Motivic correlators, cluster varieties, and Zagier’s conjecture on \zeta_{F}(4)

Motivic correlators, cluster varieties, and Zagier’s conjecture on

Alexander Goncharov, Daniil Rudenko
Abstract

We prove Zagier’s conjecture [Za90] on the value at of the Dedekind -function of a number field :

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Here is the set of all complex embeddings of , up to the conjugation. Namely, up to a standard factor, is equal to a determinant, whose entries are -linear combinations of the values of a single-valued version of the classical 4-logarithm at some numbers in , satisfying specific conditions.

For any field , we define a map of K-groups , , to the -th cohomology of the weight 4 polylogarithmic motivic complex . When is the function field of a complex variety, composing the map with the regulator map on the polylogarithmic complex to the Deligne cohomology, we get a multiple of Beilinson’s regulator. This implies that the composition , where the second map is given by , is a multiple of Borel’s regulator. This plus Borel’s theorem implies Zagier’s conjecture.

Another application is a formula expressing the value of the -function of an elliptic curve over via generalized Eisenstein-Kronecker series.

We get a strong evidence for the part of Freeness Conjecture [G91a] describing the weight four part of the motivic Lie coalgebra of via higher Bloch groups as an extension:

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Main tools are motivic correlators and a new link of cluster varieties to polylogarithms.

1 Introduction and the architecture of the proof

1.1 The classical polylogarithms and algebraic K-theory

The classical -logarithm.

The classical -logarithm function is given by the power series which are absolutely convergent on the unit disc:

It is continued analytically to a multivalued analytic function on by induction, setting

Here we integrate over a path in from to . The obtained multivalued analytic function has a single valued cousin [Za90]. Namely, set , and let

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Then the following expression is a single-valued function, well defined and continuous on :

Here are the Bernoulli numbers, and are defined via the same integration path. For example, is the Bloch-Wigner dilogarithm. For a Hodge-theoretic interpretation of the functions see [BD].

The function satisfies ”clean” functional equations.111This means that no products of polyogarithms of smaller weights are involved. For example, the dilogarithm satisfies the five-term relation. Namely, recall the cross-ratio of four points on :222It is normalized to be the negative of the so-called positive or cluster cross-ratio.

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Then for any five distinct points on we have:

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Any functional equation for the dilogarithm function follows from the five-term relation.333Precisely, any relation where are non-constant functions and , follows from (5).

Although we do not know explicitly functional equations for -logarithms for large except a trivial one , one can define the subgroup of all functional equations. This leads to higher Bloch groups, whose definition we recall now.

Higher Bloch groups and polylogarithmic motivic complexes [G91a].

Denote by the -vector space generated by a set . We denote by the generator assigned to an . Given a field , one defines inductively for each a subspace reflecting functional equations for the classical n-logarithm function, and set

We denote by the projection of the generator to the quotient .

For an abelian group , let . The subgroup is generated by the elements where , and . So . We define by induction a map

It is handy to add a generator together with the relation .

Let us define a subspace . Any expression which lies in the kernel of for the field give rise to an element . The subgroup is generated by all elements obtained this way, and .

One proves [G91b, Theorem 1.5] that there is a map

For we have a map . This means that the subgroup is indeed a subgroup of functional equations for the polylogarithm function . One can show that it contains all functional equations, that is relations depending non-trivially on a parameter.

One proves that the map induces a group homomorphism444The name Bloch group was coined by Suslin [Su82] for the kernel of the map discussed below. We use the terminology of [G91a], where the group was called the Bloch group, and its higher analogs were defined and called higher Bloch groups.

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Evidently, the following composition is zero for :

So we get a complex in the degrees , where is in the degree , called the weight polylogarithmic motivic complex:

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For example, the first four polylogarithmic motivic complexes are

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Main results.

Zagier’s conjecture [Za90] predicts that the classical regulator formula

for the residue of the Dedekind -function of a number field at has analogs for for any positive integer . It was proved for in [Za86] and for in [G91a].

Theorem 1.1 proves Zagier’s conjecture555Its original formulation does not use groups ; it uses subgroups , defined for number fields only. for .

Theorem 1.1.

Let be a number field, , and the set of all embeddings is numbered so that . Let be the discriminant of .

Then there exist elements such that

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ii) For any , the right hand side of (9) equals for some .

Zagier’s conjecture concerns number fields. For an arbitrary field , it was conjectured in [G91a] that the weight polylogarithmic motivic complexes calculate the weight pieces of the Quillen K-groups of the field modulo torsion. Preciesly, let be the Adams -filtration on Quillen’s algebraic K-theory. The conjecture states that one expects the following isomorphisms:

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This conjecture has a variant which is formulated more elementary, without reference to Quillen’s definition of algebraic K-theory and the Adams filtration, which we recall now.

