On l-adic iterated integrals, III Galois actions on fundamental groups.

Zdzisław Wojtkowiak

Abstract. We continue to study l-adic iterated integrals introduced in the first part. We shall calculate explicitly l-adic logarithm and l-adic polylogarithms. Next we shall use these results to study Galois representations on the fundamental group of P1 ; \overline{Q(μ_{n})} \ \{0, μ_{n}, ∞\}.

The plan of the paper. Part III

13. Introduction to Part III.

14. Kummer characters.

15. Applications to Galois actions on fundamental groups.

Mathematics Subject Classification 11 Number theory §13. Introduction to Part III. 13.0. In Part I and II of our paper on l-adic iterated integrals we were studying coefficients of the power series Λ_{p}(σ). We were trying to go as far as possible without explicit calculations of these coefficients. ( Only in section 7 we use explicitly the fact that the coefficients at degree one of the power series Λ_{p} are Kummer characters, because it is well known that the realization map from K^{∗} ⊗ Q to H^{1}(G_{K}; Q(1)) associates to z ∈ K the Kummer character corresponding to z. We could however use Propositions 11.0.17 and 11.1.0 to avoid explicit calculations in Proposition 7.1.0.)

We start this paper with an explicit calculations of l-adic logarithm. We show that l(z) is a Kummer character κ(z) associated to z. Next we shall calculate explicitly coefficients in degree 1 of the power series log Λ_{p}. These coefficients are also expressed by Kummer characters. We have the following result.

Theorem F. Let X = P_{K}^{1} \ {a_{1}, …, a_{n}, ∞}, let z, v ∈ X̂(K) and let p be a path from v to z. Let g_{i} : X → P_{K}^{1} \{0, ∞} be given by g_{i}(z) = z −a_{i} for i = 1, …, n. Let us fix a path q_{i} from \overrightarrow{01} to g_{i}(v) for i = 1, …, n. (If v = \overrightarrow{a_{i}a} then g_{i}(v) = (a − a_{i})\overrightarrow{01} and g_{j}(v) = a_{i} − a_{j} for i ≠ j.) If

𝔣_{p} ≡ x_{1}^{α_{1}} · . . . · x_{n}^{α_{n}} mod Γ^{2}π_{1}(XK̄; v)

then i) if z, v ∈ X(K) then α_{i} = κ(z − a_{i})_{g_{i}(p)·q_{i}} − κ(z − a_{i})_{q_{i}}; ii) if z ∈ X(K) and v = \overrightarrow{a_{i}a} then α_{j} = κ(z − a_{j})_{g_{j}(p)·q_{j}} − κ(v − a_{j})_{q_{j}} for i ≠ j and

α_{i} = κ(z − a_{i})_{g_{i}(p)·q_{i}} − κ(a − a_{i})_{q_{i}}.

We give also an explicit formula for l-adic polylogarithms following [NW]. We modify slightly the proof given in [NW] as we shall not use a free differential calculus.

Next we are discussing l-adic polylogarithms evaluated at roots of unity. Let n be a positive integer prime to l. We take K = Q(μ_{n}) as our basic field. The l-adic polylogarithms evaluated at n-th roots of 1 are cocycles if paths from \overrightarrow{01} to ξ_{n}^{i} are chosen suitable. Next we recall some results from [W2], where we relate the l-adic polylogarithms evaluated at roots of unity to Soulé classes.

We show that the l-adic polylogarithms evaluated at n-th roots of unity coincide with linear combinations of Soulé classes. We also express Soulé classes by l-adic polylogarithms evaluated at n-th roots of 1.

Using functional equations of l-adic polylogarithms we show that any l_{m+1}(ξ_{n}^{i}) is a linear combination of l_{m+1}(ξ_{n}^{k}) with 0 < k < \frac{n}{2} and (k, n) = 1. We conjecture that the cohomology classes l_{m+1}(ξ_{n}^{k}) for 0 < k < \frac{n}{2} and (k, n) = 1 are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(m + 1)). This conjecture is equivalent to the following one. Conjecture G. The cyclotomic elements of Soulé generate K_{2m−1}(Z[\frac{1}{l}](μ_{n})) ⊗ Q_{l}.

In fact we think that this is a theorem. However in the literature we found only the result concerning K_{3} (see [S3] p.246). (There is however a proof using motives in [HW].)

In section 15 we study the Galois representation

G_{Q(μ_{n})} → Autπ_{1}(P1 ; \overline{Q(μ_{n})} \ \{0, μ_{n}, ∞\}; \overrightarrow{01}).

Let us set V := P_{Q(μ_{n})}^{1}\{0, μ_{n}, ∞}. Let V = {X, Y_{0}, …, Y_{n−1}}. As in the previous sections we embed π_{1}(VQ̄; \overrightarrow{01}) into a Q_{l}-algebra Q_{l}\{\{V\}\} mapping x (loop around 0) onto e^{X} and each y_{i} (loop around ξ_{n}^{i}) onto e^{Y_{i}}. Hence we get a Galois representation

φ : G_{Q(μ_{n})} → Aut(Q_{l}{{V}}).

We recall from Part I, section 3 that

G_{m}(V, \overrightarrow{01}) := ker(G_{Q(μ_{n})} → Aut(π_{1}(VQ̄; \overrightarrow{01})/Γ^{m+1}π_{1}(VQ̄; \overrightarrow{01})))

and G_{∞}(V, \overrightarrow{01}) := ⋂_{i=1}^{∞} G_{i}(V, \overrightarrow{01}).

The filtration \{G_{m}(V, \overrightarrow{01})\}_{m∈N} of G_{1}(V, \overrightarrow{01}) induces a filtration \{Lie(G_{m}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01}))\}_{m∈N} of the Lie algebra Lie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})). Passing with the homomorphism φ to Lie algebras and then to associated graded Lie algebras we get a morphism of associated graded Lie algebras

gr(Lieφ) : gr(Lie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01}))) ⊗ Q → Der^{∗}Lie(V),

where Lie(V) is a free Lie algebra over Q_{l} on the set V,

gr(Lie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01}))) ⊗ Q := ⨁_{i=1}^{∞}(Lie(G_{i}(V, \overrightarrow{01})/G_{i+1}(V, \overrightarrow{01}))) ⊗ Q ≃

⨁_{i=1}^{∞}(G_{i}(V, \overrightarrow{01})/G_{i+1}(V, \overrightarrow{01})) ⊗ Q and

Der^{∗}Lie(V) = {D ∈ DerLie(V) | ∀k ∈ {0, …,n − 1}∃β_{k}(X, Y_{0}, …, Y_{n−1}) ∈ Lie(V),

D(X) = 0 and D(Y_{k}) = [Y_{k}, β_{k}(X, Y_{0}, …, Y_{n−1})]}

(see Part I, section 5). Theorem H. The image of the morphism of associated graded Lie algebras gr(Lieφ) is contained in the Lie algebra of derivations

Der_{Z/n}^{∗}Lie(V) := {D ∈ Der^{∗}Lie(V) | ∃β(X, Y_{0}, …, Y_{n−1}) ∈ Lie(V) ∀k ∈ {0, …,n − 1},

D(X) = 0 and D(Y_{k}) = [Y_{k}, β(X, Y_{k}, …,Y_{n−1}, Y_{0}, , …,Y_{k−1})]}.

In the next theorem l-adic polylogarithms will appear. We recall that L(V) := \underleftarrow{lim}_{i} (Lie(V)/Γ^{i}Lie(V)) is a free completed Lie algebra on the set V.

We introduce the following notation. If A and B belong to a Lie algebra then we define [[A, B]B^{0}] := [A, B], [[A, B]B^{1}] := [[A, B], B] and [[A, B]B^{m}] := [[[A, B]B^{m−1}], B] for m > 1. Theorem J. Let σ ∈ G_{m}(V, \overrightarrow{01}). If m > 1 then

(logσ)(Y_{0}) = [Y_{0}, ∑_{k=0}^{n−1} l_{m}(ξ_{n}^{n−k})(σ)[[Y_{k}, X]X^{m−2}]] mod I_{3} + Γ^{m+1}L(V),

where I_{3} is a closed Lie ideal of L(V) generated topologically by all Lie brackets which contain at least three Y ’s. If m = 1 then

(logσ)(Y_{0}) = [Y_{0}, ∑_{k=1}^{n−1} l(1 − ξ_{n}^{n−k})(σ)Y_{k}] mod Γ^{2}L(V).

Now we shall use functional equations of l-adic polylogarithms and the assumption that for each m > 0 the l-adic polylogarithms l_{m+1}(ξ_{n}^{i}) for 0 < i < \frac{n}{2} and (i, n) = 1 are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(m + 1)). We assume that n is a prime number and for traditional reason we denote it by p. We get the following results.

Theorem K. Let p be a prime number greater than 2 and different from l. Assume that for each m > 1 the l-adic polylogarithms l_{m}(ξ_{p}^{k}) for 0 < k < \frac{p}{2} are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(m)). Then there are elements σ_{1}^{k}, σ_{2}^{k}, …,σ_{n}^{k}, … (k = 1, …, \frac{p−1}{2}) in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q such that any σ_{i}^{k} is homogenous of degree i and

gr(Lieφ)(σ_{1}^{k})(Y_{0}) = [Y_{0}, Y_{k}+Y_{p−k}], l(σ_{2i}^{k})(Y_{0}) ≡ [Y_{0}, [[Y_{k}, X]X^{2i−2}]−[[Y_{p−k}, X]X^{2i−2}]] mod I_{3}

and

gr(Lieφ)(σ_{2i+1}^{k})(Y_{0}) ≡ [Y_{0}, (1−p^{2i})[[Y_{k}, X]X^{2i−1}]+(1−p^{2i})[[Y_{p−k}, X]X^{2i−1}]+2p^{2i}[[Y_{0}, X]X^{2i−1}]]

mod I_{3}.

Using Theorem K we show the following result.

Theorem L. Let p be a prime number greater than 3 and different from l. Assume that for each m > 1 the l-adic polylogarithms l_{m}(ξ_{p}^{k}) for 0 < k < \frac{p}{2} are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(m)). Let S be a subset of {1, …,p − 1} satisfying the following conditions

i) if k ∈ S then p − k ∈ S, ii) (S + S) ∩ S = ∅ (the sum of two elements of S is calculated mod p).

Then the derivations gr(Lieφ)(σ_{n}^{k_{1}}), …,gr(Lieφ)(σ_{n}^{k_{q}}) for \{k_{1}, …, k_{q}\} ⊂ S ∩ \{1, …, \frac{p−1}{2}\} and n = 1, 2, … generate a free Lie subalgebra of the image of the Lie algebra homomorphism

gr(Lieφ) : grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der_{Z/p}^{∗}Lie(V)

and moreover these derivations are free generators of this Lie subalgebra.

§14. Kummer characters. 14.0. In this subsection we shall discuss some elementary properties of Kummer characters. We shall show that an l-adic logarithm l(z) is a Kummer character corresponding to z. We also calculate coefficients in degree 1 of the power series Λ_{p}. These coefficients are expressed by Kummer characters.

Let K be a number field. First we state some well known and elementary results about Kummer characters without proofs.

Lemma 14.0.1. There is a bijection between compatible systems {ξ_{l^{n}}}_{n∈N} of primitive l^{n}-th roots of 1 and generators of π_{1}(Spec K̄[[z]][\frac{1}{z}]; \overrightarrow{01}).

Let us fix a compatible system {ξ_{l^{n}}}_{n∈N} of primitive l^{n}-th roots of 1. It corresponds to a generator x of π_{1}(Spec K̄[[z]][\frac{1}{z}]; \overrightarrow{01}). Let ζ ∈ K and let \{ζ_{n}\}_{n∈N} be a compatible system of l^{n}-th roots of ζ. We define a function

κκ_{n}(ζ) : Gal(K̄/K) → μ_{l^{n}}

in the following way. Let σ ∈ Gal(K̄/K). We set

κκ_{n}(ζ)(σ) := σ(ζ_{n})/ζ_{n}.

The family {κκ_{n}(ζ)(σ)}_{n∈N} is a compatible system of l^{n}-th roots of 1. Hence it follows from Lemma 14.0.1 that we get a function

κκ(ζ) : Gal(K̄/K) → π_{1}(Spec K̄[[z]][\frac{1}{z}]; \overrightarrow{01}).

