Multiple polylogarithms, cyclotomy and modular complexes

A.B. Goncharov

October 10, 1998

1 Introduction

1. Multiple polylogarithms. We define them by the power series expansion:

Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) = ∑_{0<k_{1}<k_{2}<...<k_{m}} \frac{x_{1}^{k_{1}}x_{2}^{k_{2}}...x_{m}^{k_{m}}}{k_{1}^{n_{1}}k_{2}^{n_{2}}...k_{m}^{n_{m}}}  (1)

Here w := n_{1} + ... + n_{m} is called the weight and m the depth.

These power series are convergent for |x_{i}| < 1, and can be continued analytically via the iterated integral presentation given by theorem 2.1.

The power series (1) generalize both the classical polylogarithms Li_{n}(x) (m=1), and multiple ζ-values (x_{1} = ... = x_{m} = 1) :

ζ(n_{1}, ..., n_{m}) := ∑_{0<k_{1}<k_{2}<...<k_{m}} \frac{1}{k_{1}^{n_{1}}k_{2}^{n_{2}}...k_{m}^{n_{m}}}  n_{m} > 1  (2)

The multiple ζ-values were invented and studied by Euler [E] and then forgotten. They showed up again in such different subjects as quantum groups [Dr] (the Drinfeld associator), Zagier’s studies [Z1-2], the Kontsevich integrals for Vassiliev knot invariants, mixed Tate motives over Specℤ [G1-2], and, recently, in computations in quantum field theory [B], [Kr].

The multiple polylogarithms were studied in [G1-4]. In this paper we investigate them at N-th roots of unity: x_{1}^{N} = ... = x_{m}^{N} = 1. Notice that Li_{1}(x) = − log(1 − x), so if ζ_{N} is a primitive N-th root of 1, then Li_{1}(ζ_{N}) is a logarithm of a cyclotomic unit in ℤ[ζ_{N}, N^{−1}]. In general the supply of numbers we get coincides with the linear combinations of multiple Dirichlet L-values

L(χ_{1}, ..., χ_{m}; n_{1}, ..., n_{m}) := ∑_{0<k_{1}<k_{2}<...<k_{m}} \frac{χ_{1}(k_{1})...χ_{m}(k_{m})}{k_{1}^{n_{1}}k_{2}^{n_{2}}...k_{m}^{n_{m}}}  (3)

They are periods of mixed Tate motives over the scheme S_{N} := Specℤ[ζ_{N}][\frac{1}{N}] (see s. 11 of [G2] and [G4]).

To study these numbers we introduce some tools from homological algebra (cyclotomic and dihedral Lie algebras, modular complex for GL_{m}(ℤ)) and geometry (a realization of the modular complex in the symmetric space SL_{m}(ℝ)/SO_{m}). To motivate them we start from a conjecture.

2. Multiple polylogarithms at roots of unity and the cyclotomic Lie algebras. Let Z_{≤w}(N) be the ℚ-vector space spaned by the numbers

\overline{L}i_{n_{1},...,n_{m}}(ζ_{N}^{α_{1}}, ..., ζ_{N}^{α_{m}}) := (2πi)^{−w}Li_{n_{1},...,n_{m}}(ζ_{N}^{α_{1}}, ..., ζ_{N}^{α_{m}})  (4)

Here we may take any branch of Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}). Then Z(N) := ∪Z_{≤w}(N) is an algebra bifiltered by the weight and by the depth. For example:

Li_{m}(x) · Li_{n}(y) = ∑_{k_{1},k_{2}>0} \frac{x^{k_{1}}y^{k_{2}}}{k_{1}^{m}k_{2}^{n}} = Li_{m,n}(x, y) + Li_{m+n}(xy) + Li_{n,m}(y, x) (5)

(To prove this split the sum over k_{1} < k_{2}, k_{1} = k_{2} and k_{1} > k_{2}).

Denote by UC_{•} the universal enveloping algebra of a graded Lie algebra C_{•}. Let UC_{•}^{∨} := ⊕_{n≥0}(UC)_{n}^{∨} be its graded dual. It is a commutative Hopf algebra.

Conjecture 1.1 There exists a graded Lie algebra C_{•}(N) over ℚ such that one has an isomorphism

Z(N) = UC_{•}(N)^{∨}  (6)

of filtered by the weight on the left and by the degree on the right algebras. b) H_{(n)}^{1}(C_{•}(N)) = K_{2n−1}(ℤ[ζ_{N}][\frac{1}{N}]) ⊗ ℚ.

c) C_{•}(1) is free graded Lie algebra.

Here H_{(n)} is the degree n part of H. Notice that H_{(n)}^{1}(C_{•}(N)) is dual to the space of degree n generators of the Lie algebra C_{•}(N).

A construction of the Lie algebra C_{•}(N) using the Hodge theory will be outlined in s. 3.2. It can be used to deduce conjecture 1.1 from some standard (but extremely dificult!) conjectures in arithmetic algebraic geometry. The simplest abelian quotient C_{1}(N) of C_{•}(N) is the group of cyclotomic units in ℤ[ζ_{N}, N^{−1}], tensored by ℚ. We call C_{•}(N) the cyclotomic Lie algebra of level N, and suggest that the ”higher cyclotomy theory” should study its properties.

Examples. i) Let N = 1. Then by the Borel theorem the only nontrivial modulo torsion K-groups are K_{4n+1}(ℤ), which have rank 1 and correspond to \overline{ζ}(2n + 1) via the regulator map.

ii) N = 2: the generators should correspond to (2πi)^{−1} log 2, \overline{ζ}(3), \overline{ζ}(5), .... iii) If N > 2, n > 1 one has dimK_{2n−1}(ℤ[ζ_{N}, N^{−1}]) ⊗ ℚ = \frac{φ(N)}{2} and the

space of generators should correspond to the span over ℚ of \overline{L}i_{n}(ζ_{N}^{α}).

Remark. Let π_{1}^{(l)}(ℙ^{1}\{0, 1, ∞}) be the l-adic completion of the fundamental group. One has canonical homomorphism

φ^{l} : Gal(\overline{ℚ}/ℚ) ⟶ Outπ_{1}^{(l)}(ℙ^{1}\\{0, 1, ∞\})  (7)

It was studied by P.Deligne, Y.Ihara and others (see [Ih] and references there). Conjecture 1.1 for N = 1 is closely related to some conjectures/questions of P.Deligne [D] about the image of the map (7) and V. Drinfeld [Dr] about the structure of the pronilpotent version of the Grothendieck-Teichmuller group.

The left hand side of (6) has an additional structure: the depth filtration. To study it we proceed as follows.

3. The dihedral Lie coalgebra and modular complexes. Let Z_{•,•}(N) be associate graded quotient with respect to the weight and the depth filtrations of the algebra Z(N). We reduce it further introducing the bigraded ℚ-space

\overline{Z}_{•,•}(N) := \frac{Z_{•,•}(N)}{(Z_{>0,>0}(N))^{2}}

The multiple polylogarithms are multivalued functions, however it is easy to show that the projection of (4) to \overline{Z}_{•,•}(N) does not depend on the branch we choose. So \overline{Z}_{w,m}(N) is the quotient of the ℚ-space generated by the numbers (4) of weight w and depth m modulo the subspace generated by the lower weight and depth numbers, and also by the products of numbers (4) of total weight w.

Here is our strategy for investigation of dim_{ℚ}\overline{Z}_{w,m}(N). Let μ_{N} be the group of N-th roots of unity. In the section 3 a Lie coalgebra D_{•,•}(μ_{N}), called the dihedral Lie coalgebra, is explicitely constructed. Namely, the generators of the ℚ-vector space D_{w,m}(μ_{N}) correspond to the projections of the numbers (4) to \overline{Z}_{w,m}(N), and the defining relations reflect the known ℚ-linear relations between these numbers. We prove that \overline{Z}_{w,m}(N) is a quotient of D_{w,m}(μ_{N}) (theorem 7.1). A really new data is the cocommutator map δ : D_{•,•}(μ_{N}) ⟶ Λ^{2}D_{•,•}(μ_{N}). The dihedral Lie coalgebra is bigraded by the weight and the depth.

