On l-adic iterated integrals, I

Analog of Zagier conjecture

Zdzisław Wojtkowiak

Abstract. We are studying some aspects of the action of Galois groups on the torsor of paths from v to z on a projective line minus a finite number of points. We obtain objects which formally behave like classical iterated integrals and polylogarithms. We formulate an analog of Zagier conjecture for these l - adic analogs of iterated integrals and polylogarithms.

The plan of the paper.

Part I

0. Introduction to Part I.

1. Torsors of paths.

2. Geometric generators of π_{1}(X(C); v).

3. Filtrations of Galois groups.

4. Coordinates on the fundamental group and on the torsor of paths.

5. l - adic iterated integrals.

6. Cocycle conditions.

7. Analog of Zagier conjecture.

8. Primitive example in the case P^{1} \ {0, 1, ∞}.

Part II

9. Introduction to Part II.

10. Functional equations. Mathematics Subject Classification 11 Number theory 11. l - adic polylogarithms.

12. Monodromy of l - adic polylogarithms.

Part III

13. Introduction to Part III.

14. Kummer characters.

15. Applications to Galois actions on fundamental groups.

§0. Introduction. 0.1. The classical complex iterated integrals appear in the study of mixed Hodge structures on fundamental groups and on torsors of paths (see [D], [BD] and [W3]). In this paper we shall study their l - adic analogs.

Let K be a number field and let X be a projective line P_{K}^{1} minus a finite number of K - points. Let z and v be two K - points or tangential base points defined over K of X. Let π_{1}(XK̄; v) be the l - completion of the étale fundamental group of XK̄ based at v. We denote by π(XK̄; z, v) the π_{1}(XK̄; v) - torsor of (l - adic) paths from v to z. The Galois group G_{K} := Gal(K̄/K) acts on the set π(XK̄; z, v). To describe this action of G_{K} we shall proceed in the following way.

Let us fix a path p from v to z. Then the map

π(XK̄; z, v) ∋ q → p^{−1}q ∈ π_{1}(XK̄; v) 0.1.1.

is a bijection. The action of G_{K} on the torsor π(XK̄; z, v) transported to an action of G_{K} on π_{1}(XK̄; v) by the map 0.1.1 is given by

π_{1}(XK̄; v) ∋ S → f_{p}(σ) · σ(S) ∈ π_{1}(XK̄; v),

where σ ∈ G_{K} and

f_{p}(σ) := p^{−1} · σ(p). 0.1.2.

The function f_{p} : G_{K} → π_{1}(XK̄; v) has the following important property.

Proposition A. (see section 1) The function f_{p} : G_{K} → π_{1}(XK̄; v) is a cocycle, i.e.,

f_{p}(τ · σ) = f_{p}(τ) · τ(f_{p}(σ)). 0.1.3.

This result was the starting point of the paper (see also Theorem A and B in [I1].) Let us write

f_{p}(σ) = x_{1}^{α_{1}(σ)} · x_{2}^{α_{2}(σ)} · . . . x_{n}^{α_{n}(σ)} · (x_{1}, x_{2})^{β_{1,2}(σ)} · (x_{1}, x_{3})^{β_{1,3}(σ)} · . . . ,

where x_{1}, …, x_{n} are generators of π_{1}(XK̄; v). The obvious question is if the exponents β_{i,j} : G_{K} → Z_{l} are also cocycles. This question is studied in sections 6 and 11.

Let X := P_{K}^{1} \ {a_{1}, …, a_{n}, ∞}. The fundamental group π_{1}(XK̄; v) we embed into the algebra Q_{l}{{X_{1}, …, X_{n}}} of non - commutative formal power series in n non - commuting variables X_{1}, …, X_{n} (n+1 is a number of points removed from P_{K}^{1}) sending a loop around a_{i} onto e^{X_{i}} for i = 1, …, n. The actions of G_{K} on the fundamental group π_{1}(XK̄; v) and on the torsor π(XK̄; z, v) we transport to linear actions of G_{K} on Q_{l}{{X_{1}, …, X_{n}}}. Hence we get representations

φ : G_{K} → Aut(Q_{l}{{X_{1}, …, X_{n}}})

in a case of the action deduced from the action on π_{1}(XK̄; v) and

ψ_{p} : G_{K} → GL(Q_{l}{{X_{1}, …, X_{n}}})

in a case of the action deduced from the action on the torsor π(XK̄; z, v).

If σ ∈ G_{K(μ_{l^{∞}})} := Gal(K̄/K(μ_{l^{∞}})) then ψ_{p}(σ) is a pro - unipotent automorphism of Q_{l}{{X_{1}, …, X_{n}}}. Hence logψ_{p}(σ) is defined and we have the following result. Proposition B. (see section 5) Let σ ∈ G_{K(μ_{l^{∞}})}. Then we have

log ψ_{p}(σ) = L_{(logψ_{p}(σ))(1)} + logφ(σ),

where for w ∈ Q_{l}{{X_{1}, …, X_{n}}}, L_{w} is a left multiplication by w.

The operator logφ(σ) is a derivation of the Q_{l} - algebra Q_{l}{{X_{1}, …, X_{n}}}. Let us fix a path γ_{i} from v to a tangential base point at a_{i} for i = 1, …, n. The generator

x_{i} := γ_{i}^{−1} · small loop around a_{i} · γ_{i}

we send to e^{X_{i}} for i = 1, …, n. Then we show the following result.

Proposition C. (see section 5) Let σ ∈ G_{K(μ_{l^{∞}})}. Then we have

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

for i = 1, …, n.

P.Deligne in [D] and Y.Ihara in [I1], [I2] have studied the Galois action on P^{1} \ {0, 1, ∞}. They got results related to our Proposition C. Their results in the case of P^{1} \ {0, 1, ∞} motivated our study of more general situations.

The power series (logψ_{p}(σ))(1) is a Lie element and its coefficients (with σ ∈ G_{K(μ_{l^{∞}})} varing) in a Hall base we shall call l - adic iterated integrals (see Definition 5.3.0). These l - adic iterated integrals are functions from G_{K(μ_{l^{∞}})} to Q_{l}. They depend on points v and z and also on a choice of a path p from v to z (compare with the classical integral ∫_{v}^{z} \frac{dz}{z} which depends on v and z and on a choice of a path p from v to z). They have all formal properties of iterated integrals on X(C). In [W1] we studied functional equations of iterated integrals. The l - adic iterated integrals have the same functional equations as classical complex iterated integrals on X(C) (see section 10). We have an analog of Zagier conjecture for l - adic iterated integrals as in [W3] (see section 7).

We would like to thank H.Nakamura for discussions of some topics considered in the paper and and for encouragements. The results of this paper were presented in unofficial talk in Berkeley and in Oberwolfach in autumn 1999, in Luminy in spring 2000 and in summer 2001, in Acquafreda di Maretea in 2001 and in Cambridge in autumn 2002.

The present paper is a rewritten version of the first six sections of [W4].

§1. Torsors of paths. 1.0. Let X be a smooth algebraic variety defined over a number field K. We denote by X̂(K) the union of K - points of X and tangential base points of X defined over K.

Let us fix a prime number l. Let z, v ∈ X̂(K). Let π_{1}(XK̄; v) be the l - completion, i.e., the maximal pro-l quotient of the étale fundamental group of XK̄ with a base point at v. We denote by π(XK̄; z, v) the profinite set of homotopy classes of (l - adic) paths from v to z. The set π(XK̄; z, v) is a π_{1}(XK̄; v) - torsor. We set G_{K} := Gal(K̄/K). The group G_{K} acts on π_{1}(XK̄; v) and on π(XK̄; z, v) and the action of G_{K} is compatible with the action of π_{1}(XK̄; v) on π(XK̄; z, v), i.e., σ(p · S) = σ(p) · σ(S) for p ∈ π(XK̄; z, v), S ∈ π_{1}(XK̄; v) and σ ∈ G_{K}.

In this section we shall study elementary properties of the action of the Galois group G_{K} on the torsor of paths π(XK̄; z, v). The set π(XK̄; z, v) is difficult to handle. We fix a path from v to z and using this path we identify the set π(XK̄; z, v) with the fundamental group π_{1}(XK̄; v). The group π_{1}(XK̄; v) is more familiar and we describe the action of G_{K} on π(XK̄; z, v) in terms of the action of G_{K} on π_{1}(XK̄; v).

Let us fix a path p ∈ π(XK̄; z, v). Then

t_{p} : π(XK̄; z, v) → π_{1}(XK̄; v)

given by t_{p}(q) := p^{−1} · q is a bijection. The map t_{p} is not G_{K} - equivariant. However using this map we shall transport the action of G_{K} on π(XK̄; z, v) into the action of G_{K} on π_{1}(XK̄; v), which is a more familiar object.

Let σ ∈ G_{K}. We set

σ_{p} := t_{p} ∘ σ ∘ t_{p}^{−1},

where σ : π(XK̄; z, v) → π(XK̄; z, v) is the map induced by σ. Definition 1.0.1. We define a function f_{p} : G_{K} → π_{1}(XK̄; v) setting

f_{p}(σ) := p^{−1} · σ(p) ∈ π_{1}(XK̄; v)

for any σ ∈ G_{K}.

Lemma 1.0.2. The action of G_{K} on π_{1}(XK̄; v) transported by the isomorphism t_{p} from the action of G_{K} on π(XK̄; z, v) is given by

σ_{p}(S) = f_{p}(σ) · σ(S),

where S ∈ π_{1}(XK̄; v) and σ ∈ G_{K}.

Proof. We have σ_{p}(S) = t_{p} ∘ σ ∘ t_{p}^{−1}(S) = t_{p}(σ(p · S)) = p^{−1} · σ(p) · σ(S) = f_{p}(σ) · σ(S).

This action of G_{K} on π_{1}(XK̄; v) transported by the isomorphism t_{p} depends on a choice of a path p from v to z. Let q ∈ π(XK̄; z, v) be another path from v to z. One easily verifies that

f_{p}(σ) = (q^{−1}p)^{−1} · f_{q}(σ) · σ(q^{−1}p) 1.0.3.

and

t_{p}(r) = t_{q}((qp^{−1}) · r) = (p^{−1}q) · t_{q}(r) 1.0.4.

for any r in π(XK̄; z, v).

The relation between actions of σ_{p} and σ_{q} is described in the next lemma.

Lemma 1.0.5. For any σ ∈ G_{K} and S ∈ π_{1}(XK̄; v) we have

σ_{p}(S) = (q^{−1}p)^{−1} · σ_{q}((q^{−1}p) · S).

Proof. The lemma follows from Lemma 1.0.2 and from 1.0.3.

We finish this section describing some elementary properties of the element f_{p}(σ).

Lemma 1.0.6. Let p be a path from v to z and let q be a path from w to v. Then we have

f_{pq}(σ) = q^{−1} · f_{p}(σ) · q · f_{q}(σ) and f_{p^{−1}}(σ) = p · (f_{p}(σ))^{−1} · p^{−1}

for any σ ∈ G_{K}.

Proof. An easy verification we left to the reader.

Proposition 1.0.7. The function f_{p} : G_{K} → π_{1}(XK̄; v) is a cocycle, i.e., for any τ and σ in G_{K} we have

f_{p}(τ · σ) = f_{p}(τ) · τ(f_{p}(σ)).

Proof. We have f_{p}(τ · σ) = p^{−1} · τ(σ(p)) = p^{−1} · τ(p) · τ(p^{−1}) · τ(σ(p)) = f_{p}(τ) · τ(f_{p}(σ)).

Corollary 1.0.8. We have

f_{p}(τ^{−1}) = τ^{−1}(f_{p}(τ)^{−1}).

Remark. Let p be a path from 01 to 10 on P^{1}(C) \ {0, 1, ∞}. The element f_{p}(σ) was used by Ihara in [I2].

§2. Geometric generators of π_{1}(X(C); v). 2.0. Let X = P_{C}^{1} \{a_{1}, …, a_{n+1}} and let x ∈ X̂(C). We shall construct a canonical family of generators of π_{1}(X(C); x). The Galois action on fundamental groups will be described in terms of these generators.

Let us choose a tangential base point v_{i} (a tangent vector) at a_{i} for i = 1, 2, …, n + 1. 2.0.1. Let us assume that x ∈ X(C). Let Γ = {γ_{k}}_{k=1,...,n+1} be a family of smooth paths from x to each v_{k} such that any two paths do not intersect, no path self-intersects and for each k, γ_{k}([0, 1[) ⊂ X(C). The indices are choosen in such a way that when we make a small circle around x in the opposite clockwise direction starting from γ_{1}, then we meet successively γ_{2}, γ_{3}, …, γ_{n+1}. The element S_{k} ∈ π_{1}(X(C); x) is defined in the following way: we move along γ_{k}, near a_{k} we make a small circle around a_{k} in the opposite clockwise direction and we return along γ_{k} to x. 2.0.2. Without loss of generality we can assume that x is a tangential base point at a_{1}. Let x′ ∈ X(C) be near a_{1} in the direction x. Let Γ = {γ′_{k}}_{k=2,...,n+1} be a family of smooth paths from x′ to each v_{k} satisfying the conditions from 2.0.1. Let S′_{k} be defined by the path γ′_{k}. Let γ be a path [0, 1] ∋ t → a_{i} + t(x′ − a_{i}) ∈ X(C). We set γ_{k} := γ′_{k} · γ and S_{k} := γ^{−1} · S′_{k} · γ for k = 2, …, n + 1. S_{1} is a small circle around a_{1} starting from x in the opposite clockwise direction.