We have

κκ(ζ)(σ) = x^{κ(ζ)(σ)}

for some element κ(ζ)(σ) ∈ Z_{l}. The function

κ(ζ) : Gal(K̄/K) → Z_{l}

we call a Kummer character associated to ζ (and corresponding to a compatible system {ζ_{n}}_{n∈N} of l^{n}-th roots of ζ). The function κ(ζ) depends on a choice of a compatible system of l^{n}-th roots of ζ.

Lemma 14.0.2. Let τ, σ ∈ G_{K}. The function κ(ζ) satisfies

κ(ζ)(τσ) = κ(ζ)(τ) + χ(τ)κ(ζ)(σ).

Hence the function κ(ζ) defines a cohomology class κ(ζ) ∈ H^{1}(G_{K}; Z_{l}(1)).

Lemma 14.0.3. Let {y_{n}}_{n∈N} be another compatible system of l^{n}-th roots of ζ. Let κ′ be the corresponding Kummer character. Then we have κ′ = κ(ζ) + a(χ − 1) for some a ∈ Z_{l}. Moreover for any a ∈ Z_{l} there is a compatible system of l^{n}-th roots of ζ such that the corresponding Kummer character is equal κ(ζ) + a(χ − 1).

Definition 14.0.4. Any function from G_{K} to Q_{l} of the form κ(ζ)+a(χ−1), where a ∈ Q_{l} we shall call also a Kummer character.

Corollary 14.0.5.

i) The function

κ(ζ)_{|Gal(K̄/K(μ_{l^{∞}}))} : Gal(K̄/K(μ_{l^{∞}})) → Z_{l}

does not depend on a choice of a compatible family of l^{n}-th roots of ζ. ii) For any τ ∈ G_{K} and σ ∈ Gal(K̄/K(μ_{l^{∞}})) we have

κ(ζ)(τ · σ · τ^{−1}) = χ(τ)κ(σ).

iii) Let α, β ∈ K. Then after the restriction to Gal(K̄/K(μ_{l^{∞}})) we have

κ(ζ)(α · β) = κ(ζ)(α) + κ(ζ)(β).

We denote by μ(K) a subgroup of roots of unity in K^{∗}.

Proposition 14.0.6. We have:

i) If κ(ζ) = 0 on Gal(K̄/K(μ_{l^{∞}})) then ζ is a root of unity. ii) The homomorphism φ : K^{∗}/μ(K) → Hom(Gal(K̄/K(μ_{l^{∞}})); Q_{l}) defined by φ(ζ) :=

κ(ζ) is injective and it extends to an injective homomorphism φ : K^{∗}/μ(K) ⊗ Z_{l} → Hom(Gal(K̄/K(μ_{l^{∞}})); Q_{l}).

We finish with a result that for a given finite number of z_{1}, …, z_{n} ∈ K^{∗} their Kummer characters from G_{K} to Q_{l}(1) can be chosen in a compatible way.

Proposition 14.0.7. Let z_{1}, …,z_{n} ∈ K^{∗}. Then we can chose Kummer characters κ(z_{1}), …,κ(z_{n}) such that any relation ∑_{i=1}^{n} z_{i} ⊗ q_{i} = 0 in K^{∗} ⊗ Q implies ∑_{i=1}^{n} q_{i}κ(z_{i}) = 0 in the group of one-cocycles Z(G_{K}; Q_{l}(1)).

Proof. We consider a vector subspace of K^{∗} ⊗ Q generated by z_{1} ⊗ 1, …,z_{n} ⊗ 1. Without lost of generality we can assume that z_{1} ⊗ 1, …,z_{p} ⊗ 1 are linearly independent. If z_{r} ⊗ 1 = ∑_{i=1}^{p} q_{i}(z_{i} ⊗ 1) then N(z_{r} ⊗ 1) = ∑_{i=1}^{p} Q_{i}(z_{i} ⊗ 1), where N and Q_{i} are integers. Observe that Nκ(z_{r}) = ∑_{i=1}^{p} Q_{i}κ(z_{i}) on Gal(K̄/K(μ_{l^{∞}})). Proposition 14.0.6 implies that (∏_{i=1}^{p} z_{i}^{Q_{i}}) · z_{r}^{−N} is a root of unity. Hence ∑_{i=1}^{p} Q_{i}κ(z_{i}) = Nκ(z_{r}) + k(χ − 1) for some k ∈ Z_{l}. We replace κ(z_{r}) by κ(z_{r}) + \frac{k}{N} (χ − 1). 14.1. We explain how a path p on P_{K}^{1} \ \{0, ∞\} from \overrightarrow{01} to ζ determines a compatible family of l^{n}-th roots of ζ. Let z be a local parameter at 0 corresponding to the tangential base point \overrightarrow{01}. We consider Puiseux elements \{z^{\frac{1}{l^{n}}} \}_{n∈N} such that at 1 they are all equal

1. By analytic continuation of Puiseux elements first from 1 to \overrightarrow{01} along the canonical path and next along p to ζ we get a compatible family of l^{n}-th roots of ζ. If ζ = \overrightarrow{0τ} is a tangential base point then a path p determines a compatible family of l^{n}-th roots of τ. We continue analytically, first from 1 to \overrightarrow{01}, next along p to ζ and finally to τ along the canonical path from \overrightarrow{0τ} to τ.

The corresponding Kummer character we denote by κ(ζ)_{p} to indicate the dependence on the path p.

In the next proposition we show that the l-adic logarithm l(ζ)_{p} coincides with the Kummer character κ(ζ)_{p}. Let x be a geometric generator of π_{1}(P^{1}(C) \ \{0, ∞\}; \overrightarrow{01}). We recall from Part II, Corollary 11.0.7 that the l-adic logarithm is defined by the following equality

p^{−1} · σ · p · σ^{−1} = 𝔣_{p}(σ) = x^{l(ζ)_{p}(σ)}.

Proposition 14.1.0. Let p be a path from \overrightarrow{01} to ζ on P_{K}^{1} \ \{0, ∞\}. Then we have

l(ζ)_{p} = κ(ζ)_{p}.

If ζ = \overrightarrow{0τ} then l(ζ)_{p} = κ(τ)_{t·p}, where t is a canonical path from \overrightarrow{0τ} to τ.

Proof. Let z be a local parameter in 0 corresponding to a tangential base point \overrightarrow{01}. The loop 𝔣_{p}(σ) = p^{−1} · σ(p) = p^{−1} · σ · p · σ^{−1} acts on z^{\frac{1}{l^{n}}} as follows:

z^{\frac{1}{l^{n}}} →^{σ^{−1}} z^{\frac{1}{l^{n}}} →^{p} ζ^{\frac{1}{l^{n}}} (1 + \frac{z − ζ}{ζ})^{\frac{1}{l^{n}}} →^{σ} ξ_{l^{n}}^{κ(ζ)_{p}}ζ^{\frac{1}{l^{n}}} (1 + \frac{z − ζ}{ζ})^{\frac{1}{l^{n}}} →^{p^{−1}} ξ_{l^{n}}^{κ(ζ)_{p}}z^{\frac{1}{l^{n}}} .

Now we assume that ζ = \overrightarrow{0τ}. Let t be a local parameter corresponding to \overrightarrow{0τ}. Then we have z = τ · t and z^{\frac{1}{l^{n}}} = τ^{\frac{1}{l^{n}}} · t^{\frac{1}{l^{n}}} (z^{\frac{1}{l^{n}}} is real positive over the interval [0, 1] and t^{\frac{1}{l^{n}}} is real positive over the interval [0, τ]). The path p^{−1} · σ · p · σ^{−1} acts on z^{\frac{1}{l^{n}}} as follows:

z^{\frac{1}{l^{n}}} →^{σ^{−1}} z^{\frac{1}{l^{n}}} →^{p} τ^{\frac{1}{l^{n}}} · t^{\frac{1}{l^{n}}} →^{σ} ξ_{l^{n}}^{κ(τ)_{p}} · τ^{\frac{1}{l^{n}}} · t^{\frac{1}{l^{n}}} →^{p^{−1}} ξ_{l^{n}}^{κ(τ)_{p}} · z^{\frac{1}{l^{n}}} .

Proposition 14.1.1. Let X = P_{K}^{1} \ {a_{1}, …, a_{n}, ∞}, let z, v ∈ X̂(K) and let p be a path from v to z. Let g_{i} : X → P_{K}^{1} \ {0, ∞} be given by g_{i}(z) = z − a_{i} for i = 1, …, n. Let us fix a path q_{i} from \overrightarrow{01} to g_{i}(v) for i = 1, …, n. (If v = \overrightarrow{a_{i}a} then g_{i}(v) = (a − a_{i})\overrightarrow{01} and g_{j}(v) = a_{i} − a_{j} for i ≠ j.) If

𝔣_{p} ≡ x_{1}^{α_{1}} · . . . · x_{n}^{α_{n}} mod Γ^{2}π_{1}(XK̄; v)

then i) if z, v ∈ X(K) then α_{i} = κ(z − a_{i})_{g_{i}(p)·q_{i}} − κ(v − a_{i})_{q_{i}}; ii) if z ∈ X(K) and v = \overrightarrow{a_{i}a} then α_{j} = κ(z − a_{j})_{g_{j}(p)·q_{j}} − κ(a_{i} − a_{j})_{q_{j}} for i ≠ j and

α_{i} = κ(z − a_{i})_{g_{i}(p)·q_{i}} − κ(a − a_{i})_{q_{i}}; iii) if z = \overrightarrow{a_{j}b} and v = \overrightarrow{a_{i}a} then α_{i} = κ(a_{j} − a_{i})_{g_{i}(p)·q_{i}}−κ(a − a_{i})_{q_{i}}, α_{j} = κ(b − a_{j})_{g_{j}(p)·q_{j}} −

κ(a_{i} − a_{j})_{q_{j}} and α_{k} = κ(a_{j} − a_{k})_{g_{k}(p)·q_{k}} − κ(a_{i} − a_{k})_{q_{k}} for k ≠ i, j; iv) if z = \overrightarrow{a_{i}b} and v = \overrightarrow{a_{i}a} then α_{i} = κ(b − a_{i})_{g_{i}(p)·q_{i}} − κ(a − a_{i})_{q_{i}} and α_{j} = 0 for j ≠ i.

Proof. The proposition follows from Proposition 11.0.17 in Part II and from Proposition 14.1.0.

Observe that we have just proved Theorem F. 14.2. In [S2] and [S3] Soulé has defined certain cohomology classes associated to compatible families of l-units. If we take compatible families of cyclotomic l-units then we get cyclotomic Soulé classes.

In this subsection we shall discuss relations of Soulé classes with l-adic polylogarithms. In [NW] we have given an arithmetic formula for l-adic polylogarithms. We present a proof of this result with some small modifications. (The related result was obtained by O.Gabber a long time ago using different methods.) Next we restrict our attention to the field K = Q(μ_{p}), where p is relatively prime to l. We express cyclotomic Soulé classes by l-adic polylogarithms evaluated at p-th roots of 1.

Let us choose a compatible family (ξ_{l^{n}})_{n∈N} of primitive l^{n}-th roots of 1. Let p be an integer prime to l and let ξ_{p} be a primitive p-th root of 1. Let (ξp^{\frac{1}{l^{n}}})_{n∈N} be a family of p-th roots of 1 such that (ξp^{\frac{1}{l^{n}}})^{l} = ξp^{\frac{1}{l^{n−1}}} , (ξp^{\frac{1}{l^{n}}})^{l^{n}} = ξ_{p} and for any n, ξp^{\frac{1}{l^{n}}} ∈ μ_{p}.

Let q be an integer. We set

u_{m,q}^{n} := ∏_{k=0}^{l^{n}−1}(1 − ξ_{l^{n}}^{k} · (ξp^{\frac{1}{l^{n}}})^{q})^{k^{m}} and v_{m,q}^{n} := ∏_{k=0 (k,l)=1}^{l^{n}−1}(1 − ξ_{l^{n}}^{k} · (ξp^{\frac{1}{l^{n}}})^{q})^{k^{m}},

where n, m ∈ N. One cheks that

u_{m,q}^{n} = v_{m,q}^{n} · (u_{m,q·l^{−1}}^{n−1})^{l^{m}}. 14.2.1.

We define Kummer characters

κ_{m,q}^{n}, κ̄_{m,q}^{n} : G_{Q(μ_{l^{n}·p})} → Z/l^{n}

by

\frac{σ((u_{m,q}^{n})^{\frac{1}{l^{n}}})}{(u_{m,q}^{n})^{\frac{1}{l^{n}}}} = ξ_{l^{n}}^{κ_{m,q}^{n}(σ)} and \frac{σ((v_{m,q}^{n})^{\frac{1}{l^{n}}})}{(v_{m,q}^{n})^{\frac{1}{l^{n}}}} = ξ_{l^{n}}^{κ̄_{m,q}^{n}(σ)}.

