Galois groups, geometry of modular varieties and graphs

A.B. Goncharov

June 23, 1999

Contents

1 Galois groups 0

2 General picture: the modular complexes and Galois symmetries of π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{1}) 4

3 Complex of colored forests and its geometric realization in the symmetric space 5

1 Galois groups

1. Galois symmetries. Let X be a regular curve, \overline{X} the corresponding projective curve, and v a tangent vector at a point x ∈ \overline{X}. Then according to Deligne ([D]) one can define the profinite fundamental group π̂_{1}(X, v) based at the vector v. If X, x and v are defined over a number field F then the group G_{F} := Gal(\overline{ℚ}/F) acts by automorphisms of π_{1}^{(l)}(X, v).

If X = ℙ^{1}\{0, μ_{N}, ∞} there is a tangent vector v_{1} corresponding to the inverse t^{−1} of the canonical coordinate t on ℙ^{1}\{0, μ_{N}, ∞}.

Let us denote by π^{(l)} the pro-l-completion of the group π̂. In this talk we are going to investigate the map

Φ_{N}^{(l)} : G_{ℚ} ⟶ Autπ_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{1})  (1)

When N = 1 this map has been studied by Grothendieck [Gr], Deligne [D], Ihara [Ih] , Drinfeld [D], ... (see [Ih]), but for N > 1 it was not investigated.

Let π_{1}^{(l)} := π_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{1}). Denote by π_{1}^{(l)}(m) the lower central series for the group π_{1}^{(l)}. Then

π_{1}^{(l)}/π_{1}^{(l)}(m) is an l-adic Lie group

Taking its Lie algebra and making the projective limit over m we get a pronilpotent Lie algebra over ℚ_{l}:

𝕃_{N}^{(l)} := lim_{⟵} Lie(\frac{π_{1}^{(l)}}{π_{1}^{(l)}(m)})

Let ℚ(ζ_{n}) be the field generated by n-th roots of unity. Set ℚ(ζ_{l^{∞}}) := ∪ℚ(ζ_{l^{n}}). We restrict map (1) to the Galois group G_{ℚ(ζ_{l^{∞}})}. Passing to Lie algebras we get a homomorphism

ϕ_{N}^{(l)} : G_{ℚ(ζ_{l^{∞}})} ⟶ Aut(𝕃_{N}^{(l)})

We would like to take the Lie algebra of the image of this map. One has to proceed as follows. Let 𝕃_{N}^{(l)}(m) be the lower central series for the Lie algebra 𝕃_{N}^{(l)}. There are homomorphisms to finite dimensional l-adic Lie groups

ϕ_{N;m}^{(l)} : G_{ℚ(ζ_{l^{∞}})} ⟶ Aut(L_{N}^{(l)}/𝕃_{N}^{(l)}(m))

The main hero of this story is the pronilpotent Lie algebra

G_{N}^{(l)} := lim_{←} Lie(Imϕ_{N;m}^{(l)})

Therefore

G_{N}^{(l)} ↪ Der𝕃_{N}^{(l)}

2. The weight and depth filtration on 𝕃_{N}^{(l)}. There are two increasing filtrations by ideals on the Lie algebra 𝕃_{N}^{(l)}, indexed by negative integers.

The weight filtration F_{•}^{W}. It coincides with the lower central series for 𝕃_{N}^{(l)}:

𝕃_{N}^{(l)} = F_{−1}^{W}𝕃_{N}^{(l)}; F_{−n−1}^{W}𝕃_{N}^{(l)} := [F_{−n}^{W}𝕃_{N}^{(l)}, 𝕃_{N}^{(l)}]

The depth filtration F_{•}^{D}. The natural inclusion

ℙ^{1}\{0, μ_{N}, ∞} ↪ ℙ^{1}\{0, ∞}

provides a morphism of the corresponding fundamental Lie algebras

p : 𝕃_{N}^{(l)} ⟶ 𝕃^{(l)}(ℙ^{1}\{0, ∞}) = ℚ_{l}(1)

Let I_{N} be the kernel of this projection Its powers give the depth filtration:

F_{0}^{D}𝕃_{N}^{(l)} = 𝕃_{N}^{(l)}, F_{−1}^{D}𝕃_{N}^{(l)} = I_{N}, F_{−n−1}^{D}𝕃_{N}^{(l)} = [I_{N}, F_{−n}^{D}𝕃_{N}^{(l)}]

The weight filtration can be defined on the Lie algebra corresponding to the pronilpotent completion of the fundamental group of an arbitrary algebraic variety. The weights on 𝕃_{N}^{(l)} are obtained by dividing by 2 the usual weights. The weight filtration admits a splitting, i.e. it is defined by a grading.

