On the Galois actions on torsors of paths I, Descent of Galois representations

Zdzisław Wojtkowiak October 29, 2007

Contents

0 Introduction 1 1 Torsors of paths 11 2 Ramification 18 3 ℓ-adic polylogarithms 20 4 Descent 22 5 Torsors of paths on ℙ1 ; ℚ̄\{0, 1, ∞} 40

Abstract We are studying representations obtained from actions of Galois groups on torsors of paths on a projective line minus a finite number of points. Using these actions on torsors of paths, we construct geometrically representations of Galois groups which realize ℓ-adically the associated graded Lie algebra of the fundamental group of the tannakian category of mixed Tate motives over Spec ℤ, Spec ℤ[i], Spec ℤ[ \frac{1}{q}], Spec O_{ℚ(\sqrt{−q})} for any prime number q (q ≠ 2 in the last case) and over Spec O_{ℚ(\sqrt{−q})}[ \frac{1}{q}] for any prime number q congruent to 3 modulo 4 and also for q = 2.

0 Introduction

0.1. In this paper we are studying actions of Galois groups on torsors of paths. Let V be an algebraic variety defined over a number field K. Let us fix two points or tangential points v and z of V defined over K. Let ℓ be a fixed prime. We denote by π_{1}(VK̄; v) the ℓ-completion of the étale fundamental group of VK̄ base at v and by π(VK̄; z, v) the π_{1}(VK̄; v)-torsor of ℓ-adic paths from v to z. The Galois group G_{K} acts on π_{1}(VK̄; v) and on π(VK̄; z, v). Therefore we have two representations

φ_{v} : G_{K} → Aut (π_{1}(VK̄; v))

and

ψ_{z,v} : G_{K} → Aut_{set}(π(VK̄; z, v)) .

We mention some examples.

Let V := ℙ_{ℚ}^{1}\{0, 1, ∞}. Then the action of G_{ℚ} on π_{1}(Vℚ̄; 01^{→}) and on

π_{1}(Vℚ̄; 01^{→})-torsor π(Vℚ̄; 10^{→}, 01^{→}) is “the same”, i.e.,

ker(G_{ℚ} → Aut (π_{1}(Vℚ̄; 01^{→}))) and ker(G_{ℚ} → Aut_{set}(π(Vℚ̄; 10^{→}, 01^{→})))

are equal. One also shows that the actions of G_{ℚ} on π_{1} (ℙ^{1}ℚ̄\{0, 1, −1, ∞}; 01^{→})

and on π(Vℚ̄; −1, 01^{→}) have equal kernels after passing to associated graded Lie

algebras (see [22]). On the other side the action of G_{ℚ} on π_{1}(Vℚ̄; 01^{→}) is unramified

outside ℓ and the action of G_{ℚ} on π(Vℚ̄; 2, 01^{→}) is ramified at ℓ and at 2.

0.2. Let a_{1}, …, a_{n+1} be K-points of the projective line ℙ_{K}^{1} and let

V := ℙ_{K}^{1}\{a_{1}, …, a_{n}, a_{n+1}} .

For simplicity we assume that a_{n+1} = ∞. Let v be a K-point of V or a tangential point of V defined over K.

Let f(T) ∈ K[T] be an irreducible polynomial and let z_{1}, …, z_{r} be all roots of f(T) in K̄. Let L := K(z_{1}, …, z_{r}) be an extension of K generated by all roots of f(T). The Galois group G_{L} acts on the disjoint union of torsors

t := ∐^{r} ; i=1 π(VK̄; z_{i}, v), hence we get a representation

ψ_{t} : G_{L} → Aut_{set} (∐^{r} ; i=1 π(VK̄; z_{i}, v)) .

Our aim is to construct from the representation ψ_{t} a representation of G_{K} which is prounipotent and pro-ℓ on G_{K(μ_{ℓ^{∞}})}.

Let x_{1}, …, x_{n} be geometric generators of π_{1}(VK̄; v) (see [18] section 2). In [18] we defined a continuous embedding

π(VK̄; z_{i}, v) → ℚ_{ℓ}{{𝕏}} ,

where 𝕏 := {X_{1}, …, X_{n}} and ℚ_{ℓ}{{𝕏}} is a ℚ_{ℓ}-algebra of non-commutative formal power series on non-commuting variables X_{1}, …, X_{n}. The action of G_{L}

on the disjoint union of torsors ∐^{r} ; i=1 π(VK̄; z_{i}, v) induces a representation

0.2.1.  ψ_{t} : G_{L} → GL (⨁_{i=1}^{r} ℚ_{ℓ}{{𝕏}}) .

In [18] section 3 we defined a filtration {H_{k}(V_{L}; z_{i}, v)}_{k∈ℕ} of G_{L} associated with the action of G_{L} on the π_{1}(VK̄; v)-torsor π(VK̄; z_{i}, v).

Let us set

H_{k} := ⋂^{r} ; i=1 H_{k}(V_{L}; z_{i}, v) for k ∈ ℕ.

Passing with the representation ψ_{t} to Lie algebras and then to associated graded Lie algebras we get a morphism of associated graded Lie algebras

gr Lie ψ_{t} : ⨁_{k=1}^{∞} (H_{k}/H_{k+1}) ⊗ ℚ → (⨁_{i=1}^{r} Lie (𝕏)) ˜× Der^{∗}Lie (𝕏) ,

where Lie (𝕏) is a free Lie algebra over ℚ_{ℓ} on 𝕏, Der^{∗}Lie (𝕏) is a Lie subalgebra

of the Lie algebra of derivations defined in [18] and (⨁^{r} ; i=1 Lie (𝕏)) ˜× Der^{∗}Lie (𝕏)

is a semi-direct product of Lie algebras.

Let G := Gal (L/K). Then the group G acts on the associated graded Lie

algebra ⨁^{∞} ; k=1 (H_{k}/H_{k+1}) ⊗ ℚ. We shall study the restriction of the morphism of

associated graded Lie algebras gr Lie ψ_{t} to the fixed part of the action of G, i.e., the morphism of Lie algebras

(gr Lie ψ_{t})^{G} : ⨁_{k=1}^{∞} (H_{k}/H_{k+1}) ⊗ ℚ^{G} → (⨁_{i=1}^{r} Lie (𝕏)) ˜× Der^{∗}Lie (𝕏) .

We give a sample indicating that something interesting is going on. In most interesting applications we shall study Galois actions on torsors of paths on Vℚ̄, where V := ℙ^{1} \ {0, 1, ∞}. Then traditionally we denote by Lie(X, Y) a free Lie algebra on two generators X and Y.

Theorem A. Let q be a prime number different from ℓ. Let V := ℙ^{1} \ {0, 1, ∞}

and let 𝔱 := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) be a disjoint union of torsors of paths, where ξ_{q} is a primitive q-th root of 1. Then the Galois group G_{ℚ(μ_{q})} acts on 𝔱 and in the image of the morphism of Lie algebras

(grLieψ_{𝔱})^{(ℤ/q)^{∗}} : ⨁_{k=1}^{∞}(H_{k}/H_{k+1} ⊗ ℚ)^{(ℤ/q)^{∗}} → ( ⨁_{0<α<q} Lie(X, Y)) ˜×Der^{∗}Lie(X, Y)

there are elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … homogenous of degree 1, 3, 5, …, 2n+ 1, … respectively and these elements generate freely a free Lie subalgebra of the image of (grLieψ_{𝔱})^{(ℤ/q)^{∗}}.

Now let us observe the followings facts:

i) the representation of G_{ℚ(μ_{q})} on 𝔱 is unramified outside prime ideals of O_{ℚ(μ_{q})} lying over prime ideals (q) or (ℓ) of ℤ; ii) the conjectural Lie algebra of the fundamental group of the tannakian

category of mixed Tate motives over Spec ℤ[\frac{1}{q}] is free, freely generated by generators d_{1}, d_{3}, d_{5}, …, d_{2n+1}, … of degree 1, 3, 5, …, 2n + 1, … respectively;

iii) the elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … are dual to \frac{1}{q−1}ℓ(q), \frac{1−q^{2}}{(q−1)q^{2}} ℓ_{3}(1),

\frac{1−q^{4}}{(q−1)q^{4}} ℓ_{5}(1), …, \frac{1−q^{2n}}{(q−1)q^{2n}} ℓ_{2n+1}(1), … respectively, i.e., to ℓ-adic polylogarithms evaluated at elements of ℤ[\frac{1}{q}]^{∗} (the point iii) we shall see in the proof of the theorem).

Now we can pose the following question. Can we construct from the representation ψ_{𝔱} a new representation θ_{𝔱}^{ℚ(μ_{q})/ℚ} of G_{ℚ} which is unramified outside prime ideals (q) and (l) of ℤ and such that the image of the Lie algebra morphism grLieθ_{𝔱}^{ℚ(μ_{q})/ℚ} is free, freely generated by elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, …?

The representation ψ_{𝔱} is an ℓ-adic realization of the mixed Tate motive associated with the geometrical object 𝔱. We can hope that the representation θ_{𝔱}^{ℚ(μ_{q})/ℚ} will be also motivic.

Let

ψ_{t} : G_{L} → GL (⨁_{i=1}^{r} ℚ_{ℓ}{{𝕏}})

be the representation 0.2.1. Below we describe the main idea how to construct the representation θ_{t}^{L/K} of G_{K} from the representation ψ_{t} of G_{L}.

0.3. Let K be a number field and let T be a finite set of prime ideals of O_{K}. Let O_{K,T} be the ring of T -integers in K. Let us denote by MM_{O_{K,T}} the conjectural tannakian category of mixed Tate motives over SpecO_{K,T} . The tannakian category MM_{O_{K,T}} is equivalent to the category of representations of an affine proalgebraic group Π(K, T) over ℚ in finite dimentional ℚ-vector spaces. The group Π(K, T) is an extension of the multiplicative group 𝔾_{m} over ℚ by an affine proalgebraic prounipotent group U(K, T) over ℚ.

Let L be a finite Galois extension of K and let G := Gal(L/K). Let S be a set of these prime ideals of O_{L} which lie over some element of T. The inclusion functor

MM_{O_{K,T}} ↪ MM_{O_{L,S}}

induces surjective morphisms of affine proalgebraic groups over ℚ

Π(L, S) → Π(K, T) and U(L, S) → U(K, T).

The group U(K, T) is free in the category of affine proalgebraic prounipotent groups over ℚ, freely generated by r_{2}(K) elements in each even positive degree, by r_{1}(K) + r_{2}(K) elements in each odd and greater than 1 degree and by dim(O_{K,T}^{∗} ⊗ ℚ) elements in degree 1.

Therefore the surjective morphism

U(L, S) → U(K, T)

has a section

s_{L,S/K,T} : U(K, T) → U(L, S).

Hence we have also a section

s_{L,S/K,T} : Π(K, T) → Π(L, S).

We assume that the representation ψ_{t} : G_{L} → GL(⊕_{i=1}^{r}ℚ_{ℓ}{{𝕏}}) is unramified outside S. The representation ψ_{t} being motivic factors through the universal map

a_{L,S,ℓ} : G_{L} → Π(L, S)(ℚ_{ℓ})

with a Zariski dense image. Hence we have a commutative diagram

G_{L} ⟶^{ψ_{t}} GL(⊕_{i=1}^{r}ℚ_{ℓ}{{𝕏}})
=↑  ψ̄_{t}↑
G_{L} ⟶^{a_{L,S,ℓ}}  Π(L, S)(ℚ_{ℓ})

where ψ̄_{t} is uniquely determined by ψ_{t}. We define the representation θ_{t}^{L/K} to be the composition

G_{K} ⟶^{a_{K,T,ℓ}} Π(K, T)(ℚ_{ℓ}) ⟶^{s_{L,S/K,T}} Π(L, S)(ℚ_{ℓ}) ⟶^{ψ̄_{t}} GL(⊕_{i=1}^{r}ℚ_{ℓ}{{𝕏}}).

The category MM_{O_{K,T}} being conjectural, we need to find substitutes for the groups Π(K, T) and U(K, T).

First however we introduce the following notation. For a number field K, we denote by V(K) the set of prime ideals of O_{K}.

Hain and Matsumoto in [7] and [8] considered a category of continous weighted Tate representations of G_{K} unramified outside T and prime ideals of O_{K} lying over ℓ, in finite dimensional ℚ_{ℓ}-vector spaces. This category is tannakian over ℚ_{ℓ}. Its fundamental group, which we denote by G(K, T, ℓ), is a proalgebraic group over ℚ_{ℓ}, an extension of the multiplicative group 𝔾_{m} by a proalgebraic prounipotent group U(K, T, ℓ). They showed that the associated graded (with respect to weight filtration) Lie algebra of U(K, T, ℓ) is free, freely generated by r_{2}(K) elements in each even positive degree, by r_{1}(K) + r_{2}(K) elements in each odd and greater than 1 degree and by dim(O_{K,T′}^{∗} ⊗ ℚ) elements in degree 1, where T′ := T ∪ {λ ∈ V(K) | λ divides ℓ}.

It follows from the result of Hain and Matsumoto that for any M ∈ MM_{O_{K,T}}, the representation φ_{M_{ℓ}} : G_{K} → Aut M_{ℓ} factors through the homomorphism

G_{K} → G(K, T, ℓ)

and the restriction of φ_{M_{ℓ}} to G_{K(μ_{ℓ^{∞}})} factors through the homomorphism

G_{K(μ_{ℓ^{∞}})} → U(K, T, ℓ).

The inclusion G_{L} ⊂ G_{K} induces surjective morphisms of affine proalgebraic groups over ℚ_{ℓ}

G(L, S, ℓ) → G(K, T, ℓ) and U(L, S, ℓ) → U(K, T, ℓ).

We shall construct a section

s_{L,S,ℓ/K,T,ℓ} : U(L, S, ℓ) → U(K, T, ℓ)

and using this section we shall define a representation θ_{t}^{L/K} to be the composition

G_{K(μ_{ℓ^{∞}})} ⟶ U(K, T, ℓ) ⟶^{s_{L,S,ℓ/K,T,ℓ}}U(L, S, ℓ)⟶^{[ψ_{t}]}GL (⨁_{i=1}^{r} ℚ_{ℓ}{{𝕏}}) ,

where the first arrow is the canonical morphism and the last arrow is induced by ψ_{t}.

Then we can prove the following result.

Theorem B. Let K be a number field and let V = ℙ_{K}^{1}\{a_{1}, a_{2}, …, a_{n}, ∞}. Let a field L, a torsor t and a representation ψ_{t} be as in 0.2. Let G = Gal(L/K). Let us assume that the representation ψ_{t} is unramified outside a finite set S of finite places of L and that S is G-invariant. Let us assume that L∩K(μ_{ℓ^{∞}}) = K and that ℓ does not divide the order of G. Then

i) the representation θ_{t}^{L/K} is unramified outside the set T = {𝔮 ∈ V(K) | ∃ 𝔭 ∈ S, 𝔮 = 𝔭 ∩ O_{K}} of finite places of K,

ii) the filtration {⊕_{j=1}^{r}I(𝕏)^{n}}_{n∈ℕ} of ⨁_{j=1}^{r} ℚ_{ℓ}{{𝕏}} by the powers of the augmentation ideal I(𝕏) of ℚ_{ℓ}{{𝕏}} is a filtration by G_{K(μ_{ℓ^{∞}})}-modules,

iii) any σ ∈ G_{K(μ_{ℓ^{∞}})} acts as the identity on ⊕_{j=1}^{r}(I(𝕏)^{n}/I(𝕏)^{n+1}) for any n ∈ ℕ.

With the representation θ_{t}^{L/K} there is associated in a standard way a filtration {F_{n}(θ_{t}^{L/K})}_{n∈ℕ} of G_{K(μ_{ℓ^{∞}})}. Passing to associated graded Lie algebras we get a morphism

gr Lie θ_{t}^{L/K} : ⨁_{k=1}^{∞} (F_{k}(θ_{t}^{L/K})/F_{k+1}(θ_{t}^{L/K}))⊗ℚ ⟶ (⨁_{j=1}^{r} L_{Lie(𝕏)}) ˜× Der^{∗} Lie(𝕏).

Choosing suitably V and a disjoint union of torsors t we get examples where the image of gr Lie θ_{t}^{L/K} is as big as possible, i.e., it is a free Lie algebra on the maximal possible number of generators depending on K and on the set of ramification places.

For example in the simplest case considered in Theorem A we have the following result. Theorem C. Let q be a prime number different from ℓ. Let V := ℙ^{1} \ {0, 1, ∞}

and let 𝔱 := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) be a disjoint union of torsors of paths, where ξ_{q} is a primitive q-th root of 1. Then the image of the morphism of associated graded Lie algebras

gr Lie θ_{𝔱}^{ℚ(μ_{q})/ℚ} : ⨁_{k=1}^{∞} (F_{k}(θ_{𝔱}^{ℚ(μ_{q})/ℚ})/F_{k+1}(θ_{𝔱}^{ℚ(μ_{q})/ℚ}))⊗ℚ ⟶ ( ⨁_{0<α<q} L_{Lie(𝕏)}) ˜× Der^{∗} Lie(𝕏).

is a free Lie algebra, freely generated by elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … from Theorem A.

Remark 0.3.1. The main problem is to show that the obtained representation θ_{t}^{L/K} is motivic. To show this one needs to show that section s_{L,S,ℓ/K,T,ℓ} : U(L, S, ℓ) → U(K, T, ℓ) is induced by a section s_{L,S/K,T} : U(L, S) → U(K, T). Unfortunately we are not able to show this as we have no U(L, S).

In order to show that θ_{t}^{K/L} is motivic one can also try to show that all coefficients of θ_{t}^{K/L} are motivic. Polylogarithmic coefficients are best understood, but in [5] we can only prove that in some special cases dilogarithm coefficients are motivic.

0.4. We indicate briefly why it is important to construct such motivic representations. Let K be a number field and let T be a finite set of prime ideals of O_{K}. Let O_{K,T} be the ring of T -integers in K.

Let us denote by MM_{O_{K,T}} the category of mixed Tate motives over Spec O_{K,T} . Then the conjectural associated graded Lie algebra of the unipotent part of the motivic fundamental group of the tannakian category MM_{O_{K,T}} is free, freely generated by r_{2}(K) elements in each even positive degree, by r_{1}(K) + r_{2}(K) elements in each odd and greater than 1 degree and by dim(O_{K,T}^{∗} ⊗ℚ) elements in degree 1.

If M ∈ MM_{O_{K,T}} then we denote by M_{ℓ} the ℓ-adic realization of M. It is a vector space over ℚ_{ℓ} equipped with a continous representation φ_{M_{ℓ}} : G_{K} → Aut (M_{ℓ}) unramified outside T and prime ideals of K lying over ℓ.

Let Rep_{ℚ_{ℓ}}(G_{K}) be the category of continous representations of G_{K} in finite dimensional vector spaces over ℚ_{ℓ}. The following conjecture seems to be universally accepted.

Conjecture 0.4.1. The functor of ℓ-adic realization

real_{ℓ} : MM_{O_{K,T}} → Rep_{ℚ_{ℓ}}(G_{K})

is faithful.

Let M ∈ Pro MM_{O_{K,T}} . We denote by M(M) a tannakian subcategory of MM_{O_{K,T}} generated by M.

Definition 0.4.2. Let M ∈ Pro MM_{O_{K,T}} . We say that M realizes the fundamental group of MM_{O_{K,T}} if the inclusion functor M(M) → MM_{O_{K,T}} induces an isomorphism of fundamental groups of the tannakian categories.

Definition 0.4.3. Let M ∈ Pro MM_{O_{K,T}} . We say that M realizes ℓ-adically the fundamental group of MM_{O_{K,T}} if the associated graded Lie algebra of the ℚ_{ℓ}-Zariski closure of the image of the Galois action homomorphism

φ_{M_{ℓ}} : G_{K(μ_{ℓ^{∞}})} → Aut (M_{ℓ})

is a free Lie algebra over ℚ_{ℓ}, freely generated by r_{2}(K) elements in each even positive degree, by r_{1}(K) + r_{2}(K) elements in each odd and greater than 1 degree and by dim(O_{K,T}^{∗} ⊗ ℚ) elements in degree 1.

Notice that the results of Hain and Matsumoto mentioned above do not imply that there is a mixed Tate motive M (in Pro MM_{O_{K,T}}) over Spec O_{K,T} which realizes the fundamental group of MM_{O_{K,T}} . They do not imply that there is a mixed Tate motive M in MM_{O_{K,T}} such that G_{K(μ_{ℓ^{∞}})} acts nontrivially on M_{ℓ}.

To study the category MM_{O_{K,T}} as well as arithmetic properties of G_{K} it is important to find objects M in ProMM_{O_{K,T}} such that the representation φ_{M_{ℓ}} realizes ℓ-adically the fundamental group of MM_{O_{K,T}} or even that it has a non-trivial image.

Here there is a sample why it is very important. Proposition 0.4.4. Let us assume that Conjecture 0.4.1 holds. If M ∈ Pro MM_{O_{K,T}} realizes ℓ-adically the fundamental group of MM_{O_{K,T}}, then M realizes the fundamental group of the tannakian category MM_{O_{K,T}} .

In a case of the category MM_{ℤ} - mixed Tate motives over Specℤ one hopes that the fundamental group of a projective line minus three points 0, 1, ∞ and based at 01^{→} realizes the fundamental group of the tannakian category MM_{ℤ}.

Looking only at ℓ-adic side one hopes that the associated graded Lie algebra of the ℚ_{ℓ}-Zariski closure of the image of

G_{ℚ(μ_{ℓ^{∞}})} → Aut (π_{1}(ℙ1 ; ℚ̄ \ {0, 1, ∞}; 01^{→}))

is free, freely generated by one element in each odd degree greater than 1. In a case of the category MM_{ℤ[ \frac{1}{2} ]}, the fundamental group π_{1}(ℙ1 ; ℚ̄\\{0, 1, −1, ∞\}; 01^{→})

as well as the torsor of paths from 01^{→} to −1 on ℙ1 ; ℚ̄ \ {0, 1, ∞} realize ℓ-adically the fundamental group of MM_{ℤ[ \frac{1}{2} ]}. This follows from a result of P.Deligne presented on the conference on Polylogarithms in Schloss Ringberg in a case of π_{1} (see also [20] where ℓ-adic version is discussed). The torsor case is considered in [22].

The fundamental group π_{1}(ℙ1 ; ℚ̄ \ ({0, ∞} ∪ μ_{4}); 01^{→}) as well as the torsor

of paths π(ℙ1 ; ℚ̄ \ {0, 1, ∞}; i , 01^{→}) (resp. π_{1}(ℙ1 ; ℚ̄ \ ({0, ∞} ∪ μ_{3}); 01^{→}) as well as the torsor of paths π(ℙ1 ; ℚ̄ \ \{0, 1, ∞\}; exp(\frac{2πi}{3}) , 01^{→}), resp. π_{1}(ℙ1 ; ℚ̄ \ (\{0, ∞\} ∪

μ_{8}); 01^{→})) realize the fundamental group of the tannakian category MM_{ℤ[i][ \frac{1}{2} ]} (resp. MM_{ℤ[exp( \frac{2πi}{3} )][ \frac{1}{3} ]}, resp. MM_{ℤ[exp( \frac{2πi}{8} )][ \frac{1}{2} ]}) (see [20] and [23]).