We want to understand the cohomology of the Lie coalgebra D_{•,•}(μ_{N}). Here is the standard cochain complex computing the cohomology of D_{•,•}(μ_{N}):

D_{•,•}(μ_{N}) ⟶^{δ} Λ^{2}D_{•,•}(μ_{N}) ⟶^{δ∧id−id∧δ} Λ^{3}D_{•,•}(μ_{N}) ⟶ ...  (8)

The first arrow is the cocommutator map, and the others obtained via the Leibniz rule. This complex is bigraded by the weight and depth. Let (Λ^{∗}D(μ_{N}))_{w,m} be the subcomplex of the weight w and depth m. It is easy to prove that

D_{w,1}(μ_{N}) = K_{2w−1}(ℤ[ζ_{N}, N^{−1}]) ⊗ ℚ ⊂ H_{(w)}^{1}(D_{•,•}(N))  (9)

We construct a certain length m − 1 complex of GL_{m}(ℤ)-modules M_{(m)}^{∗}, called the rank m modular complex. For m = 2 it is identified with the chain complex of the classical modular triangulation of the hyperbolic plane:

Let V_{m} be the standard m-dimensional representation of GL_{m}. Denote by Γ_{1}(N; m) the subgroup of GL_{m}(ℤ) consisting of matrices whose last row is congruent to (0, ..., 0, 1) modulo N.

Theorem 1.2 a) There exists a surjective morphism of complexes

μ(N)_{w,m}^{∗} : S^{w−m}V_{m} ⊗_{Γ_{1}(N;m)} M_{(m)}^{∗} ⟶ (Λ^{∗}D(μ_{N}))_{w,m}

b) It is an isomorphism if N = 1, or if N is prime and w = m.

A special case of this theorem for m = 2, w = 2 was proved in [G3]. The case N = 1, m = 2, w is arbitrary was considered in [G1] and in s. 4 of [G3].

We prove in section 6 that the modular complex of rank 3 is (essentially) quasiisomorphic to the Voronoi complex of the symmetric space SL_{3}(ℝ)/SO_{3}. Using these results we can estimate from above dim\overline{Z}_{w,m}(N) for m ≤ 3.

4. Applications to multiple ζ-values. Theorem 1.2 together with the relation between the modular and Voronoi complexes for GL_{2} and GL_{3} mentioned above lead to the following result.

Theorem 1.3

H_{(w)}^{i}(D_{•,2}(1)) = H^{i−1}(GL_{2}(ℤ), S^{w−2}V_{2}) i = 1, 2  (10)
H_{(w)}^{i}(D_{•,3}(1)) = H^{i}(GL_{3}(ℤ), S^{w−3}V_{3}) i = 1, 2, 3  (11)

This allows us to compute the Euler characterisitc of complexes (Λ^{∗}D(1))_{w,m} for m = 2, 3. Then we find dim_{ℚ}D_{w,m}(1) for m = 2, 3 by induction using as a starting point (9), which boils down to

dim_{ℚ}D_{w,1}(1) = \{ \begin{matrix}1 & w : odd \\ 0 & w : even \end{matrix}  (12)

Theorem 1.4 a) If w is odd, then dim\overline{Z}_{w,2}(1) = dimD_{w,2}(1) = 0. b) If w is even then dim\overline{Z}_{w,2}(1) ≤ dimD_{w,2}(1) = [\frac{w−2}{6}].

The part a) goes back to Euler, b) was discovered by Zagier [Z2].

Theorem 1.5 a) If w is even, then dim\overline{Z}_{w,3}(1) = dimD_{w,3}(1) = 0. b) If w is odd then

dim\overline{Z}_{w,3}(1) ≤ dimD_{w,3}(1) = [\frac{(w − 3)^{2} − 1}{48}]  (13)

Numerical calculations of multiple ζ’s by Zagier (considerebly extended by Broadhurst in [B]) suggested that the estimate (13) is exact. Our results about ”motivic” multiple ζ’s allow to deduce this from standard conjectures. The details will appear elsewhere.

Remark. The philosophy of mixed motives was the main driving force for us. However constructions and proofs of this paper are ”elementary”, i.e. do not use motives. The mixed motives/Hodge structures show up only in s. 3.2 to outline the construction of the cyclotomic Lie algebra, but we do not use s. 3.2 in the rest of the paper, so the reader may skip it.

Acknowledgment. I would like to thank the Max-Planck-Institute (Bonn) and University Paris-XI (Orsay) for hospitality and support. The support by the NSF grant DMS-9500010 is gratefully acknowledged.

I am grateful to J. Bernstein, G. Frey, G. Harder, M. Kontsevich, B.B. Venkov, V. Voevodsky, D. Zagier for useful and stimulating discussions.

2 Properties of multiple polylogarithms

1. Iterated integral presentation. Set

∫_{0}^{a_{n+1}} \frac{dt}{a_{1} − t} ∘ ... ∘ \frac{dt}{a_{n} − t} := ∫_{0≤t_{1}≤...≤t_{n}≤a_{n+1}} \frac{dt_{1}}{a_{1} − t_{1}} ∧ ... ∧ \frac{dt_{n}}{a_{n} − t_{n}}

I_{n_{1},...,n_{m}}(a_{1} : ... : a_{m} : a_{m+1}) := ∫_{0}^{a_{m+1}} \underbrace{\frac{dt}{a_{1} − t} ∘ \frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n_{1} times} ∘ ... ∘\underbrace{\frac{dt}{a_{m} − t} ∘ \frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n_{m} times}

The following theorem is the key to properties of multiple polylogarithms.

Theorem 2.1 Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) = I_{n_{1},...,n_{m}}(1 : x_{1} : x_{1}x_{2} : ... : x_{1}...x_{m}).

The proof is very easy: develope dt/(a_{i}−t) into a geometric series and integrate. If x_{i} = 1 we get the Kontsevich formula.

2. Relations. The double shuffle relations. Set

Li(x_{1}, ..., x_{m}|t_{1}, ..., t_{m}) := ∑_{n_{i}≥1} Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m})t_{1}^{n_{1}−1}...t_{m}^{n_{m}−1}

I(a_{1} : ... : a_{m} : a_{m+1}|t_{1}, ..., t_{m}) := ∑_{n_{i}≥1} I_{n_{1},...,n_{m}}(a_{1} : ... : a_{m} : a_{m+1})t_{1}^{n_{1}−1}...t_{m}^{n_{m}−1}

I^{∗}(a_{1} : ... : a_{m} : a_{m+1}|t_{1}, ..., t_{m}) := I(a_{1} : ... : a_{m} : a_{m+1}|t_{1}, t_{1}+t_{2}, ..., t_{1}+...+t_{m})

Let Σ_{k,n−k} be the subset of permutations of n letters {1, ..., n} consisting of all shuffles of {1, ..., k} and {k + 1, ..., n}. Similar to (5) we see that

Li(x_{1}, ..., x_{k}|t_{1}, ..., t_{k}) · Li(x_{k+1}, ..., x_{n}|t_{k+1}, ..., t_{n}) =  (14)

∑_{σ∈Σ_{k,n−k}} Li(x_{σ(1)}, ..., x_{σ(n)}|t_{σ(1)}, ..., t_{σ(n)}) + lower depth terms