The characters (κ_{m,q}^{n})_{n∈N} and (κ̄_{m,q}^{n})_{n∈N} satisfy

κ_{m,q}^{n+1} ≡ κ_{m,q}^{n} mod l^{n} and κ̄_{m,q}^{n+1} ≡ κ̄_{m,q}^{n} mod l^{n}

for any σ ∈ G_{Q(μ_{l^{n+1}·p})}. We define

κ_{m,q} : G_{Q(μ_{l^{∞}·p})} → Z_{l} and κ̄_{m,q} : G_{Q(μ_{l^{∞}·p})} → Z_{l}

by κ_{m,q}(σ) := (κ_{m,q}^{n}(σ))_{n∈N} and κ̄_{m,q}(σ) := (κ̄_{m,q}^{n}(σ))_{n∈N}. It follows from 14.2.1 that

κ_{m,q} = κ̄_{m,q} + l^{m}κ_{m,q·l^{−1}}. 14.2.2.

Lemma 14.2.3. Let σ ∈ G_{Q(μ_{l^{∞}·p})}. Then we have

l_{m+1}(ξ_{p}^{q})(σ) = \frac{(−1)^{m}}{m!} κ_{m,q}(σ).

Proof. The lemma follows from [NW]. We give a brief sketch of the proof following [NW].

Let π be a path from \overrightarrow{01} to z. Let x and y in π_{1}(P1 Q̄ \ \{0, 1, ∞\}; \overrightarrow{01}) be standard generators of the fundamental group. Let us set

H_{n} := ker(π_{1}(P1 Q̄ \ \{0, 1, ∞\}; \overrightarrow{01}) → Z/l^{n}),

where x → 1 and y → 0. The group H_{n} is freely generated by x^{l^{n}} and x^{i}yx^{−i} (i = 0, … l^{n} − 1). Hence we have

𝔣_{π}(σ) · x^{−κ_{z}^{0}(σ)} ≡ ∏_{i=0}^{l^{n}−1}(x^{i}yx^{−i})^{α_{i}^{n}(σ)} mod [H_{n}, H_{n}] 14.2.4.

for some α_{0}^{n}(σ), …, α_{l^{n}−1}^{n}(σ) ∈ Z_{l}. If m > n then α_{i}^{n} = ∑_{j≡i(l^{n})} α_{j}^{m}. Hence we have a mesure dα(σ) on Z_{l}.

Let us define Kummer characters β_{j,N}^{n} as follows

σ((1 − ξ_{l^{n}}^{(j−κ_{z}^{0}(σ))χ(σ^{−1})} · z^{\frac{1}{l^{n}}})^{\frac{1}{l^{N}}} ) = ξ_{l^{N}}^{β_{j,N}^{n} (σ)} · (1 − ξ_{l^{n}}^{j} · z^{\frac{1}{l^{n}}})^{\frac{1}{l^{N}}} .

We have β_{j,M}^{n} ≡ β_{j,N}^{n} mod l^{N} if M > N. We set β_{j}^{n} := (β_{j,N}^{n})_{N} ∈ Z_{l}. Studying transformations of functions (1 − ξ_{l^{n}}^{i} · z^{\frac{1}{l^{n}}})^{\frac{1}{l^{N}}} along 𝔣_{π}(σ) · x^{−κ_{z}^{0}(σ)} one shows that α_{j}^{n} = β_{j}^{n}.

We recall that k : π_{1}(P1 Q̄ \ \{0, 1, ∞\}; \overrightarrow{01}) → Q_{l}\{\{X, Y \}\} is a continous multiplicative embedding given by k(x) = e^{X} and k(y) = e^{Y} . It follows from 14.2.4 that

logk(𝔣_{π}(σ) · x^{−κ_{z}^{0}(σ)}) ≡ ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!} ( ∑_{i=0}^{l^{n}−1} i^{k}α_{i}^{n}(σ))[Y, X^{k}] 14.2.5.

modulo logarithms of elements from k([H_{n}, H_{n}]), where [Y, X^{0}] := Y and [Y, X^{k}] := [[Y, X^{k−1}], X] for k > 0.

Therefore the coefficient at [Y, X^{k}] of logk(𝔣_{π}(σ)·x^{−κ_{z}^{0}(σ)}) is congruent to \frac{(−1)^{k}}{k!} ∑_{i=0}^{l^{n}−1} i^{k}α_{i}^{n}(σ) modulo l^{n−n_{0}}, where n_{0} is a positive integer not depending on n. Hence this coefficient is equal \frac{(−1)^{k}}{k!} ∫_{Z_{l}} x^{k}dα(σ).

Let us take z = ξ_{p}^{q} and let π be a path from \overrightarrow{01} to ξ_{p}^{q} such that κ_{ξ_{p}^{q}}^{0}(σ) = 0 for any σ ∈ G_{Q(μ_{p})}. For such a path π the factor x^{−κ_{z}^{0}(σ)} in the left hand side of 14.2.5 is 1, hence we get

l_{k+1}(ξ_{p}^{q})_{π}(σ) = \frac{(−1)^{k}}{k!} ∫_{Z_{l}} x^{k}dα(σ).

Observe that ∫_{Z_{l}} x^{n−1}dα(σ) ≡ ∑_{i=0}^{l^{N} −1} i^{n−1} · α_{i}^{N}(σ) mod l^{N}. We have ∑_{i=0}^{l^{N} −1} i^{n−1} · α_{i}^{N} = ∑_{i=0}^{l^{N} −1} i^{n−1} · β_{i}^{N}. Notice that ∑_{i=0}^{l^{N} −1} i^{n−1} · β_{i}^{N} ≡ κ_{n−1,q}^{N} mod l^{N}. Hence we get ∫_{Z_{l}} x^{n−1}dα(σ) = κ_{n−1,q}(σ). This implies the lemma. 14.3. Let m be an integer prime to l and let K = Q(μ_{m}). We recall briefly the construction of the Soulé classes for the field K.

Let us set K_{n} := K(μ_{l^{n}}) and K_{∞} := K(μ_{l^{∞}}). The extension K ⊂ K_{n} is unramified outside l. Let M be a maximal Galois extension of K unramified outside l. Let us set G := Gal(M/K), G_{n} := Gal(M/K_{n}), G_{∞} := Gal(M/K_{∞}), Γ_{n} := Gal(K_{n}/K) and Γ_{∞} := Gal(K_{∞}/K). The groups G_{n} and G_{∞} are normal subgroups of G and we have G/G_{n} ≃ Γ_{n} and G/G_{∞} ≃ Γ_{∞}. Let us set R_{n} := Z[\frac{1}{l}](μ_{ml^{n}}), X_{n} := Spec R_{n}, X_{∞} := \underleftarrow{lim}_{n} X_{n} and X := Spec Z[\frac{1}{l}](μ_{m}).

The exact sequence of sheaves on X_{n}

1 → μ_{l^{n}} → G_{m} →^{l^{n}} G_{m} → 1

induces a long exact sequence of cohomology

0 → H^{0}(X_{n}, μ_{l^{n}}) → H^{0}(X_{n}, G_{m}) → H^{0}(X_{n}, G_{m}) →^{δ} H^{1}(X_{n}, μ_{l^{n}}) ….

Let (ξ_{l^{n}})_{n∈N} be a compatible family of primitive l^{n}-th roots of 1 as in 14.2. The element ξ_{l^{n}} defines an element in H^{0}(X_{n}, μ_{l^{n}}), which we also denote by ξ_{l^{n}}. Let α_{n} := ξ_{l^{n}} ∪…∪ξ_{l^{n}} (i − 1 times) and let u_{n} ∈ R_{n}^{∗} = H^{0}(X_{n}; G_{m}). Observe that α_{n} ∈ H^{0}(X_{n}, μ_{l^{n}}^{⊗(i−1)}). Let

N_{n} : H^{1}(X_{n}, μ_{l^{n}}^{⊗i}) → H^{1}(X, μ_{l^{n}}^{⊗i})

be a transfer map associated to an étale covering p_{n} : X_{n} → X. Let us set

x_{n}^{i} := N_{n}(α_{n} ∪ δ(u_{n})).

Lemma 14.3.1. Let (u_{n})_{n∈N}, u_{n} ∈ R_{n}^{∗} be a family of l-units such that N_{n+1,n}(u_{n+1}) = u_{n}, where N_{n+1,n} : R_{n+1}^{∗} → R_{n}^{∗} is a norm. Then r_{n+1,n}(x_{n+1}^{i}) = x_{n}^{i}, where r_{n+1,n} : H^{1}(X, μ_{l^{n+1}}^{⊗i}) → H^{1}(X, μ_{l^{n}}^{⊗i}) is induced by the projection μ_{l^{n+1}} → μ_{l^{n}}. Hence the family (x_{n}^{i})_{n∈N} defines an element x^{i} = (x_{n}^{i})_{n∈N} ∈ H^{1}(X, Z_{l}(i)).

Proof. One repeats the proof of Lemma 1 in [S2] replacing K-theory by cohomology.

Let us set u_{n}^{(q)} := (1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}). It is clear that the system of l-units (u_{n}^{(q)})_{n∈N} is compatible, i.e. it satisfies N_{n+1,n}(u_{n+1}^{(q)}) = u_{n}^{(q)}. Hence it defines an element x^{i}(q) ∈ H^{1}(X, Z_{l}(i)). The elements x^{i}(q) are called cyclotomic Soulé elements.

The covering p_{n} : X_{n} → X is Galois. It follows from [S1] Lemma 6 that we have a commutative diagram

(R_{n}^{∗}/(R_{n}^{∗})^{l^{n}}) ⊗ μ_{l^{n}}^{⊗(i−1)} ⟶^{δ⊗id_{μ{⊗(i−1) \atop l^{n}}}} H^{1}(X_{n}; μ_{l^{n}}^{⊗i})
↓N_{Γ_{n}}  ↓p_{n}^{∗} ∘ N_{n}
(R_{n}^{∗}/(R_{n}^{∗})^{l^{n}}) ⊗ μ_{l^{n}}^{⊗(i−1)} ⟶^{δ⊗id_{μ{⊗(i−1) \atop l^{n}}}} H^{1}(X_{n}; μ_{l^{n}}^{⊗i})

where N_{Γ_{n}}(c) = ∏_{σ∈Γ_{n}} σ(c). Observe that

N_{Γ_{n}}((1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)}) = ∏_{σ∈Γ_{n}}(1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}^{χ(σ)}) ⊗ (ξ_{l^{n}}^{χ(σ)})^{⊗(i−1)} =

∏_{0<k<l^{n}, (k,l)=1}(1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}^{k})^{k^{i−1}} ⊗ ξ_{l^{n}}^{⊗(i−1)}.

Hence the element

p_{n}^{∗}(x_{n}^{i}) ∈ H^{1}(X_{n}; μ_{l^{n}}^{⊗i}) ≈ Hom(G_{n}; μ_{l^{n}}) ⊗ μ_{l^{n}}^{⊗(i−1)}

is given by κ̄_{i−1,q}^{n}.

We summarize the above discusion in the following proposition.

Proposition 14.3.2. (see also [W2] Lemma 3.3 ) The family of l-units (1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}})_{n∈N} defines an element x^{i}(q) = (x_{n}^{i})_{n∈N} ∈ \underleftarrow{lim}_{n} H^{1}(X; μ_{l^{n}}^{⊗i}) = H^{1}(X; Z_{l}(i)). The restriction of x_{n}^{i} ∈ H^{1}(X; μ_{l^{n}}^{⊗i}) to H^{1}(X_{n}; μ_{l^{n}}^{⊗i}) ≈ Hom(G_{n}; μ_{l^{n}}) ⊗ μ_{l^{n}}^{⊗(i−1)} is equal to the homomorphism κ̄_{i−1,q}^{n}, i.e., it is equal to the Kummer character associated to the element

∏_{0<k<l^{n}, (k,l^{n})=1}(1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}^{k})^{k^{i−1}}.

Corollary 14.3.3. After the restriction to G_{K(μ_{l^{∞}}))} we have the followings identities: i). \frac{(−1)^{n}}{n!} x^{n+1}(q) = l_{n+1}(ξ_{m}^{q}) − l^{n}l_{n+1}(ξ_{m}^{q·l^{−1}}). ii). Let s be an order of l in (Z/m)^{∗}. Then we have

(1 − l^{s·n})l_{n+1}(ξ_{m}^{q}) = ∑_{i=0}^{s−1} \frac{(−1)^{n}}{n!} l^{i·n}x^{n+1}(q · l^{−i}).