On the other side π_{1}(ℙ1 ; ℚ̄ \ ({0, ∞} ∪ μ_{7}); 01^{→}) definitely does not realize the fundamental group of MM_{ℤ[exp( \frac{2πi}{7} )]} or MM_{ℤ[exp( \frac{2πi}{7} )][ \frac{1}{7} ]} (see [4]).

In this note we shall construct “geometrical objects” which realize the fundamental group of the tannakian categories MM_{ℤ}, MM_{ℤ[ \frac{1}{p}]} for any prime number p, MM_{O_{ℚ(\sqrt{−p})}} for any prime number p, MM_{ℤ[i]} and MM_{O_{ℚ(\sqrt{−p})}[ \frac{1}{p}]} for any prime number p congruent to 3 modulo 4 and also for p = 2. Observe that for all considered rings we have r_{1} + r_{2} = 1 and dim O_{K,T}^{∗} ⊗ ℚ ≤ 1.

We are working with torsors on ℙ^{1} \ {0, 1, ∞} and in the free Lie algebra on two generators there is no sufficient place to deal with examples when r_{1}+r_{2} > 1 or dim O_{K,T}^{∗} ⊗ ℚ > 1. In the future work we hope to study Galois actions on torsors of paths on ℙ^{1} \ {0, μ_{n}, ∞} and on configuration spaces.

In our study we use the idea of Deligne to work modulo a prime number presented in his lecture in Schloss Ringberg (see [3]).

Now we indicate some motivic motivations of our construction of such geometrical objects.

Let ℓ_{n}(10^{→}) : G_{ℚ} → ℚ_{ℓ} be the ℓ-adic polylogarithm evaluated at 10^{→} and

calculated along the canonical path from 01^{→} to 10^{→}. Then ℓ_{n}(10^{→}) is a cocycle with values in ℚ_{ℓ}(n) and we have a Galois representation

G_{ℚ} ∋ σ → U(σ) ∈ Aut (ℚ_{ℓ}^{2}),

where U(σ) is the following matrice

( \begin{matrix}1 & 0 \\ ℓ_{n}(10^{→}) & χ^{n}(σ) \end{matrix} )

The corresponding Hodge-De Rham realization, which we denote by E_{n}, is an extension of ℚ(0) by ℚ(n). The rational lattice is generated by

( 1 , Li_{n}(1) ) and ( 0 , (2πi)^{n}).

Let p be a prime number. We denote by F_{n} an extension of ℚ(0) by ℚ(n), whose rational lattice is generated by

( 1 , ∑_{k=1}^{p−1} Li_{n}(exp(\frac{2πik}{p}) ) and ( 0 , (2πi)^{n}).

Observe that F_{n} is over Spec ℚ though exp(\frac{2πik}{p}) ∉ ℚ for k = 1, …, p − 1. We have also an equality

(1 − p^{n−1})E_{n} = p^{n−1}F_{n}.

In fact it was observed by Zagier (see [26]) that some Galois invariant linear combinations of polylogarithms evaluated at elements of K̄ behave like linear combinations of polylogarithms evaluated at elements of K. For example in [26] we found a relation

D_{3}(\frac{1 + \sqrt{5}}{2}) + D_{3}(\frac{1 − \sqrt{5}}{2}) = \frac{1}{5}ζ(3),

where D_{3} is a univalent version of Li_{3}. Let us denote by G_{3} an extension of ℚ(0) by ℚ(3) in the category of mixed Hodge-De Rham structures over Spec ℤ, whose rational lattice is generated by

(1 , Li_{3}(\frac{1 + \sqrt{5}}{2}) + Li_{3}(\frac{1 − \sqrt{5}}{2})) and ( 0 , (2πi)^{3}).

One would like to construct representations of G_{ℚ} corresponding to mixed Hodge-De Rham structures F_{n} and G_{3}. Equivalently one would like to consider ∑_{k=1}^{p−1} ℓ_{n}(exp\frac{2πi}{p}) and ℓ_{3}(\frac{1+\sqrt{5}}{2}) + ℓ_{3}(\frac{1−\sqrt{5}}{2}) as (ℓ-adic period) functions on G_{ℚ}. Notice that a priori they are defined on G_{ℚ(μ_{p})} and on G_{ℚ(\sqrt{5})} respectively (see [19], where ℓ-adic polylogarithms ℓ_{n}(z) are defined).

In both cases considered here it is easy to find a corresponding function on G_{ℚ}. One can take \frac{(1−p^{n−1})}{p^{n−1}}ℓ_{n}(10^{→}) in the first case and \frac{1}{5}ℓ_{3}(10^{→}) in the second case. It is however far from being obvious how to do this for other examples considered in [26].

The problem is studied in [5]. Unprecisely we can formulate the main results from [5] in the following way.

Let L be a finite Galois extension of K. Assume that a formal linear combination of elements of L, ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. Assume that c_{n} := ∑_{i=1}^{N} m_{i}ℓ_{n}(z_{i})_{γ_{i}} is a cocycle on G_{L}. Then there is a cocycle s_{n} on G_{K} such that c_{n} = s_{n}_{|G_{L}} in H^{1}(G_{L}; ℚ_{ℓ}(n)).

However it is far from being obvious if s_{n} is motivic even if c_{n} is motivic. In fact we can show this only for n = 2 (see [5] Theorem 5.4).

More generally one can hope that some Galois invariant linear combinations of normalized iterated integrals (see [18] and [24]) evaluated at elements of K̄ behave like linear combinations of normalized iterated integrals evaluated at elements of K.

Coefficients of a Lie algebra representation of gr LieU(O_{K,T}) deduced from torsors of paths are given by symbols {z, v}_{e}, where z, v ∈ K (see [18] and [24], where symbols {z, v}_{e} are defined). The observation of Zagier and the hope expressed above suggest that some Galois invariant linear combinations with ℚ-coefficients of symbols {a, b}_{e} with a, b ∈ K̄ behave like linear combinations of symbols {α, β}_{e} with α, β ∈ K. Hence coefficients of a Lie algebra representation of gr LieU(O_{K,T}) should be Galois invariant linear combinations with ℚ-coefficients of symbols {a, b}_{e} with a, b ∈ K̄.

In this note we use this philosophy to construct representations of G_{K} starting from representations of G_{L}, where L is a finite Galois extension of K.

These informal motivic considerations we hope to make more precise in the second part of this paper.

During the conference in Acquafredda di Maratea I have given a talk entitle “Galois actions on torsors of paths”. Before my lecture P.Deligne told me that he has studied the product of torsors of paths on ℙ^{1} \ {0, 1, ∞} (from 01^{→} ) to ξ_{6}^{1} and ξ_{6}^{5} and that the Lie algebra (associated to the representation of G_{ℚ} or the mixed Hodge structure) is free on generators in degrees 3, 5, ..., 2n + 1, . . .. This paper is an attempt to generalize an example studied by Deligne.

Some results of this paper were presented on the conferences in Irvine 2002, in Sestri Levante 2004, in Banff 2005 (Regulators II) and on seminar talks in IHES, in MPI für Mathematik, Bonn and in Okayama University in 2004.

This paper was written during our visits to IHES and MPI für Mathematik, Bonn and we would like to thank very much both these instituts for support.

1 Torsors of paths

1.0. We recall here methods and results concerning actions of Galois groups on fundamental groups and on torsors of paths from [18].

Let K be a number field. Let a_{1}, …, a_{n+1} be K-points of a projective line ℙ^{1}. Let

V := ℙ_{K}^{1}\{a_{1}, …, a_{n+1}} .

We assume for simplicity that a_{n+1} = ∞. We denote by V̂ (K) the set of Kpoints of V and of tangential points defined over K.

Let ℓ be a fixed prime and let v ∈ V̂ (K). Let π_{1}(VK̄; v) be the pro-ℓ completion of the étale fundamental group of VK̄ based at v. Let x_{1}, …, x_{n} ∈ π_{1}(VK̄; v) be a sequence of geometric generators of the fundamental group associated with the family Γ = {γ_{i}}_{i=1}^{n+1} of paths from v to tangential base points defined over K at a_{i} for i = 1, …, n + 1.

Let 𝕏 := {X_{1}, …, X_{n}} and let

k : π_{1}(VK̄; v) → ℚ_{ℓ}{{𝕏}}

be a continuous multiplicative embedding of the fundamental group into noncommutative power series on n non-commuting variables X_{1}, …, X_{n} sending x_{i} into the power series e^{X_{i}}.

Let Lie (𝕏) be a free Lie algebra over ℚ_{ℓ} on the set 𝕏 and let L(𝕏) := \underleftarrow{lim}_{i} Lie (𝕏)/ Γ^{i} Lie (𝕏) be a completed free Lie algebra over ℚ_{ℓ} on the

set 𝕏. We identify Lie (𝕏) (resp. L(𝕏)) with Lie elements of finite length in ℚ_{ℓ}{{𝕏}} (resp. Lie elements possibly of infinite length in ℚ_{ℓ}{{𝕏}}).

The action of G_{K} on the fundamental group π_{1}(VK̄; v) induces an action of G_{K} on the ℚ_{ℓ}-algebra ℚ_{ℓ}{{𝕏}}, i.e., we get a representation

φ_{v} := φ_{V_{K},v} : G_{K} → Aut (ℚ_{ℓ}{{𝕏}}) .

The representation φ_{v} factors through

φ_{v} : G_{K} → Aut^{∗} (ℚ_{ℓ}{{𝕏}}) ,

where

Aut^{∗} (ℚ_{ℓ}{{𝕏}}) =

{f ∈ Aut (ℚ_{ℓ}{{𝕏}}) | ∃ α(f) ∈ ℚ_{ℓ}^{∗} ∀ X_{i} ∈ 𝕏 ∃ l_{i} ∈ L(𝕏), f(X_{i}) = e^{−l_{i}} · α(f)X_{i} · e^{l_{i}}} .

We also set

Aut_{1}^{∗} (ℚ_{ℓ}{{𝕏}}) =

{f ∈ Aut (ℚ_{ℓ}{{𝕏}}) | ∀ X_{i} ∈ 𝕏 ∃ l_{i} ∈ L(𝕏), f(X_{i}) = e^{−l_{i}} · X_{i} · e^{l_{i}}} .

With the action of G_{K} on π_{1}(VK̄; v) there is associated a filtration {G_{i}(V_{K}, v)}_{i∈ℕ} of G_{K} (see [18] section 3).

Passing to Lie algebras we get a morphism of Lie algebras

Lie φ_{v} : Lie (G_{1}(V_{K}, v)/G_{∞}(V_{K}, v)) → Der (ℚ_{ℓ}{{𝕏}}) ,

where G_{∞}(V_{K}, v) := ⋂^{∞} ; i=1 G_{i}(V_{K}, v).

The morphism Lie φ_{v} factors through

Lie φ_{v} : Lie (G_{1}(V_{K}, v)/G_{∞}(V_{K}, v)) → Der^{∗} L(𝕏) ,

where

Der^{∗} L(𝕏) := {D ∈ DerL(𝕏) | ∀ 1 ≤ i ≤ n ∃ A_{i} ∈ L(𝕏), D(X_{i}) = [X_{i}, A_{i}]}.

Let z ∈ V̂ (K) and let p be a path from v to z. For any σ ∈ G_{K} we set

𝔣_{p}(σ) := p^{−1} · σ(p)  and  Λ_{p}(σ) := k(𝔣_{p}(σ)) .

Let π(VK̄; z, v) be a π_{1}(VK̄; v)-torsor of ℓ-adic paths from v to z. The Galois group G_{K} acts on π(VK̄; z, v), hence we have a representation

ψ_{z,v} : G_{K} → Aut_{set}(π(VK̄; z, v)) .

We identify π(VK̄; z, v) with π_{1}(VK̄; v) sending a path q onto a loop p^{−1} · q. After the identification of π(VK̄; z, v) with π_{1}(VK̄; v) and the embedding k, the Galois group G_{K} acts also on ℚ_{ℓ}{{𝕏}}. Therefore we get a representation

ψ_{p} : G_{K} → GL(ℚ_{ℓ}{{𝕏}})

given by

ψ_{p}(σ) = L_{Λ_{p}(σ)} ∘ φ_{v}(σ) .

The representation ψ_{p} factors through

ψ_{p} : G_{K} → L_{exp L(𝕏)} ˜×Aut^{∗}(ℚ_{ℓ}{{𝕏}}) ,

where L_{exp L(𝕏)} is the subgroup of GL(ℚ_{ℓ}{{𝕏}}) of left multiplications by elements of exp L(𝕏) and L_{exp L(𝕏)} ˜×Aut^{∗}(ℚ_{ℓ}{{𝕏}}) is a subgroup of GL(ℚ_{ℓ}{{𝕏}}) generated by L_{exp L(𝕏)} and Aut^{∗}(ℚ_{ℓ}{{𝕏}}). This subgroup is a semi-direct product of these two subgroups.

It follows immediately from [18] Proposition 2.2.1 that

φ_{v}(σ)(e^{X_{i}}) = Λ_{γ_{i}}(σ)^{−1} · e^{χ(σ)X_{i}} · Λ_{γ_{i}}(σ)

for any σ ∈ G_{K}. Hence we get

1.0.0.  φ_{v}(σ)(X_{i}) = Λ_{γ_{i}}(σ)^{−1} · (χ(σ)X_{i}) · Λ_{γ_{i}}(σ)

and

1.0.1.  ψ_{p}(σ)(X_{i}) = Λ_{p}(σ) · φ_{v}(σ)(X_{i})

for any σ ∈ G_{K}.

For simplicity we shall also denote ψ_{p}(σ) by σ_{p} and φ_{v}(σ) by σ, hence σ_{p} = L_{Λ_{p}(σ)} ∘ σ.

With the action of G_{K} on the π_{1}(VK̄; v)-torsor π(VK̄; z, v) there is associated a filtration {H_{i}(V_{K}; z, v)}_{i∈ℕ} of G_{K} (see [18] section 3).

The representation ψ_{p} induces a morphism of Lie algebras

Lie ψ_{p} : Lie (H_{1}(V_{K}; z, v)/H_{∞}(V_{K}; z, v)) → End (ℚ_{ℓ}{{𝕏}}) ,

where H_{∞}(V_{K}; z, v) := ⋂^{∞} ; i=1 H_{i}(V_{K}; z, v). The morphism Lie ψ_{p} factors through

Lie ψ_{p} : Lie (H_{1}(V_{K}; z, v)/H_{∞}(V_{K}; z, v)) → L_{L(𝕏)} ˜×Der^{∗}L(𝕏) ,

where L_{L(𝕏)} is the Lie algebra of left multiplications in ℚ_{ℓ}{{𝕏}} by elements of L(𝕏). The Lie algebra Der^{∗}L(𝕏) acts on L_{L(𝕏)} via its action on L(𝕏) and L_{L(𝕏)} ˜×Der^{∗}L(𝕏) is a semi-direct product.

One shows that for any σ ∈ H_{1}(V_{K}; z, v) = G_{K(μ_{ℓ^{∞}})} we have

log σ_{p} = L_{(log σ_{p})(1)} + log σ

and

(log σ)(X_{i}) = [X_{i}, (log σ_{γ_{i}})(1)]

for i = 1, 2, …, n (see [18]).

Passing to associated graded Lie algebras with the morphisms Lie φ_{v} and Lie ψ_{p} we get morphisms of associated graded Lie algebras

Φ_{v} := gr Lie φ_{v} : gr Lie (G_{1}(V_{K}, v)/G_{∞}(V_{K}, v)) ⊗ ℚ ≈

⨁_{i=1}^{∞} G_{i}(V_{K}, v)/G_{i+1}(V_{K}, v) ⊗ ℚ → Der^{∗}Lie (𝕏)

and

Ψ_{z,v} := gr Lie ψ_{p} : gr Lie (H_{1}(V_{K}; z, v)/H_{∞}(V_{K}; z, v)) ⊗ ℚ ≈

⨁_{i=1}^{∞} H_{i}(V_{K}; z, v)/H_{i+1}(V_{K}; z, v) ⊗ ℚ → L_{Lie(𝕏)} ˜×Der^{∗}Lie(𝕏) .

The derivation D ∈ Der^{∗}Lie(𝕏) such that D(X_{i}) = [X_{i}, A_{i}] for i = 1, …, n we shall denote by D_{(A_{i})_{i=1}^{n}}.

The vector space ⨁^{n} ; i=1 Lie (𝕏)/⟨X_{i}⟩ we equip with a Lie bracket { , } given by

{(a_{i})_{i=1}^{n}, (b_{i})_{i=1}^{n}} := ([a_{i}, b_{i}] + D_{(a_{j})_{j=1}^{n}}(b_{i}) − D_{(b_{j})_{j=1}^{n}}(a_{i}))_{i=1}^{n} .

(See [11] , where this bracket is introduced in the case of π_{1}(ℙ1 ; ℚ̄\{0, 1, ∞}, 01^{→}).)

The obtained Lie algebra we denote by (⨁^{n} ; i=1 Lie (𝕏)/⟨X_{i}⟩; { , }). The Lie

algebras Der^{∗}Lie(𝕏) and (⨁^{n} ; i=1 Lie (𝕏)/⟨X_{i}⟩; { , }) are isomorphic and an iso-

morphism is given by sending D_{(A_{i})_{i=1}^{n}} to (A_{i})_{i=1}^{n}.

Hence the representation Ψ_{z,v} can be regarded as a morphism of Lie algebras

Ψ_{z,v} : ⨁_{k=1}^{∞} H_{k}(V_{K}; z, v)/H_{k+1}(V_{K}; z, v) ⊗ ℚ →

Lie (𝕏) ˜× (⨁_{i=1}^{n} Lie (𝕏)/⟨X_{i}⟩; { , }) .

If σ ∈ H_{k}(V_{K}; z, v) then the coordinates of Ψ_{z,v}(σ) are

(log Λ_{p}(σ) mod Γ^{k+1}L(𝕏), (log Λ_{γ_{i}}(σ) mod ⟨X_{i}⟩ + Γ^{k+1}L(𝕏))_{i=1}^{n}) .

We finish this section by showing that the representations φ_{v} : G_{K} → Aut(ℚ_{ℓ}{{𝕏}}) and ψ_{p} : G_{K} → GL(ℚ_{ℓ}{{𝕏}}) are ℓ-adic mixed Tate modules (see [8] section 6, where ℓ-adic mixed Tate modules are defined).

Let

I(𝕏) := ker(ε : ℚ_{ℓ}{{𝕏}} → ℚ_{ℓ})

be the augmentation ideal. Observe that the powers of the augmentation ideal {I(𝕏)^{n}}_{n∈ℕ∪{0}} (we set {I(𝕏)^{0} := ℚ_{ℓ}{{𝕏}}) define a filtration of ℚ_{ℓ}{{𝕏}}. It follows immediately from the formulas 1.0.0 and 1.0.1 that

i) φ_{v}(σ)(I(𝕏)^{n}) ⊂ I(𝕏)^{n} and ψ_{p}(σ)(I(𝕏)^{n}) ⊂ I(𝕏)^{n},

ii) φ_{v} and ψ_{p} act on I(𝕏)^{n}/I(𝕏)^{n+1} by the n-th power χ^{n} of the cyclotomic character χ : G_{K} → ℤ_{ℓ}^{∗}.

Definition 1.0.2. (see also [24]) We define a weight filtration of ℚ_{ℓ}{{𝕏}} setting

W_{−2k+1}(ℚ_{ℓ}{{𝕏}}) = W_{−2k}(ℚ_{ℓ}{{𝕏}}) := I(𝕏)^{k}

for k ∈ ℕ ∪ {0}.

Proposition 1.0.3. We have

i) ⋂_{i=0}^{∞} W_{−i}(ℚ_{ℓ}{{𝕏}}) = 0 and ⋃_{i=0}^{∞} W_{−i}(ℚ_{ℓ}{{𝕏}}) = ℚ_{ℓ}{{𝕏}},

ii) φ_{v}(σ)(W_{−i}(ℚ_{ℓ}{{𝕏}})) ⊂ W_{−i}(ℚ_{ℓ}{{𝕏}}) and ψ_{p}(σ)(W_{−i}(ℚ_{ℓ}{{𝕏}})) ⊂ W_{−i}(ℚ_{ℓ}{{𝕏}}) for any σ ∈ G_{K},

iii) φ_{v} and ψ_{p} act on W_{−2i}(ℚ_{ℓ}{{𝕏}})/W_{−2i−2}(ℚ_{ℓ}{{𝕏}}) by the i-th power χ^{i} of the cyclotomic character χ.

Proof. The proposition follows immediately from the properties of the filtration {I(𝕏)^{n}}_{n∈ℕ∪{0}} of ℚ_{ℓ}{{𝕏}}. □

Let us equip I(𝕏) with the induced weight filtration by setting

W_{−2i+1}I(𝕏) = W_{−2i}I(𝕏) := I(𝕏)^{i}

for i ∈ ℕ. Obviously the representations

φ_{v} : G_{K} → GL(I(𝕏)) and ψ_{p} : G_{K} → GL(I(𝕏))

(obtained from restriction of φ_{v} and ψ_{p} to I(𝕏)) respect the filtration of I(𝕏). We define two filtrations of G_{K} associated with φ_{v} and ψ_{p} by setting

G_{i−1}(f) := {σ ∈ G_{K} | f(σ) ≡ Id_{I(𝕏)} mod I(𝕏)^{i}}

for i ∈ ℕ and where f = φ_{v} or f = ψ_{p}.

We set also

G_{∞}(f) := ⋂_{i=0}^{∞} G_{i−1}(f)

.

One sees easily that

G_{i}(φ_{v}) = G_{i}(V_{K}, v) and G_{i}(ψ_{p}) = H_{i}(V_{K}; z, v).

1.1. Let z_{1}, …, z_{r} be K-points of V or tangential points defined over K. We

shall study the action of G_{K} on the disjoint union ∐^{r} ; j=1 π(VK̄; z_{j}, v) and on the

product ∏^{r} ; j=1 π(VK̄; z_{j}, v) of π_{1}(VK̄; v)-torsors.

We start with some linear algebra. Let f_{1}, …, f_{r} ∈ L(𝕏). Then L_{exp f_{1}} ⊕

… ⊕ L_{exp f_{r}} is an automorphism of the vector space ⨁^{r} ; j=1 ℚ_{ℓ}{{𝕏}}. The set of linear automorphisms of ⨁^{r} ; j=1 ℚ_{ℓ}{{𝕏}} of the form L_{exp f_{1}} ⊕ … ⊕ L_{exp f_{r}}, where

f_{1}, …, f_{r} ∈ L(𝕏) is a group which we denote by ⨁^{r} ; j=1 L_{exp L(𝕏)}.

Similarly L_{exp f_{1}} ⊗ … ⊗ L_{exp f_{r}} is an automorphism of the vector space ⨂^{r} ; j=1 ℚ_{ℓ}{{𝕏}}. The set of linear automorphisms of ⨂^{r} ; j=1 ℚ_{ℓ}{{𝕏}} of the form L_{exp f_{1}} ⊗ … ⊗ L_{exp f_{r}}, where f_{1}, …, f_{r} ∈ L(𝕏) is a group which we denote by

⨂^{r} ; j=1 L_{exp L(𝕏)}.