Lemma 2.0.3. The elements S_{1}, …, S_{n+1} generate π_{1}(X(C); x) and satisfy the only relation

S_{n+1} · . . . · S_{1} = 1.

Definition 2.0.4. The ordered sequence (S_{1}, …, S_{n+1}) we shall call a sequence of geometric generators of π_{1}(X(C); x) associated to a family of paths Γ. 2.1. Let F_{n+1} = F_{n+1}(x_{1}, …, x_{n+1}) be a free group on n + 1 elements (x_{1}, …, x_{n+1}). Let B_{n+1}(x_{1}, …, x_{n+1}) be a subgroup of Aut(F_{n+1}) consisting of automorphisms f such that f(x_{i}) = t_{i} · x_{μ(i)} · t_{i}^{−1} (i = 1, …, n + 1) and f(x_{n+1}) · . . . · f(x_{1}) = x_{n+1} · . . . · x_{1}, where t_{i} ∈ F_{n+1} and μ ∈ S_{n+1} is a permutation.

Let us set F_{n+1}^{∗} := F_{n+1}/ < x_{n+1} ·. . .·x_{1} > . The group B_{n+1}(x_{1}, …, x_{n+1}) acts as an automorphism group on F_{n+1}^{∗}. This automorphism group we denote by B_{n+1}^{∗}(x_{1}, …, x_{n+1}). Let B_{n+1}^{(1)∗}(x_{1}, …, x_{n+1}) := ker(π : B_{n+1}^{∗}(x_{1}, …, x_{n+1}) → Σ_{n+1}), where π is the obvious projection.

The next lemma is well known .

Lemma 2.1.1. (see [W2]) Let (S_{1}, …, S_{n+1}) be a sequence of geometric generators of π_{1}(X(C); x). Then any other sequence of geometric generators of π_{1}(X(C); x) is of the form (f(S_{1}), …, f(S_{n+1})), where f ∈ B_{n+1}^{∗}(S_{1}, …, S_{n+1}).

Definition 2.1.2. Let s = (S_{1}, …, S_{n+1}) and s′ = (S′_{1}, …, S′_{n+1}) be two sequences of geometric generators of π_{1}(X(C); x). We say that s and s′ are in the same permutation class if there is f ∈ B_{n+1}^{(1)∗}(S_{1}, …, S_{n+1}) such that f(S_{i}) = S′_{i} for each i. 2.2. Let K be a number field. Let a_{1}, …, a_{n+1} be K-points of the projective line P_{K}^{1}. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let x ∈ X̂(K). Let us choose a tangential base point v_{k} ∈ X̂(K) at a_{k} for k = 1, …, n+1. Let us fix an embedding K ⊂ C. Let Γ = {γ_{k}}_{k+1,...,n+1} be a family of paths on X(C) from x to each v_{k} and let S_{1}, …, S_{n+1} be a family of geometric generators of π_{1}(X(C); x) associated to Γ.

The geometric generators of π_{1}(X(C); x) can be interpreted as elements of π_{1}(XK̄; x). The path γ_{k} from x to v_{k} can be interpreted as an l - adic path, i.e., a natural transformation of fiber functors over x and over v_{k} from étale coverings of XK̄ to sets. A small circle around a_{k} based at v_{k} is defined in the proof of Proposition 2.2.1. However it would be very interesting to construct “geometric generators” of π_{1}(XK̄; x) in purely algebraic way.

Below we shall describe the action of G_{K} on π_{1}(XK̄; x) in terms of these generators. Let χ : G_{K} → Z_{l}^{∗} be the cyclotomic character.

Proposition 2.2.1. Let σ ∈ G_{K}. Then

σ(S_{k}) = (f_{γ_{k}}(σ))^{−1} · S_{k}^{χ(σ)} · f_{γ_{k}}(σ).

for k = 1, …, n + 1.

Proof. Without loss of generality we can assume that a_{k} = 0, a_{n+1} = ∞ and v_{k} = 01. Consider the following Galois equivariant maps

π_{1}(Spec K̄[[z, \frac{1}{z}]], 01) → π_{1}(XK̄, v_{k}) → π_{1}(Spec K̄[z, \frac{1}{z}], 01),

where K̄[[z, \frac{1}{z}]] is the algebra of formal Laurent power series. The fundamental groups on the right hand side and on the left hand side are isomorphic to Z_{l}. The group G_{K} acts on π_{1}(Spec K̄[z, \frac{1}{z}], 01) and on π_{1}(Spec K̄[[z, \frac{1}{z}]], 01) by the cyclotomic character χ : G_{K} → Z_{l}^{∗}. One can see this by looking at étale coverings p_{n} : Spec K̄[z, \frac{1}{z}] → Spec K̄[z, \frac{1}{z}], z → z^{l^{n}}. Somewhat different proof of this fact is given in [I1].

Let us fix an embedding of K̄ into C. We recall that the elements of π_{1}(Spec K̄[[z, \frac{1}{z}]], 01) act on Puisseaux elements z^{\frac{1}{l^{n}}} by analytic continuation. We define a canonical generator T of π_{1}(Spec K̄[[z, \frac{1}{z}]], 01) requiring that T (z^{\frac{1}{l^{n}}}) = e^{\frac{2πi}{l^{n}}} · z^{\frac{1}{l^{n}}}.

We denote by T_{k} the image of T in π_{1}(XK̄, v_{k}). Clearly we have σ(T_{k}) = T_{k}^{χ(σ)}. Observe that x_{k} = γ_{k}^{−1} · T_{k} · γ_{k}. Hence we get σ(S_{k}) = σ(γ_{k}^{−1}) · T_{k}^{χ(σ)} · σ(γ_{k}) = σ(γ_{k}^{−1}) · γ_{k} · (γ_{k}^{−1} · T_{k}^{χ(σ)} · γ_{k}) · (γ_{k}^{−1}) · σ(γ_{k}) = (f_{γ_{k}}(σ))^{−1} · S_{k}^{χ(σ)} · f_{γ_{k}}(σ). Remark. This result seems to be well known (see [I1] pages 51 and 52).

§3. Filtrations of G_{K} associated with the lower central series of π_{1}. 3.0. In this section we shall study various filtrations of the group G_{K} obtained from the action of G_{K} on fundamental groups and on torsors of paths. The filtrations obtained from the action on fundamental groups were already studied by Ihara (see [I1]), Nakamura and Tsunogai (see [NT]) and others.

Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let z, v ∈ X̂(K). Fix an embedding of K̄ into C. Let x = (x_{1}, …, x_{n+1}) be a sequence of geometric generators of π_{1}(X(C); v) associated with a family of paths Γ = {γ_{i}}_{i=1,...,n+1}. The action of G_{K} on π_{1}(XK̄; v) preserves Γ^{i+1}π_{1}(XK̄; v), hence G_{K} acts also on the quotient group π_{1}(XK̄; v)/Γ^{i+1}π_{1}(XK̄; v).

We set

G_{i} = G_{i}(X, v) := ker(G_{K} → Aut(π_{1}(XK̄; v)/Γ^{i+1}π_{1}(XK̄; v))).

Observe that G_{1} = Gal(K̄/K(μ_{l^{∞}})). The quotient group G_{i}/G_{i+1} is isomorphic to a finite direct sum Z_{l} (see [NT] Theorem (5.11).). This implies that G_{k}/G_{i} are l - adic Lie groups.

The group G_{K}/G_{1} ⊂ Z_{l}^{∗} acts on G_{i}/G_{i+1} and the G_{K}/G_{1} - module G_{i}/G_{i+1} is isomorphic to Z_{l}(i)^{n_{i}} (see [I1] in the special case, when X = P_{Q}^{1} \ {0, 1, ∞}). Below we shall show that this result is a corollary of a more general statement.

Let us set G_{∞} = G_{∞}(X, v) := ⋂_{i=1}^{∞} G_{i}(X, v). Then G_{1}/G_{∞} = \underleftarrow{lim}_{i} G_{1}/G_{i} is a pro l - adic Lie group.

We say that two pathes p, q ∈ π(XK̄; z, v) are Γ^{i} - equivalent if p^{−1} · q ∈ Γ^{i}π_{1}(XK̄; v). The set of Γ^{i} - equivalence classes, which we denote by π(XK̄; z, v)/Γ^{i}, is a π_{1}(XK̄; v)/Γ^{i}π_{1}(XK̄; v) - torsor. The action of G_{K} on π(XK̄; z, v) induces an action of G_{K} on π(XK̄; z, v)/Γ^{i} compatible with the structure of the π_{1}(XK̄; v)/Γ^{i}π_{1}(XK̄; v) - torsor.

We set

H_{i} = H_{i}(X; z, v) := ker(G_{i}(X, v) → Aut_{Set}(π(XK̄; z, v)/Γ^{i})).

Proposition 3.0.1. The conjugation on H_{j} by elements of G_{K} induces an action of G_{K}/G_{1} ⊂ Z_{l}^{∗} on the quotient group H_{j}/H_{j+1}. Moreover H_{j}/H_{j+1} is isomorphic to a finite direct sum Z_{l}(j)^{m_{j}} as a G_{K}/G_{1} - module.

Proof. Let us fix a path p from v to z. The map t_{p} : π(XK̄; z, v) → π_{1}(XK̄; v) is G_{K}- equivariant, if σ ∈ G_{K} acts by σ_{p} on π_{1}(XK̄; v). The map t_{p} induces a G_{K}-equivariant map π(XK̄; z, v)/Γ^{j}π(XK̄; z, v) → π_{1}(XK̄; v)/Γ^{j}π_{1}(XK̄; v). Hence we get that

H_{j} = ker(G_{j} → Aut_{Set}(π_{1}(XK̄; v)/Γ^{j}π_{1}(XK̄; v)).

Let σ ∈ H_{j}. Proposition 2.2.1 implies that σ(x_{k}) = (f_{γ_{k}}(σ))^{−1} · x_{k} · f_{γ_{k}}(σ) for k = 1, …, n, n + 1. Observe that f_{γ_{k}}(σ) ∈ Γ^{j}π_{1}(XK̄; v) for k = 1, …, n, n + 1 and f_{p}(σ) ∈ Γ^{j}π_{1}(XK̄; v). The sequence

(f_{p}(σ), f_{γ_{1}}(σ), …, f_{γ_{n}}(σ)) ∈ Γ^{j}π_{1}(XK̄; v) × (Γ^{j}π_{1}(XK̄; v))^{n}

determines the map σ_{p}. This implies that the quotient group H_{j}/H_{j+1} is isomorphic to a closed subgroup of

Γ^{j}π_{1}(XK̄; v)/Γ^{j+1}π_{1}(XK̄; v) × (Γ^{j}π_{1}(XK̄; v)/Γ^{j+1}π_{1}(XK̄; v))^{n}.

Therefore the quotient group H_{j}/H_{j+1} is isomorphic to a finite direct sum Z_{l}^{m_{j}}.

Let τ ∈ G_{K} and σ ∈ H_{j}. We shall show that τ · σ · τ^{−1} = χ(τ)^{j} · σ in H_{j}/H_{j+1}. It follows from Proposition 1.0.7 and Corollary 1.0.8 that f_{p}(τ · σ · τ^{−1}) = f_{p}(τ) · τ(f_{p}(σ)) · (τ · σ · τ^{−1})(f_{p}(τ)^{−1}). Observe that f_{p}(τ) · τ(f_{p}(σ)) · (τ · σ · τ^{−1})(f_{p}(τ)^{−1}) = τ(f_{p}(σ)) mod Γ^{j+1}π_{1}(XK̄; v) and τ(f_{p}(σ)) = χ(τ)^{j} · f_{p}(σ) mod Γ^{j+1}π_{1}(XK̄; v). This implies the proposition because we have also

f_{γ_{i}}(τ · σ · τ^{−1}) = χ(τ)^{j} · f_{γ_{i}}(σ) mod Γ^{j+1}π_{1}(XK̄; v)

for i = 1, …, n, n + 1. Corollary 3.0.2. The conjugation on G_{j} by elements of G_{K} induces an action of G_{K}/G_{1} ⊂ Z_{l}^{∗} on the quotient group G_{j}/G_{j+1}. Moreover G_{j}/G_{j+1} is isomorphic to a finite direct sum Z_{l}(j)^{n_{j}} as a G_{K}/G_{1} - module. Proof. The corollary is a special case of Proposition 3.0.1 if z = v and p is a constant path.

The element σ ∈ H_{j}/H_{j+1} is completely described by its coordinates (f_{p}(σ), f_{γ_{1}}(σ), …, f_{γ_{n}}(σ)) ∈ (Γ^{j}π_{1}(XK̄; v)/Γ^{j+1}π_{1}(XK̄; v))×(Γ^{j}π_{1}(XK̄; v)/Γ^{j+1}π_{1}(XK̄; v))^{n}. Apparentely the first coordinate f_{p}(σ) mod Γ^{j+1}π_{1}(XK̄; v) depends on a choice of a path p from v to z. However we have the following result.

Lemma 3.0.3. Let σ ∈ H_{j} and let p and q be two paths from v to z. Then f_{p}(σ) ≡ f_{q}(σ) mod Γ^{j+1}π_{1}(XK̄; v).

Proof. Let us set S = p^{−1} · q. Then f_{q}(σ) = f_{p·S}(σ) = S^{−1} · f_{p}(σ) · σ(S). Observe that σ(S) = S mod Γ^{j+1}π_{1}(XK̄; v). Hence we get that f_{q}(σ) = f_{p}(σ) mod Γ^{j+1}π_{1}(XK̄; v).

It follows from Proposition 3.0.1 that H_{k}/H_{i} are l - adic Lie groups. Let us set

H_{∞} = H_{∞}(X; z, v) := ⋂_{i=1}^{∞} H_{i}(X; z, v).