The depth filtration is more specific.

These filtrations induce two filtrations on the Lie algebra Der𝕃_{N}^{(l)} and hence on the Lie algebra G_{N}^{(l)}. Taking the associated graded we get a Lie algebra Gr_{•,•}^{D,W} G_{N}^{(l)} bigraded by the weight −w and depth −m.

Since the weight filtration admits a splitting, i.e. is defined by a grading, and both the depth filtration and the subspace G_{N}^{(l)} ⊂ Der𝕃_{N}^{(l)} are compatible with this grading, one has

Gr_{•,•}G_{N}^{(l)} ⊂ Gr_{•,•}Der𝕃_{N}^{(l)}

3. A mysterious relation. Let Γ_{1}(m; N) is the subgroup of GL_{m}(ℤ) stabilizing the vector (1, 0, ..., 0) mod N. Denote by V_{m} the standard m-dimensional representation of GL_{m}. The key point of this talk (as well as the papers [G2]- [G5]) is that

the structure of the bigraded Lie algebra ⊕_{w≥m≥1}Gr_{−m,−w}G_{N}^{(l)}

is related to the

geometry of the local system with fiber S^{w−m}V_{m} over the modular variety

Γ_{1}(m; N)\GL_{m}(ℝ)/SO_{m} · ℝ^{∗}

The Lie algebra 𝕃_{N}^{(l)} is the l-adic realization of a certain pro-Lie algebra in the category of mixed Tate motives over the scheme S_{N} := Specℤ[ζ_{N}][\frac{1}{N}] which will be denoted 𝕃_{N}^{M}. It is the “motivic fundamental group” (see [D]) of ℙ^{1}\{0, μ_{N}, ∞}. The Hodge realization of the motive 𝕃_{N}^{M} is described by multiple polylogarithms at N-th roots of unity (see [G5]).

4. Examples of this relationship. For any m ≥ 1 there is the depth ≥ −m quotient Lie algebra

Gr_{≥−m,•}G_{N}^{(l)} := \frac{Gr_{•,•}G_{N}^{(l)}}{Gr_{<−m,•}G_{N}^{(l)}}

It is a nilpotent graded Lie algebra of the nilpotence class m. In particular the depth −1 quotient Lie algebra is abelian.

Strangely enough it is more natural to describe the structure of a certain bigraded Lie algebra G̃_{N}^{(l)} which sits in the exact sequence

0 ⟶ G_{N}^{(l)} ⟶ G̃_{N}^{(l)} ⟶ ℚ_{l}(1) ⟶ 0

where ℚ_{l}(1) is a one dimensional abelian quotient of G̃_{N}^{(l)} which is of weight −1 and depth −1.

I do not know any motivic or Galois-theoretic meaning of this extension, also the added component ℚ_{l}(1) somehow corresponds to ζ(1), whatever it means.

0). Depth = −1, N = 1. The situation in this case is completely known thanks to the efforts of Soulé [So], Deligne [D], and Ihara [Ih2]. Denote by G^{(l)} the Lie algebra G_{1}^{(l)}.

Theorem 1.1

Gr_{−1,−w}G̃^{(l)} = \{ \begin{matrix}0 & w even, or w = 1 \\ ℚ_{l} & odd \end{matrix}  (2)

1). Depth = −2, N = 1. The structure of the depth ≥ −2 quotient of the Lie algebra Gr_{•,•}G̃^{(l)} is completely described by the commutator map

[,] : Λ^{2}Gr_{−1,•}G̃^{(l)} ⟶ Gr_{−2,•}G^{(l)}  (3)

Construction of the dual to the complex (3). Consider the classical modular triangulation of the hyperbolic plane:

The group GL_{2}(ℤ) acts on the hyperbolic plane H^{2} preserving the modular picture. Let

M_{(2)}^{∗} := M_{(2)}^{1} ⟶ M_{(2)}^{2}  (4)

be the chain complex of the modular triangulation. Here the group M_{(2)}^{1} is generated by the triangles, and the group M_{(2)}^{2} by the geodesics. It is a complex of GL_{2}(ℤ)-modules.

