Descent for l-adic polylogarithms

Jean-Claude Douai and Zdzisław Wojtkowiak October 29, 2007

Contents

0 Introduction 1 1 Action of a finite Galois group 3 2 l-adic polylogarithms 7 3 Action of G and l-adic polylogarithms 13 4 Zagier conjecture for l-adic dilogarithm 19 5 Action of G and Zagier conjecture for l-adic polylogarithms 24

Abstract

Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ_{l}. Let us assume that the linear combination of l-adic polylogarithms c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}} (constructed in some given way) is a cocycle on G_{L} and that the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. Then we show that c_{n} determines a unique cocycle s_{n} on G_{K}. We also prove a weak version of Zagier conjecture for l-adic dilogarithm. Finally we show that if c_{2} is “motivic”(m_{1}, …, m_{N} ∈ ℚ) then s_{2} is also “motivic”.

0 Introduction

Studying polylogarithms one meets a lot of fascinating identities. Among them there are the following ones

p^{n−1}(∑_{k=1}^{p−1} Li_{n}(e^{\frac{2πik}{p}})) = (1 − p^{n−1})Li_{n}(1),

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

and many more of the similar type (see [14] and [4]). We shall describe common features of all these identities in the following conjecture.

Conjecture 0.1. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ. Let us assume that ∑_{i=1}^{N} m_{i}Li_{n}(z_{i}) is a regulator of an extension in Ext_{MM_{Spec L}}^{1}(ℚ(0), ℚ(n)), where MM_{Spec L} is the category of mixed Tate motives over Spec L. Let us assume further that the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. Then ∑_{i=1}^{N} m_{i}Li_{n}(z_{i}) is a regulator of an extension in Ext_{MM_{Spec K}}^{1} (ℚ(0), ℚ(n)).

We recall that in [10] we have defined l-adic polylogarithms. For a given z ∈ L, an l-adic polylogarithm l_{n}(z)_{γ} is a function from G_{L} to ℚ_{l}(n), which

depends on a choice of a path on ℙ1 ; L̄ \ {0, 1, ∞} from 01^{→} to z.

The l-adic realization of mixed Tate motives over Spec L

real_{l} : Ext_{MM_{Spec L}}^{1}(ℚ(0), ℚ(n)) → H^{1}(G_{L}; ℚ(n))

associates to an extension of ℚ(0) by ℚ(n) over Spec L a cohomology class in H^{1}(G_{L}; ℚ(n)). Hence we have the obvious question when a linear combination of l-adic n-th polylogarithms evaluated at elements of L is a cocycle representing a cohomology class real_{l}(E) for some E ∈ Ext_{MM_{Spec L}}^{1}(ℚ(0), ℚ(n)).

In fact in [10] we have also asked a weaker question, when a linear combination of l-adic n-th polylogarithms evaluated at elements of L is a cocycle belonging to Z^{1}(G_{L}; ℚ_{l}(n)) and we have given an answer to this weaker question (see [10] Theorem 11.0.11 ).

The l-adic analog of Conjecture 0.1 is the following conjecture.

Conjecture 0.2. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ. Let us assume that the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. If ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}} is a cocycle representing real_{l}(E) for some E in Ext_{MM_{Spec L}}^{1}(ℚ(0), ℚ(n)) then there is F ∈ Ext_{MM_{Spec K}}^{1} (ℚ(0), ℚ(n)) such that

ι^{∗}(real_{l}(F)) = ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}}

in H^{1}(G_{L}; ℚ_{l}(n)), where ι^{∗} : H^{1}(G_{K}; ℚ_{l}(n)) → H^{1}(G_{L}; ℚ_{l}(n)) is induced by the inclusion ι : G_{L} → G_{K}.

Our main result is a weak form of Conjecture 0.2.

Theorem 0.3. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let us assume that l does not divide the order of G and that L∩K(μ_{l^{∞}}) = K. Let us assume that the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. If the linear combination ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}} constructed in some given way is a cocycle on G_{L} then there is a cocycle s_{n} on G_{K} such that

ι^{∗}(s_{n}) = ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}}

in H^{1}(G_{L}; ℚ(n)), where ι : G_{L} → G_{K} is the inclusion.

Next we are studying Zagier conjecture for l-adic dilogarithm. First we define a ℚ-vector subspace of “motivic” cocycles of the ℚ_{l}-vector space Z^{1}(G_{L}; ℚ_{l}(2)) of one cocycles on G_{L} with values in ℚ_{l}(2). Let us set

P_{2}(L) := {c_{2} := ∑_{i=1}^{N} m_{i}l_{2}(z_{i})_{γ_{i}} | z_{1}, …, z_{N} ∈ L^{∗} \ {1}, m_{1}, …, m_{N} ∈ ℚ,

∑_{i=1}^{N} m_{i}(z_{i}) ∧ (1 − z_{i}) = 0 in L^{∗} ∧ L^{∗} ⊗ ℚ, c_{2} ∈ Z^{1}(G_{L}; ℚ_{l}(2))}

Let [P_{2}(L)] be the image of P_{2}(L) in H^{1}(G_{L}; ℚ_{l}(2)). In [10] we have shown that l-adic dilogarithm satisfies the Abel functional equation. Using this result we show the following theorem.

Theorem 0.4. Let L be a number field. Then we have

dim _{ℚ}[P_{2}(L)] ≤ r_{2}(L),

where r_{2}(L) is a number of complex places of L.

Observe that if H^{1}(G_{L}; ℚ_{l}(2)) ≠ 0 then it is an infinite dimensional vector space over ℚ, so the result of the theorem is far from being obvious.

Now we state our last result.

Theorem 0.5. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z_{1}, …, z_{N} ∈ L^{∗}\{1} and let m_{1}, …, m_{N} ∈ ℚ. Let ∑_{i=1}^{N} m_{i}(z_{i})∧ (1 − z_{i}) = 0 in L^{∗} ∧ L^{∗} ⊗ ℚ. Let us assume that c_{2} := ∑_{i=1}^{N} m_{i}l_{2}(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(2)) and that formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. Then there is a cocycle s_{2} ∈ P_{2}(K) such that ι^{∗}(s_{2}) = c_{2} in H^{1}(G_{L}; ℚ_{l}(2)), where ι : G_{L} → G_{K} is the inclusion of Galois groups.

The present note has it source in the work of the second author (see [11]). In [11] the descent problem for Galois representations arising from actions of Galois groups on torsors of paths is studied.

Finally the second author would like to thank very much Herbert Gangl who pointed to him that the fact that l-adic dilogarithm satisfies 5-term Abel equation implies some form of Zagier conjecture.

1 Action of a finite Galois group

Let l be a fixed prime number. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z ∈ L. We denote by π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) the maximal pro-l quotient of the étale fundamental group of ℙ1 ; L̄ \ {0, 1, ∞} based

at 01^{→} and by π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) the π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→})-torsor of l-adic paths from 01^{→} to z. The Galois group G_{L} acts on π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) and on

π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}).

Definition 1.1. (see [11] Definitions 2.0 and 2.2, [12] Definitions 17.3 and 17.5). Let z ∈ L \ {0, 1}. (resp. Let x ∈ {0, 1} and let z = xy^{⟶} be a tangential point on ℙ_{L}^{1} \ {0, 1, ∞} defined over L.) We say that a triple (ℙ_{L}^{1} \ {0, 1, ∞}; z, 01^{⟶}) has good reduction at a prime ideal 𝔭 of O_{L} if v_{𝔭}(z) = 0 and v_{𝔭}(1 − z) = 0 (resp. v_{𝔭}(y − x) = 0), where v_{𝔭} : L^{∗} → ℤ is the valuation associated with the prime ideal 𝔭.

