}$ . Thus, the morphism (14) is injective for every  $n$ . This implies the following theorem [Reference Deligne and GoncharovDG05, Proposition 2.14].

This theorem is very helpful, since it allows one to compute in the category  $\mathsf{MT}(\mathbb{Q})$ with period matrices; in other words, a mixed Tate motive over  $\mathbb{Q}$ is uniquely determined by its period matrix.

2.5 Mixed Tate motives over  $\mathbb{Z}$

Let  $\mathsf{MT}(\mathbb{Z})$ denote the category of mixed Tate motives over  $\mathbb{Z}$ , as defined in [Reference Deligne and GoncharovDG05]. By definition, it is a full tannakian subcategory

of the category of mixed Tate motives over  $\mathbb{Q}$ , which contains the pure Tate motives  $\mathbb{Q}(-n)$ for every integer  $n$ . An object of $\mathsf{MT}(\mathbb{Q})$ is in $\mathsf{MT}(\mathbb{Z})$ if and only if it has no subquotient isomorphic to a non-split extension of $\mathbb{Q}(-n)$ by $\mathbb{Q}(-n+1)$ .

The extension groups in the category $\mathsf{MT}(\mathbb{Z})$ satisfy the following properties:

1. (1) $\text{Ext}_{\mathsf{MT}(\mathbb{Z})}^{1}(\mathbb{Q}(-1),\mathbb{Q}(0))=0$ ;

2. (2) the natural morphism $\text{Ext}_{\mathsf{MT}(\mathbb{Z})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))\rightarrow \text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))$ is an isomorphism for  $n\neq 1$ .

As in the categories  $\mathsf{MHTS}$ and  $\mathsf{MT}(\mathbb{Q})$ , the higher extension groups vanish in  $\mathsf{MT}(\mathbb{Z})$ .

For  $n\geqslant 3$ odd, there is an essentially unique non-trivial extension of  $\mathbb{Q}(-n)$ by  $\mathbb{Q}(0)$ in the category  $\mathsf{MT}(\mathbb{Q})$ , which actually lives in  $\mathsf{MT}(\mathbb{Z})$ . A period matrix for such an extension is

Apart from the case  $n=3$ (see [Reference BrownBro16, Corollary 11.3] or Proposition 4.11 below), we do not know of any geometric construction of these extensions.

3.1 The definition

Let  $n\geqslant 1$ be an integer. In the affine  $n$ -space  $X_{n}=\mathbb{A}_{\mathbb{Q}}^{n}$ we consider the hypersurfaces

The union  $A_{n}\cup B_{n}$ is almost a normal crossing divisor inside  $X_{n}$ : around the point  $P_{n}=$ $(1,\ldots ,1)$ , it looks like  $z_{1}\cdots z_{n}(z_{1}+\cdots +z_{n})=0$ (set  $x_{i}=\exp (z_{i})$ ). Let

be the blow-up at  $P_{n}$ , and  $E_{n}=\unicode[STIX]{x1D70B}_{n}^{-1}(P_{n})$ be the exceptional divisor. We denote respectively by  $\widetilde{A}_{n}$ and  $\widetilde{B}_{n}$ the strict transforms of  $A_{n}$ and  $B_{n}$ along  $\unicode[STIX]{x1D70B}_{n}$ . The union  $\widetilde{A}_{n}\cup \widetilde{B}_{n}\cup E_{n}$ is a simple normal crossing divisor inside  $\widetilde{X}_{n}$ .

There is an object  ${\mathcal{Z}}^{(n)}\in \mathsf{MT}(\mathbb{Q})$ , which we may abusively denote by

such that its Betti and de Rham realizations (12) are ( $?\in \{\text{B},\text{dR}\}$ )