Theorem 2.2

I^{∗}(a_{1} : ... : a_{k} : 1|t_{1}, ..., t_{k}) · I^{∗}(a_{k+1} : ... : a_{n} : 1|t_{k+1}, ..., t_{n}) =  (15)

∑_{σ∈Σ_{k,n−k}} I^{∗}(a_{σ(1)}, ..., a_{σ(n)} : 1|t_{σ(1)}, ..., t_{σ(n)})

Proof. It is not hard to prove the following formula

I^{∗}[a_{1} : ... : a_{m} : 1|t_{1}, ..., t_{m}] = ∫_{0}^{1} \frac{s^{−t_{1}}}{a_{1} − s}ds ∘ ... ∘ \frac{s^{−t_{m}}}{a_{m} − s}ds  (16)

The theorem follows from this and the product formula for iterated integrals. Here is the simplest case: I_{1}(x)I_{1}(y) = I_{1,1}(x, y) + I_{1,1}(y, x). Indeed,

∫_{0}^{1} \frac{dt}{t − x} · ∫_{0}^{1} \frac{dt}{t − y} = ∫_{0}^{1} \frac{dt}{t − x} ∘ \frac{dt}{t − y} + ∫_{0}^{1} \frac{dt}{t − y} ∘ \frac{dt}{t − x}

For multiple ζ’s these are precisely the relations of Zagier, who conjectured that they provide all the relations between the multiple ζ’s

Distribution relations. From the power series expansion we immediately get

Proposition 2.3 If |x_{i}| < 1 and l is a positive integer then

Li(x_{1}, ..., x_{m}|t_{1}, ..., t_{m}) = ∑_{y_{i}^{l}=x_{i}} Li(y_{1}, ..., y_{m}|lt_{1}, ..., lt_{m})  (17)

3 The dihedral Lie coalgebra of a commutative group

1. Definitions. Let G be a commutative group. We will define a bigraded Lie coalgebra D_{•,•}(G) = ⊕_{w≥m≥1}D_{w,m}(G). Let us first define a graded abelian group D̂_{•,m}(G). The group D_{•,m}(G) is its quotient.

Denote by C_{m+1} the principal homogeneous space of the cyclic group ℤ/(m+ 1)ℤ. Let ℤ[C_{m+1}] be the abelian group of ℤ-valued functions on C_{m+1}, and ℤ[C_{m+1}]_{0} is its quotient by constants. Let P ol_{•}(ℤ[C_{m+1}]_{0}) be the algebra of polynomial functions on ℤ[C_{m+1}]_{0} graded by the degree.

Let D_{m} is the dihedral group of symmetries of the (m + 1)-gon. Set

G_{0}^{C_{m+1}} := {g = (g_{1}, ..., g_{m+1}) ∈ G^{m+1}|g_{1} · ... · g_{m+1} = 1}  (18)

We think about the elements of this group as of m + 1 elements of G with product 1 located on an oriented circle. Then the group D_{m} acts on (18). Let χ_{m} : D_{m} → {±1} be the character trivial on the cyclic subgroup and sending the involution to (−1)^{m+1}. Set

D̂_{•+m,m}(G) := (ℤ[G_{0}^{C_{m+1}}] ⊗_{ℤ} P ol_{•}(ℤ[C_{m+1}]_{0}) ⊗_{ℤ} χ_{m})_{D_{m}}

Elements of the group D̂_{•,m}(G) are presented by the generating functions

{g_{0}, g_{1}, ..., g_{m}|t_{0} : ... : t_{m}} such that g_{0} · ... · g_{m} = 1 and  (19)

{g_{0}, g_{1}, ..., g_{m}|t_{0} : ... : t_{m}} = {g_{0}, g_{1}, ..., g_{m}|t + t_{0} : ... : t + t_{m}}

They satisfy the dihedral symmetry relations:

{g_{0}, ..., g_{m−1}, g_{m}|t_{0} : t_{1} : ... : t_{m}} = {g_{1}, ..., g_{m}, g_{0}|t_{1} : ... : t_{m} : t_{0}}
{g_{0}, ..., g_{m}|t_{0} : ... : t_{m}} = (−1)^{m+1}{g_{m}, ..., g_{0}|t_{m} : ... : t_{0}}

We picture the elements of D̂_{•,m}(G) as m + 1 pairs (g_{0}, t_{0}), ..., (g_{m}, t_{m}) located cyclically on an oriented circle:

One has D̂_{•,m}(G) = ⊕_{w≥m}D̂_{w,m}(G). We define elements

{g_{1}, ..., g_{m}}_{n_{1},...,n_{m}} ∈ D̂_{w,m}(G),  w = n_{1} + ... + n_{m}, n_{i} ≥ 1  (20)

generating D̂_{w,m}(G) as the coefficients of the generating function:

{g_{0}, ..., g_{m}|t_{0} : .... : t_{m}} := ∑_{n_{i}>0}{g_{0}, ..., g_{m}}_{n_{1},...,n_{m}}(t_{1} − t_{0})^{n_{1}−1}...(t_{m} − t_{0})^{n_{m}−1}

Now one can say that D̂_{w,m}(G) is generated by the elements (20) satisfying the relations imposed by the dyhedral symmetry.

We call (19) extended nonhomogeneous dihedral words in G. One can also parametrize the generators using the extended homogeneous dihedral words:

{g_{0} : g_{1} : ... : g_{m}|t_{0}, ..., t_{m}}, where t_{0} + ... + t_{m} = 0,

{g · g_{0} : ... : g · g_{m}|t_{0}, ..., t_{m}} = {g_{0} : ... : g_{m}|t_{0}, ..., t_{m}} for any g ∈ G,

and the dihedral symmetry holds. Namely, the duality between the homogeneous and nonhomogeneous extended dihedral words is given by

{g_{0} : g_{1} : ... : g_{m}|t_{0}, ..., t_{m}} ⟼ {g_{0}^{−1}g_{1}, g_{1}^{−1}g_{2}, ..., g_{m}^{−1}g_{0}|t_{0} : t_{0}+t_{1} : ... : t_{0}+...+t_{m}},

{g_{0}, g_{1}, ..., g_{m}|t_{0} : ... : t_{m}} ⟼ {g_{0} : g_{0}g_{1} : ... : g_{0}...g_{m}|t_{1} − t_{0}, t_{2} − t_{1}, ..., t_{0} − t_{m}} Definition 3.1 D_{•,m}(G) is the quotient of D̂_{•,m}(G) by the following relations:

a) The double shuffle relations (k + l = m, k ≥ 1, l ≥ 1):

∑_{σ∈Σ_{k,l}}{g_{0} : g_{σ(1)} : ... : g_{σ(m)}|t_{0}, t_{σ(1)}, ..., t_{σ(m)}} = 0,

∑_{σ∈Σ_{k,l}}{x_{0}, x_{σ(1)}, ..., x_{σ(m)}|t_{0} : t_{σ(1)} : ... : t_{σ(m)}} = 0

b) The distribution relations (l ∈ ℤ and |l| divides |G| if the group is finite.)

{x_{0}^{l}, x_{1}^{l}, ..., x_{m}^{l}|t_{0} : t_{1} : ... : t_{m}} − ∑_{y_{i}^{l}=x_{i}^{l}}{y_{0}, y_{1}, ..., y_{m}|l · t_{0} : ... : l · t_{m}} = 0

except the relation {1}_{1} = ∑_{y^{l}=1}{y}_{1}, which is not supposed to hold.

Example. The distribution relations for l = −1 are

{x_{0}^{−1}, x_{1}^{−1}, ..., x_{m}^{−1}|t_{0} : t_{1} : ... : t_{m}} = {x_{0}, x_{1}, ..., x_{m}| − t_{0} : ... : −t_{m}} (21)

Remark. The dihedral symmetry and (21) follow from the double shuffle relations, just copy the proof of theorem 4.1 below.