The group Aut^{∗} ℚ_{ℓ}{{𝕏}} acts on ⨁^{r} ; j=1 ℚ_{ℓ}{{𝕏}} and on ⨂^{r} ; j=1 ℚ_{ℓ}{{𝕏}} diago-

nally. The subgroup of GL ( ⨁^{r} ; j=1 ℚ_{ℓ}{{𝕏}}) (resp. GL ( ⨂^{r} ; j=1 ℚ_{ℓ}{{𝕏}})) gene-

rated by ⨁^{r} ; j=1 L_{exp L(𝕏)} (resp. ⨂^{r} ; j=1 L_{exp L(𝕏)}) and Aut^{∗} ℚ_{ℓ}{{𝕏}} is a semi-direct product of these two subgroups and we denote it by

(⨁_{j=1}^{r} L_{exp L(𝕏)}) ˜× Aut^{∗}ℚ_{ℓ}{{𝕏}} (resp. (⨂^{r} ; j=1 L_{exp L(𝕏)}) ˜× Aut^{∗} ℚ_{ℓ}{{𝕏}}).

The Lie algebra of

(⨁_{j=1}^{r} L_{exp L(𝕏)}) ˜× Aut_{1}^{∗} ℚ_{ℓ}{{𝕏}} (resp. (⨂^{r} ; j=1 L_{exp L(𝕏)}) ˜× Aut_{1}^{∗} ℚ_{ℓ}{{𝕏}})

is

(⨁_{j=1}^{r} L_{L(𝕏)}) ˜× Der^{∗}L(𝕏)

(resp. ( ⨁^{r} ; j=1 (Id ⊗ … ⊗ L_{L(𝕏)} ⊗ … ⊗ Id)) ˜× Der^{∗}L(𝕏)). The above facts are proved in [18] section 5 for r = 1. Proofs for r > 1 are similar and we omit them.

We return to study Galois representations on the disjoint union t := ∐^{r} ; j=1 π(VK̄; z_{j}, v)

and on the product T := ∏^{r} ; j=1 π(VK̄; z_{j}, v) of π_{1}(VK̄; v)-torsors.

We fix paths p_{j} from v to z_{j} for j = 1, …, r. We recall that k : π_{1}(VK̄; v) → ℚ_{ℓ}{{𝕏}} is a fixed continuous multiplicative embedding. Each π_{1}(VK̄; v)-torsor π(VK̄; z_{j}, v) we embed into ℚ_{ℓ}{{𝕏}} sending the path q onto k(p_{j}^{−1} · q).

It follows from the formalism presented in section 1.0 that the action of G_{K}

on ∐^{r} ; j=1 π(VK̄; z_{j}, v) (resp. ∏^{r} ; j=1 π(VK̄; z_{j}, v)) yields a Galois representation

ψ_{t} : G_{K} → (⨁_{j=1}^{r} L_{exp L(𝕏)}) ˜× Aut^{∗}ℚ_{ℓ}{{𝕏}}

(resp. ψ_{T} : G_{K} → ( ⨂^{r} ; j=1 L_{exp L(𝕏)}) ˜× Aut^{∗}ℚ_{ℓ}{{𝕏}}).

Let S := {z_{1}, …, z_{r}}. We define a filtration

{H_{i}(V_{K}; S, v)}_{i∈ℕ}

of the group G_{K} setting

H_{i}(V_{K}; S, v) := ⋂_{j=1}^{r} H_{i}(V_{K}; z_{j}, v)

for i ∈ ℕ. Observe that the subgroup

H_{∞}(V_{K}; S, v) := ⋂_{i=1}^{∞} H_{i}(V_{K}; S, v)

is the kernel of representations ψ_{t} and ψ_{T}. Passing with the representations ψ_{t} and ψ_{T} to associated graded Lie algebras we get morphisms of associated graded Lie algebras

gr Lie ψ_{t} : ⨁_{i=1}^{∞} H_{i}(V_{K}; S, v)/H_{i+1}(V_{K}; S, v) ⊗ ℚ → (⨁_{j=1}^{r} Lie(𝕏)) ˜× Der^{∗}Lie(𝕏)

and

gr Lie ψ_{T} : ⨁_{i=1}^{∞} H_{i}(V_{K}; S, v)/H_{i+1}(V_{K}; S, v) ⊗ ℚ →

(⨁_{j=1}^{r} Id ⊗ … ⊗ Id ⊗ L_{Lie(𝕏)} ⊗ Id ⊗ … ⊗ Id) ˜× Der^{∗}Lie(𝕏) .

Lemma 1.1.0. Let σ ∈ H_{k}(V_{K}; S, v). The coordinates of gr Lie ψ_{t}(σ) are

((log Λ_{p_{j}}(σ) mod Γ^{k+1}L(𝕏))_{j=1}^{r}; (log Λ_{γ_{i}}(σ) mod ⟨X_{i}⟩ + Γ^{k+1}L(𝕏))_{i=1}^{n}) .

□

We introduce the following notation. If A and B are elements of a Lie algebra then we set

[A, B^{0}] := A and [A, B^{n}] := [[A, B^{n−1}], B] for n > 0.

2 Ramification

In this section we shall study ramification properties of Galois representations on fundamental groups and on torsors of paths. (See also [23] where more detailed results are presented.) We recall that V = ℙ_{K}^{1}\{a_{1}, …, a_{n}, a_{n+1}} and v is a K-point or a tangential point defined over K. For simplicity we assume that a_{n+1} = ∞.

We denote by V(K) the set of finite places of the field K. Let 𝔭 ∈ V(K). We denote by v_{𝔭} : K^{∗} → ℤ the valuation associated with 𝔭.

Definition 2.0. Let 𝔭 be a finite place of K. We say that a pair (V, v) has good reduction at 𝔭 if

i) v_{𝔭}(a_{i}) ≥ 0 for all i ∈ {1, …, n};

ii) if v is a K-point of V then v_{𝔭}(v) ≥ 0, if v = a_{i_{0}}y^{⟶} is a tangential point defined over K then v_{𝔭}(y) ≥ 0;

iii) v_{𝔭}(a_{i} − a_{j}) = 0 for all pair i ≠ j;

iv) if v is a K-point of V then v_{𝔭}(a_{i} − v) = 0 for all i ∈ {1, …, n}, if v = a_{i_{0}}y^{⟶} then v_{𝔭}(a_{i_{0}} − y) = 0.

If any of these conditions is not satisfied then we say that a pair (V, v) has potentially bad reduction at 𝔭.

The next result generalizes Theorem from [9].

Theorem 2.1. Let S′ be a set of finite places of K where a pair (V, v) has potentially bad reduction. Let S = S′ ∪ {λ ∈ V(K) | λ divides ℓ}. The representation

φ_{v} : G_{K} → Aut (π_{1}(VK̄; v))

is unramified outside S.

Proof. We shall imitate the proof of Theorem in [9]. Let us take 𝔭 ∉ S. Let K_{𝔭} be the 𝔭-completion of K, let O_{K_{𝔭}} be the ring of integers of K_{𝔭} and let k_{𝔭} be the residue field. Let K̄_{𝔭} and k̄_{𝔭} be algebraic closures of K_{𝔭} and k_{𝔭} respectively. Let K_{𝔭}^{ur} be the maximal unramified extension of K_{𝔭} contained in K̄_{𝔭} and let O_{K_{𝔭}}^{ur} be the ring of integers of K_{𝔭}^{ur}.

We define a smooth projective scheme X over O_{K_{𝔭}} by a system of equations

T_{i}^{ℓ^{m}} − T_{j}^{ℓ^{m}} = a_{j} − a_{i}

for all pairs (i, j) with i ≠ j. The morphism p_{m} : X → ℙ_{O_{K𝔭}}^{1} given by z → T_{i}^{ℓ^{m}} +a_{i} is an étale, abelian (ℤ/ℓ^{m})^{n}-covering outside the set {a_{1}, …, a_{n}, a_{n+1}} and with ramification indices ℓ^{m} at each point a_{i}.

The K-point v is defined over O_{K_{𝔭}} and its reduction modulo 𝔭 we denote by v̄. Let us consider morphisms of pointed schemes

(Xk̄_{𝔭} := X ⊗_{O_{K𝔭}} k̄_{𝔭}, v̄) → (X ⊗_{O_{K𝔭}} O_{K_{𝔭}}^{ur}, v) ← (XK̄_{𝔭} := X ⊗_{O_{K𝔭}} K̄_{𝔭}, v) .

By the Grothendieck comparison theorem the categories of finite ℓ-coverings of Xk̄_{𝔭} and XK̄_{𝔭} are equivalent. Moreover the fiber functors defined by v̄ and v are isomorphic. Finite étale ℓ-coverings of Xk̄_{𝔭} and their morphisms are obtained from finite étale ℓ-coverings of X ⊗_{O_{K𝔭}} O_{K_{𝔭}}^{ur} and their morphisms. After tensoring with K̄_{𝔭} one gets finite étale ℓ-coverings of XK̄_{𝔭} and morphisms of these coverings over XK̄_{𝔭}. Any finite étale ℓ-covering of VK̄ is dominated by the composition of a finite étale ℓ-covering of XK̄_{𝔭} for some m and the projection p_{m} : XK̄_{𝔭} → VK̄_{𝔭}. Hence the action of Gal(K̄_{𝔭}/K_{𝔭}) on π_{1}(VK̄; v) factors through the action of Gal(K_{𝔭}^{ur}/K_{𝔭}) = Gal(k̄_{𝔭}/k_{𝔭}). □

Let z be also a K-point of V or a tangential point defined over K.

Definition 2.2. Let 𝔭 be a finite place of K. We say that a triple (V, z, v) has good reduction at 𝔭 if both pairs (V, z) and (V, v) have good reduction at 𝔭. If this condition is not satisfied then we say that a triple (V, z, v) has potentially bad reduction at 𝔭.

Proposition 2.3. Let T′ be a set of finite places of K where a triple (V, z, v) has potentially bad reduction. Let T = T′ ∪ {λ ∈ V(K) | λ divides ℓ}. The representation

ψ_{z,v} : G_{K} → Aut_{set}(π(VK̄; z, v))

is unramified outside T.

Proof. We repeat the proof of Proposition 2.1. □

If R is a set of finite places of K then we define

O_{K,R} := {x ∈ K | ∀ 𝔭 ∉ R , v_{𝔭}(x) ≥ 0} .

Corollary 2.4. The representation

φ_{v} : G_{K} → Aut(π_{1}(VK̄; v)) (resp. ψ_{z,v} : G_{K} → Aut_{set}(π(VK̄; z, v)))

factors through the epimorphism

G_{K} → π_{1}(Spec O_{K,S}; Spec K̄) (resp. G_{K} → π_{1}(Spec O_{K,T}; Spec K̄))

induced by the inclusion O_{K,S} ↪ K (resp. O_{K,T} ↪ K).

Proof. The étale fundamental group of Spec O_{K,R} is the Galois group Gal(F/K), where F is a maximal Galois extension of K unramified outside R. □

In [20] we have studied representations of Galois groups on π_{1}(Vℚ̄; 01^{→}), where V := ℙ_{ℚ(μ_{n})}^{1}\({0, ∞} ∪ μ_{n}). We have constructed a family of derivations in the image of the homomorphism

gr Lie φ01^{→} : gr Lie(G_{1}(V_{ℚ(μ_{n})}, 01^{→})/G_{∞}(V_{ℚ(μ_{n})}, 01^{→})) → Der^{∗}Lie(𝕏).

We raised a question if these derivations generate the image of gr Lie φ01^{→}. The next result is the first step to give an affirmative answer to this question (see also [23]). Corollary 2.5. Let V = ℙ_{ℚ(μ_{n})}^{1}\({0, ∞} ∪ μ_{n}). Let S be a set of finite places of ℚ(μ_{n}) which divide n or ℓ. Then the representation

φ01^{→} : G_{ℚ(μ_{n})} → Aut(π_{1}(V\overline{ℚ(μ_{n})}; 01^{→}))

is unramified outside S. The representation φ01^{→} factors through the epimorphism

G_{ℚ(μ_{n})} → π_{1}(SpecO_{ℚ(μ_{n}),S}; Spec\overline{ℚ(μ_{n})}).
□

3 ℓ-adic polylogarithms

3.0. ℓ-adic polylogarithms are very important in this paper hence we recall here their definition (see [19], [13] and [21]). In fact we shall give three slightly different definitions.

Let V := ℙ_{K}^{1}\{0, 1, ∞}. We denote by x and y standard generators of

π_{1}(VK̄; 01^{→}). Let k : π_{1}(VK̄; 01^{→}) → ℚ_{ℓ}{{X, Y }} be the standard continuous multiplicative embedding given by k(x) = e^{X} and k(y) = e^{Y}. Let z be a K-

point of V_{K} or a tangential point defined over K. Let p be a path from 01^{→} to z.

We recall that Lie (X, Y) is a free Lie algebra over ℚ_{ℓ} on X and Y and L(X, Y) := \underleftarrow{lim}_{n} Lie (X, Y)/Γ^{n} Lie (X, Y) is a completed free Lie algebra over ℚ_{ℓ}

on X and Y.

Let I_{r} be a closed Lie ideal of L(X, Y) generated by Lie brackets which contain r or more Y ’s.

Definition 3.0.1. (see [19]) We define functions ℓ(z)_{p} and ℓ_{n}(z)_{p} from G_{K} to ℚ_{ℓ} by the congruence

log Λ_{p}(σ) ≡ ℓ(z)_{p} (σ) X + ∑_{n=1}^{∞} ℓ_{n}(z)_{p} (σ) [Y, X^{n−1}] mod I_{2} .

The function ℓ(z)_{p} we call ℓ-adic logarithm and ℓ_{n}(z)_{p} we call ℓ-adic polylogarithm of degree n.

This definition is very useful to study Galois representations on fundamental groups of ℙ1 ; \overline{ℚ(μ_{n})}\(\{0, ∞\} ∪ μ_{n}) (see [20]).

The next version of ℓ-adic polylogarithms is more close to the definition given in [1].

Definition 3.0.2. We define functions ℓi_{n}(z)_{p} from G_{K} to ℚ_{ℓ} by the congruence

log(k(𝔣_{p}(σ) · x^{−ℓ(z)_{p}(σ)})) ≡ ∑_{n=1}^{∞} ℓi_{n}(z)_{p} (σ) [Y, X^{n−1}] mod I_{2} .

Now we shall give the third definition. Observe that the path p defines a compatible family {z^{1/ℓ^{n}}}_{n∈ℕ} of ℓ^{n}-th roots of z by the analytic continuation of the constant family {1}_{n∈ℕ} of ℓ^{n}-th roots of 1. Using the development of (1 − ξ_{l^{n}}^{i} z^{1/l^{n}})^{1/l^{n}} into a power series of ξ_{l^{n}}^{i} z^{1/l^{n}} for small z and analytic continuation in general case we get also a compatible family {(1 − ξ_{l^{n}}^{i} z^{1/l^{n}})^{1/l^{n}}}_{n∈ℕ} of ℓ^{n}-th roots of 1 − ξ_{l^{n}}^{i} z^{1/l^{n}}.

Definition 3.0.3. We define functions ℓ_{k}(z)_{p} (k ∈ ℕ) from G_{K} to ℤ_{ℓ} by the identity

ξ_{l^{n}}^{ℓ_{k}(z)_{p}(σ)} = \frac{σ ( ∏^{l^{n}−1} ; i=0(1 − ξ_{l^{n}}^{i} z^{1/l^{n}})^{\frac{i^{k−1}}{l^{n}}} )}{∏^{l^{n}−1} ; i=0(1 − ξ_{l^{n}}^{χ(σ)i+κ(z)(σ)} · z^{1/l^{n}})^{\frac{i^{k−1}}{l^{n}}}} ,

where κ(z) : G_{K} → ℤ_{ℓ} is the Kummer character associated with the family {z^{1/l^{n}}}_{n∈ℕ}.

Proposition 3.0.4. We have

i) ℓ_{n}(z)_{p}(σ) = ℓi_{n}(z)_{p}(σ) for σ ∈ H_{2}(V_{K}; z, 01^{→});

ii) ℓi_{n}(z)_{p}(σ) = \frac{(−1)^{n−1}}{(n−1)!} ℓ_{n}(z)_{p}(σ) for σ ∈ G_{K}.

Proof. If σ ∈ H_{2}(V_{K}; z, 01^{→}) then ℓ(z)_{p}(σ) = 0. This implies the first part of the proposition. The second part is proved in [13]. □

3.1. In the next section we shall use coinvariant group functor. We recall here its definition and its elementary properties.

We start with the following general situation. Let a group G act on a group N by automorphisms. Let I(N , G) be a normal subgroup of N generated by elements g(n) · n^{−1}, where n ∈ N and g ∈ G. Then the coinvariant group N_{G} is defined by setting

N_{G} := N /I(N , G).

The coinvariant group functor is right exact.

Let us denote by

π_{N} : N → N_{G}

the quotient map.

Proposition 3.1.1. Let

1 → C →^{i} B →^{pr} A → 1

be an exact sequence of groups. We assume that a group G acts on A, B and C by automorphisms and that i and pr are G-maps. Then i) the sequence of coinvariants

C_{G} →^{i_{G}} B_{G} →^{pr_{G}} A_{G} → 1

is exact;

ii) ker(pr_{G}) = pr^{−1}(ker π_{A})/ker π_{B};

iii) the natural map C_{G} → ker(pr_{G}) is surjective and its kernel is the group (C ∩ ker π_{B})/ker π_{C}.

Proof. The point i) is well known. The point ii) is clear. We shall show the point iii). Let a ∈ pr^{−1}(ker π_{A}). We recall that ker π_{A} is a normal subgroup of A generated by elements of the form g(α) · α^{−1} for g ∈ G and α ∈ A. Hence the assumption that the map pr : B → A is surjective implies that we can find b ∈ ker π_{B} such that pr(b) = pr(a). Then pr(a · b^{−1}) = 1. Hence a · b^{−1} ∈ C and a · b^{−1} ≡ a mod ker π_{B}. Hence we have shown that the natural map from C_{G} to ker(pr_{G}) is surjective. Obviously its kernel is the group (C ∩ ker π_{B})/ker π_{C}. □

We give an example when the sequence of coinvariants is also exact. Corollary 3.1.2. Let

1 → C →^{i} B →^{pr} A → 1

be an exact sequence of groups. We assume that a group G acts on A, B and C by automorphisms and that i and pr are G-maps. We assume also that the group G acts trivially on A and that the set H^{1}(G, C) has one element. Then the sequence of coinvariants

1 → C_{G} →^{i_{G}} B_{G} →^{pr_{G}} A → 1

is also exact.

Proof. We have ker π_{B} ⊂ C because G acts trivially on A. The subgroup ker π_{B} is a normal subgroup of B generated by elements g(b) · b^{−1} for g ∈ G and b ∈ B. Hence ker π_{B} is a subgroup of B generated by elements b_{1} · g(b) · b^{−1} · b_{1}^{−1} for g ∈ G and b, b_{1} ∈ B. Observe that b_{1} · g(b) · b^{−1} · b_{1}^{−1} = g(b′) · g(b) · b^{−1} · g(b′)^{−1} = g(b′ · b) · (b′ · b)^{−1} · b′ · g(b′)^{−1} for some b′ ∈ B. Therefore ker π_{B} is a subgroup of B generated by elements g(b) · b^{−1} for g ∈ G and b ∈ B.

Let b ∈ B. Observe that a function f : G → C given by f(g) := g(b) · b^{−1} is a cocycle. Hence there is c ∈ C such that f(g) := g(c) · c^{−1} for any g ∈ G. This implies that ker π_{B} = ker π_{C}. Now Proposition 3.1.1 implies that the sequence of coinvariants is exact. □

4 Descent

4.0. Let us fix a number field K and let a_{1}, …, a_{n+1} be K-points of ℙ_{K}^{1}. We set

V_{K} := ℙ_{K}^{1}\{a_{1}, …, a_{n+1}} .

We assume for simplicity that a_{n+1} = ∞. Let f(T) ∈ K[T] be a polynomial and let z_{1}, …, z_{r} be all roots of f(T) in the algebraic closure K̄ of K. We define L to be a field obtained from K by adjoining all roots of the polynomial f(T), i.e.,

L := K(z_{1}, …, z_{r}) .

Then L is a Galois extension of K and we set

G := Gal (L/K) .

The Galois group G permutes roots of the polynomial f(T). If g ∈ G then g(z_{i}) = z_{g(i)} for some g(i) ∈ {1, …, r}.

Let v be a fixed K-point of V_{K} or a tangential point of V_{K} defined over K. We consider a disjoint union (resp. a product) of torsors of paths

t := ∐_{j=1}^{r} π(VK̄; z_{j}, v) (resp. T := ∏_{j=1}^{r} π(VK̄; z_{j}, v)) .

We shall study actions of Galois groups G_{L} and G_{K} on the disjoint union and the product of torsors of paths.

Let S := {z_{1}, …, z_{r}}. With the action of G_{L} on the disjoint union of torsors t or on the product of torsors T there is associated a filtration {H_{k}}_{k∈ℕ} of G_{L} defined by

H_{k} := H_{k}(V_{L}; S, v) := ⋂_{j=1}^{r} H_{k}(V_{L}; z_{j}, v)

(see section 1). We recall that each subgroup H_{k} of G_{L} is a normal subgroup of G_{L}, H_{1} = G_{L(μ_{l^{∞}})}, the group Γ := Gal (L(μ_{l^{∞}})/L) ⊂ ℤ_{l}^{∗} acts on H_{k}/H_{k+1} by conjugation and we have an isomorphism of Γ-modules H_{k}/H_{k+1} ≈ ℤ_{ℓ}(k)^{m_{k}} (see [18] section 3).

Now we shall try to explore the action of G = Gal (L/K) on the tower of groups {H_{k}}_{k∈ℕ}. We shall assume that L ∩ K(μ_{l^{∞}}) = K. Then Gal (L(μ_{l^{∞}})/ K(μ_{l^{∞}})) = G. Observe that we have the following tower of groups

H_{∞} ⊂ … ⊂ H_{j+1} ⊂ H_{j} ⊂ … ⊂ H_{3} ⊂ H_{2} ⊂ H_{1} = G_{L(μ_{l^{∞}})} ⊂ G_{K(μ_{l^{∞}})} ↠ G .

We shall lift an element τ ∈ G to τ̃ ∈ G_{K(μ_{l^{∞}})}. We shall study the action of τ̃ by conjugation on the tower {H_{k}}_{k∈ℕ}.

Lemma 4.0.1. Let us assume that L ∩ K(μ_{l^{∞}}) = K. Let τ ∈ G and let τ̃ be a lifting of τ to G_{K(μ_{l^{∞}})}.

i) If σ ∈ H_{j} then τ̃ · σ · τ̃^{−1} ∈ H_{j}.

ii) The class of τ̃ · σ · τ̃^{−1} modulo H_{j+1} does not depend on a choice of lifting τ̃ of τ.

iii) The group G acts on H_{j}/H_{j+1} by conjugation, i.e., the formula τ(σ · H_{j+1}) := τ̃ · σ · τ̃^{−1} · H_{j+1} defines an action of G on H_{j}/H_{j+1}.

Proof. The element σ belongs to H_{j} if and only if σ ∈ G_{j}(V_{L}, v) and for any path γ from x to any z_{i} we have 𝔣_{γ}(σ) ≡ 1 mod Γ^{j}π_{1}(VK̄; v).