Then H_{1}/H_{∞} = \underleftarrow{lim}_{i} H_{1}/H_{i} is a pro l - adic Lie group. Definition 3.0.4. Let A and B be nilpotent groups with exponents in Z_{l}. We say that a homomorphism h : A → B of groups with exponents in Z_{l} is an f - epimorphism if for any b ∈ B there exists a positive integer n and an element a ∈ A such that h(a) = b^{l^{n}}. Remark. If A and B are Z_{l} - modules and if B is a finitely generated Z_{l} - module then h : A → B is an f - epimorphism if and only if cokerf is finite.

Proposition 3.0.5. The homomorphisms

H_{i}/H_{i+k} → G_{i}/G_{i+k}

are f - epimorphisms for any i > 0 and any k > 0.

Proof. The equality H_{1} = G_{1} implies that the homomorphism H_{1}/H_{k} → G_{1}/G_{k} is an epimorphism for any k. Let us consider the following diagram of exact sequences of nilpotent groups with exponents in Z_{l}, finitely generated as groups with exponents in Z_{l}

1 ⟶ H_{i}/H_{k} ↪ H_{1}/H_{k} ⟶ H_{1}/H_{i} ⟶ 1
↓ ↓k ↓g ↓h ↓
1 ⟶ G_{i}/G_{k} ↪ G_{1}/G_{k} ⟶ G_{1}/G_{i} ⟶ 1.

After the Malcev rational completion we obtain the following diagram of exact sequences of nilpotent groups with exponents in Q_{l}

1 ⟶ H_{i}/H_{k} ⊗ Q ↪ H_{1}/H_{k} ⊗ Q ⟶ H_{1}/H_{i} ⊗ Q ⟶ 1
↓ ↓k_{0} ↓g_{0} ↓h_{0} ↓
1 ⟶ G_{i}/G_{k} ⊗ Q ↪ G_{1}/G_{k} ⊗ Q ⟶ G_{1}/G_{i} ⊗ Q ⟶ 1.

The category of nilpotent groups with exponent in Q_{l} and the category of nilpotent Lie algebras over Q_{l} are equivalent. Passing to Lie algebras we get the following diagram of exact sequences of finite dimensional nilpotent Lie algebras over Q_{l}

1 ⟶ Lie(H_{i}/H_{k} ⊗ Q) ↪ Lie(H_{1}/H_{k} ⊗ Q) ⟶ Lie(H_{1}/H_{i} ⊗ Q) ⟶ 1
↓ ↓Lie k_{0} ↓Lie g_{0} ↓Lie h_{0} ↓
1 ⟶ Lie(G_{i}/G_{k} ⊗ Q) ↪ Lie(G_{1}/G_{k} ⊗ Q) ⟶ Lie(G_{1}/G_{i} ⊗ Q) ⟶ 1.

The construction of the Malcev rational completion and then passing to Lie algebras are functorial. The action of G_{K} on Lie algebras is linear and factors through Gal(K(μ_{l^{∞}})/K) ⊂ Z_{l}^{∗}. The standard weight arguments imply that Lie k_{0} is an epimorphism. Hence k_{0} is an epimorphism. Therefore k is an f - epimorphism.

Let us set

K_{i}(X, v) := ⋂_{z∈X̂(K)} H_{i}(X; z, v), K_{i}(X) := ⋂_{(z,v)∈X̂(K)^{2}} H_{i}(X; z, v)

and

K_{∞}(X, v) := ⋂_{i=1}^{∞} K_{i}(X, v), K_{∞}(X) := ⋂_{i=1}^{∞} K_{i}(X).

3.0.6. Observe that K_{1}(X) = K_{1}(X, v) = G_{1}(X, v) = Gal(K̄/K(μ_{l^{∞}})). We would like to show that the maps

K_{i}(X, v)/K_{k}(X, v) → H_{i}(X; z, v)/H_{k}(X; z, v)

and

K_{i}(X)/K_{k}(X) → H_{i}(X; z, v)/H_{k}(X; z, v)

are f - epimorphisms for any i and any k.

Let T be a nonempty finite subset of X̂(K)^{2}. Let us set

K_{i}^{T} (X) := ⋂_{(z,v)∈T} H_{i}(X; z, v) and K_{∞}^{T}(X) := ⋂_{i=1}^{∞} K_{i}^{T} (X).

In the same way as Proposition 3.0.5 we show the following result.

Proposition 3.0.7. Let T and S be nonempty finite subsets of X̂(K)^{2}. Assume that S ⊂ T. Then the maps

K_{i}^{T} (X)/K_{k}^{T} (X) → K_{i}^{S}(X)/K_{k}^{S}(X)

are f - epimorphisms for any i and for any k.

Lemma 3.0.8. The restriction map

H^{1}(G_{K}, Q_{l}(N)) → H^{1}(K_{N}^{T}(X), Q_{l}(N))

is injective.

Proof. Let Γ = Gal(K(μ_{l^{∞}})/K). The restriction map

H^{1}(G_{K}, Q_{l}(N)) → Hom_{Γ}(K_{1}^{T} (X)^{ab}, Q_{l}(N))

is injective. It follows from the weight arguments that the map

Hom_{Γ}(K_{1}^{T} (X)^{ab}, Q_{l}(N)) → Hom_{Γ}(K_{N}^{T}(X)^{ab}, Q_{l}(N))

is injective. This implies the lemma. Definition 3.0.9. Let C be a category whose objects are finite subsets of X̂(K)^{2} and whose morphisms are inclusions. We set

H_{C}^{1}(K_{N}(X), Q_{l}(N)) := \underrightarrow{lim}_{C} H^{1}(K_{N}^{T}(X), Q_{l}(N)).

Lemma 3.0.10. The map

H^{1}(G_{K}, Q_{l}(N)) → H_{C}^{1}(K_{N}(X), Q_{l}(N))

is injective.

Proof. The lemma follows from Lemma 3.0.8 and Proposition 3.0.7. 3.1. We shall study relations between filtrations {G_{i}}_{i∈N} and {H_{i}}_{i∈N} of G_{K} for different X.

Lemma 3.1.0. Let Y = P_{K}^{1} \ {b_{1}, …, b_{m+1}} and let g : Y → X be a non - constant morphism between affine varieties. Let y, w ∈ Ŷ (K) and let z = g(y) and v = g(w). Then we have G_{i}(Y, w) ⊂ G_{i}(X, v) and H_{i}(Y; y, w) ⊂ H_{i}(X; z, v).

Proof. Observe that the induced map g_{∗} : π_{1}(YK̄; w) → π_{1}(XK̄; v) is surjective after passing to the Malcev rational completions and it commutes with the action of G_{K}. This implies that G_{i}(Y, w) ⊂ G_{i}(X, v). Let p be a path from w to y. Then f_{g(p)}(σ) = g_{∗}(f_{p}(σ)). Hence f_{p}(σ) ∈ Γ^{i}π_{1}(YK̄; w) implies that f_{g(p)}(σ) ∈ Γ^{i}π_{1}(XK̄; v). This implies that H_{i}(Y; y, w) ⊂ H_{i}(X; z, v).

As before the weight arguments imply the following result.

Proposition 3.1.1. The induced maps

G_{i}(Y, w)/G_{i+k}(Y, w) → G_{i}(X, v)/G_{i+k}(X, v)

and

H_{i}(Y; y, w)/H_{i+k}(Y; y, w) → H_{i}(X; z, v)/H_{i+k}(X; z, v)

are f - epimorphisms for all i > 0 and all k > 0.

3.2. We recall that π_{1}(XK̄; v) is a free pro - l group on n generators x_{1}, …, x_{n}. Let Lie(X) be a free Lie algebra on n generators X_{1}, …, X_{n}. Let us fix a Hall base B of Lie(X). Let B_{i} be the set of elements of degree i in B. We fix a linear order in B_{i}. If e = [… [X_{i_{1}}, X_{i_{2}}]X_{i_{3}} …], we denote by e(x) the element (… (x_{i_{1}}, x_{i_{2}})x_{i_{3}} …) of π_{1}(XK̄; v). It is well known that any g ∈ π_{1}(XK̄; v) can be written uniquelly as an infinite convergent product

x_{1}^{α_{1}} · x_{2}^{α_{2}} · . . . · x_{n}^{α_{n}} · e_{1}^{2}(x)^{β_{1}} · e_{2}^{2}(x)^{β_{2}} · . . . · e_{1}^{3}(x)^{γ_{1}} · e_{2}^{3}(x)^{γ_{2}} · . . . · e_{1}^{p}(x)^{π_{1}} · e_{2}^{p}(x)^{π_{2}} · . . . ,

where e_{k}^{i} ∈ B_{i} and α_{k}, β_{k}, …, π_{k}, … ∈ Z_{l} and e_{1}^{i}, e_{2}^{i}, e_{3}^{i}, … is a fixed linear order in B_{i}. Definition 3.2.0. Let z ∈ X̂(K) and let p ∈ π(XK̄; z, v). For each e_{i}^{j} ∈ B_{j} we define maps

κ_{e_{i}^{j}}(p, x) : H_{j}(X; z, v) → Z_{l}(j)

by the following equations

f_{p}(σ) ≡ e_{1}^{j}(x)^{κ_{e{j \atop 1}} (p,x)(σ)} · e_{2}^{j}(x)^{κ_{e{j \atop 2}} (p,x)(σ)} · e_{3}^{j}(x)^{κ_{e{j \atop 3}} (p,x)(σ)} · . . . mod Γ^{j+1}π_{1}(XK̄; v).

Lemma 3.2.1. The map κ_{e_{i}^{j}}(p, x) : H_{j}(X; z, v) → Z_{l}(j) is a homomorphism compatible with actions of Gal(K(μ_{l^{∞}})/K). The map κ_{e_{i}^{j}}(p, x) does not depend on the choice of a path p from v to z and it does not depend on the choice of a sequence of geometric generators in the same permutation class.

Proof. Let x′ = (x′_{1}, …, x′_{n+1}) be another sequence of geometric generators of π_{1}(X(C); v) associated with a family of paths Γ′ = {γ′_{i}}_{i=1,...,n+1}. We shall assume that the automorphism of π_{1}(X(C); v) given by x_{i} → x′_{i} for i = 1, …, n + 1 is in B_{n+1}^{(1)∗}(x_{1}, …, x_{n+1}). We have

x′_{i} = f_{i}(x_{1}, …, x_{n})^{−1} · x_{i} · f_{i}(x_{1}, …, x_{n}) (i = 1, …, n + 1), 3.2.2.

where f_{i}(x_{1}, …, x_{n}) := γ_{i}^{−1} · γ′_{i} ∈ π_{1}(X(C); v). Then

f_{p}(σ) ≡ e_{1}^{j}(x′)^{κ_{e{j \atop 1}} (p,x′)(σ)} · e_{2}^{j}(x′)^{κ_{e{j \atop 2}} (p,x′)(σ)} · . . . mod Γ^{j+1}π_{1}(XK̄; v).

It follows from 3.2.2 that e_{i}^{j}(x) ≡ e_{i}^{j}(x′) mod Γ^{j+1}π_{1}(XK̄; v). Hence κ_{e_{i}^{j}}(p, x) = κ_{e_{i}^{j}}(p, x′). Let q ∈ π(XK̄; z, v) and let T := p^{−1} · q. Then it follows from 1.0.4 that

f_{q}(σ) = T^{−1} · f_{p}(σ) · σ(T).

If σ ∈ G_{j}(X, v) then σ(T) = T mod Γ^{j+1}π_{1}(XK̄; v). Hence we get f_{q}(σ) = T^{−1} · f_{p}(σ) · σ(T) = f_{p}(σ) mod Γ^{j+1}π_{1}(XK̄; v). Therefore κ_{e_{i}^{j}}(p, x) does not depend on the choice of a path p in π(XK̄; z, v).

The formula

f_{p}(τσ) = f_{p}(τ) · τ(f_{p}(σ))

( see Lemma 1.0.7) and Proposition 2.2.1 imply that κ_{e_{i}^{j}}(p, x) is a homomorphism. Let τ ∈ G_{K} and σ ∈ H_{j}(X; z, v). Then τστ^{−1} ∈ H_{j}(X; z, v) and

f_{p}(τστ^{−1}) ≡ e_{1}^{j}(x)^{κ_{e{j \atop 1}} (p,x)(τστ^{−1})} · e_{2}^{j}(x)^{κ_{e{j \atop 2}} (p,x)(τστ^{−1})} · . . . mod Γ^{j+1}π_{1}(XK̄; v).

On the other hand

f_{p}(τστ^{−1}) = f_{p}(τ) · τ(f_{p}(σ)) · τσ(f_{p}(τ^{−1})).

Working mod Γ^{j+1}π_{1}(XK̄; v) we get

f_{p}(τ)·τ(f_{p}(σ))·τσ(f_{p}(τ^{−1})) ≡ f_{p}(τ)·e_{1}^{j}(x)^{χ(τ)^{j}κ_{e{j \atop 1}} (p,x)(σ)}·e_{2}^{j}(x)^{χ(τ)^{j}κ_{e{j \atop 2}} (p,x)(σ)}·. . .· τ(f_{p}(τ^{−1})) ≡

e_{1}^{j}(x)^{χ(τ)^{j}κ_{e{j \atop 1}} (p,x)(σ)} · e_{2}^{j}(x)^{χ(τ)^{j}κ_{e{j \atop 2}} (p,x)(σ)} · . . .

because σ(f_{p}(τ^{−1})) ≡ f_{p}(τ^{−1}) mod Γ^{j+1}π_{1}(XK̄; v) and τ(f_{p}(τ^{−1})) = (f_{p}(τ))^{−1}. Hence we get that κ_{e_{i}^{j}}(p, x)(τστ^{−1}) = χ(τ)^{j}κ_{e_{i}^{j}}(p, x)(σ).