Proof. The part i) of the corollary follows from 14.2.2, Lemma 14.2.3 and Proposition 14.3.2. The part ii) is an immediate consequence of the part i).

The Soulé classes x^{n+1}(q) and the l-adic polylogarithms evaluated at m-th roots of 1 are cohomology classes in H^{1}(G_{K}; Q_{l}(m + 1)). The restriction map

H^{1}(G_{K}; Q(m + 1)) → H^{1}( ⋂_{q=0}^{m−1} H_{m+1}(P_{Q(μ_{m})}^{1} \ \{0, 1, ∞\}; ξ_{m}^{q}, \overrightarrow{01}); Q_{l}(m + 1))

is injective by Lemma 3.0.8. Let d be a divisor of m. The l-adic polylogarithms satisfy the distribution relations

d^{n}( ∑_{ξ^{d}=1} l_{n+1}(ξ)) = l_{n+1}(1)

and

d^{n}( ∑_{ξ^{d}=1} l_{n+1}(ξ · ξ_{m}^{j})) = l_{n+1}(ξ_{m}^{j·d})

for any j such that (j, m) = 1, the inversion relation

l_{n+1}(ξ_{m}^{k}) + (−1)^{n+1}l_{n+1}(ξ_{m}^{m−k}) = 0

and the equality

l_{2n}(\overrightarrow{10}) = 0

on the subgroup ⋂_{q=0}^{m−1} H_{n+1}(P_{Q(μ_{m})}^{1} \ \{0, 1, ∞\}; ξ_{m}^{q}, \overrightarrow{01}) of G_{Q(μ_{m})} (see Part II, section 11.2 and also [W2]).

Hence we get the following corollary.

Corollary 14.3.4. Let d be a divisor of m. The cyclotomic Soulé classes satisfy the distribution relations

d^{n}(∑_{i=0}^{d−1} x^{n+1}(i\frac{m}{d})) = x^{n+1}(m)

and

d^{n}(∑_{i=0}^{d−1} x^{n+1}(j + i\frac{m}{d})) = x^{n+1}(j · d)

for any j such that (j, m) = 1, the inversion relation

x^{n+1}(k) + (−1)^{n+1}x^{n+1}(m − k) = 0

and the equality

x^{2n}(0) = 0.

Corollary 14.3.5. The Q_{l}-vector subspace of H^{1}(G_{K}; Q_{l}(n + 1)) generated by the cohomology classes l_{n+1}(ξ_{m}^{k}) with 0 < k < m + 1 is generated by the cohomology classes l_{n+1}(ξ_{m}^{k}) with 0 < k < \frac{m}{2} and (k, m) = 1. This vector space coincides with the Q_{l}- vector subspace of H^{1}(G_{K}; Q_{l}(n + 1)) generated by cyclotomic Soulé classes x^{n+1}(k) with 0 < k < \frac{m}{2} and (k, m) = 1. Corollary 14.3.6. The cyclotomic Soulé classes x^{n+1}(k) in H^{1}(G_{K}; Q_{l}(n + 1)) for 0 < k < \frac{m}{2} and (k, m) = 1 are linearly independent over Q_{l} if and only if l-adic polylogarithms l_{n+1}(ξ_{m}^{k}) for 0 < k < \frac{m}{2} and (k, m) = 1 are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(n+ 1)). 14.4. We would like to show that the elements x^{i}(q) for 0 < q < \frac{m}{2} and (q, m) = 1 are linearly independent and that they generate H^{1}(X; Z_{l}(i)) ⊗ Q.

Let us set

E(i − 1) := \underleftarrow{lim}_{N_{n,n−1}⊗r_{l^{n}}}(R_{n}^{∗} ⊗ μ_{l^{n}}^{⊗(i−1)}),

where N_{n,n−1} : R_{n}^{∗} → R_{n−1}^{∗} is a norm map and r_{l^{n}} : μ_{l^{n}}^{⊗(i−1)} → μ_{l^{n−1}}^{⊗(i−1)} is a reduction mod l^{n−1}. The construction in 14.3 which to a compatible family of l-units (u_{n})_{n∈N} associates an element x in H^{1}(X; Z_{l}(i)) defines a morphism

φ : E(i − 1)_{Γ_{∞}} → H^{1}(X; Z_{l}(i)),

such that kerφ and cokerφ are finite (see [S2] p.384).

Let C(i − 1) be a subgroup of E(i − 1) generated by compatible families of cyclotonic l-units. We consider the map

C(i − 1)_{Γ_{∞}} → E(i − 1)_{Γ_{∞}}.

If K = Q then this map has a finite cokernel (see [S3] Theoreme 3).

Observe that the families ((1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)})_{n∈N} are in C(i − 1). Let us set

u(q) := ((1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)})_{n∈N}.

Lemma 14.4.1. The elements u(q) (0 < q < \frac{m}{2} and (q, m) = 1) are linearly independent over Z_{l} in C(i−1)_{Γ_{∞}}. Let C′ be a Z_{l}-submodule of C(i−1)_{Γ_{∞}} generated by elements u(q) (0 < q < \frac{m}{2} and (q, m) = 1). Then the quotient group C(i − 1)_{Γ_{∞}}/C′ is finite.

Proof. Let C_{n} be a subgroup of Q(μ_{ml^{n}})^{∗} generated by μ_{ml^{n}} and by elements 1 − ω_{n}, where ω_{n} ∈ μ_{ml^{n}} \ {1}. We recall that

C(i − 1) := \underleftarrow{lim}_{N_{n,n−1}⊗r_{l^{n}}} C_{n} ⊗ μ_{l^{n}}^{⊗(i−1)},

where N_{n,n−1} : C_{n} → C_{n−1} is a restriction of the norm map N_{n,n−1} : Q(μ_{ml^{n}})^{∗} → Q(μ_{ml^{n−1}})^{∗} and r_{l^{n}} : μ_{l^{n}}^{⊗(i−1)} → μ_{l^{n−1}}^{⊗(i−1)} is a reduction mod l^{n−1}. Let (ω_{n})_{n∈N} be a compatible system of ml^{n}-th roots of 1, i.e. ω_{n} = ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}^{α}, where α ∈ Z_{l}. Assume that α ∈ Z_{l}^{∗}. Then

((1 − ω_{n}) ⊗ ξ_{l^{n}}^{⊗(i−1)})_{n∈N} = \frac{1}{α^{i−1}} ((1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)})_{n∈N}

in C(i−1)_{Γ_{∞}}. Let k > 0 and (p, l) = 1. Then the elements (1−ξm^{\frac{q}{l^{n}}} ·ξ_{l^{n}}^{pl^{k}}) ∈ C_{n} are not in the image of the norm homomorphism N_{n+1,n} : C_{n+1}⊗Z/l^{n+1} → C_{n}⊗Z/l^{n}. Observe also that

ξm^{\frac{q}{l^{n}}} ·ξ_{l^{n}} ⊗ξ_{l^{n}}^{⊗(i−1)} = 0 in C(i−1)_{Γ_{∞}}. Hence the elements u(q) := ((1−ξm^{\frac{q}{l^{n}}} ·ξ_{l^{n}})⊗ξ_{l^{n}}^{⊗(i−1)})_{n∈N}

(m − 1 ≥ q ≥ 0) generate C(i − 1)_{Γ_{∞}} as a Z_{l}-module.

Let m = a · b. Then (1 − ξ_{m}^{ak} · ξ_{l^{n}}^{a}) = ∏_{α=0}^{a−1}(1 − ξ_{m}^{k+αb} · ξ_{l^{n}}). This implies that

(1 − ξ_{m}^{ak} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)} = a^{i−1}( ∑_{α=0}^{a−1}(1 − ξ_{m}^{k+αb} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)}),

in C(i − 1)_{Γ_{∞}}. Hence we get

u(ka) = a^{i−1} ∑_{α=0}^{a−1} u(k + αb)

in C(i − 1)_{Γ_{∞}}. The identity

1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}} = −ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}(1 − ξ_{m}^{− \frac{q}{l^{n}}} · ξ_{l^{n}}^{−1})

implies u(q) = (−1)^{i−1}u(m−q). Hence the elements u(q) (0 < q < \frac{m}{2}, (q, m) = 1) generate C(i − 1)_{Γ_{∞}} modulo torsion.

The projection of u(q) ∈ C(i − 1)_{Γ_{∞}} onto (C_{n} ⊗ μ_{l^{n}}^{⊗(i−1)})_{Γ_{∞}} is equal (1 − ξm^{\frac{q}{l^{n}}} · ξ_{l^{n}}) ⊗ ξ_{l^{n}}^{⊗(i−1)}. It follows from the Bass theorem (see [Wa] Theorem 8.9) that the elements

(1−ξm^{\frac{q}{l^{n}}} ·ξ_{l^{n}})⊗ξ_{l^{n}}^{⊗(i−1)} (0 < q < \frac{m}{2}, (q, m) = 1) are linearly independent in (C_{n}⊗μ_{l^{n}}^{⊗(i−1)})_{Γ_{∞}}.

If the map C(i − 1)_{Γ_{∞}} → E(i − 1)_{Γ_{∞}} has a finite cokernel or finite kernel then Lemma 14.4.1 will imply the following result stated below as a conjecture. Conjecture 14.4.2. The cyclotomic Soulé classes x^{n+1}(k) in H^{1}(G_{K}; Q_{l}(n + 1)) for 0 < k < \frac{m}{2} and (k, m) = 1 are linearly independent over Q_{l}. Consequently l-adic polylogarithms l_{n+1}(ξ_{m}^{q}) for 0 < q < \frac{m}{2} and (q, m) = 1 are linearly independent over Q_{l} in H^{1}(G_{K}; Q_{l}(n+ 1)) and in H^{1}(⋂_{i=0}^{m−1} H_{n+1}(P^{1} \ \{0, 1, ∞\}; ξ_{m}^{i}, \overrightarrow{01}); Q_{l}).

The referee pointed to us that the linear independence over Q_{l} of the cyclotomic Soulé classes x^{n+1}(k) for 0 < q < \frac{m}{2} and (q, m) = 1 follows from [HW]. However we still did not absorbed the motivic consideration in [HW] so we prefer to left the needed result as a conjecture. It also seems to us that a possible proof of 14.4.2 should be more elementary without using a motivic machinary.

§15. Applications to Galois actions on fundamental groups. 15.0. We shall use l-adic polylogarithms to study the Galois action on the fundamental group of the projective line minus a finite number of points.

Deligne on the conference on polylogarithms in Schloss Ringberg gave a sketch of a proof of his result concerning P^{1} \ {0, 1, −1, ∞} in the Hodge context (see [D2]). He showed that the Lie algebra associated to the mixed Hodge structures generated by the fundamental group of P^{1} \ {0, 1, −1, ∞} contains a free Lie algebra on one generator in each odd degree. Using his ideas we are trying to show the analogous result in the Galois setting as well as to study some other interesting cases. The first presentation of our results in this direction is in [W1]. In [W3] we study the Hodge analog and there the classical polylogarithms appear. We recall also that the action of G_{Q} on π_{1}(P1 Q̄ \ \{0, 1, ∞\}; \overrightarrow{01}) was studied by Y.Ihara in [I1], [I2] and P.Deligne in [D1]. The l-adic polylogarithm l_{n+1}(\overrightarrow{10}) in our notation appears in these works (see also [IS]).

Let K be a number field. 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) associated to a family Γ = {γ_{i}}_{i=1,...,n+1}. Let X be a set {X_{1}, …, X_{n}}. We embed π_{1}(XK̄; v) into Q_{l}{{X}} mapping x_{i} into e^{X_{i}}.

We recall that the action of the Galois group G_{K} on the fundamental group

G_{K} → Autπ_{1}(XK̄; v)

induces a Galois representation

φ : G_{K} → Aut(Q_{l}{{X}})

and a Lie algebra representation

Lieφ : Lie(G_{1}(X, v)/G_{∞}(X, v)) ⊗ Q → Der^{∗}L(X)

(see Part I, sections 4.0, 5.0 and 5.1).

Passing with the morphism Lieφ to associated graded Lie algebras we get a morphism of associated graded Lie algebras

gr(Lieφ) : grLie(G_{1}(X, v)/G_{∞}(X, v)) ⊗ Q → Der^{∗}Lie(X).

Let us define X_{n+1} by the equality

X_{1} + … + X_{n} + X_{n+1} = 0.