Denote by the infinite general linear group, defined as the inductive limit of the groups , sitting one in the other in the natural way. There is a canonical map

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Let be any group, and the diagonal map. Then the primitive part of the rational homology of any group is defined by

It is well known that the map (11) induces an isomorphism

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The natural filtration of the group by the subgroups induces an increasing filtration on (12), known as the rank filtration:

The stabilization theorem of Suslin [Su84] implies that

It is conjectured that the rank and the Adams filtrations have the same associate graded pieces:

Here is the main result of this paper. Denote by the weight real Deligne complex.

Theorem 1.2.

i) Let be any field. Then there are canonical homomorphisms

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Their restriction to is zero.

ii) For any complex variety , the composition

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is a non-zero rational multiple of Beilinson’s regulator map.

iii) For any regular curve over a number field, the map (14) gives rise to a non-zero rational multiple of Beilinson’s regulator map

iv) The following composition is a non-zero rational multiple of the Borel regulator map:

Theorem 1.1 follows from the part iv) of Theorem 1.2 and Borel’s theorem [B76].

The part i) of Theorem 1.2 implies that we get maps

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The map (15) for is an isomorphism due to a theorem of Suslin [Su84] relating Quillen’s and Milnor’s K-groups. In fact the integral analog of this map is an isomorphism modulo -torsion.

Conjecture 1.3.

The maps (15) are isomorphisms modulo torsion.

Theorem 1.2 gives a strong evidence to the weight parts of several ”beyond-the-standard” conjectures about mixed Tate motives, which we are going to discuss now.

The motivic Tate Lie algebra of a field .

According to Beilinson, one should have an abelian Tannakian category of mixed Tate motives over a field . It supposed to be a subcategory of the triangulated category of all mixed motives over , which is defined unconditionally, and has the expected Ext’s [V].

Conjecture 1.4.

[B] One should have

The abelian category of mixed Tate motives is available with all the expected properties when is a number field [L], [DG]. However even in this case, Conjecture 1.4 is not known in full generality.

Let us assume that the category of mixed Tate motives over a field does exist. Then it comes with a canonical fiber functor to the category of -vector spaces:

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The Lie algebra of derivations with respect to the tensor product in of the functor is a Lie algebra in the category of projective limits of -vector spaces, graded by negative integers. It is called the motivic Tate Lie algebra.

Conjecture 1.4 implies that the weight motivic complexes

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are quasi-isomorphic to the degree part of the standard cochain complex of the Lie algebra . Even a weaker claim, that the motivic complexes (17) can be realized as the standard cochain complex of a certain Lie algebra is a highly non-trivial. See [G91b] for a discussion.

Let be the graded dual to the Lie algebra . It is a Lie coalgebra in the category of inductive limits of -vector spaces, called the motivic Tate Lie coalgebra of . The weight part of Conjecture 1.4 implies that the weight motivic complex of should be quasi isomorphic to the following complex:

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The space is an ind--vector space. Denote by its linear dual, which is a pro--vector space. Conjecture 1.5 below is a part of the Freeness Conjecture, see [G91a], [G91b, Conjecture 1.20]. It describes the relation between the classical polylogarithms and all mixed Tate motives. Consider the graded ideal of the Lie algebra :

We denote by the degree part of the cohomology of a graded Lie algebra .

Conjecture 1.5.

The ideal has the cohomology given by

So the graded Lie algebra is isomorphic to the free graded Lie algebra with the generators in the degree , , given by the spaces , although this isomorphism is non-canonical.

Conjecture 1.5 predicts the following canonical isomorphisms:

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Furthermore, it predicts that the weight component is described by an extension:

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The map is identified with the component of the coproduct via the isomorphism . Therefore complex (18) looks now as follows:

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Remark.

Extension (20) does not have a functorial in splitting . Indeed, composing the map with the -component of the coproduct, we get a map

However there is no such a map given by rational functions on generators [G91b, Theorem 4.7].

1.2 Functional equations for polylogarithms and motivic Tate Lie coalgebras

The five-term relation (5) for the dilogarithm has a generalization for the trilogarithm.

Given a generic configuration of 6 points in , let us lift it to a configuration of 6 vectors in , and consider the ”triple ratio”:

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Here is the operation of skewsymmetrisation. The right hand side is independent of the choices. Then for any generic configuration of seven points in we have [G91b]:

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It is equivalent to the shorter 22-term relation for the trilogarithm from [G91a].

We denote by the group defined using these functional equations.