We now give the precise definition of  ${\mathcal{Z}}^{(n)}$ , along the lines of [Reference GoncharovGon02, Proposition 3.6]. Let us write  $Y=\widetilde{X}_{n}-\widetilde{A}_{n}$ and  $\unicode[STIX]{x2202}Y=(\widetilde{B}_{n}\cup E_{n})-(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n}$ , viewed as schemes defined over  $\mathbb{Q}$ . We have a decomposition into smooth irreducible components  $\unicode[STIX]{x2202}Y=\bigcup _{i}\unicode[STIX]{x2202}_{i}Y$ , where  $i$ runs in a set of cardinality  $2n+1$ . For a set  $I=\{i_{1},\ldots ,i_{r}\}$ of indices, we denote by  $\unicode[STIX]{x2202}_{I}Y=\unicode[STIX]{x2202}_{i_{1}}Y\cap \cdots \cap \unicode[STIX]{x2202}_{i_{r}}Y$ the corresponding intersection; it is either empty or a smooth subvariety of  $X$ of codimension  $r$ .

We thus get an object

in Voevodsky’s triangulated category  $\mathsf{DM}(\mathbb{Q})$ , see § 2.4. The differentials are the alternating sums of the natural closed immersions. One readily checks that the complex (15) lives in the triangulated subcategory  $\mathsf{DMT}(\mathbb{Q})$ . By definition, the object  ${\mathcal{Z}}^{(n)}$ in  $\mathsf{MT}(\mathbb{Q})$ is the  $n$ th cohomology group of the complex (15) with respect to the  $t$ -structure.

3.2 Betti and de Rham realizations, 1

We now give a first description of the Betti and de Rham realizations of the zeta motive  ${\mathcal{Z}}^{(n)}$ .

We let  $C_{\bullet }$ denote the functor which assigns to a topological space the complex of singular chains with rational coefficients. By definition, the dual of the Betti realization  ${\mathcal{Z}}_{\text{B}}^{(n),\vee }$ is the  $n$ th homology group of the total complex of the double complex

obtained by applying the functor  $C_{\bullet }$ to the complex (15). One readily verifies that this complex is quasi-isomorphic to the quotient complex  $C_{\bullet }(Y(\mathbb{C}))/C_{\bullet }(\unicode[STIX]{x2202}Y(\mathbb{C}))$ , classically used to define the relative homology groups  $H_{\bullet }^{\text{B}}(Y,\unicode[STIX]{x2202}Y)=H_{\bullet }^{\text{sing}}(Y(\mathbb{C}),\unicode[STIX]{x2202}Y(\mathbb{C}))$ .

We let  $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x2202}_{I}Y}^{\bullet }$ denote the complex of sheaves of algebraic differential forms on the smooth variety  $\unicode[STIX]{x2202}_{I}Y$ , extended by zero to  $Y$ . By definition, the de Rham realization  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ is the hypercohomology of the total complex of the double complex of sheaves

where the vertical arrows are the exterior derivatives and the horizontal arrows are the alternating sums of the natural restriction maps as in the complex (15).

The comparison morphism between the Betti and de Rham realizations of  ${\mathcal{Z}}^{(n)}$ is induced, after complexification, by the morphism from the double complex (17) to the double complex (16) given by integration. Note that one first has to replace (16) by the double complex of sheaves of singular cochains.

3.3 Betti and de Rham realizations, 2

We now give descriptions of the Betti and de Rham realizations of  ${\mathcal{Z}}^{(n)}$ that allow one to work directly in the affine space  $X_{n}$ and do not require to work in the blow-up  $\widetilde{X}_{n}$ . The justification of the blow-up process goes as follows. Suppose that one wants to find a motive whose periods include all absolutely convergent integrals of the form

where  $P(x_{1},\ldots ,x_{n})$ is a polynomial with rational coefficients, and  $N\geqslant 0$ is an integer. On the Betti side, we note that the boundary of  $[0,1]^{n}$ intersects the divisor  $A_{n}(\mathbb{C})$ of poles of the differential forms at the point  $P_{n}(\mathbb{C})$ . The blow-up process is thus required in order to have a class that represents the integration domain. On the de Rham side, the blow-up process is required in order to only consider absolutely convergent integrals of the form (18). This is made precise by Propositions 3.4 and 3.6 below.

We start with the Betti realization. Let us write  ${A\unicode[STIX]{x0030A}}_{n}=A_{n}-P_{n}$ and note that this is not a closed subset, but only a locally closed subset, of  $X_{n}$ .