Let us set

Der^{∗,∗}Lie(X) = {D ∈ DerLie(X) | ∀k ∈ {1, …,n + 1} ∃b_{k} ∈ Lie(X), D(X_{k}) = [X_{k}, b_{k}]}.

Proposition 15.0.1. Let σ ∈ G_{m}(X, v). Then we have

(logσ_{x})(X_{n+1}) = [X_{n+1}, (logσ_{x,γ_{n+1}})(1)] mod Γ^{m+2}L(X).

Proof. In the proof we shall omit the subscript x and we shall write σ and σ_{γ_{n+1}} instead of σ_{x} and σ_{x,γ_{n+1}}. We have x_{n+1} = x_{1}^{−1} · . . . · x_{n}^{−1}. Hence k(x_{n+1}) = e^{−X_{1}} · . . . · e^{−X_{n}} = exp(X_{n+1} + c), where c = 0 mod Γ^{2}L(X). It follows from Part I, Proposition 2.2.1 that σ(X_{n+1} + c) = Λ_{γ_{n+1}}(σ)^{−1} · χ(σ)(X_{n+1} + c) · Λ_{γ_{n+1}}(σ). Let σ ∈ G_{1}(X, v). One computes that

(logσ_{γ_{n+1}})(1) = (Λ_{γ_{n+1}}(σ) − 1) − \frac{1}{2}(Λ_{γ_{n+1}}(σ) · σ(Λ_{γ_{n+1}}(σ)) − 2Λ_{γ_{n+1}}(σ) + 1) + \frac{1}{3} ….

This implies that (logσ_{γ_{n+1}})(X_{n+1} + c) = (X_{n+1} + c) · ((logσ_{γ_{n+1}})(1)). It follows from Part I, Lemma 5.1.6 that

(logσ)(X_{n+1} + c) = [X_{n+1} + c, (logσ_{γ_{n+1}})(1)].

If σ ∈ G_{m}(X, v) then (logσ)(c) = 0 mod Γ^{m+2}L(X) and [c; (logσ_{γ_{n+1}})(1)] = 0 mod Γ^{m+2}L(X). Hence we get

(logσ)(X_{n+1}) = [X_{n+1}, logσ_{γ_{n+1}}(1)] mod Γ^{m+2}L(X).

Corollary 15.0.2. The image of the homomorphism of Lie algebras

gr(Lieφ) : grLie(G_{1}(X, v)/G_{∞}(X, v)) ⊗ Q → Der^{∗}Lie(X)

is contained in Der^{∗,∗}Lie(X). 15.1. In this subsection we shall study the action of the Galois group G_{Q(μ_{n})} on the fundamental group of P1 ; \overline{Q(μ_{n})} \ \{0, μ_{n}, ∞\}.

Let us set V := P_{Q(μ_{n})}^{1} \ {0, μ_{n}, ∞}. Let us fix an embedding Q̄ ⊂ C. Let ξ_{n} = exp(\frac{2πi}{n}). We chose a tangent vector \overrightarrow{01} as a base point of the fundamental group. At each point ξ_{n}^{k} of the projective line P_{Q(μ_{n})}^{1} we choose a tangential base point v_{k} = \overrightarrow{ξ_{n}^{k}0}. We choose a family of paths Γ = \{γ_{k}\}_{k=0,...,n−1} ∪ \{γ_{∞}\}. The path γ_{k} is a path from \overrightarrow{01} to v_{k}.

Let (x, y_{0}, …, y_{n−1}, u) be a sequence of geometric generators of π_{1}(V (C); \overrightarrow{01}) corresponding to the family Γ. We have

y_{n−1} · y_{n−2} · . . . · y_{1} · y_{0} · u · x = 1.

Let f_{k} : V → V be given by f_{k}(z) = ξ_{n}^{k} · z. Then (f_{k})_{∗}𝔣_{γ_{0}}(σ) = 𝔣_{f_{k}(γ_{0})}(σ). 15.1.1. Observe that f_{k}(γ_{0}) is a path (an interval) from \overrightarrow{0ξ_{n}^{k}} to \overrightarrow{ξ_{n}^{k}0}. Let s_{k} be a path from \overrightarrow{01} to \overrightarrow{0ξ_{n}^{k}}. We have s_{k}^{−1} · (f_{k})_{∗}(x) · s_{k} = x, s_{k}^{−1} · (f_{k})_{∗}(y_{l}) · s_{k} = y_{l+k} if k + l < n 15.1.2. and s_{k}^{−1} · (f_{k})_{∗}(y_{l}) · s_{k} = x^{−1} · y_{l+k−n} · x if k + l > n − 1. 15.1.2.

It follows from 15.1.2 that s_{k}^{−1} · (f_{k})_{∗}(𝔣_{γ_{0}}(σ)(x, y_{0}, …,y_{n−1})) · s_{k} = 15.1.3.

𝔣_{γ_{0}}(σ)(x, y_{k}, …,y_{n−1}, x^{−1} · y_{0} · x, …,x^{−1} · y_{k−1} · x)

for k = 0, 1, …, n − 1. Observe that γ_{k} = f_{k}(γ_{0}) · s_{k}. It follows from Part I, Lemma 1.0.2 and the equality 𝔣_{s_{k}}(σ) = x^{\frac{k}{n} (χ(σ)−1)} that 𝔣_{γ_{k}}(σ) = s_{k}^{−1} · 𝔣_{f_{k}(γ_{0})}(σ) · s_{k} · x^{\frac{k}{n} (χ(σ)−1)}. 15.1.4.

It rests to calculate σ(u). Let g : V → V be given by g(z) = \frac{1}{z} and let δ := g(γ_{0}^{−1}). Observe that γ_{∞} = δ · t · γ_{0}, where t is a path from \overrightarrow{10} to \overrightarrow{1∞}. One checks that

𝔣_{γ_{∞}} = γ_{0}^{−1} · t^{−1} · g_{∗}(γ_{0} · (𝔣_{γ_{0}})^{−1} · γ_{0}^{−1}) · t · γ_{0} · y0^{\frac{χ−1}{2}} · 𝔣_{γ_{0}}. 15.1.5.

Observe that γ_{0}^{−1} · t^{−1} · g_{∗}(γ_{0} · x · γ_{0}^{−1}) · t · γ_{0} = u, γ_{0}^{−1} · t^{−1} · g_{∗}(γ_{0} · y_{0} · γ_{0}^{−1}) · t · γ_{0} = y_{0} 15.1.6. and γ_{0}^{−1} · t^{−1} · g_{∗}(γ_{0} · y_{k} · γ_{0}^{−1}) · t · γ_{0} = 15.1.6.

x · y_{n−1} · . . . · y_{n−k+1} · y_{n−k} · y_{n−k−1}^{−1} · . . . · y_{n−1}^{−1} · x^{−1}

for k = 1, …,n − 1. Proposition 15.1.7. The action of G_{Q(μ_{n})} on π_{1}(P1 Q̄ \ \{0, μ_{n}, ∞\}; \overrightarrow{01}) is given by

σ(x) = x^{χ(σ)},

σ(y_{k}) = x^{− \frac{k}{n} (χ(σ)−1)} · 𝔣_{γ_{0}}(σ)(x, y_{k}, …,y_{n−1}, x^{−1} · y_{0} · x, …,x^{−1} · y_{k−1} · x)^{−1} · y_{k}^{χ(σ)}· 𝔣_{γ_{0}}(σ)(x, y_{k}, …,y_{n−1}, x^{−1} · y_{0} · x, …,x^{−1} · y_{k−1} · x) · x^{\frac{k}{n} (χ(σ)−1)} for k = 0, …,n − 1 and

σ(u) = 𝔣_{γ_{0}}(σ)(x, y_{0}, …,y_{n−1})^{−1} · y0^{\frac{1−χ(σ)}{2}} ·

𝔣_{γ_{0}}(σ)(u, y_{0}, x · y_{n−1} · x^{−1}, …,x · y_{n−1} · ... · y_{2} · y_{1} · y_{2}^{−1} · ... · y_{n−1}^{−1} · x^{−1}) · u^{χ(σ)}·

𝔣_{γ_{0}}(σ)(u, y_{0}, x · y_{n−1} · x^{−1}, …,x · y_{n−1} · ... · y_{2} · y_{1} · y_{2}^{−1} · ... · y_{n−1}^{−1} · x^{−1})^{−1} · y0^{\frac{χ(σ)−1}{2}} ·

𝔣_{γ_{0}}(σ)(x, y_{0}, …,y_{n−1}).

Proof. It follows follows from Part I, Proposition 2.2.1 and from the equalities 15.1.1 - 15.1.6 that σ(x), σ(y_{k}) and σ(u) are given by the formulas of the proposition.

15.2. Let us set V := \{X, Y_{0}, …, Y_{n−1}\}. Let k : π_{1}(P 1 ; \overline{Q(μ_{n})} \ \{0, μ_{n}, ∞\}; \overrightarrow{01}) → Q_{l}\{\{V\}\}

be a continous multiplicative embedding given by k(x) = e^{X} and k(y_{j}) = e^{Y_{i}} for j = 0, …, n − 1. We denote by Lie(V) (resp. L(V)) a free Lie algebra (resp. a completed free Lie algebra) over Q_{l} on the set V. We define U by the equality X +Y_{0} +…+Y_{n−1} +U = 0. We shall study elements Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}).

Lemma 15.2.1. Let σ ∈ G_{m}(V, \overrightarrow{01}). Then we have

log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ log(Λ_{γ_{0}}(σ)(X, Y_{k}, …,Y_{n−1}, Y_{0}, …,Y_{k−1})) mod Γ^{m+1}L(V).

Proof. We recall that Λ_{γ_{0}}(σ)(X, Y_{0}, …,Y_{n−1}) = k(𝔣_{γ_{0}}(σ)(x, y_{0}, …,y_{n−1})) and Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}) = k(𝔣_{γ_{k}}(σ)(x, y_{0}, …,y_{n−1})). It follows from 15.1.4, 15.1.1 and 15.1.3 that

Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}) = k(𝔣_{γ_{0}}(σ)(x, y_{k}, …,y_{n−1}, x^{−1}·y_{0}·x, …,x^{−1}·y_{k−1}·x)·x^{\frac{k}{n} (χ(σ)−1)}).

Hence for σ ∈ G_{1}(V, \overrightarrow{01}) we have

Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}) = Λ_{γ_{0}}(σ)(X, Y_{k}, …,Y_{n−1}, Y_{0} + ∑_{p=1}^{∞} \frac{1}{p!}[[Y_{0}, X]X^{p−1}], …,Y_{k−1}+

∑_{p=1}^{∞} \frac{1}{p!}[[Y_{k−1}, X]X^{p−1}]). If σ ∈ G_{m}(V, \overrightarrow{01}) then log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ 0 mod Γ^{m}L(V). Hence log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ log(Λ_{γ_{0}}(σ)(X, Y_{k}, …,Y_{n−1}, Y_{0}, …,Y_{k−1})) mod Γ^{m+1}L(V) for any σ ∈ G_{m}(V, \overrightarrow{01}).

Lemma 15.2.2. Let σ ∈ G_{m}(V, \overrightarrow{01}). Then we have

(logσ_{γ_{k}})(1) ≡ log(Λ_{γ_{0}}(σ)(X, Y_{k}, …,Y_{n−1}, Y_{0}, …,Y_{k−1})) mod Γ^{m+1}L(V).

Proof. Let w ∈ Q_{l}{{V}}. We recall that σ_{γ_{k}}(w) = Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}) · σ(w). Hence logσ_{γ_{k}} = L_{log(Λ_{γ_{k}} (σ)(X,Y_{0},...,Y_{n−1}))} ◯ logσ, where ◯ is the BCH multiplication. We have

logσ_{γ_{k}} = L_{log(Λ_{γ_{k}} (σ)(X,Y_{0},...,Y_{n−1}))} + logσ + \frac{1}{2}L_{−(logσ)(log(Λ_{γ_{k}} (σ)(X,Y_{0},...,Y_{n−1})))} + ….

Observe that log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) = 0 mod Γ^{m}L(V). Hence (logσ)(log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}))) = 0 mod Γ^{m+1}L(V). This implies that the term −(logσ)(log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1}))) vanishes mod Γ^{m+1}L(V). One shows that other terms also vanish mod Γ^{m+1}L(V) except the first two terms. This implies that

(logσ_{γ_{k}})(1) ≡ log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) mod Γ^{m+1}L(V).

Now the lemma follows from Lemma 15.2.1.

We recall that we have a morphism of Lie algebras

Lieφ : Lie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der^{∗}L(V)

and a morphism of associated graded Lie algebras

gr(Lieφ) : grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der^{∗}Lie(V).