Observe that the homomorphism κ_{e_{i}^{j}}(p, x) : H_{j}(X; z, v) → Z_{l}(j) depends only on (z, v) ∈ X̂(K)^{2} and on a linear order (a_{1}, …, a_{n+1}) of points removed from P_{K}^{1}. Assuming that the linear order (a_{1}, …, a_{n+1}) is fixed we set

κ_{e_{i}^{j}}(z, v) := κ_{e_{i}^{j}}(p, x).

§4. Coordinates on the fundamental group and on the torsor. 4.0. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let v ∈ X̂(K). Let x = (x_{1}, …, x_{n+1}) be a sequence of geometric generators of π_{1}(X(C); v). Let Q_{l}{{X_{1}, …, X_{n}}} be an algebra of non-commutative formal power series in n non-commuting variables X_{1}, …, X_{n}. We set X := {X_{1}, …, X_{n}}. To simplify the notation we shall write Q_{l}{{X}} instead of Q_{l}{{X_{1}, …, X_{n}}}.

We recall that Q_{l} is a topological non-archimedian field. Let I be the augmentation ideal of Q_{l}{{X}}. Observe that Q_{l}{{X}}/I^{m} is a finite dimensional topological vector space over Q_{l} and Q_{l}\{\{X\}\} = \underleftarrow{lim}_{m} Q_{l}\{\{X\}\}/I^{m}. We equip Q_{l}\{\{X\}\} with a topology of the projective limit. We recall that π_{1}(XK̄; v) is equipped with a pro - finite topology.

We define a continous embedding

k_{x} : π_{1}(XK̄; v) → Q_{l}{{X}}

setting k_{x}(x_{i}) := expX_{i} for i = 1, …, n and requiring that k_{x}(w · w′) = k_{x}(w) · k_{x}(w′). Let p ∈ π(XK̄; z, v). Composing t_{p} with k_{x} we get a continous embedding

k_{x,p} : π(XK̄; z, v) → Q_{l}{{X}}.

Let us set

Λ_{(p,x)}(σ) := k_{x}(f_{p}(σ)).

(We shall omit the subscript x if a sequence of geometric generators is fixed and we shall write Λ_{p}(σ) instead of Λ_{(p,x)}(σ).)

Let us denote by Aut(Q_{l}{{X}}) the group of continous automorphisms of the Q_{l} - algebra Q_{l}{{X}} and by GL(Q_{l}{{X}}) the group of continous linear automorphisms of the Q_{l} - vector space Q_{l}{{X}}.

The action of G_{K} on π_{1}(XK̄; v) defines a continous action of G_{K} on Q_{l}{{X}},

( )_{x} : G_{K} → Aut(Q_{l}{{X}})

given by σ_{x}(expX_{i}) := k_{x}(σ(x_{i})) for i = 1, …, n.

The action of G_{K} on π(XK̄; z, v) defines a continous action of G_{K} on Q_{l}{{X}},
( )_{x,p} : G_{K} → GL(Q_{l}{{X}})

given by σ_{x,p}(w) := Λ_{(p,x)}(σ) · σ_{x}(w). (We shall omit the subscript x if a sequence of geometric generators is fixed and we shall write σ instead of σ_{x} and σ_{p} instead of σ_{x,p}.) 4.1. The subgroups G_{i}(X, v) and H_{i}(X; z, v) of G_{K} can be describe in terms of the action of G_{K} on Q_{l}{{X}} in the following way.

Lemma 4.1.1. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let z, v ∈ X̂(K). We have

G_{i}(X; v) = ker(G_{K} → Aut(Q_{l}{{X}}/I^{i+1}))

and

H_{i}(X; z, v) = ker(G_{i}(X; v) → GL(Q_{l}{{X}}/I^{i})).

We shall omit an easy proof. 4.2. Let λ ∈ Q_{l}^{∗}. We define a continous automorphism of Q_{l} - algebras

ρ(λ) : Q_{l}{{X}} → Q_{l}{{X}}

setting ρ(λ)(w) := λ^{i}w if w is homogenous of degree i.

Let σ ∈ G_{K}. We set

φ_{x}(σ) := σ_{x} ∘ ρ(χ(σ)^{−1})

and

ψ_{x,p}(σ) := σ_{x,p} ∘ ρ(χ(σ)^{−1}).

Observe that φ_{x}(σ) (resp. ψ_{x,p}(σ)) is a pro - unipotent automorphism of Q_{l} - algebra (resp. pro - unipotent Q_{l} - linear automorphism of) Q_{l}{{X}}. Remark. If σ ∈ G_{1} then φ_{x}(σ) = σ_{x} and ψ_{x,p}(σ) = σ_{x,p}.

Lemma 4.2.1. We have

φ_{x}(τ · σ) = φ_{x}(τ) ∘ (ρ(χ(τ)) ∘ φ_{x}(σ) ∘ ρ(χ(τ)^{−1}))

and

ψ_{x,p}(τ · σ) = ψ_{x,p}(τ) ∘ (ρ(χ(τ)) ∘ ψ_{x,p}(σ) ∘ ρ(χ(τ)^{−1})).

We can interpret the equalities from Lemma 4.2.1 in the following way.

Corollary 4.2.2. Let G_{K} acts on Aut(Q_{l}{{X}}) (resp. GL(Q_{l}{{X}})) by σ(f) := ρ(χ(σ)) ∘ f ∘ ρ(χ(σ)^{−1}). Then the maps φ_{x} : G_{K} → Aut(Q_{l}{{X}}) and ψ_{x,p} : G_{K} → Aut(Q_{l}{{X}}) are 1 - cocycles.

Let λ ∈ Q_{l}^{∗}. We shall denote by f^{λ} the automorphism ρ(λ^{−1}) ∘ f ∘ ρ(λ)

§5. l - adic iterated integrals. 5.0. The purpose of this section is to introduce objects called by us l - adic iterated integrals (see Definition 5.3.0.). These l - adic iterated integrals evaluated at z are functions from the Galois group G_{K} to Q_{l}, which to σ ∈ G_{K} associate coefficients of the power series (logψ_{x,p}(σ))(1) ((logσ_{x,p})(1) if σ ∈ G_{K(μ_{l^{∞}})}). These l - adic iterated integrals correspond to suitably normalized classical complex iterated integrals.

Let a_{1}, …, a_{n+1} be K-points of the projective line P_{K}^{1}. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let v ∈ X̂(K) be a base point. Let us choose a tangential base point v_{i} at a_{i} for i = 1, 2, …, n+1. Let x = (x_{1}, …, x_{n+1}) be a sequence of geometric generators of π_{1}(X(C); v) associated with a family of paths Γ = {γ_{i}}_{i=1,...,n+1} from v to each v_{i}. It follows from section 3 that G_{1}/G_{∞} is a pro-unipotent l-adic Lie group. Hence (G_{1}/G_{∞}) ⊗ Q - the rational completion of G_{1}/G_{∞} - is a pro - unipotent Q_{l} - Lie group. Let us set 𝔤 = 𝔤(X, v) := T_{id}((G_{1}/G_{∞}) ⊗ Q) = Lie((G_{1}/G_{∞}) ⊗ Q) - the tangent space of (G_{1}/G_{∞}) ⊗ Q at the identity.

We shall denote by Der(Q_{l}{{X}}) the Lie algebra of continous derivations of the Q_{l} - algebra Q_{l}{{X}} and by End(Q_{l}{{X}}) the Lie algebra of continous automorphisms of the Q_{l} - vector space Q_{l}{{X}}.

We have the following commutative diagram:

G_{1}/G_{∞}  ⟶^{( )_{x}} Aut(Q_{l}{{X}})
↓log  ↓log
𝔤  ⟶^{Lie( )_{x}} Der(Q_{l}{{X}}).

(The upper horizontal arrow is induced by the action of G_{K} on Q_{l}{{X}}, the lower horizontal arrow is the induced map on tangent spaces, log on the right side is defined only on pro - unipotent automorphisms.)

Let z ∈ X̂(K) and let p ∈ π(XK̄; z, v). It follows from section 3 that H_{1}/H_{∞} = H_{1}(X; z, v)/H_{∞}(X; z, v) is a pro-unipotent l-adic Lie group . Hence (H_{1}/H_{∞}) ⊗ Q is a pro-unipotent Q_{l} - Lie group. Let us set 𝔥 = 𝔥(X; z, v) := T_{id}((H_{1}/H_{∞}) ⊗ Q) = Lie((H_{1}/H_{∞}) ⊗ Q) - the tangent space of (H_{1}/H_{∞}) ⊗ Q at the identity. We have the following commutative diagram:

H_{1}/H_{∞}  ⟶^{( )_{x,p}} GL(Q_{l}{{X}})
↓log  ↓log
𝔥  ⟶^{Lie( )_{x,p}} End(Q_{l}{{X}}).

Let T ⊂ X̂(K)^{2} be a finite subset containing a pair (z, v). We have epimorphisms K_{1}^{T} (X)/K_{∞}^{T}(X) → G_{1}/G_{∞} and K_{1}^{T} (X)/K_{∞}^{T}(X) → H_{1}/H_{∞} and the induced epimorphisms of Lie algebras

Lie(K_{1}^{T} (X)/K_{∞}^{T}(X) ⊗ Q) → 𝔤 and Lie(K_{1}^{T} (X)/K_{∞}^{T}(X) ⊗ Q) → 𝔥.

Hence we can consider that the homomorphisms ( )_{x} and ( )_{x,p} are defined on K_{1}^{T} (X)/K_{∞}^{T}(X) and that the morphisms of Lie algebras Lie( )_{x} and Lie( )_{x,p} are defined on Lie(K_{1}^{T} (X)/K_{∞}^{T}(X) ⊗ Q).

The image of the morphism ( )_{x} (resp. Lie( )_{x}) is contained in the ”braid - like” subgroup of Aut(Q_{l}{{X}}) (resp. Lie subalgebra of Der(Q_{l}{{X}})). We recall their definitions. We also describe subgroups and subalgebras containing images of morphisms ( )_{x,p} and Lie( )_{x,p} respectively.

5.1. We recall that X = {X_{1}, …, X_{2}}. Let Lie(X) be a free Lie algebra over Q_{l} on the set X. Let us set

L(X) := \underleftarrow{lim}_{i} Lie(X)/Γ^{i}Lie(X).

We identify L(X) with Lie elements in Q_{l}{{X}}. Definition 5.1.0. Let us define subgroups

Aut^{∗}(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}}};

Aut^{∗}L(X) := \{f ∈ AutL(X) | ∀X_{i} ∈ X ∃l_{i} ∈ L(X), f(X_{i}) = X_{i} + ∑_{m=1}^{∞} \frac{1}{m!}[..[X_{i}, l_{i}]l_{i}^{m−1}]\}

and Lie subalgebras

Der^{∗}(Q_{l}{{X}}) := {D ∈ Der(Q_{l}{{X}}) | ∀X_{i} ∈ X ∃A_{i} ∈ L(X), , D(X_{i}) = X_{i}·A_{i}−A_{i}·X_{i}};

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

and

Der^{∗}Lie(X) := {D ∈ DerLie(X) | ∀X_{i} ∈ X ∃A_{i} ∈ Lie(X), D(X_{i}) = [X_{i}, A_{i}]}.

Lemma 5.1.1. We have i). Aut^{∗}(Q_{l}{{X}}) = Aut^{∗}L(X); ii). Der^{∗}(Q_{l}{{X}}) = Der^{∗}L(X); iii). The Lie algebra of Aut^{∗}(Q_{l}{{X}}) (resp. Aut^{∗}L(X) ) is Der^{∗}(Q_{l}{{X}}) (resp.

Der^{∗}L(X)).

Proof. The first part follows from the well known formula

e^{−l_{i}} · X_{i} · e^{l_{i}} = X_{i} + ∑_{m=1}^{∞} \frac{1}{m!}[..[X_{i}, l_{i}]l_{i}^{m−1}]. 5.1.2.

The second part is obvious, so it rests to show the last statement of the lemma. It is well known that the Lie algebra of the group of automorphisms of a Q_{l} - algebra is the Lie algebra of derivations of this Q_{l} - algebra. Let D be a derivation of the Q_{l} - algebra Q_{l}{{X}}. Suppose that exp tD ∈ Aut^{∗}(Q_{l}{{X}}). Then (exp tD)(X_{i}) = e^{−l_{i}(t)} · X_{i} · e^{l_{i}(t)} for i = 1, …,n. The elements l_{i}(t) are in L(X). We can suppose that the coefficient of l_{i}(t) at X_{i} vanishes. Then we have l_{i}(0) = 0 and l_{i}(t) depends smoothly on t. Hence A_{i} := lim_{t→0} \frac{1}{t}l_{i}(t) exists and belongs to L(X). Comparing Taylor developements of (exp tD)(X_{i}) and e^{−l_{i}(t)} · X_{i} · e^{l_{i}(t)} we get D(X_{i}) = [X_{i}, A_{i}] for i = 1, …,n. Therefore D belongs to Der^{∗}(Q_{l}{{X}}).

The action of Galois groups on torsors of paths requires to introduce semi - direct products of Lie algebras. Below we give the necessary definitions.

Let L be a Lie algebra and let D be a Lie subalgebra of the algebra of Lie derivations of L. We equip the direct product L × D with a Lie bracket

[(l, D), (l_{1}, D_{1})] := ([l, l_{1}] + D(l_{1}) − D_{1}(l), [D, D_{1}]).