As a consequence of Proposition 3.4, we see that the unit  $n$ -square  $\Box ^{n}=[0,1]^{n}\subset X_{n}(\mathbb{C})-{A\unicode[STIX]{x0030A}}_{n}(\mathbb{C})$ defines a class

When viewed in  $\widetilde{X}_{n}(\mathbb{C})-\widetilde{A}_{n}(\mathbb{C})$ , it is the class of the strict transform  $\widetilde{\Box }^{n}$ , which has the combinatorial structure of an  $n$ -cube truncated at one of its vertices.

We now turn to a description of the de Rham realization of  ${\mathcal{Z}}^{(n)}$ . Instead of giving a general description in terms of algebraic differential forms on  $X_{n}-A_{n}$ , we will only give a way of defining many classes in  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ , which will turn out to be enough for our purposes.

The terminology is justified by the following proposition.

We make an abuse of notation and denote by

the class of the pullback  $\unicode[STIX]{x1D70B}_{n}^{\ast }(\unicode[STIX]{x1D714})$ for  $\unicode[STIX]{x1D714}$ integrable, so that the comparison isomorphism reads

We note the converse of Proposition 3.6, which we will not use.

3.4 The Eulerian differential forms

Recall that the family of Eulerian polynomials  $E_{r}(x)$ ,  $r\geqslant 0$ , is defined by the equation

We refer to [Reference FoataFoa10] for a survey on Eulerian polynomials. If  $r\geqslant 1$ , then (20) is equivalent to

For instance, we have  $E_{0}(x)=E_{1}(x)=1$ ,  $E_{2}(x)=1+x$ ,  $E_{3}(x)=1+4x+x^{2}$ . The Eulerian polynomials satisfy the recurrence relation

For integers  $n\geqslant 2$ and  $k=2,\ldots ,n$ , we define a differential form

Note that we have  $\unicode[STIX]{x1D714}_{n}^{(n)}=dx_{1}\cdots dx_{n}/(1-x_{1}\cdots x_{n})$ .

For every  $n\geqslant 0$ , we define  $\unicode[STIX]{x1D714}_{0}^{(n)}=dx_{1}\cdots dx_{n}$ ; we also have the class  $[\unicode[STIX]{x1D714}_{0}^{(n)}]\in {\mathcal{Z}}_{n,\text{dR}}$ , whose pairing with the class  $[\Box ^{n}]$ is

We call the differential forms  $\unicode[STIX]{x1D714}_{k}^{(n)}$ , for  $k=0,2,\ldots ,n$ , the Eulerian differential forms.

3.5 An inductive system

For  $n\geqslant 2$ there are natural morphisms

in the category  $\mathsf{MT}(\mathbb{Q})$ , that we now define. We fix the identification  $X_{n-1}=\{x_{n}=1\}\subset X_{n}$ , which implies the equality  $A_{n-1}=A_{n}\cap X_{n-1}$ . Let us set

so that we have  $B_{n}=B_{n}^{\prime }\cup X_{n-1}$ , and  $B_{n-1}=B_{n}^{\prime }\cap X_{n-1}$ .

In the blow-up  $\widetilde{X}_{n}$ , we thus get an embedding  $\widetilde{X}_{n-1}\subset \widetilde{X}_{n}$ and identifications  $\widetilde{A}_{n-1}=\widetilde{A}_{n}\cap \widetilde{X}_{n-1}$ ,  $\widetilde{B}_{n-1}=\widetilde{B}_{n}^{\prime }\cap \widetilde{X}_{n-1}$ and  $E_{n-1}=E_{n}\cap \widetilde{X}_{n-1}$ . Thus, the complex in  $\mathsf{DM}(\mathbb{Q})$ that we have used to define  ${\mathcal{Z}}^{(n-1)}$ is the subcomplex

of the complex (15) that we have used to define  ${\mathcal{Z}}^{(n)}$ , shifted by  $1$ . Taking the  $n$ th cohomology groups with respect to the  $t$ -structure gives the morphism (23).