Theorem 1.2 The depth w part of the dual to the complex (3) is canonically isomorphic to the complex

M_{(2)}^{∗} ⊗_{GL_{2}(ℤ)} S^{w−2}V_{2}  (5)

The complex (5) computes the cohomology of the group GL_{2}(ℤ) with coefficients in S^{w−2}V_{2}. Since these cohomology groups are well known, using theorems 1.1, 1.2 we immediately get the following result (compare it with [Ih3]):

Corollary 1.3

dimGr_{−2,−w}G̃^{(l)} = \{ \begin{matrix}0 & w : odd \\ [\frac{w−2}{6}] & w : even \end{matrix}  (6)

2). N = p is a prime; w = m.

2i) w = m = −1. In this case it is not dificult to show that

Gr_{−1,−1}G_{p}^{(l)} = (the group of the cyclotomic units in ℤ[ζ_{p}][\frac{1}{p}]) ⊗_{ℤ} ℚ_{l}

2ii) w = m = −2. Let Y_{1}(p) := Γ_{1}(p)\H^{2} be an open part of the modular curve. Projecting the modular triangulation of the hyperbolic plane onto Y_{1}(p) we get the modular triangulation of Y_{1}(p). The complex involution acts on the modular curve preserving the triangulation. Consider the following complex

(the chain complex of the modular triangulation of Y_{1}(p) )^{+}  (7)

Here + means the invariants of the action of the complex involution.

Theorem 1.4 The dual to the complex

[,] : Λ^{2}Gr_{−1,−1}G̃_{p}^{(l)} ⟶ Gr_{−2,−2}G̃_{p}^{(l)}

is naturally isomorphic to the complex (7).

In particular

ℚ_{l}[triangles of the modular triangulation of Y_{1}(p)]^{+} = (Gr_{−2,−2}G_{p}^{(l)})^{∨}

Remark. In [G2], [G5] we were dealing with the depth −2, weight −2 quotient of G_{p}^{(l)}, not G̃_{p}^{(l)}, and so we get only quasiisomorphism with a complex similar to (7).

2 General picture: the modular complexes and Galois symmetries of π̂_{1}^{(l)}(ℙ^{1}\{0, μ_{N}, ∞}, v_{1})

Let L_{m} be a rank m lattice. The modular complex M^{•}(L_{m}) = M_{(m)}^{•} is a complex of GL_{m}(ℤ)-modules

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

If m = 2 it coincides with complex (4). In general it is defined as follows. i) The group M_{(m)}^{1}.

Definition 2.1 An affine basis of a lattice L_{m} is an unordered (m + 1)-tuple of vectors v_{1}, ..., v_{m+1} of the lattice such that v_{1} + ... + v_{m+1} = 0 and v_{1}, ..., v_{m} is a basis.

The set A_{m} of affine basises (v_{1}, ..., v_{m+1}) of the lattice L_{m} is a principal homogeneous space of GL_{m}(ℤ).

The abelian group M^{1}(L_{m}) = M_{(m)}^{1} is generated by the elements < e_{1}, ..., e_{m+1} > where e_{1}, ..., e_{m+1} runthrough the set of all affine basises. So M_{(m)}^{l} := ℤ[A_{m}]/R_{(m)}^{l} where the subgroup of relations R_{(m)}^{l} will be defined below.