Proposition 15.2.3. For any σ ∈ grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) we have gr(Lieφ)(σ)(X) = 0 and

gr(Lieφ)(σ)(Y_{k}) = [Y_{k}, λ(σ)(X, Y_{k}, …,Y_{n−1}, Y_{0}, …,Y_{k−1})]

(k = 0, 1, …,n − 1) for some element λ(σ)(X, Y_{0}, …,Y_{n−1}) ∈ Lie(V).

Proof. If σ ∈ G_{1}(V, \overrightarrow{01}) then logσ is a derivation of the Lie algebra L(V) such that (logσ)(X) = 0 and (logσ)(Y_{k}) = [Y_{k}, (logσ_{γ_{k}})(1)] for k = 0, …, n − 1 (see Part I, Proposition 5.1.8). Hence the proposition follows from Lemma 15.2.2. Definition 15.2.4. We set

Der_{Z/n}^{∗}Lie(V) := {D ∈ Der^{∗}Lie(V) | ∃β(X, Y_{0}, …, Y_{n−1}) ∈ Lie(V) ∀k ∈ {0, …,n − 1},

D(X) = 0 and D(Y_{k}) = [Y_{k}, β(X, Y_{k}, …,Y_{n−1}, Y_{0}, , …,Y_{k−1})]}.

Lemma 15.2.5. The image of the homomorphism gr(Lieφ) is contained in Der_{Z/n}^{∗}Lie(V).

Proof. It follows from Proposition 15.2.3 that the image of gr(Lieφ) is contained in Der_{Z/n}^{∗}Lie(V). Definition 15.2.6. We set

Der_{Z/n}^{∗,∞}Lie(V) = {D ∈ Der^{∗}Lie(V) | ∃β(X, Y_{0}, …, Y_{n−1}) ∈ Lie(V) ∀k ∈ {0, …,n − 1},

D(X) = 0, D(Y_{k}) = [Y_{k}, β(X, Y_{k}, …,Y_{n−1}, Y_{0}, , …,Y_{k−1})] and D(U) = [U, −β(U, Y_{0}, Y_{n−1} …, Y_{1}) + β(X, Y_{0}, …, Y_{n−1})]}.

Lemma 15.2.7. The image of the homomorphism gr(Lieφ) is contained in Der_{Z/n}^{∗,∞}Lie(V).

Proof. The lemma follows from the formula for σ(u) in Proposition 15.1.7. The detail proof requires analogs of Lemmas 15.2.1 and 15.2.2 for γ_{∞}. We left details to the reader.

We denote by Im(gr(Lieφ)) the image of the morphism

gr(Lieφ) : grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der_{Z/n}^{∗}Lie(V).

Observe that Im(gr(Lieφ)) is a Lie subalgebra of Der_{Z/n}^{∗}Lie(V).

The derivation D ∈ Der_{Z/n}^{∗}Lie(V) such that D(Y_{0}) = [Y_{0}, β] we denote by D_{β}. Observe that Der_{Z/n}^{∗}Lie(V) ≈ Lie(V)/⟨Y_{0}⟩ as vector spaces. We introduce a new bracket { } on Lie(V) by setting

{β, β′} := [β, β′] + D_{β}(β′) − D_{β′}(β).

This new Lie algebra we denote by (Lie(V), { }). Observe that ⟨Y_{0}⟩ is a Lie ideal of (Lie(V), { }). Hence we can form a quotient Lie algebra which we denote by (Lie(V)/⟨Y_{0}⟩, { }).

Lemma 15.2.8. The Lie algebras Der_{Z/n}^{∗}Lie(V) and (Lie(V)/⟨Y_{0}⟩, { }) are isomorphic.

Proof. The isomorphism associates to D_{β} the class of β in Lie(V)/⟨Y_{0}⟩. Definition 15.2.9. We denote by I_{k} a Lie ideal of Lie(V) generated by Lie brackets which contain at least k elements (with repetitions) among Y_{0}, Y_{1}, …,Y_{n−1}. Let S be a subset of {0, …,n − 1}. We denote by I_{r}(S) a Lie ideal of Lie(V) generated by Lie brackets which contain at least r elements (with repetitions) among Y_{0}, Y_{1}, …,Y_{n−1} and at least one of these elements is Y_{k} with k ∈ S and at least one of these elements is Y_{j} with j ∉ S. We denote by A(S) a Lie subalgebra of Lie(V) generated by elements Y_{k} (k ∈ S), and by A(S)_{r} the degree r part of A(S).

Recall that Γ^{m+1}Lie(V) is the (m + 1)-st component of the lower central series of the Lie algebra Lie(V). In the following discussions, we often look at elements of Lie(V) modulo the Lie ideal I_{k} + Γ^{m+1}Lie(V).

Lemma 15.2.10. Let S be a subset of {0, …,n − 1} such that (S + S) ∩ S = ∅ (the sum of two elements of S is calculated mod n). Let w ∈ I_{r}(S), w′ ∈ I_{r′}(S) and a ∈ A(S)_{t}. Then D_{a}(w) ∈ I_{r+t}(S), D_{w}(a) ∈ I_{r+t}(S) and D_{w}(w′) ∈ I_{r+r′}(S).

15.3. We shall calculate coefficients of log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})). We shall show that they are expressed by l-adic polylogarithms evaluated at elements of μ_{n}.

Lemma 15.3.1. Let σ ∈ G_{m}(V, \overrightarrow{01}). If m > 1 then

log(Λ_{γ_{0}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ ∑_{k=0}^{n−1} l_{m}(ξ_{n}^{n−k})(σ)[[Y_{k}, X]X^{m−2}] mod I_{2} + Γ^{m+1}L(V).

If m = 1 then

log(Λ_{γ_{0}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ ∑_{k=1}^{n−1} l(1 − ξ_{n}^{n−k})(σ)Y_{k} mod Γ^{2}L(V).

Proof. It follows from the definition of l-adic polylogarithms that the coefficient of log(Λ_{γ_{0}}(σ)(X, Y_{0}, …,Y_{n−1})) at [[Y_{0}, X]X^{m−2}] is l_{m}(\overrightarrow{10})(σ) and the coefficient of log(Λ_{γ_{k}}(σ)(X, Y_{0}, …,Y_{n−1})) at [[Y_{0}, X]X^{m−2}] is l_{m}(ξ_{n}^{k})(σ) for 0 < k < n. It follows from Lemma 15.2.1 that for σ ∈ G_{m}(V, \overrightarrow{01})

log(Λ_{γ_{0}}(σ)(X, Y_{0}, …,Y_{n−1})) ≡ ∑_{k=0}^{n−1} l_{m}(ξ_{n}^{n−k})(σ)[[Y_{k}, X]X^{m−2}] mod I_{2} + Γ^{m+1}L(V).

If m = 1 then the coefficient of log(Λ_{γ_{0}}(σ)(X, Y_{0}, …, Y_{n−1})) at Y_{0} vanishes and the coefficient of log(Λ_{γ_{k}}(σ)(X, Y_{0}, …, Y_{n−1})) at Y_{0} is l_{1}(ξ_{n}^{k}). To finish the proof we observe that l_{1}(z) = l(1 − z) on G_{1}(V, \overrightarrow{01}).

Lemma 15.3.2. Let σ ∈ G_{m}(V, \overrightarrow{01}). Then

gr(Lieφ)(σ)(Y_{0}) ≡ [Y_{0}, ∑_{k=0}^{n−1} l_{m}(ξ_{n}^{n−k})(σ)[[Y_{k}, X]X^{m−2}]] mod I_{3} + Γ^{m+2}L(V)

for m > 1. If m = 1 then gr(Lieφ)(σ)(Y_{0}) ≡ [Y_{0}, ∑_{k=1}^{n−1} l(1 − ξ_{n}^{n−k})Y_{k}] mod Γ^{3}L(V). Proof. The lemma follows from Part I, Proposition 5.1.8 and from Lemmas 15.2.2 and 15.3.1. 15.4. We shall study the image of the homomorphism of Lie algebras

gr(Lieφ) : grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der_{Z/n}^{∗}Lie(V).

First we assume that n is a prime number and for traditional reason we shall denote it by p. Hence V = P_{Q(μ_{p})}^{1} \ {0, μ_{p}, ∞}. We recall that fundamental groups considered in this paper are pro-finite l-groups. Now we shall use Conjecture 14.4.2. We shall assume that p is different from l. We have not express in this paper l-adic polylogarithms evaluated at l-th roots of 1 by Soulé classes, hence we have to suppose that p ≠ l. Proposition 15.4.1. Let p be a prime number greater than 2 and different from l. Assume that for each m > 1 the cyclotomic Soulé classes x^{m}(k) in H^{1}(G_{Q(μ_{p})}; Q_{l}(m)) for k = 1, 2, …, \frac{p−1}{2} are linearly independent over Q_{l}. Then there are elements σ_{1}^{k}, σ_{2}^{k}, …,σ_{n}^{k}, … (k = 1, …, \frac{p−1}{2}) in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q such that any σ_{i}^{k} is homogenous of degree i and

gr(Lieφ)(σ_{1}^{k})(Y_{0}) = [Y_{0}, Y_{k}+Y_{p−k}], gr(Lieφ)(σ_{2i}^{k})(Y_{0}) ≡ [Y_{0}, [[Y_{k}, X]X^{2i−2}]−[[Y_{p−k}, X]X^{2i−2}]]

mod I_{3} and

gr(Lieφ)(σ_{2i+1}^{k})(Y_{0}) ≡ [Y_{0}, (1−p^{2i})[[Y_{k}, X]X^{2i−1}]+(1−p^{2i})[[Y_{p−k}, X]X^{2i−1}]+2p^{2i}[[Y_{0}, X]X^{2i−1}]]

mod I_{3}. Proof. Assume that m > 1. After the restriction of l-adic polylogarithms to G_{m}(V, \overrightarrow{01}) we have the following identities:

p^{m−1}(∑_{k=0}^{p−1} l_{m}(ξ_{p}^{k})) = l_{m}(\overrightarrow{10}) (ξ_{p}^{0} = \overrightarrow{10}) (∗)

and

l_{m}(ξ_{p}^{k}) + (−1)^{m}l_{m}(ξ_{p}^{p−k}) = 0 and l_{2k}(\overrightarrow{10}) = 0 (∗∗)

(see Part II, Corollaries 11.2.3 and 11.2.6). It follows from the identities (*) and (**) that

∑_{k=0}^{p−1} l_{m}(ξ_{p}^{p−k})(σ)[[Y_{k}, X]X^{m−2}] = 15.4.2.

∑_{k=1}^{\frac{p−1}{2}} l_{m}(ξ_{p}^{p−k})(σ)([[Y_{k}, X]X^{m−2}] + (−1)^{m−1}[[Y_{p−k}, X]X^{m−2}])+

\frac{p^{m−1}}{1 − p^{m−1}} ( ∑_{k=1}^{\frac{p−1}{2}}(l_{m}(ξ_{p}^{p−k})(σ) + (−1)^{m−1}l_{m}(ξ_{p}^{p−k})(σ)))[[Y_{0}, X]X^{m−2}].

It follows from the assumption that the cyclotomic Soulé classes x^{m}(k) in H^{1}(G_{Q(μ_{p})}; Q_{l}(m)) for k = 1, 2, …, \frac{p−1}{2} are linearly independent over Q_{l} and from Corollary 14.3.6 that there are elements σ^{1}, …,σ^{\frac{p−1}{2}} in G_{m}(V, \overrightarrow{01}) such that l_{m}(ξ_{p}^{p−k})(σ^{j}) = 0 if k ≠ j and l_{m}(ξ_{p}^{p−k})(σ^{k}) ≠ 0. Hence it follows from Lemma 15.3.2 and the equality 15.4.2 that

gr(Lieφ)(σ^{k})(Y_{0}) ≡ [Y_{0}, l_{m}(ξ_{p}^{p−k})(σ^{k})([[Y_{k}, X]X^{m−2}] + (−1)^{m−1}[[Y_{p−k}, X]X^{m−2}])+

\frac{p^{m−1}}{1 − p^{m−1}} (l_{m}(ξ_{p}^{p−k})(σ) + (−1)^{m−1}l_{m}(ξ_{p}^{p−k})(σ))[[Y_{0}, X]X^{m−2}]] mod I_{3} + Γ^{m+2}Lie(V).