The resulting Lie algebra we denote by L ˜×D and we call it a semi - direct product of L and D.

If g ∈ Q_{l}{{X}} then L_{g} denotes left multiplication by g. L_{exp(L(X))} is the set of left multiplications by elements of exp(L(X)) and L_{L(X)} is the set of left multiplications by elements of L(X).

Proposition 5.1.3. Let σ ∈ G_{K}. Then φ_{x}(σ) ∈ Aut^{∗}(Q_{l}{{X}}) and logφ_{x}(σ) ∈ Der^{∗}(Q_{l}{{X}}).

Proof. It follows from Proposition 2.2.1 that

σ_{x}(X_{i}) = (Λ_{(γ_{i},x)}(σ))^{−1} · χ(σ)X_{i} · Λ_{(γ_{i},x)}(σ) 5.1.4.

for i = 1, …, n. Hence φ_{x}(σ) ∈ Aut^{∗}(Q_{l}{{X}}). It follows from Lemma 5.1.2 that logφ_{x}(σ) ∈ Der^{∗}(Q_{l}{{X}}).

To give an explicit formula for logφ_{x}(X_{i}) we need to study Galois actions on torsors of paths.

Proposition 5.1.5. Let σ ∈ G_{K}. Then ψ_{x,p}(σ) ∈ L_{exp(L(X))} ˜×Aut^{∗}(Q_{l}{{X}}) and logψ_{x,p}(σ) ∈ L_{L(X)} ˜×Der^{∗}L(X).

Proof. Let σ ∈ G_{K} and w ∈ Q_{l}{{X}}. We have

ψ_{x,p}(σ)(w) = Λ_{(p,x)}(σ) · φ_{x}(σ)(w). 5.1.6.

It follows from 5.1.6 that ψ_{x,p}(σ) belongs to the semi - direct product

L_{exp(L(X))} ˜×Aut^{∗}(Q_{l}{{X}}).

The Lie algebra of the semi - direct product of groups L_{exp(L(X))} ˜×Aut^{∗}(Q_{l}{{X}}) is equal to a semi - direct product of Lie algebras

L_{L(X)} ˜×Der^{∗}L(X) ≈ L(X) ˜×Der^{∗}L(X).

Therefore logψ_{x,p}(σ) ∈ L_{L(X)} ˜×Der^{∗}L(X). This finishes the proof of the proposition. Below we shall calculate both components of logψ_{x,p}(σ). We denote by ◯ a product given by the Baker - Campbell - Hausdorff formula (BCH

formula).

Proposition 5.1.7. The element logψ_{x,p}(σ)(1) ∈ L(X) and we have

logψ_{x,p}(σ) = L_{(logψ_{x,p}(σ))(1)} + logφ_{x}(σ).

Proof. Let g, h ∈ L(X) and D ∈ Der^{∗}(Q_{l}{{X}}). Then [L_{g}, L_{h}] = L_{[g,h]} and [D, L_{g}] = L_{D(g)}. Hence all terms of logψ_{x,p}(σ)−logφ_{x}(σ) = L_{logΛ_{(p,x)}(σ)} ◯logφ_{x}(σ)−logφ_{x}(σ) are of the form L_{g} for some g ∈ L(X). Therefore logψ_{x,p}(σ) = L_{g} + logφ_{x}(σ) for some g ∈ L(X). Evaluating both sides of the equality at 1 we get that g = (ψ_{x,p}(σ))(1). This finishes the proof of the proposition.

Proposition 5.1.8. Let σ ∈ G_{K}. Then we have

logφ_{x}(σ)(X_{k}) = [X_{k}; (logψ_{x,γ_{k}}(σ))(1)]

for k = 1, …, n.

Proof. One computes easily that (logψ_{x,γ_{k}}(σ))(1) = (Λ_{(γ_{k},x)}(σ) − 1) − \frac{1}{2}(Λ_{(γ_{k},x)}(σ) · φ_{x}(σ)(Λ_{(γ_{k},x)}(σ)) − 2Λ_{(γ_{k},x)}(σ) + 1)+ \frac{1}{3}(Λ_{(γ_{k},x)}(σ) · φ_{x}(σ)(Λ_{(γ_{k},x)}(σ)) · φ_{x}(σ)^{2}(Λ_{(γ_{k},x)}(σ)) − 3Λ_{(γ_{k},x)}(σ) · φ_{x}(σ)(Λ_{(γ_{k},x)}(σ)) + 3Λ_{(γ_{k},x)}(σ) − 1) …. This implies that (logψ_{x,γ_{k}}(σ))(X_{k}) = X_{k} · ((logψ_{x,γ_{k}}(σ))(1)). Now it follows from Proposition 5.1.7 that logφ_{x}(σ)(X_{k}) = [X_{k}; (logψ_{x,γ_{k}}(σ))(1)]. 5.2. The free Lie algebra Lie(X) (resp. the completed free Lie algebra L(X)) has an obvious Q-structure - a free Lie algebra over Q on the set X (resp. a completed free Lie algebra over Q on the set X.) Therefore the Lie algebras of derivations DerLie(X) and Der^{∗}Lie(X) (resp. DerL(X) and Der^{∗}L(X)) have also Q-structures.

Let Lie(X)_{m} be a vector subspace of Lie(X) of homogenous elements of degree m. Let (Lie(X)_{m})^{∗} be the dual vector space of the finite dimentional vector space Lie(X)_{m}. We set

L(X)^{⋄} = Lie(X)^{⋄} := ⊕_{m=1}^{∞}(Lie(X)_{m})^{∗}.

Let ⟨X_{i}⟩ be a vector subspace of L(X) generated by X_{i} and let ⟨X_{i}⟩^{∗} be the dual vector space. We set

(L(X)/⟨X_{i}⟩)^{⋄} := ker(L(X)^{⋄} → ⟨X_{i}⟩^{∗}).

Let D ∈ Der^{∗}L(X) be such that D(X_{i}) = [X_{i}, A_{i}] for i = 1, …,n. The map

f : Der^{∗}L(X) → ⊕_{i=1}^{n}(L(X)/⟨X_{i}⟩)

given by f(D) = (A_{1}, …,A_{n}) is an isomorphism of vector spaces. The isomorphism f is compatible with Q - structures on both vector spaces. The isomorphism f identifies Der^{∗}L(X) with ⊕_{i=1}^{n}(L(X)/⟨X_{i}⟩) and we set

(Der^{∗}L(X))^{⋄} := ⊕_{i=1}^{n}(L(X)/⟨X_{i}⟩)^{⋄}.

We set

(L(X) ˜×Der^{∗}L(X))^{⋄} := L(X)^{⋄} ⊕ (Der^{∗}L(X))^{⋄}.

Definition 5.2.0. Let V be a vector space. We say that V is a Lie coalgebra if V is equipped with a linear map d : V → V ⊗ V satisfying

i) τ ∘ d + d = 0, where τ(a ⊗ b) = b ⊗ a; ii) ∑_{i=0}^{2} σ^{i} ∘ (d ⊗ id_{V}) ∘ d = 0, where σ(a ⊗ b ⊗ c) = b ⊗ c ⊗ a.

It follows from i) that d factors through d : V → V ∧ V, where

V ∧ V = {∑_{i∈I} n_{i}(a_{i} ⊗ b_{i} − b_{i} ⊗ a_{i}) ∈ V ⊗ V }.

Lemma 5.2.1.

i) If V is a Lie coalgebra then the dual vector space V^{∗} equipped with [ ] := d^{∗} : V^{∗} ⊗ V^{∗} → V^{∗} is a Lie algebra.

ii) If L is a Lie algebra then L^{∗} equipped with d := [ ]^{∗} : L^{∗} → L^{∗} ⊗L^{∗} is a Lie coalgebra.

The dual vector spaces L(X)^{∗}, (Der^{∗}L(X))^{∗} and (L(X) ˜×Der^{∗}L(X))^{∗} equipped with d := [ ]^{∗} are Lie coalgebras. Observe that d preserves L(X)^{⋄}, (Der^{∗}L(X))^{⋄} and (L(X) ˜×Der^{∗}L(X))^{⋄}. Hence these vector spaces are also Lie coalgebras. 5.3. Below we shall give the very definition of the l - adic iterated integrals. Observe that the element (logψ_{x,p}(σ))(1) is a Lie element in Q_{l}{{X}} by Proposition 5.1.7. Definition 5.3.0. Let us fix a Hall base B of Lie(X). Let σ ∈ G_{K}. We set

a_{x,p}(σ) := (logψ_{x,p}(σ))(1) = ∑_{e∈B} a_{x,p}^{e}(σ) · e.

Let ϕ ∈ Lie(X)^{⋄} be a linear form defined over Q. We set

a_{x,p}^{ϕ}(σ) := ϕ((logψ_{x,p}(σ))(1)).

The functions a_{x,p}^{e} : G_{K} → Q_{l} we shall call l - adic iterated integrals.

Theorem 5.3.1. Let e ∈ B be an element of degree i. We have: i). a_{x,p}^{e}(σ) = 0 for σ ∈ H_{i+1}. ii). a_{x,p}^{e}(τ · σ) = a_{x,p}^{e}(τ) + a_{x,p}^{e}(σ) for any τ, σ ∈ H_{i}. iii). The homomorphism a_{x,p}^{e}_{|H_{i}} : H_{i} → Q_{l}(i) is compatible with the action of Gal(K(μ_{l^{∞}})/K)

on H_{i} and Q_{l}(i). iv). The homomorphism a_{x,p}^{e}_{|H_{i}} : H_{i} → Q_{l}(i) depends only on z and v. It does not depend

on the choice of geometric generators x (in a given permutation class) and on a choice of a path p from v to z.

Proof. The point i) follows from the definition of the group H_{i} and from Lemma 4.1.1. Let τ, σ ∈ H_{i}. Then ψ_{x,p}(σ) = σ_{x,p}, ψ_{x,p}(τ) = τ_{x,p} and ψ_{x,p}(τ · σ) = (τ · σ)_{x,p} It follows from the point i) that

(logσ_{x,p})(1) = ∑_{e∈B^{i}} a_{x,p}^{e}(σ) · e + ∑_{j≥i+1} ∑_{e∈B^{j}} a_{x,p}^{e}(σ) · e. 5.3.2.

We have (τ · σ)_{x,p} = τ_{x,p} ∘ σ_{x,p}. The BCH formula implies

log(τ · σ)_{x,p} = logτ_{x,p} + logσ_{x,p} + \frac{1}{2}[logτ_{x,p}, logσ_{x,p}] + ….

Evaluating both sides of the equality at 1 we get

(log(τ · σ)_{x,p})(1) = (logτ_{x,p})(1) + (logσ_{x,p})(1) + A(τ, σ)(1),

where A(τ, σ) = \frac{1}{2}[logτ_{x,p}, logσ_{x,p}] + …. It follows from the point i) that terms of degree i of A(τ, σ)(1) vanish. Hence a_{x,p}^{e}(τ · σ) = a_{x,p}^{e}(τ) + a_{x,p}^{e}(σ) for e ∈ B^{i} and τ, σ ∈ H_{i}. The points iii) and iv) follow from Lemma 3.2.1. Definition 5.3.3. Let ε ∈ (Der^{∗}L(X))^{⋄} be a linear form of degree m. We set

ε(v) := ε ∘ log ∘ φ_{x}.

Proposition 5.3.4. There are e_{1}, …, e_{r} ∈ B_{m} and α_{k,i} ∈ Q_{l} for 0 < k < n + 1 and 0 < i < r + 1 such that

ε(v) = ∑_{k=1}^{n} ∑_{i=1}^{r} α_{k,i}a_{x,γ_{k}}^{e_{i}}.

If ε is defined over Q then α_{k,i} are in Q.

Proof. The proposition follows from Proposition 5.1.8.

We shall see later that the function a_{x,p}^{e} : G_{K} → Q_{l} depends on a choice of a path p from v to z. Assume that e is of degree m. It follows from Theorem 5.3.1 iv) that the restriction of a_{x,p}^{e} to the subgroup H_{m}(X; z, v) depends only on z and v. It does not depend on a choice of a path p. This motivate the following definition. Definition 5.3.5. Let e ∈ B be an element of degree m and let φ ∈ L(X)^{⋄} be a linear form of degree m. We set

L^{e}(z, v) := a_{x,p}^{e}_{|H_{m}(X;z,v)} and L^{φ}(z, v) := a_{x,p}^{φ}_{|H_{m}(X;z,v)}.

Let ε ∈ (Der^{∗}L(X))^{⋄} be a linear form of degree m. We set

L^{ε}(v) := ε(v)_{|H_{m}(X;z,v)}.

It follows from Proposition 5.3.4 that

L^{ε}(v) = ∑_{k=1}^{n} ∑_{i=1}^{r} α_{k,i}L^{e_{i}}(v_{k}, v).

§6. Cocycle conditions. 6.0. It follows from Proposition 1.0.7 that the function f_{p} : G_{K} → π_{1}(XK̄; v) is a cocycle. Simillarly Lemma 4.2.1 implies that the functions φ_{x} : G_{K} → Aut(Q_{l}{{X}}) and ψ_{x,p} : G_{K} → GL(Q_{l}{{X}}) are cocycles. The map ( )_{x,p} : G_{K(μ_{l^{∞}})} → GL(Q_{l}{{X}}) is a homomorphism. However coefficients of these matrix valued functions usually are not cocycles or homomorphisms.