Let S be a finite set of prime ideals of O_{L} containing all prime ideals lying over l such that the triplet (ℙ_{L}^{1} \ {0, 1, ∞}; z, 01^{→}) has good reduction outside S.

We have the following important result. Proposition 1.2. (see [11] Proposition 2.3 or [12] Proposition 17.6). The action of G_{L} on the torsor of paths π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) is unramified outside S. Proof. One repeats arguments of the Ihara proof of Theorem 1 in [3], that the action of G_{ℚ} on π_{1}(ℙ1 ; ℚ̄ \ {0, 1, ∞}; 01^{→}) is unramified outside l (see [11] or [12] for more details). □

Let us assume also that S is G-invariant.

Let M(L)_{S} be a maximal, pro-l, unramified outside S extension of L(μ_{l^{∞}}). It follows from [3] Theorem 1 and from Proposition 1.2 that the action of

G_{L} on π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) and on π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) factors through

Gal(M(L)_{S}/L).

Throughout all this paper we shall assume that

a) l does not divide the order of G;

b) L ∩ K(μ_{l^{∞}}) = K.

It follows immediately from a) and b) that

1.3.  Gal(L(μ_{l^{∞}})/K(μ_{l^{∞}})) = Gal(L/K).

Let us consider the following three exact sequences and morphisms between them:

1 ⟶ Gal(M(L)_{S}/L(μ_{l^{∞}})) ⟶ Gal(M(L)_{S}/K(μ_{l^{∞}})) ⟶^{←^{s′}} Gal(L(μ_{l^{∞}})/K(μ_{l^{∞}})) ⟶ 1
↓  ↓  =↓

1 ⟶  Gal(M(L)_{S}/L)  ⟶  Gal(M(L)_{S}/K)  ⟶^{←^{s}}  Gal(L/K)  ⟶ 1
↑  ↑  =↑
1 ⟶  G_{L}  ⟶  G_{K}  ⟶  Gal(L/K)  ⟶ 1 .

The assumption a) that l does not divide the order of G implies that the upper exact sequence has a section

s′ : G → Gal(M(L)_{S}/K(μ_{l^{∞}})).

Therefore the midle exact sequence has also a section

s : G → Gal(M(L)_{S}/K),

which is the composition of the isomorphism Gal(L/K) ≈ Gal(L(μ_{l^{∞}})/K(μ_{l^{∞}})) and s′.

Let g̃ be a lifting of s(g) to G_{K}. Then g̃ induces an isomorphism

g̃_{∗} : π_{1}(ℙ1 ; K̄ \ {0, 1, ∞}; 01^{→}) → π_{1}(ℙ1 ; K̄ \ {0, 1, ∞}; 01^{→})

and a bijection

g̃_{⋄} : π(ℙ1 ; K̄ \ {0, 1, ∞}; z, 01^{→}) → π(ℙ1 ; K̄ \ {0, 1, ∞}; g(z), 01^{→}).

Let us fix a path γ from 01^{→} to z. Let us consider the following diagram

π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) ⟶^{g̃_{⋄}} π(ℙ1 ; L̄ \ {0, 1, ∞}; g(z), 01^{→})
t_{γ} ↓  t_{g̃_{⋄}(γ)}↓

π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⟶^{g̃_{∗}} π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→})
k↓  k_{g}↓

ℚ_{l}{{X, Y }}^{∗}  ⟶^{id}  ℚ_{l}{{X, Y }}^{∗}

Diagram 1 We recall that t_{γ}(q) = γ^{−1} ·q (see [9], section 1). We choose standard generators

of π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) - x := x_{0}, y := p^{−1} · y_{0} · p (see Picture 1).

Picture 1

Then k (resp. k_{g}) is a continous multiplicative homomorphism given by k(x) = e^{X} and k(y) = e^{Y} (resp. k_{g}(g̃_{∗}(x)) = e^{X} and k_{g}(g̃_{∗}(y)) = e^{Y}). Observe that with these choices of k and k_{g} all squares of Diagram 1 commute.

Lemma 1.4. Let x_{0} (resp. y_{0}) be a small loop around 0 (resp. 1) based at 01^{→}

(resp. 10^{→} ). Then we have

g̃_{∗}(x_{0}) = x_{0} and g̃_{∗}(y_{0}) = y_{0}.

Proof. The assumptions a) and b) imply that g̃ acts as the identity on K(μ_{l^{∞}}). The action of G_{K} on π_{1}(ℙ1 ; K̄ \ {0, 1}; v) factors through Gal(K(μ_{l^{∞}})/K). Hence g̃ acts as the identity on π_{1}(ℙ1 ; K̄ \ {0, 1}; v). □

For a number field K we denote by {K/l} the set of prime ideals of O_{K} lying over l.

Lemma 1.5. Let p be a path from 01^{→} to 10^{→}, an interval [0, 1] (see Picture 1). Then we have

g̃_{⋄}(p) = p.

Proof. We recall that the section

s : G → Gal(M(L)_{S}/K)

is the composition of the isomorphism Gal(L/K) ≈ Gal(L(μ_{l^{∞}})/K(μ_{l^{∞}})) and the section s′ : Gal(L(μ_{l^{∞}})/K(μ_{l^{∞}})) → Gal(M(L)_{S}/K(μ_{l^{∞}})). Therefore the restriction of s(g) to K(μ_{l^{∞}}) is the identity. Hence it follows that the image of s(g) in Gal(M(K)_{{K/l}}/K) by the epimorphism

Gal(M(L)_{S}/K) ⟶ Gal(M(K)_{{K/l}}/K)

is trivial because Gal (M(K)_{{K/l}}/K(μ_{l^{∞}})) is a pro-l group and l does not divide the order of G.

The action of G_{K} on π(ℙ1 ; K̄\{0, 1, ∞}; 10^{→}, 01^{→}) factors through Gal(M(K)_{{K/l}}/K) because the triple (ℙ_{K}^{1} \{0, 1, ∞}; 10^{→}, 01^{→}) has good reduction at every prime ideal of O_{K}. This implies that g̃ ∈ G_{K} acts as s(g) on π(ℙ1 ; K̄ \ {0, 1, ∞}; 10^{→}, 01^{→}) and therefore it acts as the identity. Hence we have g̃_{⋄}(p) = p. □ Corollary 1.6. The isomorphism

g̃_{∗} : π_{1}(ℙ1 ; K̄ \ {0, 1, ∞}; 01^{→}) → π_{1}(ℙ1 ; K̄ \ {0, 1, ∞}; 01^{→})

is the identity. Consequently the embedding k_{g} = k. □

2 l-adic polylogarithms

We start by recalling briefly the definition of l-adic polylogarithms. First however we introduce the following notation.

Let a, b ∈ ℚ_{l}{{X, Y }}. The Lie bracket is defined by [a, b] := a · b − b · a. We set [Y, X^{0}] := Y, [Y, X^{1}] := [Y, X] and [Y, X^{n}] := [[Y, X^{n−1}], X] for n ≥ 1. We use the similar notation for successive commutators of elements of a group.

We denote by

χ : G_{L} ⟶ ℤ_{l}^{∗}

the cyclotomic character.