To list the relations we need another set of the generators:

< u_{1} : ... : u_{m+1} >:=< u_{2} − u_{1}, u_{3} − u_{2}, ..., u_{1} − u_{m+1} >

Relations. Let Σ_{k,m−k} be the set of all (k, m − k) shuffles of the ordered sets {1, ..., k} and {k + 1, ..., m}. For any 1 ≤ k ≤ m one has

∑_{σ∈Σ_{k,m−k}} < v_{σ(1)}, ..., v_{σ(m)}, v_{m+1} > = 0  (8)

∑_{σ∈Σ_{k,m−k}} < u_{σ(1)} : ... : u_{σ(m)} : u_{m+1}] = 0  (9)

ii) The group M_{(m)}^{k}. Let

L_{m} = L^{1} ⊕ ... ⊕ L^{k}  (10)

be a decomposition of the lattice L_{m} into a direct sum of sublattices. Then M_{(m)}^{k} is the product of the groups

M^{1}(L^{1}) ∧ ... ∧ M^{1}(L^{k})

over all unordered decompositions (10). In other words it is generated by the elements < A_{1} > ∧...∧ < A_{k} > where A_{i} is an affine basis of the sublattice L_{i}.

Let us set

[v_{1}, ..., v_{k}] :=< v_{1}, ..., v_{k}, v_{k+1} >, v_{1} + ... + v_{k} + v_{k+1} = 0

We define a differential

∂ : M_{(m)}^{1} ⟶ M_{(m)}^{2}  (11)

on the generators by the formula

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

where the indices are modulo m + 1 and

Cycle_{m+1}(f(v_{1}, ..., v_{m})) := ∑_{i=1}^{m+1} f(v_{i}, ..., v_{i+m})

We extend the differential to the whole complex M_{(m)}^{•} using the Leibniz rule:

∂([A_{1}] ∧ [A_{2}] ∧ ...) := ∂([A_{1}]) ∧ [A_{2}] ∧ ... − [A_{1}] ∧ ∂([A_{2}]) ∧ ... +

Theorem 2.2 There exists a canonical homomorphism of complexes

M_{(m)}^{∗} ⊗_{GL_{m}(ℤ)} (ℤ[Γ_{1}(m; N)\GL_{m}(ℤ)] ⊗_{ℤ} S^{w−m}V_{m}) ⟶  (12)
(Λ^{∗}Gr_{•,•}G̃_{N}^{(l)}^{∨})_{depth=−m,weight=−w}

Map (12) was defined on generators in [G5]. It was shown in [G5] that the second shuffle relations go to zero. What was left is a proof of the fact that the first shuffle relations go to zero. See [G4].

Conjecture 2.3 Let N = 1. Then homomorphism (12) is an isomorphism.

Conjecture 2.4 Let N = p be a prime and w = m. Then homomorphism (12) is an isomorphism.

Theorem 2.5 Conjectures 2.3 and 2.4 are true for m = 2, 3.

Remark. We almost proved they are also true for m = 4.

In general homomorphism (12) has a nontrivial kernel which contains the so-called distribution relations (see [G5]) and probably can be also described explicitely. So it should be close to be a quasiisomorphism.

The relation with Galois groups insures that the modular complex is an important mathematical object. In the next section we develope some geometry staying behind it and show how it is related with the rank m symmetric space.

3 Complex of colored forests and its geometric realization in the symmetric space

1. Voronoi’s cell decomposition of the symmetric space for GL_{m}(ℝ). Let

ℍ_{m} := GL_{m}(ℝ)/SO(m) · ℝ^{∗} = \frac{> 0 definite quadratic forms on V_{m}^{∗}}{ℝ_{+}^{∗}}

Recall that L_{m} ⊂ V_{m} is a lattice. Any vector v ∈ V_{m} defines a nonnegative definite quadratic form φ(v) :=< v, · >^{2} on V_{m}^{∗}. Take the convex hull C(L_{m}) of the forms φ(l) when l runs through all non zero primitive vectors of the lattice L_{m}. It has a structure of an infinite polyhedra. Let GL(L_{m}) ⊂ GL(V_{m}^{∗}) be the subgroup preserving the lattice L_{m}. The projection of C(L_{m}) into the closure of ℍ_{m} defines a GL(L_{m})-invariant polyhedral decomposition of ℍ_{m}.

Example. In the case m = 2 we get the modular triangulation of the hyperbolic plane.

Denote by V_{•}^{(m)} the chain complex of the Voronoi decomposition. It is called the Voronoi complex of rank m.