In the Betti and the de Rham realizations, the morphism (23) is also induced by the inclusion of double subcomplexes of (16) and (17).

We define the ind-motive

viewed as an ind-object in the category  $\mathsf{MT}(\mathbb{Q})$ , and simply call it the zeta motive.

The map  $i_{\text{B}}^{(n),\vee }:{\mathcal{Z}}_{\text{B}}^{(n),\vee }\rightarrow {\mathcal{Z}}_{\text{B}}^{(n-1),\vee }$ given by the transpose of the Betti realization of  $i^{(n)}$ satisfies

More generally and loosely speaking, if  $\unicode[STIX]{x1D70E}$ is a chain on  $\widetilde{X}_{n}(\mathbb{C})-\widetilde{A}_{n}(\mathbb{C})$ whose boundary is on  $\widetilde{B}_{n}(\mathbb{C})\cup E_{n}(\mathbb{C})$ , then  $i_{\text{B}}^{(n),\vee }([\unicode[STIX]{x1D70E}])$ is the class of ‘the component of the boundary of  $\unicode[STIX]{x1D70E}$ that lives on  $\widetilde{X}_{n-1}(\mathbb{C})$ ’. According to Proposition 3.4, one can also work with chains on  $X_{n}(\mathbb{C})-{A\unicode[STIX]{x0030A}}_{n}(\mathbb{C})$ . We note that (25) allows us to define a class

The next proposition shows that the Eulerian differential forms  $\unicode[STIX]{x1D714}_{k}^{(n)}$ are compatible with the inductive structure on the zeta motives.

Proposition 3.10 allows us to unambiguously define classes

for  $k=0,2,3,\ldots ,$ whose pairing with the class  $[\Box ]\in {\mathcal{Z}}_{\text{B}}^{\vee }$ is

3.6 A long exact sequence

We now show that the morphism  $i^{(n)}:{\mathcal{Z}}^{(n-1)}\rightarrow {\mathcal{Z}}^{(n)}$ fits into a long exact sequence. We first define objects of  $\mathsf{MT}(\mathbb{Q})$ :

and

so that  ${\mathcal{Z}}^{(n)}={\mathcal{Z}}^{(n),n}$ . We leave it to the reader to fill in the technical definitions of these objects by mimicking that of  ${\mathcal{Z}}^{(n)}$ from § 3.1.

We note that the map  ${\mathcal{Z}}^{(n-1),n-1}\rightarrow {\mathcal{Z}}^{(n),n}$ in the long exact sequence (26) is exactly  $i^{(n)}$ .

4.1 The Gysin long exact sequence

Since the divisor  $A_{n}$ is smooth, it is natural to decompose the motives  ${\mathcal{Z}}^{(n),r}$ thanks to a Gysin long exact sequence. In the next proposition, the definition of the objects  $H^{\bullet }(X_{n},B_{n})$ and  $H^{\bullet }(A_{n},B_{n}\cap A_{n})$ of  $\mathsf{MT}(\mathbb{Q})$ is similar to that of  ${\mathcal{Z}}^{(n)}$ from § 3.1.

4.2 The motives  $H^{\bullet }(X_{n},B_{n})$

The computation of the motives  $H^{\bullet }(X_{n},B_{n})$ appearing in the long exact sequence (27) is relatively easy.

4.3 The motives  $H^{\bullet }(A_{n},B_{n}\cap A_{n})$

For  $n\geqslant 1$ , we realize the  $n$ -torus as  $T^{n}=\{x_{1}\cdots x_{n+1}=1\}$ , and we have subtori  $T_{i}^{n-1}=\{x_{i}=1\}\subset T^{n}$ for  $i=1,\ldots ,n+1$ . We define

which are objects in  $\mathsf{MT}(\mathbb{Q})$ (whose definition is similar to that of  ${\mathcal{Z}}^{(n)}$ from § 3.1) and write

We then have

By mimicking the proof of Proposition 3.14, one produces a long exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

We note that we have  ${\mathcal{T}}^{(0)}=H^{0}(\text{pt},\text{pt})=0$ , so that Proposition 4.3 implies that we have