Let us define a cobracket δ : D̂_{•,•}(G) ⟶ D̂_{•,•}(G) ∧ D̂_{•,•}(G) by setting

δ{g_{0}, ..., g_{m}|t_{0} : ... : t_{m}} :=

∑_{i=1}^{m−1} ∑_{j=0}^{m}{g_{j+i+1}, ..., g_{j+m}, y_{ij}|t_{j+i+1} : ... : t_{j+m+1}}∧{x_{ij}, g_{j+1}, ..., g_{j+i}|t_{j} : ... : t_{j+i}}

where indices are modulo m + 1 and x_{ij}g_{j+1}...g_{j+i} = 1, y_{ij}g_{j+i+1}...g_{j+m} = 1. Each term of the formula corresponds to the following procedure: we choose an arc on the circle between the two neighboring distinguished points, and in addition choose a distinguished point different from the ends of the arc. Then we cut the circle in the choosen arc and in the choosen point, make two naturally oriented circles out of it, and then make the extended dihedral word on each of the circles out of the initial word in a natural way.

There is a similar formula for the homogeneous dihedral words, just exchange g’s and t’s on the circle. For example

δ{g_{0} : g_{1} : g_{2}|t_{0}, t_{1}, t_{2}} = −{g_{2} : g_{0}|t_{0}, −t_{0}} ∧ {g_{1} : g_{2}|t_{1}, −t_{1}}−

−{g_{0} : g_{1}|t_{1}, −t_{1}} ∧ {g_{2} : g_{0}|t_{2}, −t_{2}} − {g_{1} : g_{2}|t_{2}, −t_{2}} ∧ {g_{0} : g_{1}|t_{0}, −t_{0}}

Theorem 3.2 δ provides the structure of bigraded Lie coalgebra on both D̂_{•,•}(G) and D_{•,•}(G).

2. The cyclotomic Lie algebra and mixed Tate motives over S_{N}. Recall that a mixed Hodge structure is called a Hodge-Tate structure if all the Hodge numbers h^{p,q} with p ≠ q are zero. The category of mixed ℚ-Hodge-Tate structures is canonically equivalent to the category of finite dimensional comodules over a certain graded pro-Lie coalgebra L_{•}^{HT} over ℚ (see [BGSV], [G4]). One can attach to the iterated integral related to Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) by theorem 2.1 above an element Li_{n_{1},...,n_{m}}^{H}(x_{1}, ..., x_{m}) ∈ L_{w}^{HT}, the motivic multiple polylogarithm. See s.9, 11 of [G2] (or [G1]) where L_{•}^{HT} and a wframed mixed Hodge-Tate structure Li_{n_{1},...,n_{m}}^{H}(x_{1}, ..., x_{m}) were defined. The framed mixed Hodge structures corresponding to the divergent numbers (4) are defined as the limiting Hodge structures, (compare with [G1] and s.7). For instance Li_{1}^{H}(1) = 0.

Definition 3.3 C_{w}(N) is the ℚ-subspace of L_{w}^{HT} generated by the motivic multiple polylogarithms at N-th roots of unity of weight w.

Theorem 3.4 C_{•}(N) := ⊕_{w≥1}C_{w}(N) is a Lie subcoalgebra in L_{•}^{HT}.

We define the cyclotomic Lie algebra C_{•}(N) as the (graded) dual of C_{•}(N).

The hypothetical abelian category of mixed Tate motives over the scheme S_{N} is supposed to be canonically equivalent to the category of finite dimensional modules over a graded Lie algebra L(S_{N})_{•}, called the motivic Lie algebra of that scheme, (see [G4] for details). L(S_{N})_{•} is isomorphic to a free graded Lie algebra generated by the ℚ-spaces K_{2n−1}(S_{N}) ⊗ ℚ in degree n, n ≥ 1.

One can prove that the Lie algebra C_{•}(N) has the same generators, so it is a quotient of L(S_{N})_{•}. However in general it is smaller then L(S_{N})_{•} since the main result of [G3] implies that C_{•}(N) is not free when N is sufficiently big. For instance, if N = p is a prime then

H_{(2)}^{2}(C_{•}(p), ℚ) = H^{1}(X_{1}(p), ℚ)_{+} ⊕ ℤ[ζ_{p}]^{∗} ⊗ ℚ

where X_{1}(p) is the modular curve and + means the coinvariants of the complex conjugation. So it is non zero if p > 3.

The relation between the dihedral and cyclotomic Lie algberas. Let C_{•,•}(N) be the associate graded with respect to the depth filtration on the Lie coalgebra C_{•}(N).

Theorem 3.5 Assume that x_{i}^{N} = 1. Then the map

{x_{1}, ..., x_{m}}_{n_{1},...,n_{m}} ⟼ Li_{n_{1},...,n_{m}}^{H}(x_{1}, ..., x_{m})

provides a surjective morphism of bigraded Lie coalgebras D_{•,•}(μ_{N}) → C_{•,•}(N)

The subspace D_{1,1}(1) is generated by {1}_{1}. Notice that Li_{1}^{H}(1) = 0.

Conjecture 3.6 D_{•,•}(1)/D_{1,1}(1) = C_{•,•}(1).

But already C_{1,1}(N) is smaller then D_{1,1}(N) if N is not prime ( take N = 25). Let (l, N) = 1. One has canonical homomorphism

φ_{N}^{l} : Gal(\overline{ℚ}/ℚ) ⟶ Outπ_{1}^{(l)}(ℙ^{1}\\{0, \{ζ_{N}^{α}\}, ∞\})  (22)

The motivic philosophy suggests that the Lie algebra of the Zariski closure of the image of φ_{N}^{l} is isomorphic to the Lie algebra C_{•}(N)^{∨} ⊗_{ℚ} ℚ_{l}.

4 The modular complex

Let L_{m} be a lattice of rank m. The rank m modular complex is a complex of left GL_{m}(ℤ)-modules in degrees [1, m], denoted M_{(m)}^{•}(L_{m}) or simply M_{(m)}^{•}:

M_{(m)}^{1} ⟶^{∂} M_{(m)}^{2} ⟶^{∂} ... ⟶^{∂} M_{(m)}^{m}

1. The groups M_{(m)}^{l}. Let X be a set and ℤ[[X]] be the abelian group of infinite ℤ-linear combinations ∑ n_{x}{x} of the generators {x}, where x ∈ X. The set P_{m} of basises (v_{1}, ..., v_{m}) of L_{m} is a principal homogeneous space of GL_{m}(ℤ). Then M_{(m)}^{l} = ℤ[[P_{m}]]/R_{(m)}^{l}. The ”relations” R_{(m)}^{l} are defined below.

By definition the generators of the group M_{(m)}^{1} are

[v_{1}, ..., v_{m}], where (v_{1}, ..., v_{m}) is a basis of L_{m}

We will use other two notation for them:

< v_{0}, ..., v_{m} >:= [v_{1}, ..., v_{m}] where v_{0} + ... + v_{m} = 0, and
[v_{1} : ... : v_{m}] := [v_{2} − v_{1}, v_{3} − v_{2}, ..., v_{m} − v_{m−1}, −v_{m}]

Notice that if v_{1}, ..., v_{m} is a basis of L_{m} and v_{0} + ... + v_{m} = 0, then omitting any vector of v_{0}, ..., v_{m} we also get a basis.