Now we set σ_{m}^{k} := \frac{1}{l_{m}(ξ_{p}^{p−k})(σ^{k})}σ^{k} in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q if m is even and σ_{m}^{k} := \frac{1−p^{m−1}}{l_{m}(ξ_{p}^{p−k})(σ^{k})}σ^{k} in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q if m is odd.

Let us consider the case m = 1. It follows from the equality −ξ_{p}^{p−k} ·(1−ξ_{p}^{k}) = 1−ξ_{p}^{p−k} that

l(1 − ξ_{p}^{k}) − l(1 − ξ_{p}^{p−k}) = 0

on G_{1}(V, \overrightarrow{01}). It follows from [Wa] Theorem 8.9 that the p-units 1 − ξ_{p}^{k} (k = 1, …, \frac{p−1}{2}) are linearly independent in Q(μ_{p})^{∗}. Hence Proposition 14.0.6 implies that the functions l(1 − ξ_{p}^{k}) (k = 1, …, \frac{p−1}{2}) restricted to G_{Q(μ_{p·l^{∞}})} = G_{1}(V, \overrightarrow{01}) are linearly independent over Q_{l}. Now we finish the proof as for m > 1.

We shall also use the following notation. If A and B belong to a Lie algebra then we define [A, B^{0}] := A and [A, B^{k}] := [[A, B^{k−1}], B] for k > 0.

Theorem 15.4.3. Let p be a prime number greater than 3 and different from l. Assume that for each m > 1 the cyclotomic Soulé classes x^{m}(k) in H^{1}(G_{Q(μ_{p})}; Q_{l}(m)) for k = 1, 2, …, \frac{p−1}{2} are linearly independent over Q_{l}. Let S be a subset of \{1, …,p−1\} satisfying the following conditions

i) if k ∈ S then p − k ∈ S, ii) (S + S) ∩ S = ∅ (the sum of two elements of S is calculated mod p).

Then the derivations gr(Lieφ)(σ_{m}^{k_{1}}), …,gr(Lieφ)(σ_{m}^{k_{q}}) for \{k_{1}, …, k_{q}\} ⊂ S ∩ \{1, …, \frac{p−1}{2}\} and m = 1, 2, … generate a free Lie subalgebra of the image of the Lie algebra homomorphism

gr(Lieφ) : grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q → Der_{Z/p}^{∗}Lie(V)

and moreover these derivations are free generators of this Lie subalgebra.

Proof. We recall that gr(Lieφ)(σ_{m}^{k}) ∈ Der_{Z/p}^{∗}Lie(V) and gr(Lieφ)(σ_{m}^{k})(Y_{0}) = [Y_{0}, β_{m}^{k}(X, Y_{0}, …,Y_{p−1})] for some β_{m}^{k} = β_{m}^{k}(X, Y_{0}, …,Y_{p−1}) ∈ Lie(V). To show the theorem it is enough to show that a Lie subalgebra of (Lie(V), { }) generated by β_{1}^{k}, β_{2}^{k}, …,β_{n}^{k}, … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) is free and that β_{1}^{k}, β_{2}^{k}, …,β_{m}^{k}, … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) are free generators of it.

Let us set

z_{m}^{k} := [Y_{k}, X^{m−1}] + (−1)^{m−1}[Y_{p−k}, X^{m−1}].

Observe that β_{m}^{k} = z_{m}^{k} + py_{m}^{k} + ι(2)_{m}^{k}, where y_{m}^{k} ∈ I_{1} and ι(2)_{m}^{k} ∈ I_{2}. For any arrangement of brackets of length r in (Lie(V), { }) we have

{… {β_{n_{1}}^{k_{1}}, β_{n_{2}}^{k_{2}}}, … β_{n_{r}}^{k_{r}}} …} ≡ {… {z_{n_{1}}^{k_{1}} + py_{n_{1}}^{k_{1}}, z_{n_{2}}^{k_{2}} + py_{n_{2}}^{k_{2}}}, … z_{n_{r}}^{k_{r}} + py_{n_{r}}^{k_{r}}} …} 15.4.4.

mod I_{r+1}. Let us set z := {… {z_{n_{1}}^{k_{1}} + py_{n_{1}}^{k_{1}}, z_{n_{2}}^{k_{2}} + py_{n_{2}}^{k_{2}}}, …, z_{n_{r}}^{k_{r}} + py_{n_{r}}^{k_{r}}} …}. We denote by Lie(V; Z) a Lie subalgebra over Z of Lie(V) generated by the set V.

The elements z_{m}^{k} and y_{m}^{k} have integer coefficients with respect to a base of Lie(V) given by basic Lie elements in free generators X, Y_{0}, …,Y_{p−1}, hence we can view them as elements of Lie(V; Z). Therefore z has also integer coefficients with respect to this base. Observe that

15.4.5. z ≡ {… {z_{n_{1}}^{k_{1}}, z_{n_{2}}^{k_{2}}}, … z_{n_{r}}^{k_{r}}} …} mod pLie(V; Z).

The quotient Lie algebra Lie(V; Z)/pLie(V; Z) is a free Lie algebra over Z/p freely generated by the set V. We consider I_{r}(S) as a Lie ideal of Lie(V; Z)/pLie(V; Z).

It follows from Lemma 15.2.10 that in Lie(V; Z)/pLie(V; Z) we have

{… {z_{n_{1}}^{k_{1}}, z_{n_{2}}^{k_{2}}}, … z_{n_{r}}^{k_{r}}} …} ≡ [… [z_{n_{1}}^{k_{1}}, z_{n_{2}}^{k_{2}}], … z_{n_{r}}^{k_{r}}] …] mod I_{r}(S). 15.4.6.

The elements z_{1}^{k}, z_{2}^{k}, … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) are free generators of a free Lie subalgebra of Lie(V; Z)/pLie(V; Z). Hence basic Lie elements in z_{1}^{k}, z_{2}^{k} … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) are linearly independent in Lie(V; Z)/pLie(V; Z). Therefore it follows from congruences 15.4.4, 15.4.5 and 15.4.6 that basic Lie elements in elements β_{1}^{k}, β_{2}^{k}, … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) in the Lie algebra (Lie(V), { }) are linearly independent in Lie(V). Hence the elements β_{1}^{k}, β_{2}^{k}, … (k ∈ S ∩ \{1, …, \frac{p−1}{2}\}) generate a free Lie subalgebra of (Lie(V), \{ \}) and these elements are free generators of this subalgebra.

Now we shall consider the case p = 3.

Theorem 15.4.7. Let p = 3 and let l be different from 3. Assume that the cyclotomic Soulé classes x^{n}(1) ∈ H^{1}(G_{Q(μ_{3})}; Q_{l}(n)) are different from zero for n > 1. Then the derivations gr(Lieφ)(σ_{1}^{1}), …,gr(Lieφ)(σ_{n}^{1}), … generate a free Lie subalgebra of Im(gr(Lieφ)) and these derivations are free generators of this Lie subalgebra.

Proof. We must modify the proof of Theorem 15.4.3 for p = 3. Let I(Y_{0}, Y_{2}) be a Lie ideal of Lie(V; Z)/3Lie(V; Z) generated by Y_{0} and Y_{2}. Then we have

{… {z_{n_{1}}^{1}, z_{n_{2}}^{1}}, … z_{n_{r}}^{1}} …} ≡ [… [z_{n_{1}}^{1}, z_{n_{2}}^{1}], … z_{n_{r}}^{1}] …] mod I(Y_{0}, Y_{2}) 15.4.8.

and

[… [z_{n_{1}}^{1}, z_{n_{2}}^{1}], … z_{n_{r}}^{1}] …] ≡ [… [s_{n_{1}}^{1}, s_{n_{2}}^{1}], … s_{n_{r}}^{1}] …] mod I(Y_{0}, Y_{2}), 15.4.9. where s_{m}^{1} = [Y_{1}, X^{m−1}]. The congruences 15.4.8 and 15.4.9 replace the congruence 15.4.6. The rest of the proof is the same as the proof of Theorem 15.4.3. Conjecture 15.4.10. Let p be a prime number greater than 2 and different from l. The derivations gr(Lieφ)(σ_{1}^{k}), gr(Lieφ)(σ_{2}^{k}), gr(Lieφ)(σ_{3}^{k}), , …,gr(Lieφ)(σ_{n}^{k}), … for k = 1, …, \frac{p−1}{2} generate Im(gr(Lieφ)). 15.5. Now we shall consider the case when p = 2. In fact the proof will repeat the arguments from [D2] with necessary modifications for the l-adic case. Let us set V := P_{Q}^{1} \ \{0, 1, −1, ∞\}. Let \overrightarrow{01} be a base point of the fundamental group. We choose a family Γ = {γ_{0}, γ_{1}, γ_{∞}}.

Let (x, y_{0}, y_{1}, z) be a sequence of geometric generators of π_{1}(V (C); \overrightarrow{01}) corresponding to the family Γ.

Lemma 15.5.1. Let σ ∈ G_{m}(V, \overrightarrow{01}). If m > 1 then

log(Λ_{γ_{0}}(σ)(X, Y_{0}, Y_{1})) = l_{m}(\overrightarrow{10})(σ)[[Y_{0}, X]X^{m−2}] + (\frac{1}{2^{m−1}} − 1)l_{m}(\overrightarrow{10})(σ)[[Y_{1}, X]X^{m−2}]

mod I_{2} + Γ^{m+1}L(V). If m = 1 then

log(Λ_{γ_{0}}(σ)(X, Y_{0}, Y_{1})) = l(2)(σ)Y_{1} mod Γ^{2}L(V).

Proof. The lemma follows from Lemma 15.3.1 and the identity

2^{m−1}(l_{m}(\overrightarrow{10}) + l_{m}(−1)) = l_{m}(\overrightarrow{10})

(see Part II, Corollary 11.2.3).

Lemma 15.5.2. In the Lie algebra grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q there are elements σ_{1}, σ_{3}, σ_{5}, …, σ_{2m+1}, … such that any σ_{k} is homogenous of degree k, gr(Lieφ)(σ_{1})(Y_{0}) = [Y_{0}, Y_{1}] and

gr(Lieφ)(σ_{2m+1})(Y_{0}) = [Y_{0}, 2^{2m}[[Y_{0}, X]X^{2m−1}] + (1 − 2^{2m})[[Y_{1}, X]X^{2m−1}]] mod I_{3}.

Proof. The homomorphism l_{n}(\overrightarrow{10}) for n odd and greater than 1 is a generator of Hom_{Z_{l}^{∗}}(G_{Q(μ_{l^{∞}})}; Z_{l}(n)) ≃ Z_{l} (see [IS]). Hence it follows from Part I, Lemma 3.0.8 that l_{n}(\overrightarrow{10}) restricted to G_{n}(V, \overrightarrow{01}) is different from zero. Therefore there is σ(n) ∈ G_{n}(V, \overrightarrow{01}) such that l_{n}(\overrightarrow{10})(σ(n)) ≠ 0 for n odd and greater than 1. We set σ_{n} = \frac{2^{n−1}}{l_{n}(\overrightarrow{10})(σ(n))} · σ(n) in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q if n is odd and greater than 1.

The homomorphism l(2) is a Kummer character associated to 2. Hence there is σ(1) ∈ G_{1}(V, \overrightarrow{01}) such that l(2)(σ(1)) ≠ 0. We set σ_{1} = \frac{1}{l(2)(σ(1))} · σ(1) in grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q.

Theorem 15.5.3. The derivations gr(Lieφ)(σ_{1}), gr(Lieφ)(σ_{3}), …,gr(Lieφ)(σ_{2m+1}), … generate a free Lie subalgebra of Im(gr(Lieφ)) and these derivations are free generators of this Lie subalgebra.

Proof. We have gr(Lieφ)(σ_{1})(Y_{0}) = [Y_{0}, Y_{1}] and gr(Lieφ)(σ_{2n+1})(Y_{0}) = [Y_{0}, β_{2n+1}] for n > 0, where β_{2n+1} ∈ Lie(V) is homogenous of degree 2n+1. The Lie algebra Der_{Z/2}^{∗}Lie(V) is isomorphic to (Lie(V)/⟨Y_{0}⟩, { }). Hence it is enough to show that a Lie subalgebra of (Lie(V), { }) generated by β_{1}, β_{3}, …,β_{2n+1}, … is free and it is freely generated by these elements. Observe that β_{2n+1} = z_{2n+1} + y_{2n+1} + ι_{2n+1} where z_{2n+1} = [Y_{1}, X^{2n}] (z_{1} = Y_{1}), y_{2n+1} = 2^{2n}[Y_{0}, X^{2n}] − 2^{2n}[Y_{1}, X^{2n}] (y_{1} = 0) and ι_{2n+1} ∈ I_{2}.