Let φ ∈ L(X)^{⋄} be a linear form of degree m. The function a_{x,p}^{φ} : G_{K(μ_{l^{∞}})} → Q_{l}(m) (resp. a_{x,p}^{φ} : G_{K} → Q_{l}(m) ) usually is not a homomorphism (resp. a cocycle ). We are looking for conditions when a linear combination of various a_{x,p}^{φ} with Q_{l} coefficients is a homomorphism (resp. a cocycle ).

Lemma 6.0.1. Let φ ∈ L(X)^{⋄} be a linear form of degree m. Assume that

d(φ) = ∑_{k+l=m, α,β} c_{α,β}^{k,l}(e_{α}^{k})^{∗} ∧ (e_{β}^{l})^{∗} + ∑_{r+q=m, γ,δ} b_{γ,δ}^{r,q}(e_{γ}^{r})^{∗} ∧ ε_{δ}^{q}

in (L(X) ˜×Der^{∗}L(X))^{⋄}, where e_{α}^{k}, e_{β}^{l}, e_{γ}^{r} belong to a Hall base B of Lie(X) and ε_{δ}^{q} ∈ (Der^{∗}L(X))^{⋄}. Then we have

d(a_{x,p}^{φ}) = ∑_{k+l=m, α,β} c_{α,β}^{k,l}(a_{x,p}^{e_{α}^{k}}) ∧ (a_{x,p}^{e_{β}^{l}}) + ∑_{r+q=m, γ,δ} b_{γ,δ}^{r,q}(a_{x,p}^{e_{γ}^{r}}) ∧ ε_{δ}^{q}(v)

in (Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)))^{∗}, where (Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)))^{∗} := Hom_{Z_{l}}(Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)); Q_{l}) and T is a finite subset of X̂(K)^{2} containing a pair (z, v).

Proof. The lemma is an obvious consequence of the fact that the map

Lie( )_{x,p} : Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)) → L_{L(X)} ˜×Der^{∗}L(X)

is a morphism of Lie algebras.

Proposition 6.0.2. Let (z_{i}, v_{i}) ∈ X̂(K) × X̂(K), let φ_{i} ∈ L(X)^{⋄} be a linear form of degree m, let p_{i} be a path from v_{i} to z_{i} and let x_{i} be a sequence of geometric generators of π_{1}(X(C); v_{i}) for i = 1, …, N. Let n_{1}, …, n_{N} be in Q_{l}. Let T be a finite subset of X̂(K)^{2} containing pairs (z_{i}, v_{i}) for i = 1, …, N. i) Assume that d(∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}}) = 0 in (Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)))^{∗}. Then ∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}} is

a homomorphism from K_{1}^{T} (X)/K_{∞}^{T}(X) to Q_{l}(m). ii) Assume that for any τ and σ in G_{K}

∑_{i=1}^{N} n_{i}φ_{i}([...[...[logψ_{x_{i},p_{i}}(τ), logψ_{x_{i},p_{i}}(σ)^{χ(τ)^{−1}}], logψ_{x_{i},p_{i}}(τ)]...]) = 0

for all Lie brackets of lengths 2, 3, …, m. Then ∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}} is a cocycle on G_{K} with

values in Q_{l}(m).

Proof. We start with the proof of the first part of the proposition. Let τ, σ ∈ K_{1}^{T} (X). Let us set T_{i} = logτ_{x_{i},p_{i}} and S_{i} = logσ_{x_{i},p_{i}}. The equality τ_{x_{i},p_{i}} ∘ σ_{x_{i},p_{i}} = (τσ)_{x_{i},p_{i}} implies

log(τσ)_{x_{i},p_{i}} = T_{i} + S_{i} + \frac{1}{2}[T_{i}, S_{i}] − \frac{1}{12}[[T_{i}, S_{i}]T_{i}] + ….

Evaluating φ_{i} on the last equality we get

a_{x_{i},p_{i}}^{φ_{i}}(τσ) = a_{x_{i},p_{i}}^{φ_{i}}(τ) + a_{x_{i},p_{i}}^{φ_{i}}(σ) + \frac{1}{2}φ_{i}([T_{i}, S_{i}]) − \frac{1}{12}φ_{i}([[T_{i}, S_{i}]T_{i}]) + ….

Observe that φ_{i}([T_{i}, S_{i}]) = dφ_{i}(T_{i}⊗S_{i}) = da_{x_{i},p_{i}}^{φ_{i}}(logτ ⊗logσ). Hence ∑_{i=1}^{N} n_{i}φ_{i}([T_{i}, S_{i}]) = 0. Observe that φ([[T, S]R]) = ((d⊗id)∘d)(φ)(T ⊗S⊗R). This implies ∑_{i=1}^{N} n_{i}φ_{i}([[T_{i}, S_{i}]T_{i}]) = 0. We apply the same arguments to others brackets and finally we get

∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}}(τσ) = ∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}}(τ) + ∑_{i=1}^{N} n_{i}a_{x_{i},p_{i}}^{φ_{i}}(σ).

Now we assume that τ, σ ∈ G_{K}. The equality

ψ_{x_{i},p_{i}}(τσ) = ψ_{x_{i},p_{i}}(τ) ∘ ψ_{x_{i},p_{i}}(σ)^{χ(τ)^{−1}}

(see Lemma 4.2.1) implies that

logψ_{x_{i},p_{i}}(τσ) = logψ_{x_{i},p_{i}}(τ) ◯ logψ_{x_{i},p_{i}}(σ)^{χ(τ)^{−1}}

= logψ_{x_{i},p_{i}}(τ) + logψ_{x_{i},p_{i}}(σ)^{χ(τ)^{−1}} + \frac{1}{2}[logψ_{x_{i},p_{i}}(τ), logψ_{x_{i},p_{i}}(σ)^{χ(τ)^{−1}}] + ….

Now the second part of the proposition follows immediately from the assumptions ii).

6.1. We shall define filtrations of the Lie algebras Der^{∗}L(X) and L(X) ˜×Der^{∗}L(X) associated with the lower central series of L(X). Let us set

Der_{k}^{∗}L(X) := {D ∈ Der^{∗}L(X) | ∀X_{i} ∈ X ∃A_{i} ∈ Γ^{k}L(X), D(X_{i}) = [X_{i}, A_{i}]}

and

γ_{k}(L(X) ˜×Der^{∗}L(X)) := Γ^{k}L(X) ˜×Der_{k}^{∗}L(X).

Lemma 6.1.1. Der_{k}^{∗}L(X) (resp. γ_{k}(L(X) ˜×Der^{∗}L(X)))is a Lie ideal of Der^{∗}L(X) (resp. L(X) ˜×Der^{∗}L(X)). We have isomorphisms of Lie algebras

⨁_{k=1}^{∞} Der_{k}^{∗}L(X)/Der_{k+1}^{∗}L(X) = Der^{∗}Lie(X)

and

⨁_{k=1}^{∞} γ_{k}(L(X) ˜×Der^{∗}L(X))/γ_{k+1}(L(X) ˜×Der^{∗}L(X)) = Lie(X) ˜×Der^{∗}Lie(X).

Proof. The lemma follows from the fact that the graded associated Lie algebra ⨁_{k=1}^{∞} Γ^{k}L(X)/Γ^{k+1}L(X) is canonically isomorphic to Lie(X).

Let T be a finite subset of X̂(K)^{2}. We set

𝔨^{T} (X) := grLie(K_{1}^{T} (X)/K_{∞}^{T}(X)) := ⨁_{i=1}^{∞} Lie(K_{i}^{T} (X)/K_{∞}^{T}(X))/Lie(K_{i+1}^{T}(X)/K_{∞}^{T}(X)) ⊗ Q.

The homomorphism of Lie algebras

Lie( )_{x,p} : Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)) → L(X) ˜×Der^{∗}L(X)

is compatible with filtrations {Lie(K_{i}^{T} (X)/K_{∞}^{T}(X))}_{i=1}^{∞} of Lie(K_{1}^{T} (X)/K_{∞}^{T}(X)) and {γ_{i}(L(X) ˜×Der^{∗}L(X))}_{i=1}^{∞} of L(X) ˜×Der^{∗}L(X). Therefore it induces a homomorphism of associated graded Lie algebras

π_{z,v}^{T} : 𝔨^{T} (X) → Lie(X) ˜×Der^{∗}Lie(X).

Let us set

𝔨^{T} (X)^{⋄} := ⨁_{i=1}^{∞} (Lie(K_{i}^{T} (X)/K_{∞}^{T}(X))/Lie(K_{i+1}^{T}(X)/K_{∞}^{T}(X)) ⊗ Q)^{∗}.

Then 𝔨^{T} (X)^{⋄} is a Lie coalgebra and we have a homomorphism of Lie coalgebras

(π_{z,v}^{T})^{⋄} : (Lie(X) ˜×Der^{∗}Lie(X))^{⋄} → 𝔨^{T} (X)^{⋄}.

Moreover an inclusion S ⊂ T of finite subsets of X̂(K)^{2} induces a morphism of Lie coalgebras

𝔨^{S}(X)^{⋄} → 𝔨^{T} (X)^{⋄}.

Let C be a category whose objects are finite subsets of X̂(K)^{2} and whose morphisms are inclusions. We set

𝔨(X)^{⋄} := \underrightarrow{lim}_{C} 𝔨^{T} (X)^{⋄}.

The Q_{l} - vector space 𝔨(X)^{⋄} is a Lie coalgebra and morphisms (π_{z,v}^{T})^{⋄} : (Lie(X) ˜×Der^{∗}Lie(X))^{⋄} → 𝔨^{T} (X)^{⋄} induce a morphism of Lie coalgebras

π_{z,v}^{⋄} : (Lie(X) ˜×Der^{∗}Lie(X))^{⋄} → 𝔨(X)^{⋄}.

Observe that

L^{e}(z, v) = e^{∗} ∘ π_{z,v} = π_{z,v}^{⋄}(e^{∗})

and

L^{ε}(v) = ε ∘ π_{v,v} = ε ∘ π_{z,v} = π_{v,v}^{⋄}(ε) = π_{z,v}^{⋄}(ε)

for any e ∈ B and for any ε ∈ (Der^{∗}L(X))^{⋄} of degree n.

Hence we get

dL^{e}(z, v) = d(π_{z,v}^{⋄}(e^{∗})) = π_{z,v}^{⋄} ∧ π_{z,v}^{⋄}(d(e^{∗})).

Warning: π_{z,v}^{⋄} is not injective, hence we can have d(e^{∗}) ≠ 0 but d(L^{e}(z, v)) = 0. Proposition 6.1.2. If

d(e^{∗}) = ∑_{k+l=m, α,β} c_{α,β}^{k,l}(e_{α}^{k})^{∗} ∧ (e_{β}^{l})^{∗} + ∑_{p+q=m, γ,δ} b_{γ,δ}^{p,q}(e_{γ}^{p})^{∗} ∧ (ε_{δ}^{q})

in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄} then

d(L^{e}(z, v)) = ∑_{k+l=m, α,β} c_{α,β}^{k,l}L^{e_{α}^{k}}(z, v) ∧ L^{e_{β}^{l}}(z, v) + ∑_{p+q=m, γ,δ} b_{γ,δ}^{p,q}L^{e_{γ}^{p}} (z, v) ∧ L^{ε_{δ}^{q}}(v)

in 𝔨(X)^{⋄}.

§7. Analog of Zagier conjecture. 7.0. We shall present here a conjecture which is an l- adic analog of conjectures concerning iterated integrals from [W3]. These conjectures are generalizations of the Zagier conjecture for classical complex polylogarithms. The main ideas come from the Deligne - Beilinson paper.

We assume that there exists a category of mixed Tate motives over Spec K such as in [BD]. We shall denote this category by MM_{K}. The category MM_{K} is a tannakian category and it is equivalent to a category of representations of a pro - algebraic group Π_{K} defined over Q. Let U_{K} := ker(Π_{K} → G_{m}). The group U_{K} is a pro - algebraic pro - unipotent group defined over Q. We denote by Lie U_{K} its Lie algebra. This Lie algebra is equipped with the weight filtration. Let

Lie U_{K} = ⊕_{n=1}^{∞}(Lie U_{K})_{n}

be the associated graded Lie algebra. We set

(Lie U_{K})^{⋄} := ⊕_{n=1}^{∞}(Lie U_{K})_{n}^{⋄}.

(Lie U_{K})^{⋄} equipped with d - the dual of the Lie bracket - is a Lie coalgebra.

Let X be a projective line over K minus a finite number of K - points. We shall construct a Lie subcoalgebra of the Lie coalgebra (Lie U_{K})^{⋄} ( a quotient of the Lie algebra Lie U_{K}) corresponding to the pro - unipotent part of the fundamental group of the tannakian category generated by mixed motives of torsors of paths from v to z on X for all pairs (z, v) ∈ X̂(K)^{2}.

We shall construct this Lie subcoalgebra of (Lie U_{K})^{⋄} in the inductive way. This Lie subcoalgebra will be a graded Lie coalgebra. The construction in degree 1 will be clear. We shall assume that we have constructed our Lie coalgebra up to degree N, i.e., we have ⊕_{i=1}^{N−1}L^{i} and d : ⊕_{i=1}^{N−1}L^{i} → ⋀^{2}(⊕_{i=1}^{N−2}L^{i}).