The relations are given by the double shuffle relations: for any 1 ≤ k ≤ m−1

s(v_{1}, ..., v_{k}|v_{k+1}, ..., v_{m}) := ∑_{σ∈Σ_{k,m−k}}[v_{σ(1)}, ..., v_{σ(m)}] = 0 (23)

s(v_{1} : ... : v_{k}|v_{k+1} : ... : v_{m}) := ∑_{σ∈Σ_{k,m−k}}[v_{σ(1)} : ... : v_{σ(m)}] = 0 (24)

Theorem 4.1 The double shuffle relations imply the following dihedral symmetry relations: [v_{1}, ..., v_{m}] = [−v_{1}, ..., −v_{m}], and

< v_{0}, ..., v_{m} >=< v_{1}, ..., v_{m}, v_{0} >, < v_{0}, ..., v_{m} >= (−1)^{m+1} < v_{m}, ..., v_{0} >

Proof. Starting from the shuffle relation s(v_{1}|v_{2} : ... : v_{m}):

0 = [v_{1} : v_{1} + v_{2} : ... : v_{1} + ... + v_{m}] + ... + [v_{2} : v_{2} + v_{3} : ... : v_{1} + ... + v_{m}]

we rewrite it as [v_{2}, ..., v_{m}, v_{0}]+s(v_{1}|v_{3}, ..., v_{m}, v_{0})−[v_{3}, ..., v_{m}, v_{0}, v_{1}] = 0, getting the cyclic symmetry for the generators [v_{1}, ..., v_{m}]. It is easy see that

∑_{k=0}^{m}(−1)^{k}s(v_{k} : v_{k−1} : ... : v_{1}|v_{k+1} : ... : v_{m}) = 0

which means that [v_{1} : v_{2} : ... : v_{m−1} : v_{m}] = (−1)^{m+1}[v_{m} : v_{m−1} : ... : v_{2} : v_{1}]. A similar result is proved for the [v_{1}, ..., v_{m}] generators. Using this we get

[−v_{1}, −v_{2}, ..., −v_{m}] = [v_{1} + ... + v_{m} : ... : v_{m−1} + v_{m} : v_{m}] =

(−1)^{m+1}[v_{m} : v_{m−1}+v_{m} : ... : v_{1}+...+v_{m}] = (−1)^{m+1}[v_{m−1}, ..., v_{1}, v_{0}] = [v_{0}, ..., v_{m−1}]

So we proved the dihedral symmetry for the generators [v_{1}, ..., v_{m}], and thus for the generators [v_{1} : ... : v_{m}].

The generators of the group M_{(m)}^{l} are the symbols

[v_{1}, ..., v_{m_{1}}] ∧ ... ∧ [v_{m_{l−1}+1}, ..., v_{m_{l}}]  (25)

where m = k_{1} + ... + k_{l}, m_{i} := k_{1} + ... + k_{i}, (v_{1}, ..., v_{m}) is a basis of L_{m}, each of the blocks [A_{i}] := [v_{m_{i}+1}, ..., v_{m_{i+1}}] satisfy the double shuffle relations, and the blocks [A_{i}] anticommute:

... ∧ [A_{i}] ∧ [A_{i+1}] ∧ ... = −... ∧ [A_{i+1}] ∧ [A_{i}] ∧ ...

2. The differential ∂. We define a differential ∂ : M_{(m)}^{1} ⟶ M_{(m)}^{2} by

∂ : < v_{1}, ..., v_{m+1} > ⟼ −Cycle_{m+1}( ∑_{k=1}^{m−1}[v_{1}, ..., v_{k}] ∧ [v_{k+1}, ..., v_{m}])

where Cycle_{m+1}(f(v_{1}, ..., v_{m})) := ∑_{i=1}^{m+1} f(v_{i}, ..., v_{i+m}), the indices are modulo m + 1. Then we extend ∂ to the complex M_{(m)}^{•} using the Leibniz rule:

∂([A_{1}] ∧ [A_{2}] ∧ ... ∧ [A_{n}]) := ∑_{i=1}^{n}(−1)^{i}[A_{1}] ∧ ... ∧ ∂[A_{i}] ∧ ...[A_{n}]

Theorem 4.2 The differential ∂ is a well defined homomorphism of abelian groups. One has ∂^{2} = 0.

Remark. The modular complex is not the standard cochain complex of any graded Lie coalgebra.

3. Modular complexes for GL_{2} and GL_{3}. Here is explicit description of the modular complexes M_{(m)}^{•} for m = 1, 2, 3.

1. m = 1. Then [v] = [−v], and M_{(1)}^{1} = ℤ.

2. m = 2. Then the modular complex is M_{(2)}^{1} ⟶^{∂} M_{(2)}^{2}. The group M_{(2)}^{1} is generated by < v_{0}, v_{1}, v_{2} > where v_{0} + v_{1} + v_{2} = 0 and (v_{1}, v_{2}) is a basis in L_{2}. The differential is:

∂ :< v_{0}, v_{1}, v_{2} > ⟼ −[v_{1}] ∧ [v_{2}] − [v_{2}] ∧ [v_{0}] − [v_{0}] ∧ [v_{1}]  (26)

3. m = 3. The complex looks as follows: M_{(3)}^{1} ⟶^{∂} M_{(3)}^{2} ⟶^{∂} M_{(3)}^{3}. The differentials are

∂ : < v_{0}, v_{1}, v_{2}, v_{3} > ⟼
−[v_{1}, v_{2}] ∧ [v_{3}] − [v_{2}, v_{3}] ∧ [v_{0}] − [v_{3}, v_{0}] ∧ [v_{1}] − [v_{0}, v_{1}] ∧ [v_{2}]
−[v_{0}] ∧ [v_{1}, v_{2}] − [v_{1}] ∧ [v_{2}, v_{3}] − [v_{2}] ∧ [v_{3}, v_{0}] − [v_{3}] ∧ [v_{0}, v_{1}]

∂ : [v_{1}, v_{2}]∧[v_{3}] ⟼ −([v_{1}]∧[v_{2}]+[v_{2}]∧[−v_{1} −v_{2}]+[−v_{1} −v_{2}]∧[v_{1}])∧[v_{3}]

4. Modular complexes and modular cohomology. We define the modular complex MC^{∗}(Γ, V) of a subgroup Γ of GL_{m}(ℤ) with coefficients in a right GL_{m}-module V as follows:

MC^{∗}(Γ, V) := V ⊗_{Γ} M_{(m)}^{∗}

Its cohomology are called the modular cohomology MH^{∗}(Γ, V) of Γ with coefficients in V. If E is the one element group, then MC^{∗}(E, ℤ) = M_{(m)}^{∗}.

The group GL_{m}(ℤ) acts from the right on ℤ[Γ\GL_{m}(ℤ)], and thus on ℤ[Γ\GL_{m}(ℤ)] ⊗_{ℤ} V. It is easy to see (Shapiro’s lemma) that one has

MC^{∗}(Γ, V) = (ℤ[Γ\GL_{m}(ℤ)] ⊗_{ℤ} V ) ⊗_{GL_{m}(ℤ)} M_{(m)}^{∗}  (27)

5. A map from the modular complex for Γ_{1}(N; m) with coefficients in ℤ[t_{1}, ..., t_{m}] to the cochain complex of the dihedral Lie coalgebra of μ_{N}. One has

Γ_{1}(N; m)\GL_{m}(ℤ) = {(α_{1}, ..., α_{m})| α_{i} ∈ ℤ/Nℤ, g.c.d.(α_{1}, ..., α_{m}, N) = 1} (28)

Indeed, the group GL_{m}(ℤ) acts from the right on (ℤ/Nℤ)^{m} and Γ_{1}(N; m) is the stabilizer of the element (0, ..., 0, 1). The GL_{m}(ℤ)-orbit of this element is the right hand side of (28).