We denote by π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ the rationalization of the pro-l, pronilpotent group π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) and by π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) ⊗ ℚ the π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ-torsor of paths π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) (see [10], 10.3). Elements of π(ℙ1 ; L̄ \ {0, 1, ∞}; z, 01^{→}) we shall call ℚ_{l}-paths on ℙ1 ; L̄ \ {0, 1, ∞} from

z to 01^{→}.

Definition 2.0. (see [10] Definition 11.0.1). Let z be an L-point of ℙ_{L}^{1}\{0, 1, ∞} or a tangential point defined over L. Let γ be a ℚ_{l}-path on ℙ1 ; L̄ \ {0, 1, ∞} from 01^{→} to z. Let σ ∈ G_{L}. Then we set

Λ_{γ}(σ) := k(γ^{−1} · σ(γ))

and we define l-adic polylogarithms by the congruence

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

where I_{2} is an ideal of ℚ_{l}{{X, Y }} generated topologically by monomials with two or more Y ’s.

The l-adic polylogarithms depend on a choice of a path γ. We have the following result which we shall need later. Lemma 2.1. Let S ∈ π_{1}(VL̄; 01^{→}) ⊗ ℚ.

i) Let S ≡ x^{a} · y^{b} mod Γ^{2}π_{1}(VL̄; 01^{→}) ⊗ ℚ. Then

l(z)_{γ·S} = l(z)_{γ} + a(χ − 1) and l_{1}(z)_{γ·S} = l_{1}(z)_{γ} + b(χ − 1).

ii) Let n > 1 and let S ≡ (y, x^{n−1})^{a} mod Γ^{n+1}π_{1}(VL̄; 01^{→}) ⊗ ℚ. Then

l_{k}(z)_{γ·S} = l_{k}(z)_{γ} for k < n and l_{n}(z)_{γ·S} = l_{n}(z)_{γ} + a(χ^{n} − 1).

Proof. The point i) of the lemma is already prooved in [10] (see [10] Lemma 11.0.10), hence we shall prove only the second part of the lemma.

Let σ ∈ G_{L}. Then

Λ_{γ·S}(σ) = k((γ · S)^{−1} · σ(γ · S)) = k(S)^{−1} · Λ_{γ}(σ) · k(σ(S)).

Let I := ker (ℚ_{l}{{X, Y }} → ℚ_{l}) be the augmentation ideal. One shows easily by recurrence on n that k((y, x^{n−1})) ≡ 1 + [Y, X^{n−1}] mod I^{n+1} and k(σ(y, x^{n−1})) ≡ 1 + χ(σ)^{n}[Y, X^{n−1}] mod I^{n+1}. Hence we get that

Λ_{γ·S}(σ) ≡ (1 − a[Y, X^{n−1}]) · Λ_{γ}(σ) · (1 + aχ(σ)^{n}[Y, X^{n−1}]) mod I^{n+1}.

After taking logarithm we get

log Λ_{γ·S}(σ) ≡ log Λ_{γ}(σ) + a(χ(σ)^{n} − 1)[Y, X^{n−1}] mod I^{n+1}.

Hence it follows that l_{k}(z)_{γ·S} = l_{k}(z)_{γ} for k < n and l_{n}(z)_{γ·S} = l_{n}(z)_{γ} + a(χ^{n} − 1). □

In [1], assuming the existence of the category of mixed Tate motives over Spec L, one gives a condition when a linear combination of n-th polylogarithms evaluated at elements of L is a regulator of an extension of ℚ(0) by ℚ(n) over Spec L. In [9] we are presenting an l-adic version of the problem. As in a case of classical complex polylogarithms, one wants to know, when a linear combination of l-adic polylogarithms is an l-adic regulator (we should rather say “an l-adic realisation”) of an extension of ℚ(0) by ℚ(n) over Spec L. The l-adic realization induces a map

real_{l} : Ext_{MM_{L}}^{1}(ℚ(0), ℚ(n)) → H^{1}(G_{L}; ℚ_{l}(n)).

Hence we can ask a weaker question, when is a linear combination of l-adic polylogarithms evaluated at elements of L a cocycle? We recall here a result from [10] .

Proposition 2.2.(see [10] Theorem 11.0.11). Let n ≥ 2. Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ_{l}. Let Z be a subgroup of L^{∗} generated by z_{1}, …, z_{N} and 1−z_{1}, …, 1−z_{N}. Let us assume that there are ℚ_{l}-paths γ_{i} from 01^{→} to z_{i} (i = 1, …, N) such that i) the map

z_{i} → l(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

and

1 − z_{i} → l_{1}(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

is well defined on the set {z_{1}, … z_{N}, 1 − z_{1}, …, 1 − z_{N}} and it defines a homomorphism from Z to Z^{1}(G_{L}; ℚ_{l}(1));

ii) if n > 2 then ∑_{i=1}^{N} m_{i}ν_{1}(z_{i})·. . .·ν_{n−2}(z_{i})(z_{i})∧(1−z_{i}) = 0 in L^{∗} ∧L^{∗} ⊗ℚ_{l} for any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n − 2) and if n = 2 then ∑_{i=1}^{N} m_{i}(z_{i}) ∧ (1 − z_{i}) = 0 in L^{∗} ∧ L^{∗} ⊗ ℚ_{l}

iii) if n > 2 then ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}} = 0 for 2 ≤ k ≤ n − 1 and any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n − 2).

Then the map ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}} from G_{L} to ℚ_{l}(n) is a cocycle. □

We explain how this result is related to motivic considerations in [1] and also in [9]. In fact we shall repeat in l-adic situation considerations from [1].

Let L be a number field. In [1], and also in [9] in l-adic case one defines in an inductive way some groups L_{k} (k = 1, 2, …), operators d_{k} : L_{k} → ⋀^{2}(⊕_{i=1}^{k−1}L_{i}) (k = 1, 2, …) and homomorphisms φ_{k} : ker d_{k} ⟶ Ext_{MM_{L}}^{1}(ℚ(0), ℚ(n)) with the following conjectural properties.

Let ∑_{i=1}^{N} m_{i}{z_{i}}_{n} ∈ L_{n} be such that d_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n}) = 0. Then the cohomology class real_{l}(φ_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n})) in H^{1}(G_{L}; ℚ_{l}(n)) restricted to the sub-

group ⋂_{i=1}^{N} H_{n}(ℙ_{L}^{1}\{0, 1, ∞}; z_{i}, 01^{→}) of G_{L} is equal to the function ∑_{i=1}^{N} m_{i}l_{n}(z_{i})

restricted to the same subgroup of G_{L}. ( The l-adic polylogarithms restricted to this subgroup of G_{L} do not depend on choices of paths, so we drop out subscripts indicating paths along which one calculates them. Subgroups H_{n}(ℙ_{L}^{1} \

{0, 1, ∞}; z_{i}, 01^{→}) of G_{L} are defined in [9], section 3.)

Assuming motivic formalism as in [1] we shall show the following result (compare with Proposition 4.6 in [1]).

Theorem 2.3. Let ∑_{i=1}^{N} m_{i}{z_{i}}_{n} ∈ L_{n} be such that d_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n}) = 0.

Then there are ℚ_{l}-paths δ_{i} from 01^{→} to z_{i} such that ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{δ_{i}} is a cocycle representing the cohomology class real_{l}(φ_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n})).