2. The relaxed modular complex. Consider a version M̂_{(m)}^{•} of the

modular complex, called the relaxed modular complex, where the group M̂_{(m)}^{1} is defined using the same generators [v_{1}, ..., v_{m}] satisfying only the dihedral symmetry relations

< v_{2}, ..., v_{m+1}, v_{1} >=< v_{1}, ..., v_{m+1} >= (−1)^{m+1} < v_{m+1}, ..., v_{1} >

and the first shuffle relations (8). The other groups are defined in a similar way. The differentials are the same as before.

3. The geometric realization. Let us define the join ∗ as follows. Let v_{1}, ..., v_{n_{1}} and v_{n_{1}+1}, ..., v_{n_{1}+n_{2}} be two sets of vectors of the lattice such that the lattices generated by these sets of vectors has zero intersection. Then we set

φ(v_{1}, ..., v_{n_{1}}) ∗ φ(v_{n_{1}+1}, ..., v_{n_{1}+n_{2}}) := φ(v_{1}, ..., v_{n_{1}+n_{2}})

and extend the join by linearity.

Let us make a homological complex M̂_{•}^{(m)} out of the complex M̂_{(m)}^{•} by setting

M̂_{i}^{(m)} := M̂_{(m)}^{2m−1−i}

Theorem-Construction 3.1 There exists a canonical morphism of complexes

ψ̂_{•}^{(m)} : M̂_{•}^{(m)} ⟶ V_{•}^{(m)}

such that:

ψ̂_{•}^{(m)}([A_{1}] ∧ ... ∧ [A_{k}]) := ψ̂_{•}^{(m)}([A_{1}]) ∗ ... ∗ ψ̂_{•}^{(m)}([A_{k}])  (13)

Here [A_{1}] := [v_{1}, ..., v_{l_{1}}] and so on.

In particular one has ψ̂_{m−1}^{(m)}([v_{1}] ∧ ... ∧ [v_{m}]) = φ(v_{1}, ..., v_{m}).

To define such a morphism ψ̂_{•}^{(m)} one needs only to define ψ̂_{2m−2}^{(m)}([v_{1}, ..., v_{m}]) for vectors v_{1}, ..., v_{m} forming a basis of the lattice L_{m} in such a way that the dihedral and the first shuffle relations go to zero and

dψ̂_{2m−2}^{(m)}([v_{1}, ..., v_{m}]) = ψ̂_{2m−3}^{(m)}(∂[v_{1}, ..., v_{m}])

where the right hand side is computed by (13) and the formula for ∂.

4. The map ψ̂_{2m−2}^{(m)}. A plane tree is a tree without self intersections located on the plane.

Let V (E) be the ℝ-vector space generated by the edges of a tree. By definition an orientation of the tree is a choice of the connected component of det(V (E))\0. So orientations of a tree T form a ℤ/2ℤ torsor O_{T}.

Let v be a vertex of a tree T, and O_{v} the ℤ/2ℤ torsor of orientations of the link of v. It is easy to see that if T is a 3-valent tree then O_{T} = ⊗O_{v} where the tensor product is over all vertices of T. Therefore a plane 3-valent tree has a canonical orientation. Indeed, the orientation of the plane provides orientations of links of each of the vertices.

The edges of a tree consist of legs (i.e. external edges) and internal edges. In other words a leg is an edge of the tree such that one or both of its vertices have valency one. A tree may have no internal vertices; for example we allow a tree to have only one internal edge. We do not allow however vertices of valency two.

Choose once forever a lattice L_{m}. A colored tree is a tree whose legs are in a bijective correspondence with the elements of an affine basis of the lattice L_{m}.

In particular a colored 3-valent tree has 2m − 1 edges.

Examples. We visualize a colored tree as follows:

The vectors e_{0}, ..., e_{m} of an affine basis are located cyclically on an oriented circle and the legs of the tree end on the circle and labelled by e_{0}, ..., e_{m}. (The circle itself is not a part of the graph).

Construction. Each edge E of the tree T provides a vector f_{E} ∈ L_{m} defined up to a sign. Namely, the edge E determines two trees rooted at E, see the picture

The union of the incoming (i.e. different from E) legs of these rooted trees coincides with the set of all legs of the initial tree. Take the sum of all the vectors e_{i} corresponding to the incoming legs of one of these trees. Denote it by f_{E}. If we choose the second rooted tree the sum will change the sign. So the degenerate quadratic form φ(f_{E}) is well defined.