The lower central series filtration is preserved by any automorphism, hence τ̃ · σ · τ̃^{−1} ∈ G_{j}(V_{L}, v) if σ ∈ G_{j}(V_{L}, v). Let γ be a path from v to z_{i}. Then 𝔣_{γ}(τ̃ · σ · τ̃^{−1}) = τ̃(𝔣_{τ̃^{−1}(γ)}(σ)) Observe that τ̃^{−1}(γ) is a path from v to z_{τ^{−1}(i)}, hence 𝔣_{τ̃^{−1}(γ)}(σ) ≡ 1 mod Γ^{j}π_{1}(VK̄; v). This implies that 𝔣_{γ}(τ̃ · σ · τ̃^{−1}) = τ̃(𝔣_{τ̃^{−1}(γ)}(σ)) ≡ 1 mod Γ^{j}π_{1} (VK̄; v). Hence τ̃ · σ · τ̃^{−1} ∈ H_{j}.

Let τ′ = τ̃·η, where η ∈ H_{1}, be another lifting of τ. Then τ′·σ·τ^{′−1} = τ̃·η·σ· η^{−1}·τ̃^{−1} = τ̃·σ·σ_{j+1}·τ̃^{−1} ≡ τ̃·σ·τ̃^{−1} mod H_{j+1}, because η·σ·η^{−1} = σ mod H_{j+1} and τ̃ · σ_{j+1} · τ̃^{−1} ∈ H_{j+1} if σ_{j+1} ∈ H_{j+1}.

Therefore the points i) and ii) are proved. The point iii) follows immediately from i) and ii). □

We recall that ⨁^{∞} ; k=1 H_{k}/H_{k+1} ⊗ ℚ is a Lie algebra over ℚ_{ℓ}, whose Lie bracket

is induced by the commutator, i.e., if ā ∈ H_{k}/H_{k+1} and b̄ ∈ H_{j}/H_{j+1} then [ā, b̄] := \overline{a · b · a^{−1} · b^{−1}} ∈ H_{k+j}/H_{k+j+1}. It follows from Lemma 4.0.1 that the group G acts on each ℚ_{ℓ}-vector space H_{k}/H_{k+1} ⊗ ℚ. Hence we can decompose the G-module H_{k}/H_{k+1} ⊗ ℚ into a direct sum of irreducible representations of G. We get

H_{k}/H_{k+1} ⊗ ℚ ≈ (H_{k}/H_{k+1} ⊗ ℚ)^{G} ⊕ ⨁_{χ_{i}}(H_{k}/H_{k+1} ⊗ ℚ)_{χ_{i}} ,

where (H_{k}/H_{k+1} ⊗ ℚ)^{G} is a fixed part and the sum is over all non-trivial irreducible representations of G.

Lemma 4.0.2. The fixed part of the action of G, ⨁^{∞} ; k=1 (H_{k}/H_{k+1} ⊗ ℚ)^{G} is a Lie

subalgebra of a Lie algebra ⨁^{∞} ; k=1 (H_{k}/H_{k+1} ⊗ ℚ).

Proof. Let ā ∈ H_{k}/H_{k+1} and b̄ ∈ H_{j}/H_{j+1} be fixed by the action of G. Then g(a) = a·c and g(b) = b·d, where c ∈ H_{k+1} and d ∈ H_{j+1}. It follows immediately from the Witt-Hall identities that \overline{g(a) · g(b) · g(a)^{−1} · g(b)^{−1}} = \overline{a · b · a^{−1} · b^{−1}}.

□

4.1. We shall study subfields of an algebraic closure L̄ of the field L associated with the filtration {H_{k}}_{k∈ℕ} of G_{L}. We define subfields of L̄ fixed by H_{k} setting

L_{k} := L̄^{H_{k}} for k ∈ ℕ

and

L_{∞} := ⋃_{k=1}^{∞} L_{k} .

Observe that L_{∞} = L̄^{H_{∞}}. We have got a tower of fields

L ⊂ L_{1} = L(μ_{ℓ^{∞}}) ⊂ L_{2} ⊂ … ⊂ L_{n} ⊂ L_{n+1} ⊂ … ⊂ L_{∞} ⊂ L̄
∪  ∪
K ⊂ K_{1} = K(μ_{ℓ^{∞}})

such that

Gal (L̄/L_{k}) = H_{k} and Gal (L_{k}/L_{j}) = H_{j}/H_{k}

for k, j ∈ ℕ ∪ {∞} and k ≥ j.

For any k ∈ ℕ ∪ {∞} we have an exact sequence of Galois groups

1 ⟶ Gal (L_{k}/L_{1}) ⟶ Gal (L_{k}/K_{1}) ⟶^{pr_{k}} Gal (L_{1}/K_{1}) ⟶ 1 .
∥  ∥
(∗′_{k})  G_{L(μ_{ℓ^{∞}})}/H_{k}  G

We assume that ℓ does not divide the order of G. The group Gal (L_{k}/L_{1}) is a pro-ℓ group. Therefore the exact sequence (∗′_{∞}) splits, i.e., there is a homomorphism θ′_{∞} : G → Gal (L_{∞}/K_{1}) such that pr_{∞} ∘ θ′_{∞} = id_{G}. The composition of θ′_{∞} with the inclusion Gal (L_{∞}/K_{1}) → Gal (L_{∞}/K) defines a splitting θ_{∞} : G → Gal (L_{∞}/K) of the exact sequence (∗_{∞}), where

1 ⟶ Gal (L_{k}/L) ⟶ Gal (L_{k}/K) ⟶^{pr_{k}} Gal (L/K) ⟶ 1 .
∥  ∥
(∗_{k})  G_{L}/H_{k}  G

for k ∈ ℕ ∪ {∞}. Then we define a splitting θ_{k} of the sequence (∗_{k}) for k ∈ ℕ to be a composition of θ_{∞} with the projection Gal (L_{∞}/K) → Gal (L_{k}/K). Therefore the group G acts on Gal (L_{k}/L) by g(σ) := θ_{k}(g) · σ · θ_{k}(g)^{−1} for k ∈ ℕ ∪ {∞}. These actions are compatible in the sense that the projection Gal (L_{k}/L) → Gal (L_{j}/L) is a G-map for k > j. The group Gal (L_{k}/K) is a semi-direct product of Gal (L_{k}/L) by G, i.e., Gal (L_{k}/K) = Gal (L_{k}/L) ˜×G for k ∈ ℕ ∪ {∞}.

We recall that Gal (L_{k}/L)_{G} is the maximal quotient of Gal (L_{k}/L) on which G acts by the identity. Let π_{k} : Gal (L_{k}/L) → Gal (L_{k}/L)_{G} be the natural projection. Then we also have a projection

π′_{k} : Gal (L_{k}/K) = Gal (L_{k}/L) ˜×G → Gal (L_{k}/L)_{G} × G .

Observe that we have ker π_{k} = ker π′_{k}. We define subfields L′_{k} of L̄ by setting

L′_{k} := L_{k}^{ker π_{k}} .

Then the field L′_{k} is a Galois extension of L and also of K because ker π_{k} (resp. ker π′_{k}) is a normal subgroup of Gal (L_{k}/L) (resp. Gal(L_{k}/K)). Therefore we have isomorphisms

4.1.0. Gal (L′_{k}/L) = Gal (L_{k}/L)_{G} and Gal (L′_{k}/K) = Gal (L_{k}/L)_{G} × G .

Lemma 4.1.1. Let us assume that ℓ does not divide the order of G. We have

Gal (L′_{k}/L_{1}) = Gal (L_{k}/L_{1})_{G}.

Proof. The sequence of coinvariants of Galois groups of a tower of fields

L ⊂ L_{1} ⊂ L_{k}

is exact by Corollary 3.1.2. This exact sequence maps into exact sequence of Galois groups of a tower of fields

L ⊂ L_{1} ⊂ L′_{k}

(see Diagram 1).

1 ⟶ Gal(L_{k}/L_{1})_{G} ⟶ Gal(L_{k}/L)_{G} ⟶ Gal(L_{1}/L)_{G} ⟶ 1
α↓  =↓  =↓
1 ⟶ Gal(L′_{k}/L_{1}) ⟶ Gal(L′_{k}/L) ⟶ Gal(L_{1}/L) ⟶ 1 .

Diagram 1

Hence it follows that the natural map Gal(L_{k}/L_{1})_{G} → Gal(L′_{k}/L) is an isomorphism. □

Lemma 4.1.2. Let j < k. The field L′_{k} is a Galois extension of L′_{j} and

Gal (L′_{k}/L′_{j}) = pr_{k,j}^{−1}(ker π_{j})/ker π_{k},

where pr_{k,j} : Gal (L_{k}/L) → Gal (L_{j}/L) is the natural projection.

Proof. Observe that L′_{j} ⊂ L′_{k} because any element of Gal (L_{k}/L) which fixes L′_{k}, fixes also L′_{j}. The fields L′_{k} and L′_{j} are Galois extensions of L. Hence L′_{k} is a Galois extension of L′_{j}. We have an exact sequence of Galois groups

1 ⟶ Gal (L′_{k}/L′_{j}) ⟶ Gal (L′_{k}/L)  ⟶  Gal (L′_{j}/L) ⟶ 1
∥  ∥
Gal (L_{k}/L)_{G} ⟶^{(pr_{k,j})_{G}} Gal (L_{j}/L)_{G} .

It follows from Proposition 3.1.1 that

ker ((pr_{k,j})_{G}) = pr_{k,j}^{−1}( ker π_{j})/ker π_{k}.

Hence we get

Gal (L′_{k}/L′_{j}) ≈ pr_{k,j}^{−1}( ker π_{j})/ker π_{k}.
□

Let k ∈ ℕ ∪ {∞}. Let K_{k} be a subfield of L′_{k} fixed by G, i.e.,

K_{k} := L_{k}^{′G}.

Then it follows from 4.1.0 that

4.1.3. Gal (K_{k}/K) = Gal (L′_{k}/L) = Gal (L_{k}/L)_{G} and Gal (L′_{k}/K_{k}) = G .

Definition 4.1.4. We define a filtration {F_{k}(V_{L/K}; S, v)}_{k∈ℕ} of G_{K} by setting

F_{k}(V_{L/K}; S, v) := Gal (K̄/K_{k}) .

To simplify the notation we denote F_{k}(V_{L/K}; S, v) by F_{k}. We set

F_{∞} := F_{∞}(V_{L/K}; S, v) := ⋂_{k=1}^{∞} F_{k} .

Observe that F_{∞} = Gal (K̄/K_{∞}). Observe that each F_{k} is a normal subgroup of G_{K} and that for any j < k, F_{j} is a normal subgroup of F_{k}.

We indicate ramification properties of the fields L_{∞} and K_{∞}. Proposition 4.1.5. Let S′_{j} be the set of finite places of L, where the triple (V_{L}, z_{j}, v) has potentially bad reduction. Let S = ⋃^{r} ; j=1 S′_{j}∪{λ ∈ V(L) | λ divides ℓ}. Then the representation

ψ_{t} : Gal(L̄/L) → Aut_{set} (∐^{r} ; j=1 π(VL̄; z_{j}, v))

is unramified outside S. Proof. The proposition follows immediately from Proposition 2.3. □ Corollary 4.1.6. The field extension L_{∞} of L is unramified outside S. □ Corollary 4.1.7. The representation

ψ_{t} : Gal(L̄/L) → Aut_{set} (∐_{j=1}^{r} π(VL̄; z_{j}, v))

factors through the epimorphism

Gal(L̄/L) → π_{1}(Spec O_{L,S}; Spec L̄)

induced by the inclusion O_{L,S} ⊂ L. □

We recall that V(K) is the set of finite places of the field K. Let us set

T := {𝔮 ∈ V(K) | ∃ 𝔭 ∈ S , 𝔮 = 𝔭 ∩ O_{K}}.

Proposition 4.1.8. The field extension K_{∞} of K is unramified outside T.

Proof. The extension L′_{∞} of L is unramified outside S because L′_{∞} ⊂ L_{∞}. In the diagram of fields

L′_{∞} ⟵ L
↑  ↑
K_{∞} ⟵ K

we have L′_{∞} = K_{∞} L (the composition of fields K_{∞} and L), K = K_{∞} ∩ L and Gal(L/K) = Gal(L′_{∞}/K_{∞}). Hence if a finite place 𝔮 of K not belonging to T ramifies in L, then it ramifies when passing from K_{∞} to L′_{∞} with the same ramification index. Therefore 𝔮 does not ramify when passing from K to K_{∞}.

□

Corollary 4.1.9. There is an epimorphism

π_{1}(Spec O_{K,T}; Spec K̄) → Gal(K_{∞}/K) .
□

Let K(μ_{ℓ^{∞}})_{T}^{(ℓ)} be the maximal pro-ℓ extension of K(μ_{ℓ^{∞}}) unramified outside places of K(μ_{ℓ^{∞}}) lying over T.

Corollary 4.1.10. The field K_{∞} is contained in K(μ_{ℓ^{∞}})_{T}^{(ℓ)}, i.e., we have an epimorphism of Galois groups

Gal(K(μ_{ℓ^{∞}})_{T}^{(ℓ)}/K) → Gal(K_{∞}/K).
□

4.2. We shall review briefly some results from [8] and [7] and from [25] about weighted Tate completion of Galois groups.

Definition 4.2.0. Let F be a number field and let R be a finite set of finite places of F containing all finite places lying over ℓ. We denote by M(F)_{R} a maximal pro-ℓ extension of F (μ_{ℓ^{∞}}) unramified outside R, i.e., unramified outside finite places of F (μ_{ℓ^{∞}}) lying over any place of R.

In [8] and [7] Hain and Matsumoto defined weighted Tate completion of Galois groups. We shall study weighted Tate completion of the group Gal(M(F)_{R}/F).

Let G(F, R, ℓ) be the weighted Tate completion of the group Gal(M(F)_{R}/F). The group G(F, R, ℓ) is an affine proalgebraic group over ℚ_{ℓ}, an extension of the multiplicative group over ℚ_{ℓ} by an affine proalgebraic prounipotent group U(F, R, ℓ). There is a map

i_{F,R,ℓ} : Gal(M(F)_{R}/F) → G(F, R, ℓ)

of the group Gal(M(F)_{R}/F) into G(F, R, ℓ) with a Zariski dense image making the following diagram commutative

Gal(M(F)_{R}/F (μ_{ℓ^{∞}})) ⟶^{i_{F,R,ℓ}} U(F, R, ℓ)
↓  ↓

Gal(M(F)_{R}/F)  ⟶^{i_{F,R,ℓ}} G(F, R, ℓ)
↓  ↓
Γ_{F} := Gal(F (μ_{ℓ^{∞}})/F) ⟶ 𝔾_{m}(ℚ_{ℓ}).

The group G(F, R, ℓ) is equipped with the weight filtration {W_{−i}G(F, R, ℓ)}_{i∈ℕ∪{0}} such that W_{−2i+1}G(F, R, ℓ) = W_{−2i}G(F, R, ℓ) and W_{−2}G(F, R, ℓ) = U(F, R, ℓ). The group 𝔾_{m}(ℚ_{ℓ}) acts by conjugation on W_{−2i}G(F, R, ℓ)/W_{−2i−2}G(F, R, ℓ) and g ∈ ℚ_{ℓ}^{∗} acts as a multiplication by g^{i}.

Let φ : G_{F} → AutM be an ℓ-adic mixed Tate module unramified outside R. Then it follows from the definition of the field M(F)_{R} that φ factors through Gal(M(F)_{R}/F), i.e., we have a commutative diagram

G_{F}  ⟶^{φ} AutM
↓  id↓
Gal(M(F)_{R}/F) ⟶^{φ} AutM,

where the map from Gal(M(F)_{R}/F) to AutM induced by φ we denote also by φ. It follows from the universal property of weighted Tate completion that φ : Gal(M(F)_{R}/F) → AutM factors through G(F, R, ℓ), i.e., we have a commutative diagram

Gal(M(F)_{R}/F) ⟶^{φ} AutM
i_{F,R,ℓ}↓  id↓
G(F, R, ℓ)  ⟶^{[φ]} AutM,

where [φ] is the unique morphism making the diagram commutative.

If t is a coefficient of the representation φ then we denote by [t] the corresponding coefficient of the representation [φ].

Now we shall review the results from [25]. There we are studying relations between weighted Tate completions of Galois groups of two different fields.

4.2.1. Let L be a finite Galois extension of K and let G := Gal(L/K). Let T be a finite set of finite places of K containing all places over ℓ. Let us define a set S of finite places of L by

S := { 𝔭 ∈ V(L) | ∃ 𝔮 ∈ T, 𝔭 ∩ O_{K} = 𝔮 }.

Observe that the set S is G-invariant.

Let us assume also that

i) ℓ does not divide the order of G;

ii) K(μ_{ℓ^{∞}}) ∩ L = K.

Lemma 4.2.2. (see [25] section 2). Let us assume 4.2.1. Then the group G acts on Gal(M(L)_{S}/L) by conjugations.

Proof. The assumptions i) and ii) imply that the exact sequence of Galois groups

1 → Gal(M(L)_{S}/L(μ_{ℓ^{∞}})) → Gal(M(L)_{S}/K(μ_{ℓ^{∞}})) → G → 1

has a section s′ : G → Gal(M(L)_{S}/K(μ_{ℓ^{∞}})). Composing s′ with the inclusion Gal(M(L)_{S}/K(μ_{ℓ^{∞}})) → Gal(M(L)_{S}/K) we get a section s : G → Gal(M(L)_{S}/K) of the exact sequence of Galois groups

1 → Gal(M(L)_{S}/L) → Gal(M(L)_{S}/K) → G → 1.

Hence we get an action of G by conjugations (σ → s(g)·σ·s(g)^{−1}) on Gal(M(L)_{S}/L). □

The construction of weighted Tate completion is functorial hence the action of G on Gal(M(L)_{S}/L) implies that G acts on G(L, S, ℓ) preserving weight filtration. This implies that G acts by Lie algebra automorphisms on the associated graded Lie algebra

grLieU(L, S, ℓ) := ⨁_{i=1}^{∞} W_{−2i}U(L, S, ℓ)/W_{−2i−2}U(L, S, ℓ).

Let us set

grLieU(L, S, ℓ)_{i} := W_{−2i}U(L, S, ℓ)/W_{−2i−2}U(L, S, ℓ).

Definition 4.2.3. We denote by D_{i} the image of the Lie bracket

[ , ] : ⋀^{2} ( ⨁_{k=1}^{i−1} grLieU(L, S, ℓ)_{k}) → grLieU(L, S, ℓ)_{i}

and we call D_{i} the space of decomposable elements in degree i of grLieU(L, S, ℓ). Observe that the subspace D_{i} of grLieU(L, S, ℓ)_{i} is G-invariant.

Definition 4.2.4. We denote by I_{i} the G-invariant complement of D_{i} in grLieU(L, S, ℓ)_{i} and we call I_{i} the space of indecomposable elements in degree i of grLieU(L, S, ℓ). We denote by W_{i} the fixed part of I_{i} and by V_{i} the G-invariant complement of W_{i} in I_{i}, i.e., we have

W_{i} := I_{i}^{G} and I_{i} = W_{i} ⊕ V_{i}

as G-modules.

If it will be necessary to indicate dependence on the field L then we shall write D_{i}(L), I_{i}(L), W_{i}(L) and V_{i}(L).

Now we shall define two Lie algebras associated with the action of G on grLieU(L, S, ℓ).

Definition 4.2.5. We denote by grLieU(L, S, ℓ)^{[G]} a Lie subalgebra of grLieU(L, S, ℓ) generated by ⊕_{i=1}^{∞}W_{i}.

Proposition 4.2.6. The Lie subalgebra grLieU(L, S, ℓ)^{[G]} of grLieU(L, S, ℓ) is free, freely generated by a base of ⊕_{i=1}^{∞}W_{i}.

Proof. The associated graded Lie algebra grLieU(L, S, ℓ) is free, freely generated by a base of ⊕_{i=1}^{∞}I_{i}. Hence it follows from the Sirsov-Witt theorem (see [10] p. 331) that grLieU(L, S, ℓ)^{[G]} is free, freely generated by a base of ⊕_{i=1}^{∞}W_{i}. □

Definition 4.2.7. Let I(⊕_{i=1}^{∞}V_{i}) be a Lie ideal of the Lie algebra grLieU(L, S, ℓ) generated by ⊕_{i=1}^{∞}V_{i}. Let us set

grLieU(L, S, ℓ)_{[G]} := grLieU(L, S, ℓ)/I(⊕_{i=1}^{∞}V_{i}).

We denote by

α : grLieU(L, S, ℓ)^{[G]} → grLieU(L, S, ℓ)

the inclusion of the Lie algebra grLieU(L, S, ℓ)^{[G]} into grLieU(L, S, ℓ). Let

pr : grLieU(L, S, ℓ) → grLieU(L, S, ℓ)_{[G]}

be the projection and let

β := pr ∘ α : grLieU(L, S, ℓ)^{[G]} → grLieU(L, S, ℓ)_{[G]}

be the composition of the inclusion α with the projection pr.

Observe that G acts trivially on grLieU(K, T, ℓ). Hence the natural morphism of associated graded Lie algebras

p_{G} : grLieU(L, S, ℓ) → grLieU(K, T, ℓ)

induced by the inclusion G_{L} ⊂ G_{K} induces a morphism of Lie algebras

p̄_{G} : grLieU(L, S, ℓ)_{[G]} → grLieU(K, T, ℓ)

which we denote also by p_{K,T}^{L,S} .

Proposition 4.2.8. We have i) the Lie algebra grLieU(L, S, ℓ)_{[G]} is free, freely generated by a base of

⊕_{i=1}^{∞}I_{i}/V_{i};

ii) the morphism of Lie algebras

β : grLieU(L, S, ℓ)^{[G]} → grLieU(L, S, ℓ)_{[G]}

is an isomorphism;

iii) the morphism of Lie algebras

p̄_{G} : grLieU(L, S, ℓ)_{[G]} → grLieU(K, T, ℓ)

is an isomorphism.

Proof. Let us choose a base of ⊕_{i=1}^{∞}W_{i} and a base of ⊕_{i=1}^{∞}V_{i}. Let us write a Hall base of a vector space grLieU(L, S, ℓ) in terms of these two chosen bases. Then it is clear that we have a direct sum decomposition

grLieU(L, S, ℓ) = grLieU(L, S, ℓ)^{[G]} ⊕ I(⊕_{i=1}^{∞}V_{i}).

This implies immediately points i) and ii).

The inclusion of Galois groups G_{L} ⊂ G_{K} induces a map of cohomology groups

H^{1}(G_{K}; ℚ_{ℓ}(i)) → H^{1}(G_{L}; ℚ_{ℓ}(i))

for i > 1. It follows from the Lyndon spectral sequence of the exact sequence of Galois groups 1 → G_{L} → G_{K} → G → 1 that

H^{1}(G_{K}; ℚ_{ℓ}(i)) ≃ H^{1}(G_{L}; ℚ_{ℓ}(i))^{G}

for i > 1. Hence we get that

I(K)_{i} ≃ (I(L)_{i})_{G} = I(L)_{i}/V (L)_{i}

for i > 1.

For i = 1 we use the Lyndon spectral sequence of the exact sequence of Galois groups

1 → π_{1}(Spec O_{L,S}; SpecL̄) → π_{1}(Spec O_{K,T}; SpecL̄) → G → 1.

We can also observe that the isomorphism I(K)_{1} ≃ (I(L)_{1})_{G} follows from the equality

O_{K,T}^{∗} ⊗ ℚ ≃ (O_{L,S}^{∗} ⊗ ℚ)^{G}.

