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

# Motivic correlators, cluster varieties, and Zagier’s conjecture on ζF(4)

Alexander Goncharov, Daniil Rudenko
###### Abstract

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

 ζF(4)=π4(r1+r2)|dF|−1/2⋅det(L4(σi(yj))),    1≤i,j≤r2. (1)

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:

 0⟶B4(F)⟶L4(F)⟶Λ2B2(F)⟶0. (2)

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 n-logarithm.

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

 Lin(z)=∞∑k=1zkkn,    |z|<1.

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

 Lin(z)=∫z0Lin−1(z)dlogz,    n>1.

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

 πn:C⟶C/R(n)=R(n−1),    z∈C⟼{Re(z)n=2k+1Im(z)n=2k.. (3)

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

 Ln(z):=πn(n−1∑k=02kBkk!Lin−k(z)logk|z|),  n>1.

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.

 [s1,s2,s3,s4]:=(s1−s2)(s3−s4)(s4−s1)(s2−s3),    [∞,−1,0,z]=−z. (4)

Then for any five distinct points on we have:

 5∑i=1L2([si,si+1,si+2,si+3])=0,    i∈Z/5Z. (5)

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

 Bn(F):=Q[F]Rn(F).

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

 Q[F]\lx@stackrelδn⟶{Bn−1(F)⊗F×Qn>2,F∗Q∧F×Qn=2,    {x}⟼{{x}n−1⊗xn>2,(1−x)∧xn=2.  δ2{1}=δ2{0}=0.

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

 Ln:Bn(C)⟶R,    {z}n⟼Ln(z),  n>1.

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.

 δn:Bn(F)⟶{Bn−1(F)⊗F×Qn>2,F×Q∧F×Qn=2. (6)

Evidently, the following composition is zero for :

 Bn(F)\lx@stackrelδn⟶Bn−1(F)⊗F×Q\lx@stackrelδn−1⊗Id⟶Bn−2(F)⊗Λ2F×Q.

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

 (7)

For example, the first four polylogarithmic motivic complexes are

 B∙(F;1):    B1(F)=F×Q.B∙(F;2):    B2(F)\lx@stackrel⟶Λ2F×Q.B∙(F;3):    B3(F)\lx@stackrel⟶B2(F)⊗F×Q\lx@stackrel⟶Λ3F×Q.B∙(F;4):    B4(F)\lx@stackrel⟶B3(F)⊗F×Q\lx@stackrel⟶B2(F)⊗Λ2F∗Q⟶Λ4F×Q. (8)

#### Main results.

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

 Ress=1ζF(s)=2r1+r2πr2RFhFwF√|dF|

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

 ζF(4)=π4(r1+r2)|dF|−1/2⋅det(L4(σi(yj))),    1≤i,j≤r2. (9)

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:

 grnγK2n−i(F)Q\lx@stackrel?=HiB∙(F;n),    i>0. (10)

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

 Kn(F)⟶πm(BGL(F)+)\lx@stackrelHurevicz⟶Hm(BGL(F)+,Z)=Hn(GL(F),Z). (11)

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

 PrimHn(G,Q):={X∈Hn(G,Q) | Δ∗(X)=X⊗1+1⊗X}.

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

 Kn(F)Q\lx@stackrel∼⟶PrimHn(GL(F),Q). (12)

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

 FrkmKn(F)Q:=Im(PrimHn(GLm(F),Q)⟶PrimHn(GL(F),Q)).

The stabilization theorem of Suslin [Su84] implies that

 FrknKn(F)Q=Kn(F)Q.

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

 grrkiKn(F)Q\lx@stackrel?=grγiKn(F)Q.

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

 K8−i(F)⟶HiB∙(F;4),    i=1,2,3,4. (13)

Their restriction to is zero.

ii) For any complex variety , the composition

 K8−i(C(X))⟶HiB∙(C(X);4)⟶Hi(Spec(C(X)),RD(4)) (14)

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

 K8−i(X)⟶Hi(X,RD(4)).

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

 K7(C)⟶H1B∙(C;4)\lx@stackrelL4⟶R.

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

 grrk4K8−i(F)\lx@stackrel⟶HiB∙(F;4),    i=1,2,3,4. (15)

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 F.