Consider the right GL_{m}-module structure on ℤ[t_{1}, ..., t_{m}] given by t_{i} · g := ∑_{j=1}^{m}(g^{−1})_{ij}t_{j}. We will construct a canonical morphism of complexes

μ(N)_{•,m}^{∗} : MC^{∗}(Γ_{1}(N; m), ℤ[t_{1}, ..., t_{m}]) ⟶ Λ^{∗}(D_{•,•}(μ_{N}))depth = m

where the left hand side is graded by (degree of a polynomial in t_{i}) +m, and the right hand side by the weight.

Let S^{k}[t_{1}, ..., t_{m}] be the abelian group of degree k polynomials in t_{1}, ..., t_{m} with integer coefficients. We will use (27). Let us define first maps

μ_{w,m}^{1} : MC^{1}(Γ_{1}(N; m), S^{w−m}[t_{1}, ..., t_{m}]) ⟶ D_{m,w}(μ_{N})

Choose a basis (v_{1}, ..., v_{m}) in V_{m}. Let α_{0} + α_{1} + ... + α_{m} = 0. Set

μ(N)_{•,m}^{1} : ∑_{n_{i}>0}(α_{1}, ..., α_{m}) ⊗ t_{1}^{n_{1}−1}...t_{m}^{n_{m}−1} ⊗ [v_{1}, ..., v_{m}] ⟼

{ζ_{N}^{α_{0}}, ζ_{N}^{α_{1}}, ..., ζ_{N}^{α_{m}}|0 : t_{1} : ... : t_{m}} := ∑_{n_{i}>0}{ζ_{N}^{α_{0}}, ζ_{N}^{α_{1}}, ..., ζ_{N}^{α_{m}}}_{n_{1},...,n_{m}}t_{1}^{n_{1}−1}...t_{m}^{n_{m}−1}

Lemma 4.3 The map μ_{w,m}^{1} ⊗ ℚ is a surjective homomorphism .

Proof. It is well defined thanks to the definitions of the modular complex and the dihedral Lie coalgebra. It is surjective because of the distribution relations.

Define a map

μ(N)_{•,m}^{l} : MC^{l}(Γ_{1}(N; m), ℤ[t_{1}, ..., t_{m}]) ⟶ D_{m,•}(μ_{N})

as follows. Choose a primitive N-th root of unity ζ_{N}. Then

μ(N)_{w,m}^{l} : ∑_{n_{i}>0}(α_{1}, ..., α_{m})⊗t_{1}^{n_{1}−1}...t_{m}^{n_{m}−1}⊗[v_{1}, ..., v_{n_{1}}]∧...∧[v_{n_{l−1}+1}, ..., v_{n_{l}}] ⟼

{ζ_{N}^{β_{1}}, ζ_{N}^{α_{1}}, ..., ζ_{N}^{α_{n_{1}}} |0 : t_{1} : ... : t_{n_{1}}}∧...∧{ζ_{N}^{β_{l}}, ζ_{N}^{α_{n_{l−1}+1}}, ..., ζ_{N}^{α_{n_{l}}} |0 : t_{n_{l−1}+1} : ... : t_{n_{l}}}

where by definition β_{i} + α_{n_{i−1}+1} + ... + α_{n_{i}} = 0 for i = 1, ..., l.

Proof of theorem 1.2. By the very definitions the map μ(N)_{•,•}^{∗} is a well defined morphism of complexes. It is surjective thanks to lemma 4.3. If N = 1 we have the distribution relations only for l = −1, and they follow from the double shuffle relations by theorem 4.1. So the map μ(1)_{•,•}^{∗} is an isomorphism.

If N is a prime and w = m > 1 there is one additional distribution relation, {1, ..., 1}_{1,...,,1} = ∑_{x_{i}^{p}=1}{x_{1}, ..., x_{m}}_{1,...,,1}. But the shuffle relations give ∑_{σ∈S_{m}}{x_{σ(1)}, ..., x_{σ(m)}}_{1,...,1} = 0, which imply the distribution relation.

5 The Voronoi complex

1. Voronoi’s cell decomposition of SL_{n}(ℝ)/SO_{n}. Let Q(V_{m}) be the space of quadratic forms in an m-dimensional vector space V_{m} over ℝ. Denote by P(V_{m}) (resp. \overline{P}(V_{m})) the cone of positive definite (resp. non negative definite) quadratic forms in V_{m}. Then

ℍ_{m} := SL_{m}(ℝ)/SO(m) = P(V_{m})/ℝ_{+}^{∗}

and \overline{P}(V_{m})/ℝ_{+}^{∗} is its compactification. For example ℍ_{2} is the hyperbolic plane.

Any vector f ∈ V_{m}^{∗} defines a degenerate non negatively definite quadratic form φ(f) := (f, x)^{2}. Choose a lattice L_{m} ⊂ V_{m}^{∗}. Let GL(L_{m}) ⊂ GL(V_{m}) be the subgroup preserving the lattice L_{m}. Take the convex hull C(L_{m}) of the vectors φ(l) in the cone \overline{P}(V_{m}) when l runs through all non zero primitive vectors of the lattice L_{m}. It is of codimension 1 in Q(V_{m}) and has a structure of an infinite polyhedra. Its faces are certain convex polyhedras with vertices φ(l_{1}), ..., φ(l_{n}), l_{i} ∈ L_{m}. Projecting it onto P(V_{m})/ℝ_{+}^{∗} we get a GL(L_{m})-invariant polyhedral decomposition of ℍ_{m} called Voronoi’s cell decomposition. Set

φ(l_{1}, ..., l_{n}) := {λ_{1} · φ(l_{1}) + ... + λ_{n} · φ(l_{n})}/ℝ^{∗}, λ_{i} ≥ 0, λ_{1} + ... + λ_{n} = 1

The cells of the projection of C(L_{m}) are polyhedras φ(l_{1}, ..., l_{n}) for certain vectors l_{1}, ..., l_{n} ∈ L_{m}. They satisfy the condition rk < l_{1}, ..., l_{n} >= m.

The non zero vectors of the lattice L_{m} minimizing the values of a form F on V_{m}^{∗} on L_{m}\0 are called the minimal vectors of F. A quadratic form F in V_{m}^{∗} is called perfect if the number of minimal vectors of F is at least \frac{m(m+1)}{2} = dimQ(V_{m}^{∗}).

Let s be a codimension 1 face of C(L_{m}). Let h(s) be the codimension 1 subspace in Q(V_{m}) parallel to the face s.

Voronoi’s lemma. F ∈ Q(V_{m}^{∗}) is orthogonal to the subspace h(s) if and only if F is a perfect quadratic form. In this case {±l_{1}, ..., ±l_{n}} is the set of minimal vectors for F.

Proof. One has (F, φ(l)) = F (l). Let (F, x) = c be the equation of the hyperplane h(s). Since C(L_{m}) is a convex hull it is located in just one of the subspaces (F, x) < c or (F, x) > c. Since (F, φ(l)) = F (l) could be arbitrary big, the domain {x|(F, x) < c} does not intersect C(L_{m}). Further, (F, φ(l)) = c for any vertex φ(l) of the face φ, so such l’s are minimal vectors for F. Since the face φ is of codimension 1, the number of its vertices is at least dimQ(V_{m}). So the form F is perfect. The lemma is proved.

Let v_{1}, ..., v_{m+1} vectors of L_{m} such that v_{1} + ... + v_{m+1} = 0 and v_{1}, ..., v_{m} is a basis of L. Set

v_{i,j} := v_{i} + v_{i+1} + ... + v_{j−1} + v_{j},  1 ≤ i, j ≤ m + 1, i ≠ j − 1 (29)

and indices are modulo m+1. The configuration of vectors v_{i,j} in (29) is linearly equivalent to the configuration of the roots of the root system A_{m}. See fig. 4 for the configuration of points in P^{2} corresponding to the root system A_{2}.

The convex hull of φ(v_{i,j}) is a Voronoi cell, called the cell of type A_{m}, and the correseponding perfect form is the quadratic form of the root system A_{m} with the the set of mimimal vectors given by the roots.