This finishes the proof of the proposition. □

Definition 4.2.9. Let L = ⨁_{i=1}^{∞} L_{i} be a graded Lie algebra. We define a completion c(L) of L setting

c(L) := \underleftarrow{lim}_{m}(L/ ⊕_{i=m}^{∞} L_{i}).

The completed Lie algebra c(L) equipped with a group law given by the Baker-Campbell-Hausdorff formula we denote by g(L).

Proposition 4.2.10. We have i) c(grLieU(L, S, ℓ)) = LieU(L, S, ℓ);

ii) the isomorphism β induces an isomorphism of Lie algebras

c(β) : c(grLieU(L, S, ℓ)^{[G]}) → c(grLieU(L, S, ℓ)_{[G]}) ;

iii) the isomorphism p̄_{G} induces an isomorphism of Lie algebras

c(p̄_{G}) : c(grLieU(L, S, ℓ)_{[G]}) → c(grLieU(K, T, ℓ)) = LieU(K, T, ℓ).

Proof. The point i) follows from the fact that the Lie algebra grLieU(L, S, ℓ) is free. The points ii) and iii) are obvious. □

Definition 4.2.11. Let us set

U(L, S, ℓ)_{[G]} := g(grLieU(L, S, ℓ)_{[G]})

and

U(L, S, ℓ)^{[G]} := g(grLieU(L, S, ℓ)^{[G]}).

Proposition 4.2.12. Let us assume 4.2.1. Then we have:

i) there is a natural isomorphism of affine proalgebraic prounipotent groups

a_{L,S} : g(grLieU(L, S, ℓ)) ≃ U(L, S, ℓ);

ii) there is a natural isomorphism of affine proalgebraic prounipotent groups

U(L, S, ℓ)_{[G]} ≃ (g(grLieU(L, S, ℓ)))_{G};

iii) the isomorphism p̄_{G} : grLieU(L, S, ℓ)_{[G]} → grLieU(K, T, ℓ) induces an isomorphism of affine proalgebraic prounipotent groups

g(p̄_{G}) : U(L, S, ℓ)_{[G]} ≃ g(grLieU(K, T, ℓ)) ;

iv) the isomorphism β : grLieU(L, S, ℓ)^{[G]} → grLieU(L, S, ℓ)_{[G]} induces an isomorphism of affine proalgebraic prounipotent groups

g(β) : U(L, S, ℓ)^{[G]} → U(L, S, ℓ)_{[G]}.

Proof. We shall identify the Lie algebras c(grLieU(L, S, ℓ)) and LieU(L, S, ℓ). Hence we have the identity isomorphism between the group g(grLieU(L, S, ℓ)) and LieU(L, S, ℓ) equipped with a group law given by the Baker-Campbell-Hausdorff formula. On the other side there is a canonical isomorphism between the Lie algebra LieU(L, S, ℓ) equipped with a group law given by Baker-Campbell-Hausdorff formula and the affine proalgebraic prounipotent group U(L, S, ℓ). The isomorphism a_{L,S} is the composition of these two isomorphisms.

Closed ideals of the Lie algebra c(grLieU(L, S, ℓ)) are exactly closed normal subgroups of the Lie group g(grLieU(L, S, ℓ)). Observe that the closed Lie ideal of c(grLieU(L, S, ℓ)) generated by elements gv − v is exactly the closed normal subgroup of g(grLieU(L, S, ℓ)) generated by elements (gv) · v^{−1}. This implies the point ii) of the proposition.

The points iii) and iv) are obvious. □

Definition 4.2.13. We set

s_{K,T}^{L,S} := g(α) ∘ g(β)^{−1} ∘ g(p̄_{G})^{−1}.

Lemma 4.2.14. The morphism

s_{K,T}^{L,S} : g(grLieU(K, T, ℓ)) → g(grLieU(L, S, ℓ))

is a section of the natural projection

g(p_{G}) : g(grLieU(L, S, ℓ)) → g(grLieU(K, T, ℓ)).

Proof. Observe that g(p_{G}) ∘ s_{K,T}^{L,S} = g(p_{G}) ∘ g(α) ∘ g(β)^{−1} ∘ g(p̄_{G})^{−1} = g(p̄_{G}) ∘ g(pr)∘g(α)∘g(β)^{−1}∘g(p̄_{G})^{−1} = g(p̄_{G})∘g(β)∘g(β)^{−1}∘g(p̄_{G})^{−1} = Id_{g(grLieU(K,T,ℓ))}. □

Definition 4.2.15. Let us set

S_{K,T}^{L,S} := a_{L,S} ∘ s_{K,T}^{L,S} ∘ a_{K,T}^{−1} .

We need to understand the morphism induced by S_{K,T}^{L,S} on associated graded Lie algebras.

Lemma 4.2.16. We have

i) grLie(a_{L,S}) = Id_{grLieU(L,S,ℓ)};

ii) grLie(g(f)) = f.

□

Definition 4.2.17. Let ρ : G_{L} → AutM be an ℓ-adic mixed Tate module unramified outside S. We define a representation θ(ρ) of G_{K(μ_{l^{∞}})} by setting

θ(ρ) := [ρ] ∘ S_{K,T}^{L,S} ∘ i_{K,T,ℓ}

(see Diagram 2).

ρ : G_{L(μ_{ℓ^{∞}})} ⟶^{i_{L,S,ℓ}}  U(L, S, ℓ)  ⟶^{[ρ]} AutM
a_{L,S}↑
g(grLieU(L, S, ℓ))
s_{K,T}^{L,S} ↑
g(grLieU(K, T, ℓ))
a_{K,T}^{−1} ↑

G_{K(μ_{ℓ^{∞}})}  ⟶^{i_{K,T,ℓ}}  U(K, T, ℓ)
Diagram 2.

We would like to express coefficients of the representation θ(ρ) by the coefficients of ρ.

Proposition 4.2.18. Let ρ : G_{L} → AutM be an ℓ-adic mixed Tate module unramified outside S which is an extension of ℚ_{ℓ}(0) by ℚ_{ℓ}(n), i.e.,

ρ(σ) = \begin{matrix}1 & t(σ) \\ 0 & χ^{n}(σ) \end{matrix}

for some t : G_{L} → ℚ_{ℓ}. Then

[θ(ρ)](τ) = \begin{matrix}1 & \frac{1}{|G|} ∑_{g∈G}([t] ∘ g)(τ̃) \\ 0 & 1 \end{matrix}

where τ̃ ∈ U(L, S, ℓ) is a lifting of τ ∈ U(K, T, ℓ).

Proof. We recall that [t] is a homomorphism from U(L, S, ℓ) to ℚ_{ℓ} iduced by t : G_{L} → ℚ_{ℓ}. Observe that [t] and \frac{1}{|G|} ∑_{g∈G}([t] ∘ g) restricted to U(L, S, ℓ)^{[G]} coincide. The homomorphism \frac{1}{|G|} ∑_{g∈G}([t] ∘ g) : U(L, S, ℓ) → ℚ_{ℓ} is G-invariant and, when viewed as a homomorphism from grLieU(L, S, ℓ), it vanishes on decomposable elements. Hence it induces a homomorphism from U(L, S, ℓ)_{[G]} to ℚ_{ℓ}, i.e., a homomorphism from U(K, T, ℓ) to ℚ_{ℓ}. □

In [25] we have shown that Gal(M(L)_{S}/L)_{G} ≃ Gal(M(K)_{T} /K). Hence we get the following corollary.

Corollary 4.2.19. Let ρ : G_{L} → AutM be an ℓ-adic mixed Tate module unramified outside S, which is an extension of ℚ_{ℓ}(0) by ℚ_{ℓ}(n), i.e.,

ρ(σ) = \begin{matrix}1 & t(σ) \\ 0 & χ^{n}(σ) \end{matrix}

for some t : G_{L} → ℚ_{ℓ}. Then

θ(ρ)(τ) = \begin{matrix}1 & \frac{1}{|G|} ∑_{g∈G}(t ∘ g)(τ̃) \\ 0 & 1 \end{matrix}

where τ ∈ G_{K(μ_{ℓ^{∞}})} and τ̃ ∈ Gal(M(L)_{S}/L(μ_{ℓ^{∞}})) is a lifting to Gal(M(L)_{S}/L(μ_{ℓ^{∞}})) of the image of τ in Gal(M(K)_{T} /K(μ_{ℓ^{∞}})).

Proof. The corollary follows immediately from [25] Corollary 2.5. □

Proposition 4.2.20. Let ρ : G_{L} → AutM be an ℓ-adic mixed Tate module unramified outside S. Let α ∈ (End(M))^{∗} be such that α∘grLie[ρ] vanishes on decomposable elements of grLieU(L, S, ℓ). Then α∘grLie[θ(ρ)](τ) = \frac{1}{|G|} ∑_{g∈G}(α∘ grLie[ρ] ∘ g)(τ̃), where τ̃ ∈ grLieU(L, S, ℓ) is a lifting of τ ∈ grLieU(K, T, ℓ) to grLieU(L, S, ℓ).

Proof. The linear forms α ∘ grLie[ρ] and \frac{1}{|G|} ∑_{g∈G}(α ∘ grLie[ρ] ∘ g) coincide when restricted to grLieU(L, S, ℓ)^{[G]}. The homomorphism \frac{1}{|G|} ∑_{g∈G}(α ∘ grLie[ρ]∘g) : grLieU(L, S, ℓ) → ℚ_{ℓ} is G-invariant and it vanishes on decomposable elements. Hence it induces a homomorphism from grLieU(L, S, ℓ)_{[G]}, i.e., from grLieU(K, T, ℓ) to ℚ_{ℓ}. This homomorphism is of course α ∘ grLie[θ(ρ)]. □

4.3. Let us choose a sequence of geometric generators x_{1}, …, x_{n}, x_{n+1} of π_{1}(VK̄; v) associated with a family Γ = {γ_{i}}_{i=1}^{n+1} of paths from v to (a tangent vector defined over K at) each a_{i}. Let 𝕍 = {X_{1}, …, X_{n}}. As usual we define a continuous multiplicative embedding

k : π_{1}(VK̄; v) → ℚ_{ℓ}{{𝕍}}

setting k(x_{i}) = e^{X_{i}} for i = 1, …, n.

We fix paths p_{j} from v to z_{j} for j = 1, …, r. The action of G_{L} on the

disjoint union of torsors t = ∐^{r} ; j=1 π(VK̄; z_{j}, v) induces a representation

ψ_{t} : G_{L} → (⨁_{j=1}^{r} L_{exp L(𝕍)}) ˜× Aut^{∗} ℚ_{ℓ}{{𝕍}} .

We assume that the representation ψ_{t} is unramified outside the finite set S of finite places of L. We assume that S contains all finite places lying over (ℓ) and that S is G-invariant.

We assume also that

i) ℓ does not divide the order of G;

ii) K(μ_{ℓ^{∞}}) ∩ L = K.

Let us set

T := {𝔮 ∈ V(K) | ∃ 𝔭 ∈ S, 𝔮 = 𝔭 ∩ O_{K}}.

We recall from section 4.2 that the representation

θ(ψ_{t}) : G_{K(μ_{ℓ^{∞}})} → ⊕_{j=1}^{r}GL(ℚ_{ℓ}{{𝕍}})

is defined by the formula

θ(ψ_{t}) := [ψ_{t}] ∘ S_{K,T}^{L,S} ∘ i_{K,T,ℓ}

(see Definition 4.2.17). Further we shall also denote the representation θ(ψ_{t}) by θ_{t}^{L/K}.

The representation θ_{t}^{L/K} respects the filtration {⊕_{j=1}^{r}I(𝕍)^{m}}_{m∈ℕ∪{0}} of ⊕_{j=1}^{r}ℚ{{𝕍}}. We define a filtration {F_{m}(θ_{t}^{L/K})}_{m∈ℕ∪{0}} of G_{K(μ_{ℓ^{∞}})} by setting

F_{m}(θ_{t}^{L/K}) := {σ ∈ G_{K(μ_{ℓ^{∞}})} | θ_{t}^{L/K}(σ) induces the identity of ⊕_{j=1}^{r}(I(𝕍)/I(𝕍)^{m+1})}.

We state below elementary properties of the representation θ_{t}^{L/K}.

Proposition 4.3.0. The representation θ_{t}^{L/K} has the following properties:

i) it is compatible with the filtration {F_{m}(θ_{t}^{L/K})}_{m∈ℕ∪{0}} of G_{K(μ_{ℓ^{∞}})} and the filtration induced by the powers of the augmentation ideal I(𝕍) on ⊕_{j=1}^{r}GL(ℚ_{ℓ}{{𝕍}});

ii) it is unramified outside the set T of finite places of K;

iii) any σ ∈ G_{K(μ_{ℓ^{∞}})} acts on ⊕_{j=1}^{r}(I(𝕍)^{m}/I(𝕍)^{m+1}) by the identity.

□

The main object of our study is the representation induced by θ_{t}^{L/K} on the associated graded Lie algebras, i.e., the representation

grLieθ_{t}^{L/K} : ⊕_{i=1}^{∞}F_{i}(θ_{t}^{L/K})/F_{i+1}(θ_{t}^{L/K}) ⊗ ℚ → ( ⊕_{j=1}^{r} L_{Lie(𝕍)}) ˜×Der^{∗}Lie(𝕍).

Let us set

Θ_{t}^{Gal(L/K)} := grLieθ_{t}^{L/K}

To calculate the image of the Lie algebra representation Θ_{t}^{Gal(L/K)} first we need to calculate coordinates of the Lie algebra homomorphism Ψ_{t}. We recall that

Ψ_{t} := grLieψ_{t} : ⊕_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ → ( ⊕_{j=1}^{r} L_{Lie(𝕍)}) ˜×Der^{∗}Lie(𝕍).

and Ψ_{t}^{G} is the restriction of Ψ_{t} to ⊕_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ^{G}.

We recall that the representation ψ_{t} induces

[ψ_{t}] : U(L, S, ℓ) → ( ⊕_{j=1}^{r} L_{expL(𝕍)}) ˜×Aut^{∗}(ℚ_{ℓ}{{𝕍}}).

We set

[Ψ_{t}] := grLie[ψ_{t}]

and we denote by [Ψ_{t}]^{[G]} the morphism [Ψ_{t}] restricted to grLieU(L, S, ℓ)^{[G]}.

Let σ ∈ H_{k}/H_{k+1}. We recall that the coordinates of Ψ_{t}(σ) are

((log Λ_{p_{j}}(σ) mod Γ^{k+1} L(𝕍))_{j=1,...,r}; (log Λ_{γ_{i}}(σ) mod ⟨X_{i}⟩ + Γ^{k+1} L(𝕍))_{i=1,...,n}) .

Definition 4.3.1. We denote by J a closed Lie ideal of L(𝕍) generated by Lie bracket which contain at least three different X_{i}’s and by Lie brackets containing two different X_{i}’s at least twice.

In the next proposition we shall calculate coefficients of log Λ_{p_{j}}(σ) modulo the Lie ideal J.

Proposition 4.3.2. Let σ ∈ H_{k}. If k > 2 then

log Λ_{p_{j}}(σ) ≡ ∑_{α≠β} (ℓ_{k} (\frac{z_{j} − a_{α}}{a_{β} − a_{α}} ) (σ) − ℓ_{k} (\frac{v − a_{α}}{a_{β} − a_{α}} ) (σ))
[… [X_{β}, X_{α}], X_{α}^{k−2}] mod J + Γ^{k+1}L(𝕍) .

If k = 2 then

log Λ_{p_{j}}(σ) ≡ ∑_{α<β} (ℓ_{k} (\frac{z_{j} − a_{α}}{a_{β} − a_{α}} ) (σ) − ℓ_{k} (\frac{v − a_{α}}{a_{β} − a_{α}} ) (σ)) [X_{β}, X_{α}] mod Γ^{3}L(𝕍) .

If k = 1 then

log Λ_{p_{j}}(σ) ≡ ∑_{α=1}^{n} ℓ (\frac{z_{j} − a_{α}}{v − a_{α}} ) (σ) X_{α} mod Γ^{2}L(𝕍) .

Proof. Let f : V → ℙ_{K}^{1}\\{0, 1, ∞\} be given by f(z) = \frac{z−a_{α}}{a_{β}−a_{α}}. It follows from [19] Proposition 11.0.15 that

log Λ_{f(p_{j})}(σ) ≡ (ℓ_{k} (\frac{z_{j} − a_{α}}{a_{β} − a_{α}} ) (σ) − ℓ_{k} (\frac{v − a_{α}}{a_{β} − a_{α}} ) (σ)) [Y, X^{k−1}]
mod J + Γ^{k+1}L(X, Y) .

Observe that f(a_{α}) = 0, f(a_{β}) = 1 and f(∞) = ∞. This implies that f_{∗}(X_{α}) = X, f_{∗}(X_{β}) = Y and f_{∗}(X_{i}) = 0 for i ≠ α, β. Now the proposition follows from the equality f_{∗} log Λ_{p_{j}}(σ) = log Λ_{f(p_{j})}(σ). □

Let B be a Hall base of a free Lie algebra Lie(𝕍) and let B_{k} be elements of degree k in B. Let p be a path from v to a K-point z of V. Let σ ∈ H_{k}(V_{K}; z, v). We recall that in [18] section 5 we defined functions L^{e}(z, v) from H_{k}(V_{K}; z, v) to ℚ_{ℓ} by the following congruence

log Λ_{p}(σ) ≡ ∑_{e∈B_{k}} L^{e}(z, v) (σ) · e mod Γ^{k+1}L(𝕍) .

Proposition 4.3.3. Let σ ∈ H_{k} and let τ̃ ∈ G_{K(μ_{ℓ^{∞}})} be a lifting of τ ∈ G = Gal(L(μ_{ℓ^{∞}})/K(μ_{ℓ^{∞}})). Then for any e ∈ B_{k} we have

L^{e}(z_{j}, v) (τ̃ · σ · τ̃^{−1}) = L^{e}(z_{τ^{−1}(j)}, v) (σ) .

Proof. After calculations we get

𝔣_{p_{j}}(τ̃ · σ · τ̃^{−1}) = τ̃(𝔣_{τ̃^{−1}(p_{j})}(σ)) ,

where τ̃^{−1}(p_{j}) is a path from v to z_{τ̃^{−1}(j)}. Hence

log Λ_{p_{j}}(τ̃ · σ · τ̃^{−1}) = τ̃(log Λ_{τ̃^{−1}(p_{j})}(σ)) .

The assumption σ ∈ H_{k} implies that

τ̃(log Λ_{τ̃^{−1}(p_{j})}(σ)) ≡ log Λ_{τ̃^{−1}(p_{j})}(σ) mod Γ^{k+1}L(𝕍) .

Therefore we get the congruence

log Λ_{p_{j}}(τ̃ · σ · τ̃^{−1}) ≡ log Λ_{τ̃^{−1}(p_{j})}(σ) mod Γ^{k+1}L(𝕍)

which implies the lemma. □

Corollary 4.3.4. Let σ ∈ H_{k} and let τ̃ ∈ G_{K(μ_{ℓ^{∞}}}) be a lifting of τ ∈ G. Then

ℓ_{k} (\frac{z_{j} − a_{α}}{a_{β} − a_{α}} ) (τ̃ · σ · τ̃^{−1}) = ℓ_{k} (\frac{z_{τ^{−1}(j)} − a_{α}}{a_{β} − a_{α}} ) (σ)

for k > 1 and

ℓ (\frac{z_{j} − a_{α}}{v − a_{α}} ) (τ̃ · σ · τ̃^{−1}) = ℓ (\frac{z_{τ^{−1}(j)} − a_{α}}{v − a_{α}} ) (σ)

for k = 1. □

4.4. It is well known that

dim_{ℚ_{ℓ}} H^{1}(Spec O_{K,T}; ℚ_{ℓ}(i)) = r_{1} + r_{2} for i > 1 and odd;
dim_{ℚ_{ℓ}} H^{1}(Spec O_{K,T}; ℚ_{ℓ}(i)) = r_{2} for i > 0 and even;

and

dim_{ℚ_{ℓ}} H^{1}(Spec O_{K,T}; ℚ_{ℓ}(1)) = dim_{ℚ}(O_{K,T}^{∗} ⊗ ℚ) .

Generators of H^{1}(Spec O_{K,T}; ℚ_{ℓ}(1)) are given by Kummer classes of generators of the group O_{K,T}^{∗}. Generators of H^{1}(Spec O_{K,T}; ℚ_{ℓ}(i)) for i > 1 are given by Soulé classes (see [15]). For K = ℚ(μ_{n}) Soulé classes can be expressed by ℓ-adic polylogarithms (see [21]). In general case one can hope that Soulé classes can be expressed by linear combinations of ℓ-adic polylogarithms (surjectivity in Zagier conjecture for polylogarithms) or by linear combination of ℓ-adic iterated integrals or at least by Galois invariant linear combination of ℓ-adic polylogarithms or ℓ-adic iterated integrals.

In most applications L = ℚ(μ_{n}) and S is the set of finite places of ℚ(μ_{n}) which divide the product n · ℓ. Then we have

dim_{ℚ_{ℓ}} H^{1}(Spec O_{ℚ(μ_{n}),S}; ℚ_{ℓ}(i)) = \frac{1}{2}φ(n)

(φ(n) is the order of ℤ/n^{∗}) for i > 1 and

dim_{ℚ_{ℓ}} H^{1}(Spec O_{ℚ(μ_{n}),S}; ℚ_{ℓ}(1)) = dim_{ℚ} (O_{ℚ(μ_{n}),S}^{∗} ⊗ ℚ).

Conjecture 4.4.1. Let i be greater than 1. The ℓ-adic polylogarithms ℓ_{i}(ξ_{n}^{k}) for 0 < k < \frac{n}{2} and (k, n) = 1 are linearly independent over ℚ_{ℓ} and generate H^{1}(G_{ℚ(μ_{n})}; ℚ_{ℓ}(i)) = H^{1}(Spec O_{ℚ(μ_{n}),S}; ℚ_{ℓ}(i)).

5 Torsors of paths on ℙ1 ; ℚ̄\{0, 1, ∞}

5.0. In this section we shall study torsors of paths on

V := ℙ_{ℚ}^{1}\{0, 1, ∞} .

First we fix a notation. We denote by ℚ_{ℓ}{{X, Y }} a ℚ_{ℓ}-algebra of non-commuta-tive formal power series on non-commuting variables X and Y. The Lie algebra Lie (X, Y) is a free Lie algebra over ℚ_{ℓ} on X and Y and L(X, Y) is a completion of Lie (X, Y) with respect to lower central series. If R is a commutative ring we denote by Lie (X, Y; R) a free Lie algebra over R on X and Y.

We assume that ℚ̄ ⊂ ℂ. Let ξ_{q} = e^{\frac{2πi}{q}} be a primitive q-th root of 1. We shall

study Galois actions on a disjoint union and on a product of π_{1}(Vℚ̄; 01^{→})-torsors

t := ∐_{{(α,q)=1 \atop 0<α<q}} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) and T := ∏_{{(α,q)=1 \atop 0<α<q}} π(Vℚ̄; ξ_{q}^{α}, 01^{→})) .

Let us set

H_{i} := ⋂_{{(α,q)=1 \atop 0<α<q}} H_{i}(V_{ℚ(μ_{q})}; ξ_{q}^{α}, 01^{→}) .