The next two lemmas are technical. We need these lemmas in order to show Theorem 2.3. In the first lemma we show that we can always choose ℚ_{l}-paths in such a way that the assumption i) of Proposition 2.2 is satisfied. The second lemma shows that in the motivic situation described above the assumption iii) of Proposition 2.2 will be satisfied.

Lemma 2.4. Let z_{1}, …, z_{N} ∈ L^{∗} \{1} and let Z be a subgroup of L^{∗} generated

by z_{1}, …, z_{N} and 1 − z_{1}, …, 1 − z_{N}. Then there are ℚ_{l}-paths γ_{i} from 01^{→} to z_{i} (i = 1, …, N) such that the map

z_{i} → l(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

and

1 − z_{i} → l_{1}(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

is well defined on the set {z_{1}, … z_{N}, 1 − z_{1}, …, 1 − z_{N}} and it defines a homomorphism from Z to Z^{1}(G_{L}; ℚ_{l}(1)).

Proof. The subgroup Z of L^{∗} is a finitely generated abelian group, hence

Z = Z_{0} ⊕ torsion(Z),

where Z_{0} is a finitely generated free abelian group and torsion(Z) ⊂ μ(L).

Let x_{1}, …, x_{r} ∈ L^{∗} be free generators of Z_{0}. The group torsion(Z) is cyclic. Let ξ be a generator of torsion(Z). For ξ we choose a compatible family of l^{n}-th roots (ξ^{\frac{1}{l^{n}}})_{n∈ℕ} such that the corresponding Kummer character is zero. We also choose a compatible family of l^{n}-th roots (xi^{\frac{1}{l^{n}}})_{n∈ℕ} for any x_{i} (i = 1, …, r). If z_{j} (resp. 1 − z_{j}) is equal ξ^{k} · ∏_{i=1}^{r} x_{i}^{k_{i}} then we choose compatible family of l^{n}-th

roots of z_{j} (resp. 1 − z_{j}) by taking ((ξ^{\frac{1}{l^{n}}})^{k} · ∏_{i=1}^{r}(xi^{\frac{1}{l^{n}}})^{k_{i}})_{n∈ℕ}.

The choice of a compatible family of l^{n}-th roots of z_{j} and 1 − z_{j} determines

a ℚ_{l}-path γ_{z_{j}} from 01^{→} to z_{j} modulo Γ^{2}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ). The l-adic logarithm l(z_{j})_{γ_{j}} (resp. l_{1}(z_{j})_{γ_{j}}) is the Kummer character corresponding to given above compatible family of l^{n}-roots of z_{j} (resp. 1 − z_{j}). □

Lemma 2.5. Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ_{l}. Let n > r be two natural numbers greater than 2. Let γ_{i} be ℚ_{l}-paths on ℙ1 ; L̄ \ {0, 1, ∞} from

01^{→} to z_{i} for i = 1, …, N such that

i) Lemma 2.4 holds for paths γ_{i};

ii) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}} = 0 for 2 ≤ k ≤ r − 1 and for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − 2);

iii) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−r}(z_{i})l_{r}(z_{i})_{γ_{i}} ∈ B^{1}(G_{L}; ℚ_{l}(r)) for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − r).

Then there are ℚ_{l}-paths γ̄_{i} from 01^{→} to z_{i} (i = 1, …, N) such that

1) γ_{i} ≡ γ̄_{i} mod Γ^{r}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ) for i = 1, …, N;

2) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})γ̄_{i} = 0 for 2 ≤ k ≤ r and for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − 2).

Proof. In a ℚ_{l}-vector space Maps(G_{L}; ℚ_{l}) we consider a subspace A_{r} generated by l_{r}(z_{i})_{γ_{i}} (i = 1, …, N) and by χ^{r} −1. Let {x_{1}, …, x_{p}} ⊂ {z_{1}, …, z_{N}} be such that l_{r}(x_{j})_{γ_{j}} (j = 1, …, p) and χ^{r} − 1 is a base of A_{r}. Then for any 1 ≤ i ≤ N we have

2.5.1.  l_{r}(z_{i})_{γ_{i}} = ∑_{j=1}^{p} a_{j}^{i}l_{r}(x_{j})_{γ_{j}} + a_{i}(χ^{r} − 1)

for some a_{j}^{i} (j = 1, …, p) and a_{i} in ℚ_{l}.

Let γ̄_{i} := γ_{i} · (y, x^{r−1})^{−a_{i}}. Then it is clear that

γ̄_{i} ≡ γ_{i} mod Γ^{r}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ).

It follows from Lemma 2.1 that

l_{k}(z_{i})_{γ_{i}} = l_{k}(z_{i})γ̄_{i}

for k < r and i = 1, …, N. This implies that 2) holds for 2 ≤ k < r.

Let us observe that

l_{r}(z_{i})γ̄_{i} = l_{r}(z_{i})_{γ_{i}} − a_{i}(χ^{r} − 1)

by Lemma 2.1. Hence it follows from 2.5.1 that the point 2) also holds for k = r. □

Proof of Theorem 2.3. Let ∑_{i=1}^{N} m_{i}{z_{i}}_{n} ∈ L_{n} be such that d_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n}) = 0. By the very definition of d_{n} we get

∗_{n−1}  d_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n}) = ∑_{i=1}^{N} m_{i}(z_{i}) ⊗ {z_{i}}_{n−1} = 0.

in L_{1} ⊗ L_{n−1}.

Let 1 ≤ k < n − 2. Let us suppose that

∗_{n−k}  ∑_{i=1}^{N} m_{i}(z_{i})^{⊗k} ⊗ {z_{i}}_{n−k} = 0

in (⨂_{i=1}^{k} L_{1}) ⊗ L_{n−k}. Let d be the prolongation of d_{k} on tensor products. Then

0 = d(∑_{i=1}^{N} m_{i}(z_{i})^{⊗k} ⊗ {z_{i}}_{n−k}) = ∑_{i=1}^{N} m_{i}(z_{i})^{⊗(k+1)} ⊗ {z_{i}}_{n−k−1}

in (⨂_{i=1}^{k+1} L_{1}) ⊗ L_{n−k−1}. Hence we have got the equality ∗_{n−k−1}. Therefore the equality ∗_{n−k} holds for 1 ≤ k ≤ n − 2.

Let k = n − 2. Then we get

0 = d(∑_{i=1}^{N} m_{i}(z_{i})^{⊗(n−2)} ⊗ {z_{i}}_{2}) = ∑_{i=1}^{N} m_{i}(z_{i})^{⊗(n−2)} ⊗ ((z_{i}) ∧ (1 − z_{i})) = 0

in (⨂_{i=1}^{n−2} L_{1}) ⊗ (L_{1} ∧ L_{1}). Hence we have

∗_{n−1}  ∑_{i=1}^{N} m_{i}(z_{i})^{⊗(n−2)} ⊗ ((z_{i}) ∧ (1 − z_{i})) = 0.

The equalities ∗_{n−k} (1 ≤ k ≤ n − 2) and ∗_{n−1} are equivalent to the equalities

∗∗_{n−k}  ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{k}(z_{i}){z_{i}}_{n−k} = 0

for 1 ≤ k ≤ n − 2 and

∗∗_{n−1}  ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})(z_{i}) ∧ (1 − z_{i}) = 0

for any homomorphisms ν_{1}, …, ν_{n−2} from Z to ℚ_{l}.

We chose ℚ_{l}-paths γ_{i} (i = 1, …, N) from 01^{→} to z_{i} such as in Lemma 2.4. It follows from Proposition 22 ii) and the equality ∗∗_{n−1} that

∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})l_{2}(z_{i})_{γ_{i}}

is a cocycle. In L_{2} we have the equality ∗∗_{n−2}, i.e., the equality

∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i}){z_{i}}_{2} = 0.