In the next proposition, we will prove that the weight filtration of  ${\mathcal{T}}^{(n)}$ actually splits in  $\mathsf{MT}(\mathbb{Q})$ . For that we introduce the involution  $\unicode[STIX]{x1D70F}$ which acts on the tori  $T^{n}$ by

This induces an involution, still denoted by  $\unicode[STIX]{x1D70F}$ , on the objects  ${\mathcal{T}}^{(n),r}$ and  $^{\prime }{\mathcal{T}}^{(n),r}$ of  $\mathsf{MT}(\mathbb{Q})$ , such that all the maps in the long exact sequence (28) commute with  $\unicode[STIX]{x1D70F}$ .

4.4 The structure of the zeta motives

We can now determine the structure of the zeta motives  ${\mathcal{Z}}^{(n)}$ , for  $n\geqslant 1$ .

We have already noted that the classes  $v_{k}^{(n)}$ are compatible with the inductive system of the zeta motives. By the uniqueness statement in Theorem 4.9, this is also the case for the classes  $\unicode[STIX]{x1D711}_{k}^{(n)}$ , and the zeta motive  ${\mathcal{Z}}$ has an infinite period matrix

4.5 The odd zeta motive

Let us write  ${\mathcal{T}}^{(n-1)}={\mathcal{T}}_{+}^{(n-1)}\oplus {\mathcal{T}}_{-}^{(n-1)}$ for the direct sum decomposition into its invariant and anti-invariant parts with respect to  $\unicode[STIX]{x1D70F}$ , and let us write  $p^{(n)}:{\mathcal{Z}}^{(n)}\rightarrow {\mathcal{T}}^{(n-1)}(-1)$ for the surjection appearing in the short exact sequence (30).

We obviously have a short exact sequence

with

We note that there are morphisms

such that  $i^{(2n),\text{odd}}$ is an isomorphism for every integer  $n$ . The limit

is an ind-object in  $\mathsf{MT}(\mathbb{Q})$ that we simply call the odd zeta motive.

Proposition 4.11 implies that the odd zeta motive  ${\mathcal{Z}}^{\text{odd}}$ has an infinite period matrix (6). In particular, ${\mathcal{Z}}^{(3),\text{odd}}$ is the essentially unique non-trivial extension of $\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$ in $\mathsf{MT}(\mathbb{Q})$ .

We finish by proving that all the objects in  $\mathsf{MT}(\mathbb{Q})$ considered earlier actually live in the full subcategory  $\mathsf{MT}(\mathbb{Z})$ .

4.6 An elementary computation of the motives  ${\mathcal{T}}^{(n)}$

We give an elementary proof of Proposition 4.5, which only uses basic algebraic topology. The proof is Hodge-theoretic, and the only drawback is that we have to use the full faithfulness of the Hodge realization (Theorem 2.6). Let us consider the relative homology group

By homotopy invariance, one may replace every  $\mathbb{C}^{\ast }$ by the unit circle  $S^{1}=\{|x|=1\}{\hookrightarrow}\mathbb{C}^{\ast }$ and the divisor  $\{x_{1}\cdots x_{n}=1\}$ by its intersection with  $(S^{1})^{n}$ , and we get

Let us look at the projection  $[0,1]^{n}\rightarrow (S^{1})^{n},(t_{1},\ldots ,t_{n})\mapsto (e^{2\unicode[STIX]{x1D70B}it_{1}},\ldots ,e^{2\unicode[STIX]{x1D70B}it_{n}})$ . Then by excision we can write

This is simply the singular homology of the unit hypercube  $[0,1]^{n}$ relative to the union of its faces  $\{t_{i}=0\}$ and  $\{t_{i}=1\}$ , for  $1\leqslant i\leqslant n$ , and the hyperplanes  $\{t_{1}\,+\cdots +\,t_{n}=k\}$ for  $k=0,1,\ldots ,n$ . We note that these hyperplanes cut the unit hypercube into polytopes

for  $k=0,\ldots ,n-1$ . We note that  $\unicode[STIX]{x1D6E5}(n,0)$ is the usual  $n$ -simplex; the polytopes  $\unicode[STIX]{x1D6E5}(n,k)$ are usually called hypersimplices.