Voronoi’s theorem [V], [M]. For m = 2, 3 any cell of top dimension in the Voronoi decomposition of ℍ_{m} is GL_{m}(L_{m})-equivalent to a cell of type A_{m}.

2. The Voronoi complex. Let

(V_{•}^{(m)}, d) = (V_{•}(L_{m}), d) := V_{\frac{m(m+1)}{2} −1}^{(m)} ⟶^{d} V_{\frac{m(m+1)}{2} −2}^{(m)} ⟶^{d} ... ⟶^{d} V_{m−1}^{(m)}

be the complex of (infinite) chains with closed supports associated with the Voronoi decomposition of ℍ_{m}. We call it the Voronoi complex of the lattice L_{m}. An isomorphism between lattices lifts to an isomorphism between the corresponding Voronoi complexes, justifying name the Voronoi complex for GL_{m}.

6 Relating the modular and Voronoi complexes for GL_{2} and GL_{3}

The modular complex M_{(m)}^{•} is a cohomological complex placed in degrees [1, m+ 1]. Let us cook up out of him a homological complex sitting in degrees [2m, m] by setting M_{•}^{(m)} := M_{(m)}^{2m+1−•}.

1. An isomorphism between the modular and Voronoi complexes for GL_{2}. The Voronoi complex for GL_{2} looks as follows: V_{•}^{(2)} := V_{2}^{(2)} ⟶^{d} V_{1}^{(2)}. It is the chain complex of the classical modular triangulation of the hyperbolic plane, see the figure on page 2

Define a map of GL_{2}(ℤ)-modules ψ^{(2)} : M_{•}^{(2)} ⟶ V_{•}^{(2)} as follows. Let v_{1}, v_{2} be a basis of L_{2}, v_{1} + v_{2} + v_{3} = 0. Set

[v_{1}, v_{2}] ⟼ φ(v_{1}, v_{2}, v_{3}), [v_{1}] ∧ [v_{2}] ⟼ φ(v_{1}, v_{2})

Theorem 6.1 a) The map ψ^{(2)} is an isomorphism of complexes M_{•}^{(2)} ⟶ V_{•}^{(2)}.

b) Let Γ be a subgroup of GL_{2}(ℤ). Then for any GL_{2}-module V there are canonical isomorphisms HM_{(2)}^{i}(Γ, V) ⊗ ℚ = H^{i−1}(Γ, V) ⊗ ℚ.

Proof. a) When (v_{1}, v_{2}) run through all basises of the lattice L_{2}, the triangles φ(v_{1}, v_{2}, v_{3}) where v_{1} + v_{2} + v_{3} = 0 are cells of type A_{2}, and so by Voronoi’s theorem produce all the 2-cells of Voronoi’s complex for GL_{2}. Since ψ^{(2)} commutes with the differentials, we obviously get an isomorphism of complexes.

b) Voronoi’s complex V_{•}^{(2)} is a resolution of the trivial GL_{2}(ℤ)-module ℤ[2]. This resolution is free over a certain finite index subgroup Γ ⊂ GL_{2}(ℤ).

2. A quasiisomorphism between the modular and truncated Voronoi complexes for GL_{3}. The Voronoi complex for GL_{3} looks as follows:

(V_{•}^{(3)}, d) := V_{5}^{(3)} ⟶^{d} V_{4}^{(3)} ⟶^{d} V_{3}^{(3)} ⟶^{d} V_{2}^{(3)}

We will suppose that v_{1}, v_{2}, v_{3} is a basis in L_{3} and v_{1} + v_{2} + v_{3} + v_{4} = 0, v_{12} := v_{1} + v_{2}, v_{23} := v_{2} + v_{3}, v_{13} := v_{1} + v_{3}, ...

By Voronoi’s theorem the GL_{3}(ℤ)-orbits of the 5-symplex φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}, v_{23}) and its faces provide all cells of the Voronoi decomposition for GL_{3}.

Define a map ψ^{(3)} : M_{•}^{(3)} ⟶ V_{•}^{3}/dV_{5}^{(3)} as follows:

[v_{1}] ∧ [v_{2}] ∧ [v_{3}] ⟼ φ(v_{1}, v_{2}, v_{3})

[v_{1}, v_{2}] ∧ [v_{3}] ⟼ φ(v_{1}, v_{2}, −v_{1} − v_{2}, v_{3}),
[v_{1}, v_{2}, v_{3}] ⟼ φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}) − φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{23})  (30)

Theorem 6.2 a) The map ψ^{(3)} provides an injective morphism of complexes of GL_{3}(ℤ)-modules M_{•}^{(3)} ⟶ V_{•}^{(3)}/dV_{5}^{(3)}. It is a quasiisomorphisms.

b) Let Γ be a subgroup of GL_{3}(ℤ). Then for any GL_{3}-module V there are canonical isomorphisms HM_{(3)}^{i}(Γ, V) ⊗ ℚ = H^{i}(Γ, V) ⊗ ℚ, i ≥ 1.

Proof. b) => a). It is similar to the proof of theorem 6.1 a).

a) Let us show that ψ^{(3)} is a well defined morphism of complexes. The map ψ^{(3)} sends the first shuffle relation to zero already in the group V_{4}^{(3)}:

ψ^{(3)} : s(v_{1}|v_{2}, v_{3}) =

< v_{1}, v_{2}, v_{3}, v_{4} > + < v_{2}, v_{1}, v_{3}, v_{4} > + < v_{2}, v_{3}, v_{1}, v_{4} > ⟼
φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}) − φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{23}) + φ(v_{2}, v_{1}, v_{3}, v_{4}, v_{12})−

−φ(v_{2}, v_{1}, v_{3}, v_{4}, v_{13}) + φ(v_{2}, v_{3}, v_{1}, v_{4}, v_{23}) − φ(v_{2}, v_{3}, v_{1}, v_{4}, v_{13}) = 0 The second shuffle relation looks as follows:

s(u_{1}|u_{2} : u_{3}) := [u_{1} : u_{2} : u_{3}] + [u_{2} : u_{1} : u_{3}] + [u_{2} : u_{3} : u_{1}] = [u_{2} − u_{1}, u_{3} − u_{2}, −u_{3}] + [u_{1} − u_{2}, u_{3} − u_{1}, −u_{3}] + [u_{3} − u_{2}, u_{1} − u_{3}, −u_{1}]

Changing the variables v_{1} := u_{2} − u_{1}, v_{2} := u_{3} − u_{2}, v_{3} := −u_{3} we get

< v_{1}, v_{2}, v_{3}, v_{4} > + < −v_{1}, v_{12}, v_{3}, v_{41} > + < v_{2}, −v_{12}, −v_{4}, v_{41} > The maps ψ^{(3)} sends it to the boundary of Voronoi’s 5-simplex φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}, v_{23}):

φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}) − φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{23}) + φ(v_{1}, v_{12}, v_{3}, v_{23}, v_{2})−

−φ(v_{1}, v_{12}, v_{3}, v_{23}, v_{4}) + φ(v_{2}, v_{12}, v_{4}, v_{23}, v_{1}) − φ(v_{2}, v_{12}, v_{4}, v_{23}, v_{3}) =
dφ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}, v_{23})

The other components of the map ψ^{(3)} are obviously group homomorphisms. The map ψ^{(3)} respects the differentials, and it is clearly injective. It is an isomorphism in all the degrees except 3 and 4. One has

Coker(ψ^{(3)}) = \frac{V_{4}^{(3)}}{ψ^{(3)}(M_{4}^{(3)})} ⟶^{∂} \frac{V_{3}^{(3)}}{ψ^{(3)}(M_{3}^{(3)})}  (31)

Denote by {v_{1}, v_{2}, v_{3}, v_{4}} the set of all unordered 4-tuple of vectors (v_{1}, v_{2}, v_{3}, v_{4}) in L_{3} such that v_{1} + v_{2} + v_{3} + v_{4} = 0 and (v_{1}, v_{2}, v_{3}) is a basis of L_{3}.