Applying real_{l} to the element φ_{2}(∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i}){z_{i}}_{2}) we get 0 in H^{1}(G_{L}; ℚ_{l}(2)). Hence the cocycle ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})l_{2}(z_{i})_{γ_{i}} is a coboundary.

Observe that the conditions i), ii)(trivially) and iii) of Lemma 2.5 are satisfied. Hence there are ℚ_{l}-paths γ_{i}^{(2)} such that

1) γ_{i} ≡ γ_{i}^{(2)} mod Γ^{2}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ) for i = 1, 2, …, N;

2) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i})·. . .·ν_{n−2}(z_{i})l_{2}(z_{i})_{γ_{i}^{(2)}} = 0 for any homomorphisms ν_{1}, …, ν_{n−2} from Z to ℚ_{l}.

Let r ≤ n − 1 be such that

∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}^{(r−1)}} = 0

for 2 ≤ k < r ≤ n − 1. Then Proposition 2.2 implies that

∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−r}(z_{i})l_{r}(z_{i})_{γ_{i}^{(r−1)}}

is a cocycle. Applying real_{l} to the element φ_{r}(∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · ν_{2}(z_{i}) · . . . · ν_{n−r}(z_{i}){z_{i}}_{r}) we get that the cocycle ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−r}(z_{i})l_{r}(z_{i})_{γ_{i}^{(r−1)}} is a coboundary because of the equality ∗ ∗_{n−r} . Lemma 2.5 implies that there are ℚ_{l}-paths γ_{i}^{(r)} (i = 1, …, N) such that

1)_{r} γ_{i}^{(r)} ≡ γ_{i}^{(r−1)} mod Γ^{r}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ) for i = 1, 2, …, N;

2)_{r,k} ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}^{(r)}} = 0 for 2 ≤ k ≤ r for any homomorphisms ν_{1}, …, ν_{n−2} from Z to ℚ_{l}.

Hence after a finite number of steps we get equations 2)_{n−1,k} for all 2 ≤ k ≤ n − 1. Then Proposition 2.2 implies that

∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}^{(n−1)}}

is a cocycle. We recall that the map

H^{1}(G_{L}; ℚ_{l}(n)) → Hom(⋂_{i=1}^{N} H_{n}(ℙ_{L}^{1} \ {0, 1, ∞}; z_{i}, 01^{→}); ℚ_{l}(n))

is injective. Hence it follows from the motivic formalism developped in [9], section 7 that the cohomology class real_{l}(φ_{n}(∑_{i=1}^{N} m_{i}{z_{i}}_{n})) is represented by the cocycle ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}^{(n−1)}}. □

3 Action of G and l-adic polylogarithms

Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). In this section we shall study a G-equivariant version of results from section 2. We start with a relation between l-adic polylogarithms evaluated at z and g(z). We denote by g̃ a lifting of g ∈ G to G_{K}.

Proposition 3.1. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z be an L-point of ℙ_{L}^{1} \ {0, 1, ∞} and let γ be a path from

01^{→} to z. Then we have

l_{n}(z)_{γ}(g̃^{−1} · τ · g̃) = l_{n}(g(z))_{g̃(γ)}(τ)

for any τ ∈ G_{L} and any g ∈ G.

Proof. Let σ ∈ G_{L} and let g ∈ G. Then it follows from Corollary 1.6 that Λ_{γ}(σ) = k(γ^{−1} · σ(γ)) = k(g̃(γ^{−1} · σ(γ))) = k(g̃(γ)^{−1} · g̃(σ(γ))) = k(g̃(γ)^{−1} · (g̃ · σ · g̃^{−1})(g̃(γ))) = Λ_{g̃(γ)}(g̃ · σ · g̃^{−1}). For σ = g̃^{−1} · τ · g̃ we get Λ_{γ}(g̃^{−1} · τ · g̃) = Λ_{g̃(γ)}(τ). The proposition follows immediately from the definition of l-adic polylogarithms. □

Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ_{l}. We shall assume that the set {z_{1}, …, z_{N}} and the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] are G-invariant. We shall show that the family of paths {γ_{i} | 1 ≤ i ≤ N} from Lemmas 2.4 and 2.5 and from Proposition 2.2 can be chosen G-invariant.

Let S be a finite set of prime ideals of O_{L} containing all prime ideals lying over l and such that for any 1 ≤ i ≤ N a triplet (ℙ_{L}^{1} \ {0, 1, ∞}; z_{i}, 01^{→}) has good reduction for any prime ideal of O_{L} not belonging to S. We assume also that S is G-invariant.

Observe that then the action of G_{L} on a disjoint union of torsors of paths

∐_{i=1}^{N} π(ℙ1 ; L̄ \ {0, 1, ∞}; z_{i}, 01^{→}) factors through Gal(M(L)_{S}/L).

Lemma 3.2. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). We assume that

a) l does not divide the order of G,

b) K(μ_{l^{∞}}) ∩ L = K.

Let z_{1}, …, z_{N} ∈ L^{∗} \ {1}. We assume that the set {z_{1}, …, z_{N}} is G-invariant.

Then there are paths γ_{z_{i}} from 01^{→} to z_{i} for i = 1, …, N and liftings g̃ of g ∈ G to G_{K} such that

g̃(γ_{z_{i}}) = γ_{g(z_{i})}

for 1 ≤ i ≤ N and for any g ∈ G.

Proof. The action of G_{K} on the disjoint union of torsors of paths ∐_{i=1}^{N} π(ℙ1 ; L̄ \

{0, 1, ∞}; z_{i}, 01^{→}) factors through Gal(M(L)_{S}/K). The assumptions a) and b) imply that the exact sequence

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

has a section s : G → Gal(M(L)_{S}/K). Let g̃ be a lifting of s(g) to G_{K}. Then

in Aut(∐_{i=1}^{N} π(ℙ1 ; L̄ \ {0, 1, ∞}; z_{i}, 01^{→})) we have g̃_{1} · g̃_{2} = g̃_{1}g_{2} for any g_{1}, g_{2} ∈ G.

Let z ∈ L and let K(Gz) be a subfield of L generated by elements g(z) for g ∈ G. It is clear that K(Gz) is a Galois extension of K.

Let us fix a path γ_{z_{1}} from 01^{→} to z_{1}. If K(Gz_{1}) = L then we take γ_{g(z_{1})} := g̃(γ_{z_{1}}) for g ∈ G. So let us assume that K(Gz_{1}) = L′ ≠ L. Observe that

i) in the commutative diagram

Gal(M(L)_{S}/K) ⟶^{←^{s}} G = Gal(L/K)
π↓  pr↓
Gal(M(L′)_{S}/K) ⟶ G′ := Gal(L′/K)

we have ker (π ∘ s) = ker pr;

ii) the action of G_{K} on ∐_{g∈G} π(ℙ1 ; L̄ \ {0, 1, ∞}; g(z_{1}), 01^{→}) factors through Gal(M(L′)_{S}/K).

Hence we set γ_{g(z_{1})} := π(s(g))(γ_{z_{1}}) for g ∈ G.

Now we take an element with the smallest index in

{z_{1}, …, z_{N}} \ {g(z_{1}) | g ∈ G}

and we repeat the construction. It is clear that the constructed family of paths is G-invariant. □

From now on we denote γ_{z_{i}} by γ_{i} and γ_{g(z_{i})} by γ_{g(i)}.