Relation (23) played a central role in relating the trilogarithmic motivic complex to the algebraic K-theory for any field , and proving Zagier’s conjecture for [G91a]. Yet it is unclear how to generalize relations (5) and (23) to the classical n-logarithms to implement a similar strategy for for .

Our strategy is different. Given a field , we look for an explicit construction of the weight part of the motivic Tate Lie coalgebra rather then groups . The two problems for turned out to coincide - a deep fact on its own, consistent with the Freeness Conjecture.

The existence of the Lie coalgebra for any field is unknown yet. So we aim at a functorial in explicit construction of a Lie coalgebra , which is conjecturally isomorphic to . We define a Lie coalgebra by using simple and uniform in relations between weight iterated integrals, .

The relation is just the Abel five-term relation.

The relation not only implies the 22-term relation, and hence the relation (23) for the trilogarithm, but allows also to express any weight 3 iterated integral via the classical trilogarithm. So it is a new way to present the trilogarithm story.

Constructing the Lie coalgebra .

Recall the moduli space parametrizing configurations of distinct points on modulo the diagonal action of the group . Recall the cross-ratio , see (4). Consider the following regular function on :666Mind the minus sign.

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We use the cyclic summation notation, where the indices are modulo :

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Before we proceed with the definitions, let us make few comments applicable to all of them.

In Definitions 1.6-1.8 we assume that is any field, and include, by default, all relations obtained by specializations of the ”generic” relation . Precisely, given any curve , we add the specialization at of the relation to the list of relations.

Definition 1.6.

The -vector space is generated by symbols for satisfying , and the pentagon relation:777Relations follow from the pentagon relation by the specialization.

  • : for any configuration the following cyclic sum is zero:

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By the very definition we have , where is the traditional Bloch group.

Figure 1: Geometry of relation . The diagonal divides the hexagon into quadrangles. The arguments of are the cross-ratios assigned to them, and is their ratio. Elements and correspond to the quadrangles in the two hexagons on the right.
Definition 1.7.

The -vector space is generated by symbols where and where , satisfying relations888These relations also follow from relation by the specialization.

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plus the following one, together with all its specializations:

  • For any distinct points on the following cyclic sum is zero:

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Figure 2: Geometry of relation . The diagonal divides the septagon into a quadrangle and a pentagon. Deleting one of three vertices of the pentagon we get another quadrangle: on the picture we deleted . The arguments of are cross-ratios assigned to these quadrangles.

Specialising the relation to the divisor in we get the following key relation:

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Substituting (29) to the relation we get the -term relation for trilogarithm from [G91a]. Therefore the map induces an isomorphism

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The relation (29) has the following geometric interpretation. Take five points on , where the last four points are ordered cyclically. Then

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Definition 1.8.

The -vector space is generated by elements , where , and where , obeying the following relations:

  1. The generators satisfy the 4-logarithmic relations ;

  2. Specialization relations:999Specialisation relations (32) could be deduced from relation , but this would require long calculations.

    (32)
  3. For any configuration the following cyclic sum is zero:

    (33)
Conjecture 1.9.

Relation and its specializations imply the tetralogarithm relations .

Let us define the coproduct maps

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First, we define them on the generators: the coproduct is given by formula (6), and101010 Formulas (35) coincide with the map given by formulas (5) and (6) in [G91a].

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Let us give a motivic interpretation of elements and their coproduct formula (35).

Motivic correlators and the Lie coalgebra .

Motivic correlators are elements of the motivic Lie coalgebra introduced in [G08]. They are defined whenever we have the abelian category of mixed Tate motives, e.g. when is a number field, or in realizations.

Weight motivic correlators are determined by a cyclically ordered set of points on , a point different from , and a non-zero tangent vector at , see Figure 3:

If , they do not depend on the tangent vector .

Figure 3: A data defining a weight 5 motivic correlator.

To state the properties of motivic correlators, let us recall that a framed mixed Tate motive is a mixed Tate motive plus non-zero maps and . It provides an element .

Let . The following properties explain ubiquity of motivic correlators:

  • Motivic correlators describe the structure of the motivic fundamental group . In particular, any element given by a framed subquotient of is a linear combination of motivic correlators.

  • The coproduct of motivic correlators is given by a simple formula, illustrated on Figure 7.

  • The Hodge realization of motivic correlators is described by a Feynman type integral.

Motivic correlators with arguments in a field are closed under the coproduct. So they form a graded Lie coalgebra, denoted by . It is a Lie subcalgebra of . Then Universality Conjecture [G94, Conjecture 17a] can be restated as follows.

Conjecture 1.10.

The canonical embedding is an isomorphism.

In Section 2 we show that the elements can be materialized by motivic correlators:

Theorem 1.11.

a) There is a map of Lie coalgebras

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defined on generators as