Proposition 6.3 One has canonical isomorphisms

\frac{V_{4}^{(3)}}{ψ^{(3)}(M_{4}^{(3)})} = ℤ[[\{v_{1}, v_{2}, v_{3}, v_{4}\}]] = \frac{V_{3}^{(3)}}{ψ^{(3)}(M_{3}^{(3)})}  (32)

It transforms the differential in (31) to the identity map on ℤ[[{v_{1}, v_{2}, v_{3}, v_{4}}]]. Therefore the complex Coker(ψ^{(3)}) is acyclic.

Proof. The following observations about the Voronoi cell decomposition are easy to see from figure 4.

1) A 3-dimensional cell of Voronoi’s decomposition is GL_{3}(ℤ)-equivalent to one of the following two: a 3-cell φ(v_{1}, v_{2}, −v_{12}, v_{3}), called special 3-cell, or a 3-cell φ(v_{1}, v_{2}, v_{3}, v_{4}), called generic 3-cell. So generic 3-cells are parametrized by the set {v_{1}, v_{2}, v_{3}, v_{4}}.

2) A 5-simplex containing generic 3-cell φ(v_{1}, v_{2}, v_{3}, v_{4}) is determined by the dihedral order of the vectors v_{1}, v_{2}, v_{3}, v_{4}. So a given generic 3-cell is contained in three 5-simplices. For the 3-cell φ(v_{1}, v_{2}, v_{3}, v_{4}) these are

φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}, v_{23}), φ(v_{2}, v_{1}, v_{3}, v_{4}, v_{21}, v_{13}), φ(v_{2}, v_{3}, v_{1}, v_{4}, v_{23}, v_{31})

3) The elements (30) are in bijective correspondence with the pairs {generic 3-cell, a 5-cell containing it}. The right hand side of (30) is the sum of the 4-cells containing a given generic 3-cell and contained in a given 5-cell.

The first shuffle relation means that the sum of the elements of type (30) corresponding to 5-cells containing a given generic 3-cell is zero.

4) A given 5-simplex has only three generic 3-cells. For the 5-simplex φ(v_{1}, v_{2}, v_{3}, v_{4}, v_{12}, v_{23}) they are

φ(v_{1}, v_{2}, v_{3}, v_{4}), φ(v_{12}, v_{4}, v_{23}, −v_{2}), φ(v_{12}, v_{3}, −v_{23}, −v_{1})

The sum of the elements (30) corresponding to generic 3-cells of a given 5simplex is the second shuffle relation. It is the boundary of that 5-simplex.

The observation 1) implies the second isomorphism in (32). The observations 2) and 3) lead to the first isomorphism in (32). It is easy to check that the differential is the identity map. The proposition is proved.

7 Applications to multiple ζ-values

1. Regularization and the map D_{w,m}(N) ⟶ \overline{Z}_{w,m}(N). The iterated integral I_{n_{1},...,n_{m}}(a_{1} : ... : a_{m} : 1) is divergent if and only if n_{m} = 1, a_{m} = 1. The power series Li_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) are divergent if and only if n_{m} = 1, x_{m} = 1.

Theorem 7.1 a) Let |x_{i}| ≤ 1. Then both the power series

Li_{n_{1},...,n_{l},1,...,1}(x_{1}, ..., x_{l}, 1 − ε, ..., 1 − ε)  (33)

and the power series Li_{n_{1},...,n_{l},1,...,1}(x_{1}, ..., x_{l}, 1, ..., 1, 1 − ε) corresponding to

I_{n_{1},...,n_{l},1,...,1}(a_{1} : ... : a_{l} : 1 : ... : 1 : 1 − ε)  (34)

admit asymptotic expansions which are polynonials in log ε. Their coefficients are explicitely computable sums of multiple polylogarithms.

b) The constant terms of these expansions differ by lower depth multiple polylogarithms, and the other terms are of lower depth. In particular if x_{i}^{N} = 1 they define the same element RegLi_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) ∈ \overline{Z}_{•,•}(N).

c) The map {x_{1}, ..., x_{m}}_{n_{1},...,n_{m}} ⟼ RegLi_{n_{1},...,n_{m}}(x_{1}, ..., x_{m}) provides a surjective linear map D_{w,m}(N) ⟶ \overline{Z}_{w,m}(N).

Proof. Direct integration gives us

I_{1,...,1}(1 : ... : 1 : 1 − ε) = ∫_{0<t_{1}<...<t_{m}<1−ε} \frac{dt_{1}}{1 − t_{1}} ∧ ... ∧ \frac{dt_{m}}{1 − t_{m}} = \frac{(− log ε)^{m}}{m!}

Computing (∑_{k>0} \frac{(1−ε)^{k}}{k})^{m} and using then induction in m − l we get

\frac{(− log ε)^{m}}{m!} = Li_{1,...,1}(1−ε, ..., 1−ε) + ∑_{0<i<m} (lower depth multiple ζ’s)(log ε)^{i} Assume n_{l} ≠ 1 or x_{l} ≠ 1. Applying the power series product formula to

Li_{n_{1},...,n_{l}}(x_{1}, ..., x_{l}) · Li_{1,...,1}(1 − ε, ..., 1 − ε)

and then to

Li_{n_{1},...,n_{l}}(x_{1}, ..., x_{l}) · Li_{1,...,1}(1, ..., 1, 1 − ε),

and using the induction on m − l, we get parts a) and b) of the theorem.

c) The power series (resp. the iterated integral) product formulas clearly hold for the asymptotic expansions (33) (resp. (34)). So the shuffle relations are valid. The distribution relations for l > 0 hold for (33). The distribution relations for l = −1 follows from the shuffle relations and theorem 4.1.

Corollary 7.2 dim\overline{Z}(1)_{w,m} = 0 if w + m is odd.

Proof. Indeed, (21) implies that dimD(1)_{w,m} = 0 if w + m is odd.

2. Proof of theorem 1.3. It follows from theorems 1.2, 6.1 and 7.1.

3. Proof of theorem 1.4. If w > 2 then H^{0}(GL_{2}(ℤ), S^{w−2}V_{2}) = 0, and

∑_{w} dimH^{1}(GL_{2}(ℤ), S^{w−2}V_{2}) · t^{w} = \frac{1}{(1 − t^{4})(1 − t^{6})} − 1  (35)

So by (10) the generating function for the Euler characteristic of the complex D_{•,2} ⟶ Λ^{2}D_{•,1} is given by (35). Using formula (12) we get the result.

Proof of theorem 1.5. Since S^{w−3}V_{3} for w > 3 is not a self dual GL_{3}- module, the kernel of the restriction of H^{i}(GL_{3}(ℤ), S^{w−3}V_{3}) to the boundary of the Borel-Serre bordification vanishes by a theorem of Borel [BW]. Computing the boundary contribution to the cohomology we get

H^{i}(GL_{3}(ℤ), S^{w−3}V_{3}) = \{ \begin{matrix}0 & i = 1, 2 \\ H_{cusp}^{1}(GL_{2}(ℤ), S^{w−2}V_{3}) & i = 3 \end{matrix}  (36)

Combining formulas (11) and (36) we get the cohomology of the complex D_{•,3} ⟶ D_{•,2}⊗D_{•,1} ⟶ Λ^{3}D_{•,1}. Using theorems 1.4 and 7.1, formula (12), and the Euler characteristic argument, we obtain the theorem.

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