Let us denote by φ(v_{1}, ..., v_{k}) the convex hull of the forms φ(v_{1}), ..., φ(v_{k}) in the space of quadratic forms. Set

ψ̂_{2m−2}^{(m)}(< e_{0}, e_{1}, ..., e_{m} >) :=
∑ ; plane 3-valent trees sgn(E_{1} ∧ ... ∧ E_{2m−1}) · φ(f_{E_{1}}, ..., f_{E_{2m−1}})  (14)

Here the sum is over all plane 3-valent trees colored by e_{0}, ..., e_{m} and the sign is taken with respect to the canonical orientation of the plane 3-valent tree.

Examples. a) For m = 2 there is one plane 3-valent tree colored by e_{0}, e_{1}, e_{2}, so we get the triangle φ(e_{0}, e_{1}, e_{2}).

b) Let f_{ij} := e_{i} + e_{j}. For m = 3 there are two plane 3-valent trees colored by e_{0}, e_{1}, e_{2}, e_{3}, see the picture, so the chain is

φ(e_{0}, e_{1}, e_{2}, e_{3}, f_{01}) − φ(e_{0}, e_{1}, e_{2}, e_{3}, f_{12})

c) For m = 4 there are five plane 3-valent trees colored by e_{0}, e_{1}, e_{2}, e_{3}, e_{4}. We showed one of them on the picture below; the rest are obtained by rotations of the tree. Therefore the chain is

Cycle_{5}φ(e_{0}, e_{1}, e_{2}, e_{3}, e_{4}, f_{12}, f_{34})

5. The complex of colored forests. Terminology. A forest is a graph without loops. Any forest T is a union if its connected components, i.e. trees: T = T_{1} ∪ ... ∪ T_{k}.

A colored forest is a forest T equipped with the following data. Let T = T_{1} ∪ ... ∪ T_{k} be the decomposition on trees, and L^{(1)} ⊕ ... ⊕ L^{(k)} = L_{m} be a decomposition into a direct sum of k nonzero sublattices such that the number of legs of the tree T_{i} is equal to rkL^{(i)} + 1. Choose an affine basis in each of the sublattices L^{(i)} and label the legs of the tree T_{i} by the vectors of this basis (i.e. choose a bijection between the legs and the vectors of the basis). Then we will say that the forest T is colored (by the vectors of the affine basises of the lattices L^{(i)}).

Here is an example of a colored forest:

We define the ℤ/2ℤ torsor of orientations of a forest T by O_{T} = ⊗_{i=1}^{k}O_{T_{i}}. Let F_{(m)}^{i} be the abelian group generated by the isomorphism classes of pairs

(T, Or_{T}) where T is a colored forest with 2m−i edges, and Or_{T} is an orientation of the forest. The only relation is this: if we change the orientation of T, the generator is multiplied by −1.

Example. The group F_{(m)}^{1} is generated by colored 3-valent trees with m+1 legs.

Let us define a differential d : F_{(m)}^{i} ⟶ F_{(m)}^{i+1}. One has d = d_{I} + d_{L}.

The map d_{I} is defined as follows. Let (T, Or_{T}) be a generator of F_{(m)}^{i} and E an internal edge of T. Let T/E be the tree obtained by contraction of the edge E. So it has one less edge, and one less vertex. The orientation Or_{T} of the graph T induces a natural orientation Or_{T/E} of the graph T/E. Namely, if Or_{T} is represented by E ∧ E_{1} ∧ E_{2} ∧ ... then Or_{T/E} is given by E_{1} ∧ E_{2} ∧ ... . Notice that contructing an internal edge we do not touch the coloring. So d_{I} is essentially the differential used by Boardman and Kontsevich.

Let us define the map d_{L}. Let L be a leg of T. Let us remove this leg together with a little neighborhood of its vertices. If both of the vertices of L has valency 1 this means that L is a connected component of T, so we just removed this connected component. Otherwise one of the vertices has the valency v ≥ 3, and the connected component of the tree containing the leg L is replaced by v − 1 connected trees. Namely, each of the internal edges sharing the vertex with L produces a new tree. The coloring of the tree T provides a natural coloring of the new tree, see the picture.