Lemma 3.3. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). We assume that

a) l does not divide the order of G;

b) K(μ_{l^{∞}}) ∩ L = K.

Let z_{1}, …, z_{N} ∈ L^{∗}\{1}. Let us assume that the set {z_{1}, …, z_{N}} is G-invariant. Let Z be a subgroup of L^{∗} generated by z_{1}, …, z_{N} and 1−z_{1}, …, 1−z_{N}. Then there are ℚ_{l}-paths γ_{z_{i}} from 01^{→} to z_{i} such that

i) the map

z_{i} → l(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1)),
1 − z_{i} → l_{1}(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

is well defined on the set {z_{1}, …, z_{N} , 1 − z_{1}, …, 1 − z_{N}} and it defines a homomorphism from Z to Z^{1}(G_{L}; ℚ_{l}(1));

ii) for any 1 ≤ i ≤ N and any g ∈ G we have

g̃(γ_{z_{i}}) = γ_{g(z_{i})}.

Proof. It follows from Lemma 3.2 that there are paths γ_{i} from 01^{→} to z_{i} such that

g̃(γ_{i}) = γ_{g(i)}

for 1 ≤ i ≤ N and g ∈ G. We shall modify these paths in such a way that the condition i) is satisfied.

Let V := ⟨l(z_{i})_{γ_{i}}, l_{1}(z_{i})_{γ_{i}}, χ − 1 | 1 ≤ i ≤ N ⟩ be a ℚ_{l}-vector subspace of Maps(G_{L}, ℚ_{l}) generated by l(z_{i})_{γ_{i}}, l_{1}(z_{i})_{γ_{i}} (1 ≤ i ≤ N) and by χ − 1. It follows from Proposition 3.1 that V is also a ℚ_{l}[G]-module. Let I(G) be the augmentation ideal of ℚ_{l}[G]. Then

V = I(G)V ⊕ V^{G}

is a decomposition into a direct sum of two ℚ_{l}[G]-modules. Let v_{1}, …, v_{s}, χ − 1 be a base of V^{G}. Then for any 1 ≤ i ≤ N we have

l(z_{i})_{γ_{i}} = w_{i}^{0} + u_{i}^{0} + a_{i}(χ − 1)

and

l_{1}(z_{i})_{γ_{i}} = w_{i}^{1} + u_{i}^{1} + b_{i}(χ − 1),

where w_{i}^{0}, w_{i}^{1} ∈ I(G)V and u_{i}^{0}, u_{i}^{1} belong to a subspace generated by v_{1}, …, v_{s}. Observe that

3.3.1.  a_{g(i)} = a_{i} and b_{g(i)} = b_{i}

for any 1 ≤ i ≤ N and g ∈ G.

If ∏_{i=1}^{N} z_{i}^{p_{i}}(1 − z_{i})^{q_{i}} = 1 is a multiplicative relation between z_{1}, …, z_{N}, 1 − z_{1}, …, 1 − z_{N} then

∑_{i=1}^{N}(p_{i}l(z_{i})_{γ_{i}} + q_{i}l_{1}(z_{i})_{γ_{i}}) = α(χ − 1)

for some α ∈ ℤ_{l}. Hence we get ∑_{i=1}^{N}(p_{i}a_{i} + q_{i}b_{i}) = α. Let us set γ̄_{i} := γ_{i} · y^{−b_{i}} · x^{−a_{i}} for 1 ≤ i ≤ N. It follows from Lemma 2.1 that l(z_{i})γ̄_{i} = l(z_{i})_{γ_{i}} − a_{i}(χ − 1) and l_{1}(z_{i})γ̄_{i} = l_{1}(z_{i})_{γ_{i}}−b_{i}(χ−1). Hence we get that ∑_{i=1}^{N}(p_{i}l(z_{i})γ̄_{i}+q_{i}l_{1}(z_{i})γ̄_{i}) = 0 for any multiplicative relation ∏_{i=1}^{N} z_{i}^{p_{i}} ·(1−z_{i})^{q_{i}} = 1. It follows from Corollary 1.6 and from 3.3.1 that g̃(γ̄_{i}) = γ̄_{g(i)} for 1 ≤ i ≤ N. □

Lemma 3.4. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ. Let us assume that the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] is G-invariant. Let n > r be two natural

numbers. Let γ_{i} be ℚ_{l}-paths on ℙ1 ; L̄ \ {0, 1, ∞} from 01^{→} to z_{i} for i = 1, …, N such that

i) the family of paths {γ_{i} | 1 ≤ i ≤ N} is G-invariant;

ii) Lemma 2.4 holds for the family of paths {γ_{i}; | 1 ≤ i ≤ N};

iii) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}} = 0 for 2 ≤ k ≤ r − 1 and for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − 2);

iv) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−r}(z_{i})l_{r}(z_{i})_{γ_{i}} ∈ B^{1}(G_{L}; ℚ_{l}(r)) for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − r).

Then there are ℚ_{l}-paths γ̄_{i} from 01^{→} to z_{i} (i = 1, …, N) such that

1) γ_{i} ≡ γ̄_{i} mod Γ^{r}(π_{1}(ℙ1 ; L̄ \ {0, 1, ∞}; 01^{→}) ⊗ ℚ) for i = 1, …, N;

2) ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})γ̄_{i} = 0 for 2 ≤ k ≤ r and for any homomorphisms ν_{s} : Z → ℚ_{l} (s = 1, …, n − 2);

3) the family of paths {γ̄_{i} | 1 ≤ i ≤ N} is G-invariant.

Proof. Let A be a ℚ_{l}-vector subspace of Maps(G_{L}; ℚ_{l}) generated by l_{r}(z_{i})_{γ_{i}} for 1 ≤ i ≤ N and by χ^{r} − 1. It follows from Proposition 3.1 that A_{r} is G-invariant. Let us decompose A_{r} into a direct sum

A_{r} = I(G)A_{r} ⊕ (V ⊕ ⟨χ^{r} − 1⟩),

where A_{r}^{G} = V ⊕ ⟨χ^{r} − 1⟩. Hence we can write

l_{r}(z_{i})_{γ_{i}} = I_{z_{i}} + v_{i} + α_{i}(χ^{r} − 1),

where I_{i} ∈ I(G)A_{r} and v_{i} ∈ V. It follows from Proposition 3.1 that

3.4.1.  l_{r}(g(z_{i}))_{g̃(γ_{i})} = I_{g(z_{i})} + v_{i} + α_{i}(χ^{r} − 1)

for g ∈ G.

Let us set

γ̄_{i} := γ_{i} · (y, x^{r−1})^{−α_{i}}

for 1 ≤ i ≤ N. It follows from Corollary 1.6 that the family of paths {γ̄_{i} | 1 ≤ i ≤ N} is G-invariant. It follows from Lemma 2.1 that

l_{r}(z_{i})γ̄_{i} = I_{z_{i}} + v_{i} and l_{r}(g(z_{i}))_{g̃(γ̄_{i})} = I_{g(z_{i})} + v_{i}.

Hence it follows immediately from 3.4.1 that conditions 1 and 2 of the lemma are satisfied. □

Now we shall formulate our main result.