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

 Ext∙MT(F)(Q(0),Q(n))=Ext∙DMF(Q(0),Q(n)).

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:

 ω:MT(F)⟶VectQ,    ω(M):=⊕n∈ZHom(Q(−n),grW2nM). (16)

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

 RHomDMF(Q(0),Q(n)) (17)

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:

 L4(F)⟶L3(F)∧L1(F)⨁Λ2L2(F)⟶L2(F)⊗Λ2L1(F)⟶Λ4L1(F). (18)

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 :

 I∙(F)=⊕n>1L−n(F).

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

###### Conjecture 1.5.

The ideal has the cohomology given by

 H1(n)I∙(F)=Bn(F)Q,    Hi(n)I∙(F)=0,  ∀i,n>1.

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:

 L1(F)=F×Q,    L2(F)=B2(F),    L3(F)=B3(F). (19)

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

 0⟶B4(F)⟶L4(F)\lx@stackrelp⟶Λ2B2(F)⟶0. (20)

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

 L4(F)⟶B3(F)∧F×Q⨁Λ2B2(F)⟶B2(F)⊗Λ2F×Q⟶Λ4F×Q. (21)

#### Remark.

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

 Λ2B2(F)⟶B3(F)∧F×Q.

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”:

 r3(x1,x2,x3,x4,x5,x6):=Alt6{ω(l1,l2,l4)ω(l2,l3,l5)ω(l1,l3,l6)ω(l1,l2,l5)ω(l2,l3,l6)ω(l1,l3,l4)}3∈Q[F]. (22)

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]:

 7∑i=1(−1)iL3(r3(x1,…,ˆxi,…,x7))=0. (23)

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 L≤4(F).

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.

 [x1,x2,…,x6]:=−(x1−x2)(x3−x4)(x5−x6)(x2−x3)(x4−x5)(x6−x1). (24)

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

 CycnF(x1,…,xn):=∑k∈Z/nZF(xk+1,xk+2,…,xk+n). (25)

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:

 (26)

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

###### Definition 1.7.

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

 (27)

plus the following one, together with all its specializations:

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

 Cyc6({[x1,x2,x3,x4],[x4,x5,x6,x1]}2,1−{[x1,x2,x4,x5]}3+2⋅{[x1,x3,x4,x5]}3)−4⋅{[x1,x2,x3,x4,x5,x6]}3+6⋅{1}3=0. (28)

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

 {x,y}2,1={1−x−1}3+{1−y−1}3+{yx}3+{1−y1−x}3−{1−y−11−x−1}3−{1}3. (29)

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

 B3(F)\lx@stackrel∼⟶L3(F). (30)

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

 {x,y}2,1={[∞,0,x,1]}3+{[∞,1,y,0]}3+{[∞,y,0,x]}3+{[∞,x,1,y}3−{[0,x,1,y]}3−{1}3. (31)
###### 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.

 {x,0}3,1:=Spt→0{x,t}3,1=−{x}4,{x,1}3,1=−{1−x−1}4−{1−x}4+{x}4. (32)
3. For any configuration the following cyclic sum is zero:

 Cyc7(−{[x1,x2,x3,x4],[x4,x6,x7,x1]}3,1+{[x1,x2,x3,x4],[x4,x5,x7,x1]}3,1−{[x1,x2,x3,x4],[x4,x5,x6,x1]}3,1+{[x1,x2,x4,x6]}4+{[x1,x2,x3,x4,x5,x6]}4)=0. (33)
###### Conjecture 1.9.

Relation and its specializations imply the tetralogarithm relations .

Let us define the coproduct maps

 δ:L2(F)⟶F×∧F×,δ:L3(F)⟶L2(F)⊗F×,δ:L4(F)⟶L3(F)⊗F× ⨁ L2(F)∧L2(F). (34)

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].

 (35)

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

#### Motivic correlators and the Lie coalgebra L≤4(F).

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:

 CorMa,v(b1,…,bm+1)∈Lm(F).

If , they do not depend on the tangent vector .

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

 L≤4(F)⟶L≤4(F), (36)

defined on generators as

 {x,y}2,1⟼CorM∞(0,x,1,y).