For any arrangement of brackets of length r in (Lie(V), { }) we have

{… {β_{n_{1}}, β_{n_{2}}}, … β_{n_{r}}} …} ≡ {… {z_{n_{1}} + y_{n_{1}}, z_{n_{2}} + y_{n_{2}}}, … z_{n_{r}} + y_{n_{r}}} …} 15.5.4.

mod I_{r+1}. Let us set z := {… {z_{n_{1}} + y_{n_{1}}, z_{n_{2}} + y_{n_{2}}}, … z_{n_{r}} + y_{n_{r}}} …}. The elements z_{m} and y_{m} have integer coefficients with respect to a base of Lie(V) given by basic Lie elements in free generators X, Y_{0} and Y_{1}. Hence z has integer coefficients with respect to this base. We consider these elements as elements of Lie(V; Z), where Lie(V; Z)is a Lie subalgebra over Z of Lie(V) generated by X, Y_{0} and Y_{1}. Observe that

15.5.5. z ≡ {… {z_{n_{1}}, z_{n_{2}}}, … z_{n_{r}}} …} mod 2Lie(V; Z).

The quotient Lie algebra Lie(V; Z)/2Lie(V; Z) is a free Lie algebra over Z/2 on free generators X, Y_{0} and Y_{1}. Let S = {1}. We recall that I_{r}(S) is a Lie ideal of Lie(V; Z)/2Lie(V; Z) generated by Lie brackets which contain at least r Y ’s and at least one of them is Y_{1} and at least one of them is Y_{0}.

Observe that

{… {z_{n_{1}}, z_{n_{2}}}, … z_{n_{r}}} …} ≡ [… [z_{n_{1}}, z_{n_{2}}], … z_{n_{r}}] …] mod I_{r}(S). 15.5.6.

The elements z_{1}, z_{3}, …, z_{2n+1}, … are free generators of a free Lie subalgebra of Lie(V; Z)/2Lie(V; Z). Hence basic Lie elements of degree r in z_{1}, z_{3}, …,z_{2n+1}, … are linearly independent in Lie(V; Z)/2Lie(V; Z). Therefore it follows from congruences 15.5.4, 15.5.5 and 15.5.6 that basic Lie elements of degree r in elements β_{1}, β_{3}, …,β_{2n+1}, … in the Lie algebra (Lie(V), { }) are linearly independent in Lie(V). Hence the elements β_{1}, β_{3}, …,β_{2n+1}, … generate a free Lie subalgebra of (L(V), { }) and these elements are free generators of this Lie subalgebra. (This proof is essentialy the repetition of the proof given by Deligne for the Hodge realization (see [D2]).) Conjecture 15.5.4. The derivations l(σ_{1}), l(σ_{3}), …,l(σ_{2n+1}), … generate Im(gr(Lieφ)). 15.6. Now we shall assume that n is a power of a prime number. Let p be a prime number different from l and let n = p^{r}. Let V = P_{Q(μ_{n})}^{1} \ {0, μ_{n}, ∞}.

Proposition 15.6.1. Assume that for each i > 1 the cyclotomic Soulé classes x^{i}(k) ∈ H^{1}(G_{Q(μ_{n})}; Q_{l}(i)) for 0 < k < \frac{n}{2} and (k, p) = 1 are linearly independent over Q_{l}. Then in the Lie algebra grLie(G_{1}(V, \overrightarrow{01})/G_{∞}(V, \overrightarrow{01})) ⊗ Q there are elements σ_{m+1}^{k} for m = 1, 2, … and for 0 < k < \frac{n}{2} and (k, p) = 1 such that σ_{m+1}^{k} is homogenous of degree m + 1 and gr(Lieφ)(σ_{m+1}^{k})(Y_{0}) = [Y_{0}, β_{m+1}^{k}], where

β_{m+1}^{k} = β_{m+1}^{k}(X, Y_{0}, …,Y_{n−1}) = (1 − p^{rm})([Y_{k}, X^{m}] + (−1)^{m}[Y_{n−k}, X^{m}])+

p^{m} ∑_{j=1 (j,p)≥p}^{n−1} d_{m+1,k}^{j}[Y_{j}, X^{m}] + p^{rm}d_{m+1,k}^{0}[Y_{0}, X^{m}] mod I_{2}

and where d_{m+1,k}^{j} are integers.

Proposition 15.6.2. Assume that for each i > 1 the cyclotomic Soulé classes x^{i}(k) ∈ H^{1}(G_{Q(μ_{n})}; Q_{l}(i)) for 0 < k < \frac{n}{2} and (k, p) = 1 are linearly independent over Q_{l}. Let S be a subset of {i | 0 < i < n and (i, p) = 1} satisfying the following conditions

i) if k ∈ S then n − k ∈ S, ii) (S + S) ∩ S = ∅ (the sum of two elements of S is calculated mod n).

Then the derivations gr(Lieφ)(σ_{m+1}^{k_{1}}), …,gr(Lieφ)(σ_{m+1}^{k_{q}}) for \{k_{1}, …, k_{q}\} ⊂ S ∩[1, \frac{n}{2}] and for m = 1, 2, … generate a free Lie subalgebra of Im(gr(Lieφ)) and these derivations are free generators of this Lie subalgebra.

Corollary 15.6.3. Let n = 2^{r}. Assume that for each i > 1 the cyclotomic Soulé classes x^{i}(k) ∈ H^{1}(G_{Q(μ_{n})}; Q_{l}(i)) for 0 < k < \frac{n}{2} and k odd are linearly independent over Q_{l}. Then the derivations gr(Lieφ)(σ_{m+1}^{k}) for m = 1, 2, … and for 0 < k < \frac{n}{2} and k odd generate a free Lie subalgebra of Im(gr(Lieφ)) and these derivations are free generators of this Lie subalgebra.

The proofs of 15.6.1 - 15.6.3 are the same as proofs of 15.4.2 and 15.4.3. We do not include in our considerations derivations in degree 1 because working modulo p we are not able to eliminate Y_{j} for (j, p) ≥ p.

Corollary 15.6.4. Let n = 4. Assume that for each m > 0 the cyclotomic Soulé class x^{2m}(1) ∈ H^{1}(G_{Q(i)}; Q_{l}(2m)) does not vanish. Then the derivations gr(Lieφ)(σ_{m}^{1}) (m = 1, 2, 3 …) are free generators of a free Lie subalgebra of Im(gr(Lieφ)).

Proof. We have (1 − i)(1 + i) = 2 and i(1 − i) = 1 + i. This implies that l(2) = 2l(1 − i). Hence we can include also the derivations in degree 1 in our considerations.

Proposition 15.6.5. Let n = 8. Assume that for any m > 1 the l-adic polylogarithms l_{m}(ξ_{8}) and l_{m}(ξ_{8}^{3}) are linearly independent. Then the derivations gr(Lieφ)(σ_{m}^{1}), gr(Lieφ)(σ_{m}^{3}) for m = 1, 2, 3, … are free generators of a free Lie subalgebra of Im(gr(Lieφ)).

Proof. It follows from Corollary 15.6.3 that the most interesting derivations are in degree 1. Let σ ∈ G_{1}(V, \overrightarrow{01}). Lemma 15.3.1 implies that

log(Λ_{γ_{0}}(σ)(X, Y_{0}, Y_{1}, …, Y_{7})) ≡ ∑_{i=1}^{7} l(1 − ξ_{8}^{8−k})(σ)Y_{k} mod Γ^{2}L(V).

Observe that (1 − ξ_{8}^{2}) = (1 − ξ_{8}^{1}) · (1 − ξ_{8}^{5}), (1 − ξ_{8}^{6}) = (1 − ξ_{8}^{3}) · (1 − ξ_{8}^{7}), (1 − ξ_{8}^{4}) = (1 − ξ_{8}^{1}) · (1 − ξ_{8}^{3}) · (1 − ξ_{8}^{5}) · (1 − ξ_{8}^{7}) and −ξ_{8}^{8−k} · (1 − ξ_{8}^{k}) = (1 − ξ_{8}^{8−k}). The 2-units (1 − ξ_{8}^{1}) and (1 − ξ_{8}^{3}) are linearly independent in Q(μ_{8})^{∗}. Hence in degree 1 we have two linearly independent derivations gr(Lieφ)(σ_{1}^{1}) and gr(Lieφ)(σ_{1}^{3}) such that

gr(Lieφ)(σ_{1}^{1})(Y_{0}) = [Y_{0}, Y_{1} + Y_{7} + Y_{2} + Y_{6} + 2Y_{4}]

and

gr(Lieφ)(σ_{1}^{3})(Y_{0}) = [Y_{0}, Y_{3} + Y_{5} + Y_{2} + Y_{6} + 2Y_{4}].

After passing to Lie algebra (Lie(V; Z)/2Lie(V; Z), { }) we need to study a Lie subalgebra generated by elements z_{1}^{1} := Y_{1}+Y_{7}+Y_{2}+Y_{6}, z_{1}^{3} := Y_{3}+Y_{5}+Y_{2}+Y_{6} and z_{m}^{1} := [Y_{1}, X^{m−1}]+ (−1)^{m−1}[Y_{7}, X^{m−1}], z_{m}^{3} := [Y_{3}, X^{m−1}] + (−1)^{m−1}[Y_{5}, X^{m−1}] for m = 2, 3, 4, ….

Let Lie(X, Y_{1}, Y_{5}; Z/2) be a Lie subalgebra of Lie(V; Z)/2Lie(V; Z) generated by X, Y_{1} and Y_{5}. Let J be a Lie ideal of Lie(V; Z)/2Lie(V; Z) generated by Y_{0}, Y_{2}, Y_{3}, Y_{4}, Y_{6} and Y_{7}. Observe that

Lie(V; Z)/2Lie(V; Z) ≃ Lie(X, Y_{1}, Y_{5}; Z/2) ⊕ J.

Let us set

s_{m}^{1} := [Y_{1}, X^{m−1}] and s_{m}^{3} := [Y_{5}, X^{m−1}]

for m = 1, 2, 3, …. One shows that for any Lie bracket of elements z_{m}^{1}, z_{m}^{3} for m = 1, 2, 3, … of length r we have

{… {z_{m_{1}}^{i_{1}}, z_{m_{2}}^{i_{2}}}, …, z_{m_{r}}^{i_{r}}} ≡ [… [s_{m_{1}}^{i_{1}}, s_{m_{2}}^{i_{2}}], …, s_{m_{r}}^{i_{r}}] mod J.

The elements s_{m}^{1} and s_{m}^{3} for m = 1, 2, 3, … are free generators of a free Lie subalgebra of Lie(X, Y_{1}, Y_{5}; Z/2). This implies that the elements z_{m}^{1}, z_{m}^{3} for m = 1, 2, 3, … are free generators of a free Lie subalgebra of (Lie(V; Z)/2Lie(V; Z), { }). This implies the proposition. Remark 15.6.6. According to [G] p.427, the results for n = 3 and n = 4 are also given in Deligne’s letter to Goncharov.

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[D2] P.Deligne, talk on the conference in Schloss Ringberg, 1998.

[G] A.Goncharov, The dihedral Lie algebra and Galois symmetries of π_{1}^{(l)}(P^{1} \ ({0, ∞} ∪ μ_{N})), Duke Math. J. 110 (2001), 397-486.

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Deligne, Documenta Mathematica 3 (1998), 27-113.

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[IS] H.Ichimura, K.Sakaguchi, The Non-Vanishing of a Certain Kummer Character χ_{m} (after Soulé) and Some Related Topics, Advanced Studies in Pure Mathematics 12, 1987, pp.53-64.

[NW] H.Nakamura, Z.Wojtkowiak, On explicit formulae for l-adic polylogarithms.

[S1] Ch.Soulé, K-théorie des anneaux d‘entiers de corps de nombres et cohomologie etale, Inventiones math. 55, 251-295 (1979).

[S2] Ch.Soulé, On higher p-adic regulators, Springer Lecture Notes, N 854 (1981), 372-401.

[S3] Ch.Soulé, Eléments Cyclotomiques en K-Théorie, Asterisque, 147 - 148, 1987, 225-258.

[Wa] L.C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag 1997.

[W1] Z. Wojtkowiak, Monodromy of iterated integrals and Galois action on the fundamental group of a projective line minus a finite number of points, Prépublication n 578, Université de Nice, 2000.

[W2] Z. Wojtkowiak, A note on functional equations of l-adic polylogarithms, to appear in Journal of the Inst. of Math. Jussieu.

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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