Theorem 3.5. Let n ≥ 2. Let z_{1}, …, z_{N} ∈ L^{∗} \ {1} and let m_{1}, …, m_{N} ∈ ℚ_{l}. Let Z be a subgroup of L^{∗} generated by z_{1}, …, z_{N} and 1 − z_{1}, …, 1 − z_{N}. Let us assume that there are ℚ_{l}-paths γ_{i} from 01^{→} to z_{i} (i = 1, …, N) such that

i) the map

z_{i} → l(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

and

1 − z_{i} → l_{1}(z_{i})_{γ_{i}} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

is well defined on the set {z_{1}, … z_{N}, 1 − z_{1}, …, 1 − z_{N}} and it defines a homomorphism from Z to Z^{1}(G_{L}; ℚ_{l}(1));

ii) if n > 2 then ∑_{i=1}^{N} m_{i}ν_{1}(z_{i})·. . .·ν_{n−2}(z_{i})(z_{i})∧(1−z_{i}) = 0 in L^{∗} ∧L^{∗} ⊗ℚ_{l} for any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n − 2) and if n = 2 then ∑_{i=1}^{N} m_{i}(z_{i}) ∧ (1 − z_{i}) = 0 in (L^{∗} ∧ L^{∗}) ⊗ ℚ_{l};

iii) if n > 2 then ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})_{γ_{i}} = 0 for 2 ≤ k ≤ n − 1 and any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n − 2).

Let us assume further that the set {z_{1}, z_{2}, …, z_{N}} and the formal sum ∑_{i=1}^{N} m_{i}[z_{i}]

are G-invariant. Then there are ℚ_{l}-paths γ̄_{i} from 01^{→} to z_{i} (i = 1, …, N) such that 1) the family of paths { γ̄_{i} | 1 ≤ i ≤ N } is G-invariant;

2) the map

z_{i} → l(z_{i})γ̄_{i} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

and

1 − z_{i} → l_{1}(z_{i})γ̄_{i} ∈ Z^{1}(G_{L}; ℚ_{l}(1))

is well defined on the set {z_{1}, … z_{N}, 1 − z_{1}, …, 1 − z_{N}} and it defines a homomorphism from Z to Z^{1}(G_{L}; ℚ_{l}(1));

3) if n > 2 then ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−k}(z_{i})l_{k}(z_{i})γ̄_{i} = 0 for 2 ≤ k ≤ n − 1 and any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n − 2).

For the family of paths { γ̄_{i} | 1 ≤ i ≤ N }, c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})γ̄_{i} is a cocycle on G_{L} satisfying c_{n}(g̃^{−1}·τ ·g̃) = c_{n}(τ) for any τ ∈ G_{L} and any g ∈ G. Consequently there exists a cocycle s_{n} : G_{K} → ℚ_{l}(n) such that

s_{n} ∘ ι = c_{n},

in H^{1}(G_{L}; ℚ_{l}(n)), where ι : G_{L} → G_{K} is the inclusion.

Proof. It follows from Lemmas 3.2 and 3.3 that there exists a family of paths { δ_{i} | 1 ≤ i ≤ N } satisfying conditions 1) and 2).

The assumption ii) and Proposition 2.2 imply that ∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})l_{2}(z_{i})_{δ_{i}} is a cocycle for any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n− 2). The assumption iii) implies that this cocycle is a coboundary. It follows from Lemma 3.4 that there is a family of paths {δ_{i}^{(2)} | 1 ≤ i ≤ N} satisfying conditions 1) and 2) and such that

∑_{i=1}^{N} m_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})l_{2}(z_{i})_{δ_{i}^{(2)}} = 0

for any homomorphisms ν_{j} : Z → ℚ_{l} (j = 1, …, n−2). Repeating this reasoning for 3, 4, …, n−1 we finally get a family of paths {δ_{i}^{(n−1)} | 1 ≤ i ≤ N} satisfying conditions 1), 2) and 3).

We set γ̄_{i} := δ_{i}^{(n−1)} for 1 ≤ i ≤ N. It follows from Proposition 2.2 that

c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})γ̄_{i}

is a cocycle on G_{L}. It follows from Proposition 3.1 that for any τ ∈ G_{L} and any g ∈ G we have c_{n}(g̃^{−1}·τ·g̃) = ∑_{i=1}^{N} m_{i}l_{n}(z_{i})γ̄_{i}(g̃^{−1}·τ·g̃) = ∑_{i=1}^{N} m_{i}l_{n}(g(z_{i}))_{g̃(γ̄_{i})}(τ) = ∑_{i=1}^{N} m_{i}l_{n}(z_{i})γ̄_{i}(τ) = c_{n}(τ), i.e.,

c_{n}(g̃^{−1} · τ · g̃) = c_{n}(τ),

because the family of paths {γ̄_{i} | 1 ≤ i ≤ N} is G-invariant.

Therefore the cohomology class [c_{n}] belongs to H^{1}(G_{L}; ℚ_{l}(n))^{G}. The inclusion of Galois groups ι : G_{L} → G_{K} induces a homomorphism of Galois cohomology ι^{∗} : H^{1}(G_{K}; ℚ_{l}(n)) → H^{1}(G_{L}; ℚ_{l}(n)). It follows from the Lyndon spectral sequence of the exact sequence of Galois groups

1 → G_{L} → G_{K} → G → 1

that

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

Hence there exists a cocycle s_{n} ∈ Z^{1}(G_{K}; ℚ_{l}(n)) such that

s_{n} ∘ ι = c_{n}

in H^{1}(G_{L}; ℚ_{l}(n)). □

4 Zagier conjecture for l-adic dilogarithm

In this section we shall prove a part of Zagier conjecture for l-adic dilogarithm. We shall use F instead of L to denote a number field.

Definition 4.0. Let n be a natural number greater than 1. Let F be a number field. We set

P_{n}(F) := {c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}} | z_{1}, …, z_{N} ∈ F^{∗}, m_{1}, …, m_{N} ∈ ℚ, c_{n} ∈ Z^{1}(G_{F}; ℚ_{l}(n))}.

We define [P_{n}(F)] to be the image of P_{n}(F) in H^{1}(G_{F}; ℚ_{l}(n)).

It is clear from the definition that [P_{n}(F)] is a ℚ-vector subspace of H^{1}(G_{F}; ℚ_{l}(n)). Observe that if H^{1}(G_{F}; ℚ_{l}(n)) is non null then it is an infinite dimensional vector space over ℚ, because already ℚ_{l} is infinite dimensional over ℚ. Hence [P_{n}(F)] considered as a vector space over ℚ can have a priori an infinite dimension over ℚ. Below we state a conjecture that [P_{n}(F)] is a finite dimensional vector space over ℚ. We view this conjecture as an analogue of Zagier conjecture for polylogarithms (see [14] and [1]).

We denote by r_{1}(F) (resp. r_{2}(F)) the number of real (resp. complex) places of F.

Conjecture 4.1. Let F be a number field. Then

dim_{ℚ}[P_{n}(F)] ≤ r_{2}(F) if n is even

and

dim_{ℚ}[P_{n}(F)] ≤ r_{1}(F) + r_{2}(F) if n is odd and n > 1.

Definition 4.2. We define P_{n}(F) to be the set of c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}}, where z_{1}, …, z_{N} ∈ F^{∗}, m_{1}, …, m_{N} ∈ ℚ and c_{n} is a cocycle obtained by the way of Proposition 2.2. We define [P_{n}(F)] to be the image of P_{n}(F) in H^{1}(G_{F}; ℚ_{l}(n)).