We construct a candidate L′_{N} in degree N and d′_{N} : L′_{N} → ⋀^{2}(⊕_{i=1}^{N−1}L^{i}). Our construction should be motivic hence we should have the following commutative diagram

kerd′_{N}  ⟶^{Φ′_{N}}  kerd_{N}
↓  ↓
L′_{N}  ⟶^{Φ̄_{N}}  (Lie U_{K})_{N}^{⋄}
↓d′_{N}  ↓d_{N}
⋀^{2}(⊕_{i=1}^{N−1}L^{i}) \underrightarrow{⋀^{2}(⊕_{i=1}^{N−1}Φ_{i})} ⋀^{2}(⊕_{i=1}^{N−1}(Lie U_{K})_{i}^{⋄})

Then it is clear that L_{N} = L′_{N}/kerΦ̄_{N}. Observe that L′_{N}/kerΦ̄_{N} = L′_{N}/kerΦ′_{N}. In fact we shall conjecture that we have a map

Φ′_{N} : kerd′_{N} → kerd_{N} = Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q

and we shall set L_{N} = L′_{N}/kerΦ′_{N}. This is a short motivic justification of our next steps. We recall that in the category MM_{K}

Ext_{MM_{K}}^{1} (Q(0), Q(1)) ⊗ Q = K^{∗} ⊗ Q

7.1. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}}. We assume for simplicity that a_{n+1} = ∞. Let us choose a tangential base point v_{i} (a tangent vector) at a_{i} for i = 1, 2, …, n + 1. Let B be a Hall base of Lie(X) and let B_{m} be elements of degree m in B.

For k = 1 we set L_{1} := K^{∗} ⊗ Q, d_{1} = 0 : L_{1} → 0. We define symbols {z, v}_{X_{i}} ∈ L_{1} in the following way. If (z, v) ∈ X(K)^{2} then \{z, v\}_{X_{i}} := \frac{z−a_{i}}{v−a_{i}} ⊗ 1 ∈ L_{1}, if z ∈ X(K) and v = a⃗_{i}x then \{z, v\}_{X_{i}} := \frac{z−a_{i}}{x−a_{i}} ⊗ 1 ∈ L_{1} and \{z, v\}_{X_{j}} := \frac{z−a_{j}}{a_{i}−a_{j}} ⊗ 1 ∈ L_{1}, if z = a⃗_{k}x′ and v = a⃗_{l}x then \{z, v\}_{X_{i}} := \frac{a_{k}−a_{i}}{a_{l}−a_{i}} ⊗ 1 ∈ L_{1} for i ≠ k, l, \{z, v\}_{X_{k}} := \frac{x′−a_{k}}{a_{l}−a_{i}k} ⊗ 1 ∈ L_{1} and \{z, v\}_{X_{l}} := \frac{a_{k}−a_{l}}{x−a_{l}} ⊗ 1 ∈ L_{1}.

We define a map

φ_{1} : L_{1} → Ext_{MM_{K}}^{1} (Q(0), Q(1)) ⊗ Q = K^{∗} ⊗ Q

by φ_{1}(z ⊗ 1) := z ⊗ 1. We define

ψ_{1} : L_{1} → H^{1}(K_{1}(X), Q_{l}(1))

by ψ_{1}({z, v}_{X_{i}}) := L^{X_{i}}(z, v). (We recall that K_{1}(X) = Gal(K̄/K(μ_{l^{∞}})).)

Proposition 7.1.0. The diagram

L_{1}  ⟶^{φ_{1}} Ext_{MM_{K}}^{1} (Q(0), Q(1)) ⊗ Q
↓ψ_{1}  ↓realization
H^{1}(K_{1}(X), Q_{l}(1)) ↩  H^{1}(G_{K}, Q_{l}(1)),

commutes, where realization associates to z ⊗ 1 ∈ K^{∗} ⊗ Q = Ext_{MM_{K}}^{1} (Q(0), Q(1)) ⊗ Q the Kummer character corresponding to z.

Proof. Let (z, v) ∈ X(K)^{2} and let p be a path from v to z. Let us take σ ∈ G_{K(μ_{l^{∞}})}. We shall calculate the coefficients of (logσ_{x,p})(1) at X_{i}. This coefficient is equal to the exponent of f_{p}(σ) at x_{i}. Let ζ be a coordinate on P_{K}^{1}. The loop f_{p}(σ) = p^{−1} · σ · p · σ^{−1} transforms (ζ − a_{i})^{\frac{1}{l^{n}}} into

\frac{σ^{−1}((v − a_{i})^{\frac{1}{l^{n}}})}{(v − a_{i})^{\frac{1}{l^{n}}}} · \frac{σ((z − a_{i})^{\frac{1}{l^{n}}})}{(z − a_{i})^{\frac{1}{l^{n}}}} · (ζ − a_{i})^{\frac{1}{l^{n}}}

This finishes the proof of the proposition when (z, v) ∈ X(K)^{2}. The cases when one or both elements are tangential base points we left to the readers.

Let N > 1. Assume that the groups L_{k}, the symbols {z, v}_{e} ∈ L_{k} for e ∈ B_{k}, the homomorphisms d_{k} : L_{k} → ⊕_{i+j=k}L_{i} ∧ L_{j}, φ_{k} : kerd_{k} → Ext_{MM_{K}}^{1} (Q(0), Q(k)) ⊗ Q and ψ_{k} : L_{k} → H_{C}^{1}(K_{k}(X), Q_{l}(k)) are defined for k < N. Assume that for k < N the diagram

kerd_{k}  ⟶^{φ_{k}} Ext_{MM_{K}}^{1} (Q(0), Q(k)) ⊗ Q
↓ψ_{k}  ↓realization
H_{C}^{1}(K_{k}(X), Q_{l}(k)) ↩  H^{1}(G_{K}, Q_{l}(k))

commutes.

Let φ ∈ Lie(X)^{⋄} be a linear form of degree k defined over Q. If φ = ∑_{i} a_{i}(e_{i}^{k})^{∗} then we set {z, v}_{φ} := ∑_{i} a_{i}{z, v}_{e_{i}^{k}}.

Let ε ∈ (Der^{∗}Lie(X))^{⋄} = ⊕_{i=1}^{n}(Lie(X)/⟨X_{i}⟩)^{⋄} be a linear form of degree k defined over Q. Assume that ε = (φ_{1}, …,φ_{n}), where φ_{i} ∈ (L(X)/⟨X_{i}⟩)^{⋄}. Then we set

{v}_{ε} := ∑_{i=1}^{n}{v_{i}, v}_{φ_{i}}.

Observe that L^{ε}(v) = ∑_{i=1}^{n} L^{φ_{i}}(v_{i}, v).

Let B_{N} = {e_{i}^{N}}_{i∈I}. For each e_{i}^{N} we set

L^{e_{i}^{N}} := ⊕_{{z,v}∈X̂(K)^{2}}Q{z, v}′_{e_{i}^{N}} − a vector space over Q on symbols {z, v}′_{e_{i}^{N}}

and

L′_{N} := ⊕_{i∈I}L^{e_{i}^{N}} .
Let e ∈ B_{N}. We define
d′_{N} : L′_{N} → ⊕_{i+j=N}L_{i} ∧ L_{j}

setting

d′_{N}({z, v}′_{e}) = ∑_{k+l=N, α,β} c_{α,β}^{k,l}{z, v}_{e_{α}^{k}} ∧ {z, v}_{e_{β}^{l}} + ∑_{p+q=N, γ,δ} b_{γ,δ}^{p,q}{z, v}_{e_{γ}^{p}} ∧ {v}_{ε_{δ}^{q}}

if

d(e^{∗}) = ∑_{k+l=N, α,β} c_{α,β}^{k,l}(e_{α}^{k})^{∗} ∧ (e_{β}^{l})^{∗} + ∑_{p+q=N, γ,δ} b_{γ,δ}^{p,q}(e_{γ}^{p})^{∗} ∧ ε_{δ}^{q}

in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄}. Conjecture D_{N}. There is a homomorphism

φ′_{N} : kerd′_{N} → Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q

such that the diagram

kerd′_{N}  ⟶^{φ′_{N}} Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q
↓ψ′_{N}  ↓realization
H_{C}^{1}(K_{N}(X), Q_{l}(N)) ↩  H^{1}(G_{K}, Q_{l}(N)),

commutes, where the map ψ′_{N} is given by ψ′_{N}({z, v}′_{e_{i}^{N}}) := L^{e_{i}^{N}} (z, v).

If the conjecture is true then we set L_{N} := L′_{N}/kerφ′_{N}. The maps d_{N}, ψ_{N} and φ_{N} are defined by passing to quotient. The symbol {z, v}_{e_{i}^{N}} is the image of {z, v}′_{e_{i}^{N}} in L_{N}. Definition 7.1.1. We set

L^{K}(X) := ⊕_{N=1}^{∞}L^{N}.

We define d : L^{K}(X) → L^{K}(X) ∧ L^{K}(X) by setting d |_{L^{N}}:= d_{N}.

Proposition 7.1.2. The Q - vector space L^{K}(X) equipped with the homomorphism d : L^{K}(X) → L^{K}(X) ∧ L^{K}(X) is a Lie coalgebra.

Proof. It is enough to show that

∑_{i=0}^{2} σ^{i} ∘ (d ⊗ id_{L^{K}(X)}) ∘ d = 0, 7.1.3.

where σ(a ⊗ b ⊗ c) = b ⊗ c ⊗ a. In the Lie coalgebra (Lie(X) ˜×Der^{∗}Lie(X))^{⋄} we obviously have

7.1.4.  ∑_{i=0}^{2} σ^{i} ∘ (d ⊗ id_{(Lie(X) ˜×Der^{∗}Lie(X))^{⋄}}) ∘ d = 0.

Hence the equality 7.1.3 should follow from 7.1.4. The fact that it is not obvious is due to our “artificial ” definition of symbols {v}_{ε}. Alternatively we could add symbols {v}_{ε} in the definition of L′_{N} and then express d({v}_{ε}) in terms of {v}_{ε′}. Next we should identify {v}_{ε} with ∑_{i=1}^{n}{v_{i}, v}_{φ_{i}} for some linear forms φ_{1}, …, φ_{n} ∈ Lie(X)^{⋄} and show that d({v}_{ε}) = d(∑_{i=1}^{n}{v_{i}, v}_{φ_{i}}). We shall show this last equality.

We recall that

Der^{∗}Lie(X) = {D ∈ DerLie(X) | ∀X_{i} ∈ X ∃A_{i} ∈ Lie(X), D(X_{i}) = [X_{i}, A_{i}]}.

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

Der^{∗}Lie(X) = ⨁_{i=1}^{n} Lie(X)/⟨X_{i}⟩

sending D_{(A_{1},...,A_{n})} to a sequence (A_{1}, …, A_{n}). We have

[D_{(V_{k})_{k=1,...,n}}, D_{(W_{k})_{k=1,...,n}}] = D_{([V_{k},W_{k}]+D_{(V_{j})_{j}} (W_{k})−D_{(W_{j})_{j}} (V_{k}))_{k=1,...,n}}. 7.1.5.

Let ε ∈ (Der^{∗}Lie(X))^{⋄}. Then ε = ∑_{i=1}^{n}(∑_{e∈B} n_{e,i}(e)^{i∗}), where (e)^{i∗} is a composition of e^{∗} with the projection ⊕_{k=1}^{n}Lie(X)/⟨X_{k}⟩ → Lie(X)/⟨X_{i}⟩. We shall compare d(e^{∗}) with d((e)^{i∗}) in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄}. Observe that e^{∗} ∈ Lie(X)^{⋄} and (e)^{i∗} ∈ (Der^{∗}Lie(X))^{⋄}.

It follows from the definition of the Lie bracket in the semi - direct product Lie(X) ˜×Der^{∗}(Lie(X)) that

d(e^{∗}) = ∑_{α,β} n(e)_{α,β}e_{α}^{∗} ∧ e_{β}^{∗} + ∑_{k=1}^{n} ∑_{γ,δ} m(e)_{γ,δ}e_{k,δ}^{∗} ∧ (e_{δ})^{k∗}. 7.1.6.

On the other side it follows from 7.1.5 that

d((e)^{i∗}) = ∑_{α,β} n(e)_{α,β}(e_{α})^{i∗} ∧ (e_{β})^{i∗} + ∑_{k=1}^{n} ∑_{γ,δ} m(e)_{γ,δ}(e_{k,δ})^{i∗} ∧ (e_{δ})^{k∗}. 7.1.7.

Hence we get

d({v_{i}, v}_{e}) = ∑_{α,β} n(e)_{α,β}{v_{i}, v}_{e_{α}} ∧ {v_{i}, v}_{e_{β}} + ∑_{k=1}^{n} ∑_{γ,δ} m(e)_{γ,δ}{v_{i}, v}_{e_{k,δ}} ∧ {v}_{(e_{δ})^{k∗}} 7.1.8.

and

d({v}_{(e)^{i∗}}) = ∑_{α,β} n(e)_{α,β}{v}_{(e_{α})^{i∗}} ∧ {v}_{(e_{β})^{i∗}} + ∑_{k=1}^{n} ∑_{γ,δ} m(e)_{γ,δ}{v}_{(e_{k,δ})^{i∗}} ∧ {v}_{(e_{δ})^{k∗}}. 7.1.9.

We recall that we have defined {v}_{(e)^{i∗}} := {v_{i}, v}_{e}. Hence the two way of calculation of d({v}_{(e)^{i∗}}) are compatible. The calculation of d({z, v}_{e}) (corresponding to e^{∗} in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄}) involves only symbols {z, v}_{e_{1}} (corresponding to e_{1}^{∗} in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄}) and {v_{i}, v}_{e_{2}} = {v}_{(e_{2})^{i∗}} (corresponding to (e_{2})^{i∗} in (Lie(X) ˜×Der^{∗}Lie(X))^{⋄}). Therefore 7.1.4 implies 7.1.3.