The action of G_{ℚ(μ_{q})} on t and T yields Lie algebra representations of associated graded Lie algebras

Ψ_{t} : ⨁_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ → ( ⨁_{{(α,q)=1 \atop 0<α<q}} Lie(X, Y)) ˜× Der^{∗} Lie(X, Y)

and

Ψ_{T} : ⨁_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ →

( ⨁_{{(α,q)=1 \atop 0<α<q}} Id ⊗ … ⊗ Id ⊗ L_{Lie(X,Y)} ⊗ … ⊗ Id) ˜× Der^{∗} Lie(X, Y) ,

where

Der^{∗} Lie(X, Y) := {D ∈ Der Lie(X, Y) | ∃ A(X, Y) ∈ Lie(X, Y), D(X) = 0

and D(Y) = [Y, A(X, Y)]} .

A derivation D ∈ Der^{∗} Lie(X, Y) such that D(Y) = [Y, A] we denote by D_{A}. The vector space Lie(X, Y)/⟨Y ⟩ we equip with a Lie bracket { } given by

{A, B} = [A, B] + D_{A}(B) − D_{B}(A).

The obtained Lie algebra we denote by (Lie(X, Y)/⟨Y ⟩, { }). The map

Der^{∗} Lie(X, Y) ⟶ (Lie(X, Y)/⟨Y ⟩, { })

sending D_{A} to A is an isomorphism of Lie algebras.

Hence we get a Lie algebra representation

Ψ_{t} : ⨁_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ → ( ⨁_{{(α,q)=1 \atop 0<α<q}} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, \{ \}) .

We denote by I_{r} the vector subspace of Lie(X, Y) generated by Lie brackets in X and Y which contain at least r Y ’s.

Let us set

I_{r} := ( ⨁_{{(α,q)=1 \atop 0<α<q}} I_{r}) ⊕ I_{r} .

I_{r} is a vector subspace of ( ⨁_{{(α,q)=1 \atop 0<α<q}}Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, \{ \}) and one can

easily verify that I_{r} is a Lie ideal of the Lie algebra ( ⨁_{{(α,q)=1 \atop 0<α<q}}Lie(X, Y)) ˜× (Lie(X, Y)/⟨Y ⟩, \{ \}).

To simplify the notation we denote the Lie algebra

( ⨁_{{(α,q)=1 \atop 0<α<q}}Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, \{ \}) by L.

Let S be a set of finite places of ℚ(μ_{q}) which divide q or ℓ.

Lemma 5.0.1. We have:

i) Let σ ∈ H_{k}. The coordinates of Ψ_{t}(σ) calculated modulo I_{2}+Γ^{k+1}L(X, Y) on each component are

((ℓ_{k}(ξ_{q}^{α})(σ) [Y, X^{k−1}])_{{(α,q)=1 \atop 0<α<q}}; ℓ_{k}(1)(σ) [Y, X^{k−1}])

for k > 1 and

((ℓ_{1}(ξ_{q}^{α})(σ)Y)_{{(α,q)=1 \atop 0<α<q}}; 0)

for k = 1.

ii) Let σ ∈ grLieU(L, S, ℓ)_{k}. The coordinates of grLie[ψ_{t}](σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are

(([ℓ_{k}(ξ_{q}^{α})](σ) [Y, X^{k−1}])_{{(α,q)=1 \atop 0<α<q}}; [ℓ_{k}(1)](σ) [Y, X^{k−1}])

for k > 1 and

(([ℓ_{1}(ξ_{q}^{α})](σ)Y)_{{(α,q)=1 \atop 0<α<q}}; 0)

for k = 1.

Proof. The first part of the lemma follows from Lemma 1.1.0 and from Definition 3.0.1 of ℓ-adic polylogarithms.

The universal property of the weighted Tate completion and the fact that all morphisms are strict imply that we have a surjective morphism of Lie algebras

grLieU(L, S, ℓ) → ⨁_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ.

Hence it follows immediately the second part of the lemma. □

Further we shall apply formalism developed in section 4 in order to get representations of Galois groups of ℚ and other fields K intermediary between ℚ and ℚ(μ_{q}) with prescribed ramifications. In some cases we shall be able to show that the image of the morphism Θ_{t}^{Gal(ℚ(μ_{q})/K)} is a free Lie algebra on a maximal possible, depending on K and ramifications, number of generators.

We finish this section with some technical definitions and results. First we recall that

Gal(ℚ(μ_{q})/ℚ) = ℤ/q^{∗}.

The group ℤ/q^{∗} = Gal(ℚ(μ_{q})/ℚ) acts by conjugation on the abelianization of the Galois group G_{ℚ(μ_{q})(μ_{ℓ^{∞}})}. Hence it acts also on Hom_{ℤ_{ℓ}}((G_{ℚ(μ_{q})(μ_{ℓ^{∞}})})^{ab}; ℚ_{ℓ}(m)).

Lemma 5.0.2. Let τ ∈ ℤ/q^{∗} and let τ̃ be a lifting of τ to G_{ℚ(μ_{ℓ^{∞}})}. Then for any σ ∈ G_{ℚ(μ_{q})(μ_{ℓ^{∞}})} we have

ℓ_{m}(ξ_{q}^{α})(τ̃ · σ · τ̃^{−1}) = ℓ_{m}(ξ_{q}^{τ^{−1}·α})(σ).

Proof. Let γ be a path from 01^{→} to ξ_{q}^{α} and let σ′ := τ̃ · σ · τ̃^{−1}. Then σ′ · γ · σ′^{−1} transforms

f(z) = ∏^{ℓ^{n}−1} ; i=0(1 − ξ_{ℓ^{n}}^{i} z^{1/ℓ^{n}})^{\frac{i^{m−1}}{ℓ^{n}}}

into ξ_{ℓ^{n}}^{ℓ_{m}(ξ_{q}^{α})(σ′)} · f(z).

On the other side applying terms of the product τ̃ · σ · τ̃^{−1} · γ · τ̃ · σ^{−1} · τ̃^{−1} one after another to f(z), we get that the product transforms f(z) into

ξ_{ℓ^{n}}^{ℓ_{m}(ξ_{q}^{τ^{−1}·α})(σ)} · f(z). Hence we get that

ℓ_{m}(ξ_{q}^{α})(τ̃ · σ · τ̃^{−1}) = ℓ_{m}(ξ_{q}^{τ^{−1}·α})(σ).

Observe that ℓ(ξ_{q}^{α})(σ) = 0 for σ ∈ G_{ℚ(μ_{q})(μ_{ℓ^{∞}})}. Hence the lemma follows from Proposition 3.0.4. □

For each m we define a ℤ/q^{∗}-module V_{m}(q) in the following way. Let m > 1. In the group ring ℚ_{ℓ}[ℤ/q] we consider a ℚ_{ℓ}-vector subspace R_{m}(q) generated by elements

[α] + (−1)^{m}[−α] for [α] ∈ ℤ/q

and

r^{m−1} (∑_{i=0}^{r−1}[ip]) − [0] , p^{m−1} (∑_{j=0}^{p−1}[k + jr]) − [pk] if q = p · r.

(The class of α in ℤ/q we denote by [α].)

If m = 1 we denote by R_{1}(q) a ℚ_{ℓ}-vector subspace of ℚ_{ℓ}[ℤ/q] generated by elements

[0] and [α] − [−α] for [α] ∈ ℤ/q

and

(∑_{i=0}^{r−1}[j + ip]) − [rj] with j ≢ 0 mod p if q = p · r.

We set

V_{m}(q) := ℚ_{ℓ}[ℤ/q]/R_{m}(q) .

The group ℤ/q^{∗} acts on ℚ_{ℓ}[ℤ/q] permuting elements of ℤ/q. Observe that the vector subspace R_{m}(q) of ℚ_{ℓ}[ℤ/q] is preserved by the action of ℤ/q^{∗}, hence the quotient space V_{m}(q) is also a ℤ/q^{∗}-module.

Lemma 5.0.3. There is a ℤ/q^{∗}-equivariant map

s_{m}^{q} : V_{m}(q) → H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(m))

for m > 1 and

s_{1}^{q} : V_{1}(q) → H^{1}(SpecO_{ℚ(μ_{q}),S}; ℚ_{ℓ}(1))

for m = 1 such that s_{m}^{q}([α]) = ℓ_{m}(ξ_{q}^{α}).

Proof. It is clear that the formula s̄_{m}^{q}([α]) := ℓ_{m}(ξ_{q}^{α}) defines a ℚ_{ℓ}-linear map s̄_{m}^{q} from V_{m}(q) to Hom_{ℤ_{ℓ}}((G_{ℚ(μ_{q})(μ_{ℓ^{∞}})})^{ab}; ℚ_{ℓ}(m)) for m > 1 and from V_{1}(q) to Hom_{ℤ_{ℓ}}(π_{1}(SpecO_{ℚ(μ_{q})(μ_{ℓ^{∞}}),S}; Specℚ̄)^{ab}; ℚ_{ℓ}(1)) for m = 1 because generators of R_{m}(q) correspond to the distribution and inversion relations of ℓ-adic polylogarithms.

Observe that we have ℤ/q^{∗}-isomorphisms

t_{m} : Hom_{ℤ_{ℓ}}((G_{ℚ(μ_{q})(μ_{ℓ^{∞}})})^{ab}; ℚ_{ℓ}(m)) ⟶ H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(m))

for m > 1 and

t_{1} : Hom_{ℤ_{ℓ}}(π_{1}(SpecO_{ℚ(μ_{q})(μ_{ℓ^{∞}}),S}; Specℚ̄)^{ab}; ℚ_{ℓ}(1)) → H^{1}(Spec O_{ℚ(μ_{q}),S}; ℚ(1))

for m = 1.

We set s_{m}^{q} := t_{m} ∘ s̄_{m}^{q}. It follows from Lemma 5.0.2 that the map s̄_{m}^{q} is ℤ/q^{∗}-equivariant hence s_{m}^{q} is also ℤ/q^{∗}-equivariant. □

Lemma 5.0.4. Let us assume that Conjecture 4.4.1 holds. Then we have

i) the ℤ/q^{∗}-equivariant map

s_{m}^{q} : V_{m}(q) → H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(m))

is an isomorphism for m > 1;

ii) the ℤ/q^{∗}-equivariant map

s_{1}^{q} : V_{1}(q) → H^{1}(SpecO_{ℚ(μ_{q}),S}; ℚ_{ℓ}(1))

is injective.

Proof. The result of Soulé implies that dimH^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(m)) = \frac{1}{2}φ(q) for m > 1 (see [15]). It follows from [12] that dimV_{m}(q) = \frac{1}{2}φ(q) for m > 1. We have assumed that Conjecture 4.4.1 holds. This implies that the map s_{m}^{q} is an isomorphism for m > 1.

Now we assume that m = 1. Let us denote by C_{ℚ(μ_{q})} a subgroup of ℚ(μ_{q})^{∗} generated by elements of the form 1 − ξ_{q}^{α} and by μ_{2q}. It follows from the Bass theorem (see [17] Theorem 8.9) that the only relations between these elements are

−ξ_{q}^{−α}(1 − ξ_{q}^{α}) = 1 − ξ_{q}^{−α}

and

∏_{i=0}^{r−1}(1 − ξ_{q}^{j+ip}) = 1 − ξ_{q}^{rj} if q = rp.

These relations correspond of course to generators of R_{1}(q).

On the other side the homomorphism

O_{ℚ(μ_{q}),S}^{∗}/μ_{2q} ⟶ H^{1}(SpecO_{ℚ(μ_{q}),S}; ℚ_{ℓ}(1))

which to a unite u associates a corresponding Kummer character is injective. This implies that the map s_{1}^{q} is injective. □

Lemma 5.0.5. Let us assume that Conjecture 4.4.1 holds. Let G be a subgroup of Gal(ℚ(μ_{q})/ℚ) = ℤ/q^{∗}. Then the map s_{m}^{q} induces

i) an isomorphism

(s_{m}^{q})^{G} : V_{m}(q)^{G} → H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(m))^{G}

for m > 1;

ii) an injective map

(s_{1}^{q})^{G} : V_{1}(q)^{G} → H^{1}(SpecO_{ℚ(μ_{q}),S}; ℚ_{ℓ}(1))^{G}

for m = 1.

Proof. The lemma follows immediately from Lemma 5.0.4. □

5.1. Let q be a prime number. In this section we shall construct geometrically ℓ-adic realization of the associated graded Lie algebra of the fundamental group of the tannakien category of the mixed Tate motives over Spec ℤ[\frac{1}{q}]. We recall that conjecturally this Lie algebra is free, freely generated by elements in degree 1, 3, 5, …, 2n + 1, …. Generator in degree 1 corresponds to log q and generator in degree 2n + 1 corresponds to ζ(2n + 1) for n > 0.

We assume that q is a prime number different from ℓ. We assume also that ℓ does not divide q − 1. Then

ℚ(μ_{q}) ∩ ℚ(μ_{ℓ^{∞}}) = ℚ

and ℓ does not divide the order of the group

ℤ/q^{∗} = Gal(ℚ(μ_{q})/ℚ) .

We consider actions of Galois groups on

t := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) and T := ∏_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) .

We shall apply formalism from section 4 for

K = ℚ and L = ℚ(μ_{q}).

We recall that the Galois group ℤ/q^{∗} = Gal(ℚ(μ_{q})/ℚ) acts on (H_{k}/H_{k+1}) ⊗ ℚ and that ((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}} is a fixed point subspace.

Observe that the triple (V, ξ_{q}^{α}, 01^{→}) has good reduction at every finite place of ℚ(μ_{q}) not dividing q. Let

S := {𝔭 ∈ V(ℚ(μ_{q})) | 𝔭 divides q or ℓ}.

Then it follows from Proposition 2.3 that the representation ψ_{t} is unramified outside S.

Lemma 5.1.1. We have:

i) Let σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}}. The coordinates of Ψ_{t}^{ℤ/q^{∗}}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are

((\frac{1 − q^{k−1}}{(q − 1)q^{k−1}} ℓ_{k}(1)(σ) [Y, X^{k−1}])_{0<α<q}; ℓ_{k}(1)(σ) [Y, X^{k−1}])

for k > 1 and odd;

( ( 0 )_{0<α<q}; 0 )

for k > 1 and even;

((\frac{1}{q − 1} ℓ(q)(σ) Y )_{0<α<q}; 0)

for k = 1.

ii) Let σ ∈ (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{k}. The coordinates of [Ψ_{t}]^{[ℤ/q^{∗}]}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are given by the same formulas as in the point i) if we replace ℓ_{k}(1) and ℓ(q) by [ℓ_{k}(1)] and [ℓ(q)] respectively.

Proof. Let σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}} and let 0 < α < q. It follows from Lemma 5.0.1 that the α coordinate of Ψ_{t}^{ℤ/q^{∗}}(σ) is l_{k}(ξ_{q}^{α})(σ)[Y, X^{k−1}] modulo I_{2} +Γ^{k+1}L(X, Y). It follows from Corollary 4.3.4 or Lemma 5.0.2 that l_{k}(ξ_{q}^{α})(τ̃· σ·τ̃^{−1}) = l_{k}(ξ_{q}^{τ^{−1}·α})(σ), where τ̃ is a lifting of τ ∈ ℤ/q^{∗} to G_{ℚ(μ_{ℓ^{∞}})}. Hence we get that l_{k}(ξ_{q}^{α})(σ) = l_{k}(ξ_{q})(σ). The distribution relation for ℓ-adic polylogarithms (see [19] Corollaries 11.2.3 or [21] Theorems 2.1) and the vanishing of l_{2n}(1) (see

[19] Corollary 11.2.11) imply that for any σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}} and any 0 < α < q we have

l_{k}(ξ_{q}^{α})(σ) = \frac{1 − q^{k−1}}{(q − 1)q^{k−1}} ℓ_{k}(1)(σ) for k > 1 and odd,

l_{k}(ξ_{q}^{α})(σ) = 0 for k > 1 and even,

l(1 − ξ_{q}^{α})(σ) = \frac{1}{q − 1} ℓ(q)(σ) for k = 1.

This implies the first part of the lemma.

The functions [ℓ_{k}(ξ_{q}^{α})] ( for k > 1 and 0 ≦ α < q) and [ℓ_{1}(ξ_{q}^{α})] ( for 0 < α < q) from grLieU(L, S, ℓ) to ℚ_{ℓ} vanish on the subspace of decomposable elements D_{k} of grLieU(L, S, ℓ)_{k}. The restriction of [ℓ_{k}(ξ_{q}^{α})] ( for 0 < α < q) to W_{k} := I_{k}^{ℤ/q^{∗}} is equal \frac{1}{q−1} ∑_{i∈ℤ/q^{∗}}[ℓ_{k}(ξ_{q}^{iα})] = \frac{1}{q−1} ∑_{i=1}^{q−1}[ℓ_{k}(ξ_{q}^{i})]. It follows from the inversion relation for ℓ-adic polylogarithms that the last expresion vanishes for k even.

For k > 1 and odd it follows from the distribution relations that \frac{1}{q−1} ∑_{i=1}^{q−1}[ℓ_{k}(ξ_{q}^{i})] =

\frac{1−q^{k−1}}{(1−q)q^{k−1}} [ℓ_{k}(1)].

Finally it follows from the equality ∏_{i=1}^{q−1}(1−ξ_{q}^{i}) = q that \frac{1}{q−1} ∑_{i=1}^{q−1}[ℓ_{1}(ξ_{q}^{i})] = [ℓ_{1}(q)].

The universal property of the weighted Tate completion implies that the morphism grLie[ψ_{t}] factors through a surjective morphism of Lie algebras

grLieU(L, S, ℓ) → ⊕_{k=1}^{∞}(H_{k}/H_{k+1}) ⊗ ℚ.

Hence the morphism [Ψ_{t}]^{[ℤ/q^{∗}]} factors through

grLieU(L, S, ℓ)^{[ℤ/q^{∗}]} → ⊕_{k=1}^{∞}((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}}.

This implies the second part of the lemma. □

Proposition 5.1.2. In the image of the morphism of Lie algebras

[Ψ_{t}]^{[ℤ/q^{∗}]} : grLieU(L, S, ℓ)^{[ℤ/q^{∗}]} → ( ⨁_{0<α<q} Lie(X, Y)) ˜× (Lie(X, Y)/⟨Y ⟩, { })

there are elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … homogeneous of degree 1, 3, 5, …, 2n + 1, … respectively whose coordinates calculated modulo I_{2} on each component are

(((1 − q^{2n}) [Y, X^{2n}])_{0<α<q}; (q − 1) q^{2n}[Y, X^{2n}])

for k = 2n + 1 and greater than 1 and

((Y)_{0<α<q}; 0)

for k = 1.

Proof. The ℓ-adic polylogarithm ℓ_{2n+1}(1) = ℓ_{2n+1}(10^{→}) is a rational non-zero multiple of the Soulé class – the generator of H^{1}(G_{ℚ}; ℚ_{ℓ}(2n+1)). Hence ℓ_{2n+1}(1) restricted to H^{1}(G_{L}; ℚ_{ℓ}(2n + 1)) is also non-zero. Therefore [ℓ_{2n+1}(1)] is a nonzero homomorphism from grLieU(ℚ, {ℓ, q}, ℓ)_{2n+1} to ℚ_{ℓ} and hence also from grLieU(L, S, ℓ)_{2n+1} to ℚ_{ℓ}. It follows from Proposition 4.2.8 ii) and iii) that [ℓ_{2n+1}(1)] restricted to (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{2n+1} is non-zero. Therefore there is σ_{2n+1} ∈ (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{2n+1} such that [ℓ_{2n+1}(1)](σ_{2n+1}) = 1. We set

D_{2n+1} := (q − 1) q^{2n} [Ψ_{t}]^{[ℤ/q^{∗}]} (σ_{2n+1})

for 2n + 1 > 1.

Now we consider the case m = 1. The cohomology class of the cocycle ℓ(q) is one of generators of H^{1}(Specℤ[\frac{1}{q}, \frac{1}{ℓ}]; ℚ_{ℓ}(1)). Therefore the homomorphism [ℓ(q)] : grLieU(ℚ, {ℓ, q}, ℓ)_{1} → ℚ_{ℓ} is non-zero. It follows from Proposition 4.2.8 ii) and iii) that the homomorphism [ℓ(q)] : grLieU(L, S, ℓ)_{1} → ℚ_{ℓ} restricted to (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{1} is non-zero. Hence we can find σ_{1} ∈ (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{1} such that [ℓ(q)](σ_{1}) = 1. Then we set

D_{1} := (q − 1) [Ψ_{t}]^{[ℤ/q^{∗}]} (σ_{1}) .

It is clear from the construction and from Lemma 5.1.1 that the elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … have the required coordinates. □

Proposition 5.1.3. The Lie algebra Im ([Ψ_{t}]^{[ℤ/q^{∗}]}) is free, freely generated by elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, ….

Proof. First we shall show that elements D_{1}, D_{3}, D_{5}, … generate freely a free Lie subalgebra of the image of the morphism [Ψ_{t}]^{[ℤ/q^{∗}]}. To show this it is sufficient to show that the basic Lie elements in D_{1}, D_{3}, D_{5}, … are linearly independent over ℚ_{ℓ}.

We recall that a Lie bracket in the Lie algebra ( ⨁_{0<α<q} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩,

{ }), which further we denote by L, is given by the formula

[((f_{α})_{0<α<q}; φ), ((g_{α})_{0<α<q}; ψ)]
= (([f_{α}, g_{α}] + D_{φ}(g_{α}) − D_{ψ}(f_{α}))_{0<α<q}; [φ, ψ] + D_{φ}(ψ) − D_{ψ}(φ)) .

Let us set

z_{2n+1} := (((1 − q^{2n}) [Y, X^{2n}])_{0<α<q}; (q − 1) q^{2n}[Y, X^{2n}])

for n > 0 and let

z_{1} := ((Y)_{0<α<q}; 0) .

We recall that I_{r} := ( ⨁_{0<α<q} I_{r}) ⊕ I_{r} is a Lie ideal of the Lie algebra L. For any

Lie bracket of length r in elements D_{1}, D_{3}, D_{5}, … in the Lie algebra L we have

5.1.4.  [… [D_{i_{1}}, D_{i_{2}}], … D_{i_{r}}] ≡ [… [z_{i_{1}}, z_{i_{2}}], … z_{i_{r}}] mod I_{r} .

We recall that Lie (X, Y; ℤ) denote a free Lie algebra over ℤ freely generated by X and Y. The elements z_{i} have integer coefficients hence they belong to the semi-direct product ( ⨁_{0<α<q} Lie (X, Y; ℤ)) ˜×(Lie (X, Y; ℤ)/⟨Y ⟩, { }) of Lie

algebras over ℤ, which we denote by L(ℤ). Observe that qL(ℤ) is a Lie ideal of the Lie algebra L(ℤ). The quotient Lie algebra L(ℤ)/qL(ℤ) is ( ⨁_{0<α<q} Lie (X, Y; ℤ/q)) ˜× (Lie (X, Y; ℤ/q)/⟨Y ⟩, { }). The classes modulo q, z̄_{1} =

((Y)_{0<α<q}; 0) and z̄_{2n+1} = (([Y, X^{2n}])_{0<α<q}; 0) for n > 0, generate freely a free Lie subalgebra of the Lie algebra ( ⨁_{0<α<q} Lie (X, Y; ℤ/q)) ˜×(Lie (X, Y; ℤ/q)/⟨Y ⟩, { }).