The Eulerian numbers are the coefficients of the Eulerian polynomials and are denoted by symbols  $\langle \!\begin{smallmatrix}n\\ k\end{smallmatrix}\!\rangle$ :

They satisfy many beautiful identities, including the following recursion, which can be deduced from (21):

The following lemma is a classical result due to Laplace [Reference FoataFoa77, § 2].

Recall from Remark 4.6 that, for every integer  $n\geqslant 1$ ,  ${\mathcal{T}}_{\text{dR}}^{(n)}$ has a basis  $(w_{1}^{(n)},\ldots ,w_{n}^{(n)})$ which is compatible with the weight grading and with the morphisms  $j_{\text{dR}}^{(n)}:{\mathcal{T}}_{\text{dR}}^{(n-1)}\rightarrow {\mathcal{T}}_{\text{dR}}^{(n)}$ . We let  $P_{n}$ be the period matrix of  ${\mathcal{T}}^{(n)}$ with respect to the  $w$ -basis and the  $\unicode[STIX]{x1D6E5}$ -basis from Lemma 4.14. The first period matrix  $P_{1}$ is simply the  $1\times 1$ matrix  $(2\unicode[STIX]{x1D70B}i)$ . Let us introduce the following  $n\times n$ integer matrix encoding the family of Eulerian numbers:

We note that the period matrices  $P_{n}$ are not block upper-triangular. This is because the  $\unicode[STIX]{x1D6E5}$ -basis is not compatible with the weight filtration. We thus have to introduce a change of basis. Let  $(Q_{n})_{n\geqslant 1}$ be the family of matrices (with rational entries) defined by  $Q_{1}=(1)$ and the recurrence relation

The first terms are

Let us put

We view  $\unicode[STIX]{x1D6F4}_{k}^{(n)}$ as a relative cycle with rational coefficients. The change of indexing is here to remind the reader that  $\unicode[STIX]{x1D6F4}_{k}^{(n)}$ lives in weight ${\leqslant}2k$ . We have thus proved the following result.

By using Theorem 2.6, we thus get an alternate (Hodge-theoretic) proof of Proposition 4.5.

5.1 Integral formulas for the coefficients

5.2 The Ball–Rivoal integrals

We apply Theorems 5.1 and 5.5 to a special family of integrals.

Corollary 5.6 applies in particular to the special case

for some integer parameters  $r,m\geqslant 0$ satisfying  $n(m+1)\geqslant (2r+1)m+3$ . We then recover the integrals considered by Ball and Rivoal [Reference Ball and RivoalBR01, Lemme 2]. The vanishing of the coefficients is [Reference Ball and RivoalBR01, Lemme 1]. The notations  $(a,n,r)$ in [Reference Ball and RivoalBR01] correspond to our notations  $(n-1,m,r)$ .

The integrals (41) can be expressed as generalized hypergeometric series

If  $2u_{i}+v_{i}=N+1$ , then the corresponding generalized hypergeometric series is said to be well-poised.

5.3 Weight drop

In the context of Theorem 5.1, we say that the integral  $\int _{[0,1]^{n}}\unicode[STIX]{x1D714}$ has weight drop if the highest weight coefficient  $a_{n}(\unicode[STIX]{x1D714})$ vanishes. This amounts to saying that the class  $[\unicode[STIX]{x1D714}]$ actually lives in the step  $W_{2(n-1)}{\mathcal{Z}}_{n,\text{dR}}$ of the weight filtration, hence the terminology. We give a sufficient condition for this phenomenon to happen.

Note that Proposition 5.8 applies in particular if for every  $i$ ,  $2u_{i}+v_{i}=N+1$ . This gives in particular a geometric interpretation of the weight drop in the Ball–Rivoal integrals [Reference RivoalRiv00, Reference Ball and RivoalBR01], which comes from the representations as hypergeometric series (42). Note that a careful analysis of the degree of the polynomial  $P(t)$ in Lemma 5.7 can lead to sufficient conditions for the vanishing of the subleading coefficients.