More precisely, the coloring of the legs of T different from L remains unchanged; each of the new trees has one new leg, and the vector attached to it is the minus sum of the vectors atached to the rest of the legs of this tree.

By definition the component of the differential corresponding to a leg L which itself forms a tree is zero.

It is easy to see that we get a complex, denoted F_{(m)}^{•}. We call it the complex of colored forests attached to a lattice L_{m}. It is different from the complex of trees/graphs considered by Boardman and Kontsevich.

6. A geometric realization of the complex of colored forests. Let us make a homological complex F_{•}^{(m)} out of F_{(m)}^{•} by setting F_{i}^{(m)} := F_{(m)}^{2m−i−1}.

Theorem-Construction 3.2 There exists a canonical injective homomorphism of complexes

τ_{•}^{(m)} : F_{•}^{(m)} ⟶ V_{•}^{(m)}

Proof. Let (T, Or_{T}) be a generator of F_{i}^{(m)}.

Let E_{1}, ..., E_{2m−i} be the set of all edges of T. Let sgn(E_{1} ∧...∧E_{2m−i}) = +1 if Or_{T} = E_{1} ∧ ... ∧ E_{2m−i}, and −1 otherwise. Set

τ_{i}^{(m)} : (T, Or_{T}) ⟼ sgn(E_{1} ∧ ... ∧ E_{2m−i}) · φ(f_{E_{1}}, ..., f_{E_{2m−i}}) ∈ V_{i}^{(m)}

We get a morphism of complexes. Indeed, the cell φ(f_{E_{1}}, ..., f_{E_{2m−i}}) is a simplex, so its boundary is a union of simplices corresponding to the edges of the tree. The contribution of an internal edge E is precisely the cell corresponding to (T/E, Or_{T/E}). Similarly the contribution to the boundary of a leg L of the tree T is given by the d_{i} component of the differential.

Theorem 3.3 The complex of colored forests is a resolution of the relaxed modular complex.

Proof. The complex of the colored forests has a filtration by the number of connected components of the forest. The E_{1}-term of the corresponding spectral sequence is a sum (over affine basises) of the tree complexes. By the Ginzburg-Kapranov theorem [GK] the tree complex with m + 1 legs is a resolution of the space of m-ar operations in the cyclic Lie cooperad CLie_{m+1}^{∗}. It remains to notice that

M^{1}(L_{m}) = ⊕_{affine basises}CLie_{m+1}^{∗}

In particular the relaxed modular complex is a subcomplex of colored forests. Combining this with the geometric realization given above we get the map ψ̂_{•}^{(m)}.

Example. The rank 3 complex of colored forests F_{•}^{(3)} is isomorphic to the 4-sceleton of the Voronoi complex for GL_{3}. Indeed, the list of the cells of the Voronoi decomposition of ℍ_{3} considered up to GL_{3}(ℤ) equivalence is the following (see [G3]):

4 − cell : φ(v_{1}, v_{2}, v_{3}, v_{4}, f_{12})
3 − cells : φ(v_{1}, v_{2}, v_{3}, v_{4}), φ(v_{1}, v_{2}, f_{12}, v_{3})
2 − cell : φ(v_{1}, v_{2}, v_{3}),

They corresponds presisely to the colored forests listed on the picture.

A similar but easier investigation shows that F_{•}^{(2)} is isomorphic to the Voronoi complex for GL_{2}.

7. Modular complexes and the Voronoi complexes. Here is one of the essential ingrediets of the proof of theorem 2.5. Its proof useses the machinery developed in this section.

Theorem 3.4 a) There exist isomorphisms of complexes

M_{•}^{(2)} ⟶ V_{•}^{(2)}, M_{•}^{(3)} ⟶ τ_{[4,2]}(V_{•}^{(3)})

b) There exist a quasiisomorphism of complexes

M_{•}^{(4)} ⟶ τ_{[6,3]}(V_{•}^{(4)})

The part a) was proved in [G3]. For the part b) see [G4].

REFERENCES

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