This follows from Shirshow-Witt Theorem applied to a free Lie algebra Lie (X, Y; ℤ/q) over a field ℤ/q (see [10] page 331).

Hence basic Lie elements of length r in z̄_{1}, z̄_{3}, z̄_{5}, … are linearly independent over ℤ/q. This implies that basic Lie elements of length r in z_{1}, z_{3}, z_{5}, … are linearly independent over ℤ, hence also over ℚ and ℚ_{ℓ}. It follows from the congruence 5.1.4 that basic Lie elements of length r in D_{1}, D_{3}, D_{5}, … are linearly independent.

Assume that a non-zero linear combination of basic Lie elements of length r in D_{1}, D_{3}, D_{5}, … is equal to a linear combination of basic Lie elements of length greater than r. Working modulo the Lie ideal I_{r} one sees immediately that this is impossible. Therefore basic Lie elements in D_{1}, D_{3}, D_{5}, … are linearly independent, hence the elements D_{1}, D_{3}, D_{5}, … generate freely a free Lie subalgebra of Im ([Ψ_{t}]^{[ℤ/q^{∗}]}).

It follows from results of section 4 (Proposition 4.2.6) that the Lie algebra grLieU(ℚ(μ_{q}), S, ℓ)^{[ℤ/q^{∗}]} is free, freely generated by elements σ_{3}, σ_{5}, …, σ_{2n+1}, … constructed in the proof of Proposition 5.1.2 and by two elements in degree 1. One of these elements is σ_{1} and the other is mapped to zero by [Ψ_{t}]^{[ℤ/q^{∗}]}. This implies that the elements D_{1}, D_{3}, D_{5}, …, D_{2n+1}, … generate Im ([Ψ_{t}]^{[ℤ/q^{∗}]}). □

Theorem 5.1.5. Let q be a prime number different from ℓ. Let V := ℙ^{1} \ {0, 1, ∞} and let

t := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}).

The representation

θ_{t}^{ℚ(μ_{q})/ℚ} : Gal(ℚ̄/ℚ(μ_{ℓ^{∞}})) ⟶ ⨁_{0<α<q} GL(ℚ_{ℓ}{{X, Y }})

have the following properties:

i) it is unramified outside finite places of ℚ(μ_{ℓ^{∞}}) lying over ℓ or q;

ii) it respects the filtration {⨁_{0<α<q} I(ℚ_{ℓ}{{X, Y }})^{i}}_{i∈ℕ} of ⨁_{0<α<q} ℚ_{ℓ}{{X, Y }};

iii) any σ ∈Gal(ℚ̄/ℚ(μ_{ℓ^{∞}})) acts on ⨁_{0<α<q} (I(ℚ_{ℓ}{{X, Y }})^{i}/I(ℚ_{ℓ}{{X, Y }})^{i+1}) as the identity; iv) the image of the associated graded Lie algebra representation

Θ_{t}^{Gal(ℚ(μ_{q})/ℚ)} : ⨁_{i=1}^{∞} (F_{i}(θ_{t}^{ℚ(μ_{q})/ℚ})/F_{i+1}(θ_{t}^{ℚ(μ_{q})/ℚ}))⊗ℚ ⟶ ⨁_{0<α<q} End(ℚ_{ℓ}{{X, Y }})

is a free Lie algebra, freely generated by D_{1}, D_{3}, D_{5}, …, D_{2n+1}, ….

Proof. The representation ψ_{t} is unramified outside S, where S = {𝔭 ∈ V(ℚ(μ_{q})) | 𝔭 divides q or 𝔭 divides ℓ }. Let us set T := {q, ℓ}. Hence the representation θ_{t}^{ℚ(μ_{q})/ℚ}, which is equal by definition [ψ_{t}]∘S_{K,T}^{L,S} ∘i_{K,T,ℓ}, is unramified outside finite places of ℚ(μ_{ℓ^{∞}}) lying over ℓ or q.

The points ii) and iii) follow immediately from Proposition 4.3.0.

By Proposition 5.1.3 the image of [Ψ_{t}]^{[ℤ/q^{∗}]} is a free Lie algebra, freely generated by D_{1}, D_{3}, D_{5}, …, D_{2n+1}, …. This implies that the image of the composition [Ψ_{t}]^{[ℤ/q^{∗}]} ∘β^{−1} ∘(p̄_{ℤ/q^{∗}})^{−1}, and hence also the image of grLie[θ_{t}^{ℚ(μ_{q})/ℚ}], is a free Lie algebra, freely generated by D_{1}, D_{3}, D_{5}, …, D_{2n+1}, …. The morphism

grLie[θ_{t}^{ℚ(μ_{q})/ℚ}] : grLieU(K, T, ℓ) ⟶ ⨁_{0<α<q} GL(ℚ_{ℓ}{{X, Y }})

factors through a surjective morphism

grLieU(K, T, ℓ) ⟶ ⨁_{i=1}^{∞}(F_{i}(θ_{t}^{ℚ(μ_{q})/ℚ})/F_{i+1}(θ_{t}^{ℚ(μ_{q})/ℚ})) ⊗ ℚ.

Hence the image of Θ_{t}^{ℤ/q^{∗}} = grLieθ_{t}^{ℚ(μ_{q})/ℚ} is a free Lie algebra, freely generated by D_{1}, D_{3}, D_{5}, .. ., D_{2n+1}, …. □

5.2. We assume as in section 5.1 that q is a prime number different from ℓ and that ℓ does not divide q − 1.

We assume also that q ≡ 3 mod 4. Then the field ℚ(i \sqrt{q}) is a subfield of ℚ(μ_{q}). The field ℚ(i \sqrt{q}) is fixed by the kernel of the homomorphism

h_{q} : ℤ/q^{∗} → {1, −1}

given by h_{q}(x) = x^{\frac{q−1}{2}} . Let us set

G(q) := ker h_{q} .

As in section 5.1 we are studying actions of Galois groups on

t := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}) and T := ∏_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→})) .

We shall apply formalism from section 4 for

K = ℚ(i \sqrt{q}), L = ℚ(μ_{q}) and the Galois group G(q) = Gal(ℚ(μ_{q})/ℚ(i \sqrt{q})).

We recall that S = { 𝔭 ∈ V(ℚ(μ_{q})) | 𝔭 divides q or ℓ } and therefore in the case considered now

T := \{ 𝔭 ∈ V(ℚ(i\sqrt{q})) | 𝔭 divides q or ℓ \}.

We shall construct geometrically ℓ-adic realization of the associated graded Lie algebra of the fundamental group of the tannakien category of the mixed Tate motives over Spec O_{ℚ(i \sqrt{q})[ \frac{1}{q}]} .

Lemma 5.2.1. We have:

i) Let σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{G(q)}. The coordinates of Ψ_{t}^{G(q)}(σ) calculated modulo I_{2} + Γ^{k+1} L(X, Y) on each component are

((\frac{1 − q^{k−1}}{(q − 1)q^{k−1}} ℓ_{k}(1)(σ)[Y, X^{k−1}])_{0<α<q}; ℓ_{k}(1)(σ)[Y, X^{k−1}])

for k > 1 and odd;

(((\frac{2}{q − 1} ∑_{j∈G(q)} ℓ_{k}(ξ_{q}^{j})(σ))[Y, X^{k−1}])_{α∈G(q)},

((−\frac{2}{q − 1} ∑_{j∈G(q)} ℓ_{k}(ξ_{q}^{j})(σ))[Y, X^{k−1}])_{α∉G(q)}; 0)

for k > 1 and even;

(((\frac{1}{q − 1} ∑_{0<j<q} ℓ(1 − ξ_{q}^{j})(σ)) Y )_{0<α<q}; 0)

for k = 1.

ii) Let σ ∈ (grLieU(L, S, ℓ))_{k}^{[G(q)]}. The coordinates of [Ψ_{t}]^{[G(q)]}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are given by the same formula as in point i) if we replace ℓ_{k}(1), ℓ_{k}(ξ_{q}^{j}) and ℓ(1 − ξ_{q}^{j}) by [ℓ_{k}(1)], [ℓ_{k}(ξ_{q}^{j})] and [ℓ(1 − ξ_{q}^{j})] respectively.

Proof. The lemma is proved in the same way as Lemma 5.1.1. □

Proposition 5.2.2. Let us assume that Conjecture 4.3.1 holds. Let q be a prime number satisfying q ≡ 3 mod 4. In the image of the morphism of Lie algebras

[Ψ_{t}]^{[G(q)]} : grLieU(L, S, ℓ)^{[G(q)]} → ( ⨁_{0<α<q} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, { })

there are elements D_{1}, D_{2}, D_{3}, …, D_{m}, … homogeneous of degree 1, 2, 3, …, m, . … respectively whose coordinates calculated modulo I_{2} are

(((1 − q^{m−1})[Y, X^{m−1}])_{0<α<q}; (q − 1) q^{m−1} [Y, X^{m−1}])

for m > 1 and odd;

((([Y, X^{m−1}])_{α∈G(q)}, (−[Y, X^{m−1}])_{α∉G(q)}); 0)

for m > 1 and even;

((Y)_{0<α<q}; 0)

for m = 1.

Proof. Observe that for all m ∈ ℕ we have dim _{ℚ_{ℓ}}V_{m}(q)^{G(q)} = 1. The space V_{m}(q)^{G(q)} is generated by [0] if m > 1 and odd, by ∑_{α∈G(q)} [α] if m > 1 and even

and by ∑_{α∈ℤ/q^{∗}} [α] if m = 1.

Let us assume that k is even. It follows from Lemma 5.0.5 that ∑_{j∈G(q)} ℓ_{k}(ξ_{q}^{j}) is a non-zero element of H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(k))^{G(q)}. Hence there is a unique, non-zero element t_{k} of H^{1}(G_{ℚ(i\sqrt{q})}; ℚ_{ℓ}(k)) such that t_{k} restricted to H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(k)) is equal ∑_{j∈G(q)} ℓ_{k}(ξ_{q}^{j}). The homomorphism [t_{k}] from grLieU(ℚ(i\sqrt{q}), T, ℓ)_{k} to ℚ_{ℓ} is non-zero because t_{k} is non-zero. Therefore it follows from Proposition 4.2.8 ii) and iii) that [t_{k}] restricted to (grLieU(L, S, ℓ)^{[G(q)]})_{k} is non-zero. Hence the restriction of ∑_{j∈G(q)}[ℓ_{k}(ξ_{q}^{j})] to (grLieU(L, S, ℓ)^{[G(q)]})_{k} is non-zero. Hence

there is σ_{k} ∈ grLieU(L, S, ℓ)_{k}^{[G(q)]} such that ∑_{j∈G(q)}[ℓ_{k}(ξ_{q}^{j})](σ_{k}) = 1. We set

D_{k} := \frac{q − 1}{2}[Ψ_{t}]^{[G(q)]}(σ_{k})

for k > 0 and even.

Let us assume that k = 1. It follows from Lemma 5.0.5 that the element ∑_{0<j<q} ℓ(1 − ξ_{q}^{j}) is a non-zero element of H^{1}(Spec O_{ℚ(μ_{q}),S}; ℚ_{ℓ}(1))^{G(q)}. Observe that ∑_{0<j<q} ℓ(1 − ξ_{q}^{j}) and 2ℓ(i\sqrt{q}) define the same cohomology class in H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(1)). The element 2ℓ(i\sqrt{q}) is non-zero in H^{1}(G_{ℚ(i\sqrt{q})}; ℚ_{ℓ}(1)). In the same way as before, it follows from Proposition 4.2.8 that ∑_{0<j<q}[ℓ(1−ξ_{q}^{j})] restricted to (grLieU(L, S, ℓ)^{[G(q)]})_{1} is non-zero. Hence there is σ_{1} ∈ (grLieU(L, S, ℓ)^{[G(q)]})_{1} such that ∑_{0<j<q}[ℓ(1 − ξ_{q}^{j})](σ_{1}) = 1. We set

D_{1} := (q − 1)[Ψ_{t}]^{[G(q)]}(σ_{1}).

If k > 1 and odd we can find σ_{k} ∈ grLieU(L, S, ℓ)_{k}^{[G(q)]} such that [ℓ_{k}(1)](σ_{k}) = 1. Then we set

D_{k} := \frac{(q − 1)q^{k}}{1 − q^{k−1}} [Ψ_{t}]^{[G(q)]}(σ_{k}).

One verifies easily that the constructed elements have required properties. □

Proposition 5.2.3. The Lie algebra Im ([Ψ_{t}]^{[G(q)]}) is free, freely generated by elements D_{1}, D_{2}, D_{3}, …, D_{m}, …. Proof. The proof of the proposition repeats arguments of the proof of Proposition 5.1.3 and we omit it. □

Theorem 5.2.4. Let us assume that Conjecture 4.3.1 holds. Let q be a prime number different from ℓ and congruent to 3 modulo 4. We assume also that ℓ does not divide q − 1. Let V := ℙ^{1} \ {0, 1, ∞} and let

t := ∐_{0<α<q} π(Vℚ̄; ξ_{q}^{α}, 01^{→}).

The representation

θ_{t}^{ℚ(μ_{q})/ℚ(i\sqrt{q})} : Gal (\overline{ℚ(i\sqrt{q})}/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) → ⨁_{0<α<q} GL (ℚ_{ℓ}\{\{X, Y \}\})

have the following properties:

i) it is unramified outside finite places of ℚ(i\sqrt{q})(μ_{ℓ^{∞}}) dividing ℓ or q,

ii) the representation θ_{t}^{ℚ(μ_{q})/ℚ(i\sqrt{q})} respects the filtration \{⨁_{0<α<q} I(ℚ_{ℓ}\{\{X, Y \}\})^{i}\}_{i∈ℕ} of ⨁_{0<α<q} ℚ_{ℓ}{{X, Y }};

iii) any σ ∈Gal(ℚ̄/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) acts on ⨁_{0<α<q} (I(ℚ_{ℓ}\{\{X, Y \}\})^{i}/I(ℚ_{ℓ}\{\{X, Y \}\})^{i+1}) as the identity;

iv) the image of the associated graded Lie algebra representation

Θ_{t}^{Gal(ℚ(μ_{q})/ℚ)} : ⨁_{i=1}^{∞} (F_{i}(θ_{t}^{ℚ(μ_{q})/ℚ(i\sqrt{q})})/F_{i+1}(θ_{t}^{ℚ(μ_{q})/ℚ(i\sqrt{q})})) ⟶ ⨁_{0<α<q} End (ℚ_{ℓ}\{\{X, Y \}\})

is a free Lie algebra, freely generated by elements D_{1}, D_{2}, D_{3}, …, D_{m}, ….

Proof. The proof of the theorem is the same as the proof of Theorem 5.1.5. □

The ℓ-adic representation from Theorem 5.2.4 represents geometrically the ℓ-adic realization (for ℓ ≠ q) of the associated graded Lie algebra of the fundamental group of the tannakien category of the mixed Tate motives over Spec O_{ℚ(i\sqrt{q})}[\frac{1}{q}].

Now we shall construct geometrically an ℓ-adic realization of the fundamental group of the tannakien category of mixed Tate motives over Spec O_{ℚ(i\sqrt{q})}.

Let us set

τ := ∐_{0<j<q} π(Vℚ̄; −ξ_{q}^{j}, 01^{→}) and T := ∏_{0<j<q} π(Vℚ̄; −ξ_{q}^{j}, 01^{→}) .

Observe that the triple (V_{ℚ(i\sqrt{q})}; −ξ_{q}^{j}, 01^{→}) has good reduction at every finite place of ℚ(μ_{q}). Therefore we set

S := \{𝔭 ∈ V(ℚ(μ_{q})) | 𝔭 divides ℓ\} and T := \{𝔭 ∈ V(ℚ(i\sqrt{q})) | 𝔭 divides ℓ\}.

We shall apply formalism from section 4 to

K = ℚ(i\sqrt{q}), L = ℚ(μ_{q}) and G(q) = Gal (ℚ(μ_{q})/ℚ(i\sqrt{q})).

Lemma 5.2.7. We have

i) Let σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{G(q)}. The coordinates of Ψ_{τ}^{G(q)}(σ) calculated modulo I_{2} + Γ^{k+1} L(X, Y) on each component are

((\frac{1 − q^{k−1} − 2^{k−1} + 2^{k−1} · q^{k−1}}{(q − 1) · 2^{k−1} · q^{k−1}} ℓ_{k}(1)(σ)[Y, X^{k−1}])_{0<α<q}; ℓ_{k}(1)(σ)[Y, X^{k−1}])

for k > 1 and odd;

((\frac{2}{q − 1}( ∑_{j∈G(q)} ℓ_{k}(−ξ_{q}^{j})(σ))[Y, X^{k−1}])_{α∈G(q)}, ( − \frac{2}{q − 1}( ∑_{j∈G(q)} ℓ_{k}(−ξ_{q}^{j})(σ))[Y, X^{k−1}])_{α∉G(q)}; 0 )

for k > 1 and even;

( ( 0 )_{0<α<q}; 0 )

for k = 1.

ii) Let σ ∈ (grLieU(L, S, ℓ))_{k}^{[G(q)]}. The coordinates of [Ψ_{t}]^{[G(q)]}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are given by the same formula as in point i) if we replace ℓ_{k}(1) and ℓ_{k}(−ξ_{q}^{j}) by [ℓ_{k}(1)] and [ℓ_{k}(−ξ_{q}^{j})] respectively.

Proof. First we show the point i) of the lemma. Observe that the set {−ξ_{q}^{j}}_{0<j<q} is the set of primitive 2q-th roots of 1. Hence we can apply formalism from section 5.0. In particular we can apply Lemma 5.0.1 to calculate coordinates of Ψ_{τ} (σ).

First we suppose that k is odd. We have the following relations between ℓ-adic polylogarithms

2^{k−1}(ℓ_{k}(−ξ_{q}^{j}) + ℓ_{k}(ξ_{q}^{j})) = ℓ_{k}(ξ_{q}^{2j});

q^{k−1}(∑_{j=1}^{q−1} ℓ_{k}(ξ_{q}^{j})) = (1 − q^{k−1})ℓ_{k}(1);

ℓ_{k}(ξ_{q}^{j}) = ℓ_{k}(ξ_{q}^{−j}) and ℓ_{k}(−ξ_{q}^{j}) = ℓ_{k}(−ξ_{q}^{−j}).

Observe that (ℤ/2q)^{∗} = ℤ/q^{∗}, hence G(q) acts on V_{k}(2q).

The symbol \frac{1}{q−1} ∑_{j=1}^{q−1}[q +2j] ∈ V_{k}(2q) is invariant under the action of G(q).

Observe that s_{q}^{2q} (\frac{1}{q−1} ∑_{j=1}^{q−1}[q + 2j]) = \frac{1}{q−1} ∑_{j=1}^{q−1} ℓ_{k}(−ξ_{q}^{j}). Using the relations between ℓ-adic polylogarithms we get that

\frac{1}{q − 1} ∑_{j=1}^{q−1} ℓ_{k}(−ξ_{q}^{j}) = \frac{1 − q^{k−1} − 2^{k−1} + 2^{k−1} · q^{k−1}}{(q − 1) · 2^{k−1} · q^{k−1}} ℓ_{k}(1).

Then the formula for k odd follows from Corollary 4.3.4 or Lemma 5.0.2. Now we suppose that k is even. We recall that for k even

ℓ_{k}(−ξ_{q}^{j}) = −ℓ_{k}(−ξ_{q}^{−j}) and ℓ_{k}(1) = 0.

We have also that

ℤ/q^{∗} = G(q) ∪ (−1)G(q).

The symbols \frac{2}{q−1} ∑_{j∈G(q)} j([1]) and \frac{2}{q−1} ∑_{j∈G(q)} j([−1]) are non-trivial in V_{k}(2q) and invariant under the action of G(q). Moreover we have

\frac{2}{q − 1} ∑_{j∈G(q)} j([1]) + \frac{2}{q − 1} ∑_{j∈G(q)} j([−1]) = 0

in V_{k}(2q). This implies the formula for k even.

Now we assume that k = 1. We have

(1 + ξ_{q}^{j}) = ξ_{q}^{j} · (1 + ξ_{q}^{−j}) and ∏_{j=1}^{q−1}(1 + ξ_{q}^{j}) = 1.

Therefore the formula for k = 1 follows from Corollary 4.3.4. One can also use the fact that V_{1}(2q)^{G(q)} = 0.

The point ii) of the lemma is proved in the same way as the point ii) of Lemma 5.1.1. □

Let

α(k) := v_{q}(1 − 2^{k−1})

be the exponent of the highest power of q which divides 1 − 2^{k−1}. It is clear that α(k) < k − 1 and we have

1 − 2^{k−1} = a(k) · q^{α(k)},

where a(k) is not divisible by q.

Proposition 5.2.8. Let us assume that Conjecture 4.3.1 holds. Let q be a prime number congruent to 3 modulo 4. In the image of the morphism of Lie algebras

[Ψ_{τ}]^{[G(q)]} : grLieU(L, S, ℓ)^{[G(q)]} → ( ⨁_{0<α<q} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, { })

there are elements D_{2}, D_{3}, …, D_{k}, … homogeneous of degree 2, 3, …, k, … respectively whose coordinates calculated modulo I_{2} are

((a(k) · (1 − q^{k−1})[Y, X^{k−1}])_{0<α<q}; (q − 1) · 2^{k−1} · q^{k−1−α(k)} [Y, X^{k−1}])

for k > 1 and odd;

((([Y, X^{k−1}])_{α∈G(q)}, (−[Y, X^{k−1}])_{α∉G(q)}); 0)

for m > 1 and even.

Proof. First we consider the case when k is odd. The ℓ-adic polylogarithm ℓ_{k}(1) is a generator of H^{1}(G_{ℚ}; ℚ_{ℓ}(k)). Hence ℓ_{k}(1) restricted to H^{1}(G_{L}; ℚ_{ℓ}(k)) is non-zero. Therefore [ℓ_{k}(1)] is a non-zero, G(q)-equivariant homomorphism from grLieU(L, S, ℓ)_{k} to ℚ_{ℓ} vanishing on decomposable elements. This implies that the restriction of [ℓ_{k}(1)] to grLieU(L, S, ℓ)_{k}^{[G(q)]} is non-zero. Hence there is σ_{k} ∈ grLieU(L, S, ℓ)_{k}^{[G(q)]} such that [ℓ_{k}(1)](σ_{k}) ≠ 0. We set

D_{k} := [Ψ_{τ}]^{[G(q)]} (\frac{(q − 1) · 2^{k−1} · q^{k−1−α(k)}}{ℓ_{k}(1)(σ_{k})} · σ_{k}) .

Let us assume that k is even. The symbol ∑_{j∈G(q)} j([1]) is non-trivial in V_{k}(2q) and belongs to V_{k}(2q)^{G(q)}. Hence the symbol ∑_{j∈G(q)} j([1]) is a generator of V_{k}(2q)^{G(q)}. Hence it follows from Lemma 5.0.5 that ∑_{j∈G(q)} ℓ_{k}(−ξ_{q}^{j}) is a generator of H^{1}(G_{ℚ(μ_{q})}; ℚ_{ℓ}(k))^{G(q)}. Therefore the restriction of ∑_{j∈G(q)}[ℓ_{k}(−ξ_{q}^{j})] to

grLieU(L, S, ℓ)_{k}^{[G(q)]} is non-zero. Hence there is σ_{k} ∈ grLieU(L, S, ℓ)_{k}^{[G(q)]} such that ∑_{j∈G(q)}[ℓ_{k}(−ξ_{q}^{j})](σ_{k}) ≠ 0. We set

D_{k} := [Ψ_{τ}]^{[G(q)]} (\frac{q − 1}{2} · \frac{1}{∑_{j∈G(q)} ℓ_{k}(−ξ_{q}^{j})(σ_{k})} · σ_{k}) .