Proposition 7.1.10. Assume that Conjectures D_{N} are true for all N. Assume that for all N the maps realization : Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q → H^{1}(G_{K}, Q_{l}(N)) are injective. Let n_{i} ∈ Q_{l} for i = 1, …, m. If ∑_{i=1}^{m} n_{i}L^{e_{i}^{N}} (z_{i}, v_{i}) = 0 then ∑_{i=1}^{m} n_{i}{z_{i}, v_{i}}_{e_{i}^{N}} = 0 in L_{N} ⊗ Q_{l}.

Proof. It is well known that the restriction map H^{1}(G_{K}, Q_{l}(1)) → H^{1}(G_{K(μ_{l^{∞}})}, Q_{l}(1)) is injective. Hence it follows from Proposition 7.1.0 that the proposition is true for k = 1. Let us assume that it is true for k < N.

Let ∑_{i=1}^{m} n_{i}L^{e_{i}^{N}} (z_{i}, v_{i}) = 0. This implies that d(∑_{i=1}^{m} n_{i}L^{e_{i}^{N}} (z_{i}, v_{i})) = ∑_{k+l=N} ∑_{α,β} c_{α,β}^{k,l}L^{e_{α}^{k}}(z_{α}, v_{α}) · L^{e_{β}^{l}}(z_{β}, v_{β}) = 0 in 𝔱(X)^{⋄} ∧ 𝔱(X)^{⋄}. Hence for any σ ∈ K_{k}^{T} (X)/K_{k+1}^{T}(X) we have ∑_{α,β} c_{α,β}^{k,l}L^{e_{α}^{k}}(z_{α}, v_{α})(σ) · L^{e_{β}^{l}}(z_{β}, v_{β}) = 0 for T sufficiently big. Hence by the induction hypothesis we have ∑_{α,β} c_{α,β}^{k,l}L^{e_{α}^{k}}(z_{α}, v_{α})(σ) · {z_{β}, v_{β}}_{e_{β}^{l}} = 0. Let f : L → Q_{l} be a homomorphism. We get that for all σ ∈ K_{k}^{T} (X)/K_{k+1}^{T}(X), ∑_{α,β} c_{α,β}^{k,l}L^{e_{α}^{k}}(z_{α}, v_{α})(σ) · f({z_{β}, v_{β}}_{e_{β}^{l}}) = 0. The induction hypothesis implies that for any homomorphism f : L → Q_{l} we have ∑_{α,β} c_{α,β}^{k,l}{z_{α}, v_{α}}_{e_{α}^{k}} · f({z_{β}, v_{β}}_{e_{β}^{l}}) = 0. This implies that d(∑_{i=1}^{m} n_{i}{z_{i}, v_{i}}_{e_{i}^{N}}) = 0. The assumption that the realization and the restriction are injective implies that ∑_{i=1}^{m} n_{i}{z_{i}, v_{i}}_{e_{i}^{N}} = 0 in L^{K}.

Corollary 7.1.11. Assume that Conjectures D_{N} are true for all N. Assume that for all N the maps realization : Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q → H^{1}(G_{K}, Q_{l}(N)) are injective. Let q_{i} ∈ Q for i = 1, …, m.

i) We have a relation ∑_{i=1}^{m} q_{i}L^{e_{i}}(z_{i}, v_{i}) = 0 if and only if ∑_{i=1}^{m} q_{i}{z_{i}, v_{i}}_{e_{i}} = 0 in L^{K}(X). ii) The vector space of linear relations between functions L^{e}(z, v) is defined over Q.

Proof. The first part follows immediately from Proposition 7.1.10. Observe that a vector space of linear relations between elements {z, v}_{e} is generated by relations with Qcoefficients. This implies the second part of the corollary.

Proposition 7.1.12. Assume that Conjectures D_{N} are true for all N. Assume that for all N the maps realization : Ext_{MM_{K}}^{1} (Q(0), Q(N)) ⊗ Q → H^{1}(G_{K}, Q_{l}(N)) are injective. Then the Lie coalgebras (L^{K}(X) ⊗ Q_{l}, d) and (𝔨(X)^{⋄}, d) are isomorphic.

Proof. Let us define a map

r_{l} : L^{K}(X) ⊗ Q_{l} → 𝔨(X)^{⋄}

by r_{l}({z, v}_{e} ⊗ 1) := L^{e}(z, v). The vector space 𝔨(X)^{⋄} is generated over Q_{l} by linear forms L^{e}(z, v) ( (z, v) ∈ X̂(K) × X̂(K), e ∈ B). Corollary 7.1.11 implies that the map r_{l} is an isomorphism of vector spaces over Q_{l}. It follows from the definition of d in L^{K}(X) that r_{l} is an isomorphism of Lie co - algebras over Q_{l}.

§8. Primitive example in the case P^{1} \ {0, 1, ∞}. 8.0. We shall show here that the functions a_{x,p}^{φ} are generalizations of characters considered by Soulé, Deligne, Ihara (see [S1], [S2], [D] and [I1]).

Let V = P_{Q}^{1} \ {0, 1, ∞}. Let us fix a path p from 01 to 10. We recall that π_{1}(VQ̄, 01) is a free group on x - a small loop around 0, and y - a loop around 1. (One goes from 01 to 10 along p, makes a small loop around 1 and returns to 01 along p.)

The action of σ ∈ G_{Q(μ_{l^{∞}})} is given by

σ(x) = x and σ(y) = f_{p}(σ)^{−1} · y · f_{p}(σ).

Let us set π′_{1} := [π_{1}(VQ̄, 01), π_{1}(VQ̄, 01)] and π′′_{1} := [π′_{1}, π′_{1}]. The element f_{p}(σ) belongs to π′_{1}. Assume that

f_{p}(σ) = ∏_{i,j≥1}(y^{j−1}(x^{i−1}(x, y)..)..)^{α_{i,j}(σ)} mod π′′_{1}. 8.0.1.

It implies that

σ((x, y)) = (x, y) ∏_{i,j≥1}(y^{j}(x^{i}(x, y)..)..)^{α_{i,j}(σ)} mod π′′_{1}. 8.0.2.

Ihara shows that π′_{1}/π′′_{1} is a free Z_{l}[[u, v]] - module generated by (x, y), where (u + 1) · z = x · z · x^{−1} and (v + 1) · z = y · z · y^{−1} for any z ∈ π′_{1}/π′′_{1} (see [I1] Theorem 2). It follows from 8.0.2 that σ((x, y)) = h_{σ}(u, v)·(x, y), where h_{σ}(u, v) := 1+∑_{i,j≥1} α_{i,j}(σ)u^{i}v^{j}. Coefficients β_{i,j} : G_{Q(μ_{l^{∞}})} → Q_{l}(i + j) are defined by the equality

logh_{σ}(e^{U} − 1, e^{V} − 1) = ∑_{i,j≥1} \frac{β_{i,j}(σ)}{i!j!}U^{i}V^{j} 8.0.3.

(see [I1] pages 96 and 105). We shall compare these coefficients with l - adic iterated integrals defined by us.

The inclusion k of π_{1}(XQ̄, 01) into Q_{l}{{X, Y }} given by k(x) = e^{X} and k(y) = e^{Y} induces an action of σ on Q_{l}{{X, Y }} given by

σ(X) = X and σ(Y) = Λ_{p}(σ)^{−1} · Y · Λ_{p}(σ).

The logarithm of σ, logσ ∈ Der^{∗}(Q_{l}{{X, Y }}) and

(logσ)(X) = 0, (logσ)(Y) = [Y, L(X, Y)(σ)]

for some element L(X, Y)(σ) ∈ [L(X, Y), L(X, Y)]. Let L′ := [L(X, Y), L(X, Y)] and L′′ := [L′, L′]. Then

L(X, Y)(σ) = ∑_{n=2}^{∞} ∑_{i+j=n i>0,j>0} a_{ij}(σ)[… [… [Y, X]X^{i−1}]Y^{j−1}] mod L′′,

where a_{ij} : G_{Q(μ_{l^{∞}})} → Q_{l}(i + j). Hence

(logσ)(Y) = ∑_{n=2}^{∞} ∑_{i+j=n i>0,j>0} a_{ij}(σ)[… [… [X, Y]X^{i−1}]Y^{j}] mod L′′. 8.0.4.

We shall calculate the coefficients a_{i,j}. Lemma 8.0.5. We have k((y^{b−1}(x^{a−1}(x, y)..)..) = e^{r_{a,b}(X,Y)}, where

r_{a,b}(X, Y) = ∑_{i_{a},..,i_{1},j_{b},..,j_{1}≥1} \frac{(−1)^{i_{a}+..+i_{1}+j_{b}+..+j_{1}−1}}{i_{a}! · .. · i_{1}! · j_{b} · .. · j_{1}!} [..[..[Y, X]X^{i_{a}+..+i_{1}−1}]Y^{j_{b}+..+j_{1}−1}] mod L′′.

Proof. First one calculates r_{1,1}(X, Y) and next by induction r_{a,b}(X, Y) for any pair (a, b). Lemma 8.0.6. There is a continous bijection of vector spaces

L′/L′′ ≈ Q_{l}[[s, t]]

given by [..[..[Y, X]X^{i−1}]Y^{j−1}] → s^{i}t^{j}. The element r_{a,b}(X, Y) ∈ L′/L′′ corresponds to a power series −(e^{−s} − 1)^{a}(e^{−t} − 1)^{b}.

Observe that Λ_{p}(σ) = e^{φ_{σ}(X,Y)}, where φ_{σ}(X, Y) ∈ L′. The action of σ on Q_{l}{{X, Y }} induces

σ : L(X, Y)/L′′ → L(X, Y)/L′′

given by σ(X) = X and σ(Y) = Y + [Y, φ_{σ}(X, Y)] mod L′′. It follows from 8.0.1 that

φ_{σ}(X, Y) = ∑_{i,j≥1} α_{i,j}(σ)r_{i,j}(X, Y) mod L′′. 8.0.7.

We shall calculate (logσ)(Y), where σ : L(X, Y)/L′′ → L(X, Y)/L′′.

Proposition 8.0.8. The element (logσ)(Y) ∈ L′/L′′ corresponds to the power series

tlog(1 + ∑_{i,j≥1} α_{i,j}(σ)(e^{−s} − 1)^{i}(e^{−t} − 1)^{j}) ∈ Q_{l}[[s, t]].

Proof. Let F_{σ}(s, t) ∈ Q_{l}[[s, t]] corresponds to φ_{σ}(X, Y) ∈ L′/L′′. Then the series −tF_{σ}(s, t) corresponds to (σ − Id)(Y), the series tF_{σ}(s, t)^{2} corresponds to (σ − Id)^{2}(Y), the series t(−F_{σ}(s, t))^{n} corresponds to (σ − Id)^{n}(Y). Hence (logσ)(Y) corresponds to the series tlog(1 − F_{σ}(s, t)). It follows from Lemma 8.0.6 that F_{σ}(s, t) = − ∑_{i,j≥1} α_{i,j}(σ)(e^{−s} − 1)^{i}(e^{−t} − 1)^{j}.

It follows from 8.0.4 and Proposition 8.0.8 that the coefficient a_{i,j}(σ) is equal to the coefficient of the power series −log(1+∑_{i,j≥1} α_{i,j}(σ)(e^{−s} −1)^{i}(e^{−t} −1)^{j}) at s^{i}t^{j}. It follows from 8.0.3 that \frac{β_{i,j}(σ)}{i!j!} is the coefficient of the series log(1+∑_{i,j≥1} α_{i,j}(σ)(e^{U} −1)^{i}(e^{V} −1)^{j}) at U^{i}V^{j}. Hence we get that \frac{β_{i,j}(σ)}{i!j!} = (−1)^{i+j−1}a_{i,j}(σ). It follows from Proposition 5.1.8 that (logσ)(Y) = [Y, (logσ_{p})(1)]. We recall that (logσ_{p})(1) = ∑_{e∈B} a_{p}^{e}(σ)e, where B is a Hall base of Lie(X, Y). Hence we get that a_{i,j}(σ) = a_{p}^{[...[...[Y,X]X^{i−1}]Y^{j−1}]}(σ). Therefore we have proved the following result.

Proposition 8.0.9. We have

\frac{β_{i,j}(σ)}{i!j!} = (−1)^{i+j−1}a_{p}^{[[[Y,X]X^{i−1}]Y^{j−1}]}(σ).

References

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[I1] Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Annals of Math. 123 (1986), 43 – 106.

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[NT] H.Nakamura, H. Tsunogai, Some finiteness theorems on Galois centralizers in pro - l mapping class groups, Journal fur die reine und angewandte Mathematik, 441 (1993), 115 - 144.

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[W1] Z. Wojtkowiak, The Basic Structure of Polylogarithmic Functional Equations, in Structural Properties of Polylogarithms, Editor L.Lewin, Mathematical Surveys and Monographs, Vol 37, 205-231.

[W2] Z. Wojtkowiak, Monodromy of iterated integrals and non-abelian unipotent periods, in Geometric Galois Actions London Math. Soc. L.N. Series 243 Cambridge University Press, 219 – 289.

[W3] Z. Wojtkowiak, Mixed Hodge Structures and Iterated Integrals, Prépublication n 507, Université de Nice, 1997 (revised version Prépublication n 546, Université de Nice, 1999).

[W4] Z. Wojtkowiak, On l-adic iterated integrals, Prépublication n 603, Université de Nice, 2000.

Université de Nice-Sophia Antipolis Département de Mathématiques Laboratoire Jean Alexandre Dieudonné U.R.A. au C.N.R.S., No 168 Parc Valrose - B.P.N^{∘} 71 06108 Nice Cedex 2, France E-mail address wojtkow@math.unice.fr fax number 04 93 51 79 74