It is clear that the constructed elements D_{2}, D_{3}, D_{4}, …, D_{n}, … have the required properties. □

Proposition 5.2.9. The Lie algebra Im ([Ψ_{τ}]^{[G(q)]}) is free, freely generated by the elements D_{2}, D_{3}, …, D_{m}, …. Proof. One only need to notice that α(k) < k − 1. We left the details, which are the same as in the proof of Proposition 5.1.3, to readers. □

Theorem 5.2.10. Let us assume that Conjecture 4.3.1 holds. Let q be a prime number different from ℓ and congruent to 3 modulo 4. We assume that ℓ does not divide q − 1. Let V := ℙ^{1} \ {0, 1, ∞} and let

τ := ∐_{0<j<q} π(Vℚ̄; −ξ_{q}^{j}, 01^{→}).

The representation

θ_{τ}^{ℚ(μ_{q})/ℚ(i\sqrt{q})} : Gal (\overline{ℚ(i\sqrt{q})}/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) → ⨁_{0<α<q} GL (ℚ_{ℓ}\{\{X, Y \}\})

have the following properties:

i) it is unramified outside finite places of ℚ(i\sqrt{q})(μ_{ℓ^{∞}}) lying over ℓ;

ii) the representation θ_{τ}^{ℚ(μ_{q})/ℚ(i\sqrt{q})} respects the filtration \{⨁_{0<α<q} I(ℚ_{ℓ}\{\{X, Y \}\})^{i}\}_{i∈ℕ} of ⨁_{0<α<q} ℚ_{ℓ}{{X, Y }};

iii) any σ ∈Gal(ℚ̄/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) acts on ⨁_{0<α<q} (I(ℚ_{ℓ}\{\{X, Y \}\})^{i}/I(ℚ_{ℓ}\{\{X, Y \}\})^{i+1}) as the identity;

iv) the image of the associated graded Lie algebra representation

Θ_{τ}^{Gal(ℚ(μ_{q})/ℚ(i\sqrt{q}))} : ⨁_{i=1}^{∞} (F_{i}(θ_{τ}^{ℚ(μ_{q})/ℚ(i\sqrt{q})})/F_{i+1}(θ_{τ}^{ℚ(μ_{q})/ℚ(i\sqrt{q})})) ⟶ ⨁_{0<α<q} End (ℚ_{ℓ}\{\{X, Y \}\})

is a free Lie algebra, freely generated by elements D_{2}, D_{3}, …, D_{m}, ….

Proof. We omit the detailed proof as it is the same as the proofs of Theorems 5.1.5 and 5.2.4. We only notice that the Galois representation

G_{ℚ(i\sqrt{q})(μ_{ℓ^{∞}})} → Aut_{set}(τ)

is unramified outside finite places of ℚ(i\sqrt{q})(μ_{ℓ^{∞}}) lying over ℓ. This implies that there is no generator in degree 1. □

5.3. We assume that q is a prime number different from ℓ and that ℓ does not divide q − 1. We assume also that q ≡ 1 mod 4.

We shall construct geometrically ℓ-adic realization of the associated graded Lie algebra of the fundamental group of the tannakien category of the mixed Tate motives over Spec O_{ℚ(i\sqrt{q})}.

Observe that the Galois group

Gal (ℚ(μ_{4q})/ℚ) = Gal (ℚ(i)/ℚ) × Gal (ℚ(μ_{q})/ℚ) = ℤ/4^{∗} × ℤ/q^{∗}.

Let s ∈ Gal (ℚ(μ_{q})/ℚ) = ℤ/q^{∗} be a generator and let c ∈ Gal (ℚ(i)/ℚ) = ℤ/4^{∗} be the complex conjugation.

We denote by H(q) the subgroup of Gal (ℚ(μ_{4q})/ℚ) generated by (c, s) ∈ ℤ/4^{∗} × ℤ/q^{∗}. Then the subfield of ℚ(μ_{4q}) fixed by H(q),

ℚ(μ_{4q})^{H(q)} = ℚ(i\sqrt{q}).

We shall study the action of G_{ℚ(μ_{4q})} on the disjoint union of torsors of paths

𝔱 := ∐_{{0<j<4q \atop (j,4q)=1}} π(Vℚ̄; ξ_{4q}^{j}, 01^{→}).

We set

L = ℚ(μ_{4q}) and K = ℚ(i\sqrt{q}).

Observe that a triple (V_{ℚ(μ_{4q})}; ξ_{4q}^{j}, 01^{→}) (for 0 < j < 4q and (j, 4q) = 1) has good reduction everywhere, hence we set

S = { 𝔭 ∈ V(ℚ(μ_{4q})) | 𝔭 divides ℓ }.

Lemma 5.3.1. We have

i) Let σ ∈ (H_{n}/H_{n+1}) ⊗ ℚ^{H(q)}. The coordinates of Ψ_{𝔱}^{H(q)}(σ) calculated modulo I_{2} + Γ^{n+1}L(X, Y) on each component are

(((\frac{1}{q − 1} ∑_{k=1}^{q−1} ℓ_{n}((−1)^{k}iξ_{q}^{s^{k}})(σ))[Y, X^{n−1}])_{t∈H(q)}, ((\frac{−1}{q − 1} ∑_{k=1}^{q−1} ℓ_{n}((−1)^{k}iξ_{q}^{s^{k}})(σ))[Y, X^{n−1}])_{t∉H(q)} ; 0)

for n > 1 and even;

((\frac{1 − q^{n−1}}{(1 − q)q^{n−1}} · \frac{1 − 2^{n−1}}{2^{2n−1}} ℓ_{n}(1)(σ)[Y, X^{n−1}])_{t∈(ℤ/4q)^{∗}} ; ℓ_{n}(1)(σ)[Y, X^{n−1}])

for n > 1 and odd;

(( 0 )_{t∈(ℤ/4q)^{∗}} ; 0 )

for n = 1.

ii) Let σ ∈ (grLieU(L, S, ℓ))_{n}^{[H(q)]}. The coordinates of [Ψ_{t}]^{[H(q)]}(σ) calculated modulo I_{2} + Γ^{n+1}L(X, Y) on each component are given by the same formula as in point i) if we replace ℓ_{n}(1) and ℓ_{n}((−1)^{k}iξ_{q}^{s^{k}}) by [ℓ_{n}(1)] and [ℓ_{n}((−1)^{k}iξ_{q}^{s^{k}})] respectively.

Proof. Let n = 1. We have the identity

\frac{x^{4q} − 1}{x − 1} = \frac{x^{q} − 1}{x − 1} · (x^{q} + 1) · (x^{2} + 1) · ∏_{k=0}^{q−2}(x − (−1)^{k}iξ_{q}^{s^{k}}) · ∏_{k=0}^{q−2}(x + (−1)^{k}iξ_{q}^{s^{k}}).

Hence we get

∏_{k=0}^{q−2}(1 − (−1)^{k}iξ_{q}^{s^{k}}) · ∏_{k=0}^{q−2}(1 + (−1)^{k}iξ_{q}^{s^{k}}) = 1.

Therefore ∏_{t∈H(q)}(1−t(iξ_{q})) = ∏_{k=0}^{q−2}(1−(−1)^{k}iξ_{q}^{s^{k}}) and ∏_{t∈H(q)}(1−t(−iξ_{q})) =

∏_{k=0}^{q−2}(1 + (−1)^{k}iξ_{q}^{s^{k}}) are units of the ring O_{ℚ(i\sqrt{q})}. The only units of the ring O_{ℚ(i\sqrt{q})} are roots of unity. This implies the lemma for n = 1.

Let us assume that n > 1. Observe that −iξ_{q}^{−1} does not belong to the H(q)- orbit of iξ_{q}. Hence the set of primitive 4q-th roots of 1 is a union of H(q)-orbits of iξ_{q} and −iξ_{q}^{−1}. Observe also that all primitive q-th roots of 1 belong to one orbit of H(q). Now the first part of the lemma follows from Corollary 4.3.4 and from distribution and inversion relations for ℓ-adic polylogarithms and we leave it for readers.

The second part of the lemma follows in the same way as in Lemma 5.1.1. □ Proposition 5.3.2. Let q be a prime number different from ℓ and congruent to 1 modulo 4. We assume that ℓ does not divide q − 1. We assume also that Conjecture 4.3.1 holds. In the image of the morphism of Lie algebras

[Ψ_{𝔱}]^{[H(q)]} : grLieU(L, S, ℓ)^{[H(q)]} → ( ⨁_{{0<α<4q \atop (α,4q)=1}} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, \{ \})

there are elements D_{2}, D_{3}, …, D_{m}, … homogeneous of degree 2, 3, …, m, … respectively whose coordinates calculated modulo I_{2} are

((([Y, X^{n−1}])_{t∈H(q)}, (−[Y, X^{n−1}])_{t∈(ℤ/4q)^{∗}\H(q)}) ; 0 )

for n > 1 and even;

((a(n)(1 − q^{n−1})[Y, X^{n−1}])_{t∈(ℤ/4q)^{∗}} ; (1 − q) · q^{n−1−α(n)} · 2^{2n−1}[Y, X^{n−1}])

for n > 1 and odd;

( ( 0 )_{t∈(ℤ/4q)^{∗}} ; 0 )

for n = 1. Proof. The proposition follows from Lemma 5.3.1. The details which are similar to the proof of Proposition 5.1.2 we omit. □ Proposition 5.3.3. The Lie algebra Im ([Ψ_{𝔱}]^{[H(q)]}) is free, freely generated by the elements D_{2}, D_{3}, …, D_{m}, . . .. Proof. As in the proof of Proposition 5.2.9 one need to notice that n−1−α(n) > 0. The details of the proof we left to readers. □

Theorem 5.3.4. Let us assume that Conjecture 4.3.1 holds. Let q be a prime number different from ℓ and congruent to 1 modulo 4. We assume that ℓ does not divide q − 1. Let V := ℙ^{1} \ {0, 1, ∞} and let

𝔱 := ∐_{{0<j<4q \atop (j,4q)=1}} π(Vℚ̄; ξ_{4q}^{j}, 01^{→}).

θ_{𝔱}^{ℚ(μ_{4q})/ℚ(i\sqrt{q})} : Gal (\overline{ℚ(i\sqrt{q})}/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) → ⨁_{{0<α<4q \atop (α,4q)=1}} GL (ℚ_{ℓ}\{\{X, Y \}\})

have the following properties:

i) it is unramified outside finite places of ℚ(i\sqrt{q})(μ_{ℓ^{∞}}) lying over ℓ;

ii) it respects the filtration \{⨁_{{0<α<4q \atop (α,4q)=1}} I(ℚ_{ℓ}\{\{X, Y \}\})^{i}\}_{i∈ℕ} of ⨁_{{0<α<q \atop (α,4q)=1}} ℚ_{ℓ}\{\{X, Y \}\};

iii) any σ ∈Gal(ℚ̄/ℚ(i\sqrt{q})(μ_{ℓ^{∞}})) acts on ⨁_{{0<α<q \atop (α,4q)=1}} (I(ℚ_{ℓ}\{\{X, Y \}\})^{i}/I(ℚ_{ℓ}\{\{X, Y \}\})^{i+1}) as the identity;

iv) the image of the associated graded Lie algebra representation

Θ_{𝔱}^{Gal(ℚ(μ_{4q})/ℚ(i\sqrt{q}))} : ⨁_{i=1}^{∞} (F_{i}(θ_{𝔱}^{ℚ(μ_{4q})/ℚ(i\sqrt{q})})/F_{i+1}(θ_{𝔱}^{ℚ(μ_{4q})/ℚ(i\sqrt{q})}))⊗ℚ ⟶ ⨁_{{0<α<q \atop (α,4q)=1}} End(ℚ_{ℓ}\{\{X, Y \}\})

is a free Lie algebra, freely generated by D_{2}, D_{3}, D_{4}, …, D_{n}, ….

□

5.4. We continue to study Galois actions on torsors of paths. In this section we shall construct geometrically an ℓ-adic realization of the associated graded Lie algebra of the motivic fundamental group of the category of mixed Tate motives over Spec ℤ.

Let

V := ℙ^{1} \ {0, 1, ∞}.

Let q be a prime number greater than 2 and different from ℓ. Let ξ_{q} = exp (\frac{2πi}{q}) be a primitive q-th root of 1. We shall study actions of Galois groups on a disjoint union and on a product of π_{1}(Vℚ̄; 10^{→})-torsors

t_{q} := ∐_{0<α<q} π_{1}(Vℚ̄; −ξ_{q}^{α}, 10^{→}) and T_{q} := ∏_{0<α<q} π_{1}(Vℚ̄; −ξ_{q}^{α}, 10^{→}).

We have ℚ(μ_{2q}) = ℚ(μ_{q}). Observe that the Galois group G_{ℚ(μ_{q})} acts on t_{q} and

on T_{q}. Observe that a triple (V_{ℚ(μ_{q})}; −ξ_{q}^{α}, 01^{→}) has good reduction everywhere, hence we set

S := {𝔭 ∈ V(ℚ(μ_{q})) | 𝔭 divides ℓ}.

We shall apply formalism from section 4 to

L := ℚ(μ_{2q}), K := ℚ and ℤ/q^{∗} = Gal(ℚ(μ_{2q})/ℚ).

The action of G_{ℚ(μ_{q})} on t_{q} yields a Lie algebra representation

Ψ_{t_{q}} : ⨁_{k=1}^{∞} ((H_{k}/H_{k+1}) ⊗ ℚ) → ( ⨁_{0<α<q} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, { }),

where

H_{k} := ⋂_{0<α<q} H_{k}(V_{ℚ(μ_{q})}; −ξ_{q}^{α}, 10^{→})

for k ∈ ℕ.

Lemma 5.4.0. We have:

i) Let σ ∈ H_{k}. The coordinates of Ψ_{t_{q}} calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are

((ℓ_{k}(−ξ_{q}^{α})(σ)[Y, X^{k−1}])_{0<α<q}; ℓ_{k}(1)(σ)[Y, X^{k−1}])

for k > 1 and

((ℓ_{1}(−ξ_{q}^{α})(σ)Y )_{0<α<q}; 0 )

for k = 1;

ii) Let σ ∈ (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{k}. The coordinates of [Ψ_{t_{q}}]^{[ℤ/q^{∗}]}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are given by the same formulas as in the point i) if we replace ℓ_{k}(1), ℓ_{k}(−ξ_{q}^{α}) and ℓ_{1}(−ξ_{q}^{α}) by [ℓ_{k}(1)], [ℓ_{k}(−ξ_{q}^{α})] and [ℓ_{1}(−ξ_{q}^{α})] respectively.

Proof. The lemma follows from Lemma 1.1.0 and Definition 3.0.1 of ℓ-adic polylogarithms. □

The group ℤ/q^{∗} = Gal (ℚ(μ_{q})/ℚ) acts on the associated graded Lie algebra grLieU(L, S, ℓ). We shall study the Lie algebra homomorphism

[Ψ_{t_{q}}]^{[ℤ/q^{∗}]} : (grLieU(L, S, ℓ))^{[ℤ/q^{∗}]} → ( ⨁_{0<α<q} Lie(X, Y)) ˜×(Lie(X, Y)/⟨Y ⟩, { }).

Lemma 5.4.1. Let q be a prime number greater than 2 and different from ℓ. We assume that ℓ does not divide q − 1. We have

i) Let σ ∈ ((H_{k}/H_{k+1}) ⊗ ℚ)^{ℤ/q^{∗}}. The coordinates of Ψ_{t_{q}}^{ℤ/q^{∗}}(σ) calculated modulo I_{2} + Γ^{k+1}L(X, Y) on each component are

( (\frac{(1 − 2^{k−1}) · (1 − q^{k−1})}{2^{k−1} · q^{k−1} · (q − 1)}ℓ_{k}(1)(σ)[Y, X^{k−1}] )_{0<α<q}; ℓ_{k}(1)(σ)[Y, X^{k−1}] )

for k > 1 and odd;

( ( 0 ) )_{0<α<q}; 0 )

for k > 1 and even;

( ( 0 ) )_{0<α<q}; 0 )

for k = 1. ii) Let σ ∈ (grLieU(L, S, ℓ))_{k}^{[ℤ/q^{∗}]}. The coordinates of [Ψ_{t}]^{[ℤ/q^{∗}]}(σ) calculated

modulo I_{2} + Γ^{k+1}L(X, Y) on each component are given by the same formula as in point i) if we replace ℓ_{k}(1) by [ℓ_{k}(1)].

Proof. The first part of the lemma follows from Lemma 5.4.0, Corollary 4.3.4 and the distribution and inversion relations for ℓ-adic polylogarithms. For k = 1 we use the relation ∏_{k=1}^{q−1}(1 + ξ_{q}^{k}) = 1.

The second part of the lemma is proved in the same way as part ii) of Lemma 5.1.1. □

We recall that

1 − 2^{k−1} = a(k) · q^{α(k)},

where a(k) is an integer not divisible by q. Proposition 5.4.2. In the image of the morphism [Ψ_{t_{q}}]^{[ℤ/q^{∗}]} there are elements D_{3}, D_{5}, D_{7}, …, D_{2n+1}, … homogenous of degree 3, 5, 7, …, 2n + 1, … respectively whose coordinates calculated modulo I_{2} are

( (a(2n + 1) · (1 − q^{2n})[Y, X^{2n}] )_{0<α<q}; 2^{2n} · q^{2n−α(2n+1)} · (q − 1)[Y, X^{2n}] )

for n = 1, 2, 3, ….

Proof. The ℓ-adic polylogarithm ℓ_{2n+1}(10^{→}) is a rational non-zero multiple of the

Soule class - the generator of H^{1}(G_{ℚ}; ℚ_{ℓ}(2n+1)). Therefore ℓ_{2n+1}(10^{→}) restricted to H^{1}(G_{L}; ℚ_{ℓ}(2n + 1)) is also non-zero. It follows from Proposition 4.2.8 that [ℓ_{2n+1}(1)] restricted to (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{2n+1} is non-zero. Therefore there

is σ_{2n+1} ∈ (grLieU(L, S, ℓ)^{[ℤ/q^{∗}]})_{2n+1} such that [ℓ_{2n+1}(10^{→})](σ_{2n+1}) = 1. We set

D_{2n+1} := (q − 1) · 2^{2n} · q^{2n−α(2n+1)} · [Ψ_{t_{q}}]^{[ℤ/q^{∗}]}(σ_{2n+1}).

□

Proposition 5.4.3. The Lie algebra Im ([Ψ_{t_{q}}]^{[ℤ/q^{∗}]}) is free freely generated by elements D_{3}, D_{5}, …, D_{2n+1}, …. Proof. The proof is identical to the proof of Proposition 5.1.3 and we omit it. □ Theorem 5.4.4. Let q be a prime number different from 2 and from ℓ. We assume also that ℓ does not divide q − 1. Let V = ℙ^{1} \ {0, 1, ∞} and let

t_{q} := ∐_{0<α<q} π_{1}(Vℚ̄; −ξ_{q}^{α}, 10^{→}).

Then we have:

i) the representation

θ_{t_{q}}^{ℚ(μ_{2q})/ℚ} : Gal (\overline{ℚ}/ℚ(μ_{ℓ^{∞}})) ⟶ ⨁_{0<j<q} GL (ℚ_{ℓ}\{\{X, Y \}\})

is unramified outside finite places of ℚ(μ_{ℓ^{∞}}) lying over ℓ; ii) the representation θ_{t_{q}}^{ℚ(μ_{2q})/ℚ} respects the filtration \{⨁_{{0<α<4q \atop (α,4q)=1}} I(ℚ_{ℓ}\{\{X, Y \}\})^{i}\}_{i∈ℕ}

of ⨁_{{0<α<q \atop (α,4q)=1}} ℚ_{ℓ}\{\{X, Y \}\};

iii) any σ ∈Gal(ℚ̄/ℚ(μ_{ℓ^{∞}})) acts on ⨁_{{0<α<q \atop (α,4q)=1}} (I(ℚ_{ℓ}\{\{X, Y \}\})^{i}/I(ℚ_{ℓ}\{\{X, Y \}\})^{i+1}) as the identity;

iv) the image of the associated graded Lie algebra representation gr Lie θ_{t_{q}}^{ℚ(μ_{2q})/ℚ} is a free Lie algebra, freely generated by D_{3}, D_{5}, …, D_{2n+1}, ….

Proof. Observe that −ξ_{q} ≡ ξ_{q} mod 2 and −ξ_{q} ≡ −1 mod q. Hence the triple

(ℙ^{1} \{0, 1, ∞}; ξ_{q}^{j}, 10^{→}) has good reduction everywhere. Proposition 4.1.7 implies that the representation

ψ_{t_{q}} : Gal (\overline{ℚ(μ_{2q})}/ℚ(μ_{2q})) ⟶ Aut_{set}(t_{q})

is unramified outside finite places of ℚ(μ_{2q}) dividing ℓ. The rest of the proof repeats arguments of the proof of Theorem 5.1.5. □

The representation of G_{ℚ} on π_{1}(Vℚ̄; 10^{→}) was studied by Ihara, Deligne, Grothendieck and others. This representation is unramified outside ℓ (see [9]). We do not know if the image of the associated graded Lie algebra representation is free, freely generated by single generators in degrees 3, 5, 7, …, 2n + 1, …. The representation of G_{ℚ} on π_{1}(Vℚ̄; 10^{→}) is isomorphic to the representation of

G_{ℚ} on π_{1}(Vℚ̄; 10^{→})-torsor of paths π(Vℚ̄; 10^{→}, 01^{→}) (see [22]).

We hope that Theorem 5.4.4 will help to understand the representation of G_{ℚ} on π (Vℚ̄; 10^{→}, 01^{→}) and hence also on π_{1}(Vℚ̄; 10^{→}). Notice that for any prime q greater than 2 the image of the morphism gr Lie θ_{t_{q}}^{ℚ(μ_{2q})/ℚ} from Theorem 5.4.4 is free, freely generated by elements D_{3}, D_{5}, …, D_{2n+1}, … of degrees 3, 5, …, 2n + 1, … respectively.

To realize ℓ-adically the associated graded Lie algebra of the motivic fundamental group of the tannakian category of mixed Tate motives over Spec ℤ[i] (resp. Spec ℤ[i\sqrt{2}], resp. Spec ℤ[i\sqrt{2}][\frac{1}{2}]) one needs to study Galois action

on torsors of paths ∐_{k=1,5,7,11} π(Vℚ̄; ξ_{12}^{i}, 01^{→}) (resp. ∐_{{0<j<8 \atop j odd}} π(Vℚ̄; ξ_{8}^{j}, 01^{→}), resp.

∐_{{0<j<24 \atop (j,24)=1}} π(Vℚ̄; ξ_{24}^{j}, 01^{→})).

References

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Université de Nice-Sophia Antipolis Département de Mathématiques Laboratoire Jean Alexandre Dieudonné U.R.A. au C.N.R.S., N^{o} 168 Parc Valrose – B.P. N^{o} 71 06108 Nice Cedex 2, France E-mail address wojtkow@math.unice.fr Fax number 04 93 51 79 74