Observe that for n = 2 we have

P_{2}(F) := {∑_{i=1}^{N} m_{i}l_{2}(z_{i})_{γ_{i}} | z_{1}, …, z_{N} ∈ F^{∗}, m_{1}, …, m_{N} ∈ ℚ, ∑_{i=1}^{N} m_{i}(z_{i})∧(1−z_{i}) = 0

in F^{∗}∧F^{∗}⊗ℚ and the paths γ_{i} (i = 1, …, N) are such that l(z_{i})_{γ_{i}} and l_{1}(z_{i})_{γ_{i}}

respect all multiplicative relations between z_{1}, …, z_{N}, 1 − z_{1}, …, 1 − z_{N}.}

Observe that Conjecture 4.1 implies the following conjecture.

Conjecture 4.3. Let F be a number field. Then

dim_{ℚ}[P_{n}(F)] ≤ r_{2}(F) if n is even

and

dim_{ℚ}[P_{n}(F)] ≤ r_{1}(F) + r_{2}(F) if n is odd and n > 1.

We recall here the definition of the Bloch group (see [6]). Let D(F) be a ℚ-vector space with basis [x] (x ∈ F^{∗} \ {1}.) Let

λ : D(F) → (F^{∗} ∧ F^{∗}) ⊗ ℚ

be the homomorphism [x] → (x∧(1−x))⊗1. Let R(F) be the ℚ-vector subspace of D(F) generated by elements of the form

[x] − [y] + [\frac{y}{x}] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [\frac{1 − x}{1 − y}]

(x ≠ y belong to F^{∗} \ {1}). We define the Bloch group by setting

B(F) := ker λ/R(F).

Remark 4.3.1. The group B(F) defined here is equal to the corresponding group in [6] tensored by ℚ.

We define a tower of fields

F ⊂ F_{1} := F (μ_{l^{∞}}) ⊂ F_{2} := F_{1}(x^{\frac{1}{l^{∞}}} | x ∈ F)

and a corresponding tower of Galois groups

G_{F_{2}} ⊂ G_{F_{1}} ⊂ G_{F} .

Let us set

Γ := Gal (F (μ_{l^{∞}})/F).

Proposition 4.4. The restriction homomorphism

H^{1}(G_{F}; ℚ_{l}(2)) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2))

is injective.

Proof. We have an isomorphism

H^{1}(G_{F}; ℚ_{l}(2)) → Hom_{Γ}(G_{F (μ_{l^{∞}})}; ℚ_{l}(2))

induced by the inclusion G_{F (μ_{l^{∞}})} ⊂ G_{F} . Hence it is enough to show that the restriction map

Hom_{Γ}(G_{F (μ_{l^{∞}})}; ℚ_{l}(2)) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2))

is injective.

Let c_{2} : G_{F (μ_{l^{∞}})} → ℚ_{l}(2) be a Γ-homomorphism such that the composition

G_{F_{2}} → G_{F (μ_{l^{∞}})} → ℚ_{l}(2)

is the zero map. Passing to the quotient group G_{F (μ_{l^{∞}})}/G_{F_{2}} = Gal(F_{2}/F_{1}) we get a Γ-homomorphism c̄_{2} : Gal(F_{2}/F_{1}) → ℚ_{l}(2). Let x ∈ F^{∗} be not a root of 1. Observe that then we have an isomorphism of Γ-modules Gal(F (μ_{l^{∞}})(x^{\frac{1}{l^{∞}}})/F (μ_{l^{∞}})) ≃ ℤ_{l}(1). Therefore the map c̄_{2} must be zero because weights are incompatible. Hence c_{2} is also the zero map. □

For any z ∈ F we denote by L_{2}(z) the restrictin of l_{2}(z)_{γ} to the subgroup G_{F_{2}} of G_{F} . Observe that L_{2}(z) does not depend on a choice of a path γ and that L_{2}(z) is a Γ-homomorphism from G_{F_{2}} to ℚ_{l}(2) (see [9] Lemma 3.2.1).

We define a homomorphism

D : D(F) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2))

by setting

D(∑_{i=1}^{N} m_{i}[z_{i}]) := ∑_{i=1}^{N} m_{i}L_{2}(z_{i}).

Lemma 4.5. The subgroup R(F) of D(F) is contained in ker D.

Proof. We must show that the l-adic polylogarithm L_{2}(z) satisfies the functional equation

4.5.1.  L_{2}(x) − L_{2}(y) + L_{2}(\frac{y}{x}) − L_{2}(\frac{1 − x^{−1}}{1 − y^{−1}} ) + L_{2}(\frac{1 − x}{1 − y}) = 0.

This is of course one of the forms of the Abel functional equation.

Let f : ℙ^{1} \ \{0, 1, ∞\} ⟶ ℙ^{1} \ \{0, 1, ∞\} be given by f(z) = \frac{z}{z−1}. Then one

has id_{∗} + f_{∗} = 0 in Hom(Γ^{2}π/Γ^{3}π; Γ^{2}π/Γ^{3}π), where π := π_{1}(ℙ1 ; F̄ \ {0, 1, ∞}; v).

It follows from [10] Theorem 11.2.1 that

L_{2}(z) + L_{2}(\frac{z}{z − 1}) = 0.

In [10] we have shown that the l-adic dilogarithm satisfies the functional equation

L_{2}(\frac{(1 − t)x}{x − 1}) − L_{2}(tx) + L_{2}(\frac{(x − 1)t}{1 − t}) − L_{2}(\frac{t}{t − 1}) + L_{2}(x) = 0.

Substituting t by y^{−1} and using functional equations L_{2}(z) + L_{2}(z^{−1}) = 0 (see [10] Corollary 11.2.6) and L_{2}(z) + L_{2}(\frac{z}{z−1}) = 0 we get the equation 4.5.1. □

Theorem 4.6. Let F be a number field. Then we have

dim_{ℚ}[P_{2}(F)] ≤ r_{2}(F).

Proof. It follows from Lemma 4.5 that D : D(F) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2)) induces a homomorphism

D̄ : ker λ/R(F) = B(F) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2)).

It is well known that B(F) ≃ K_{3}(F) ⊗ ℚ (see [7] Theorem 5.2.). It follows from the Borel theorem that dim_{ℚ}(K_{3}(F) ⊗ ℚ) = r_{2}(F) (see [2] Proposition 12.2.). Therefore we get that

4.6.1.  dim_{ℚ}(Image D̄) ≤ r_{2}(F).

It follows from Proposition 2.2 that to any element x of ker λ we can associate a cocycle in Z^{1}(G_{F}; ℚ_{l}(2)). It follows from Proposition 4.4 that the corresponding cohomology class in H^{1}(G_{F}; ℚ_{l}(2)) is uniquely determined by the class of x in B(F). Let us denote by cl_{F} (x) this cohomology class. Observe that we have a commutative diagram

B(F) ⟶^{cl_{F}}  H^{1}(G_{F}; ℚ_{l}(2))
=↓  restriction↓

B(F) ⟶^{D̄} Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2)).

It follows from Proposition 4.4 and the already proved inequality 4.6.1 that dim_{ℚ}[P_{2}(F)] ≤ r_{2}(F). □

Conjecture 4.7. For a number field F we have

dim_{ℚ}[P_{2}(F)] = r_{2}(F).

Remark 4.8. To show Conjecture 4.7 we only need to show that the following diagram commmutes (perhaps up to a multiplication by a constant in ℚ_{l}^{∗})

B(F) ⟶^{cl_{F}} H^{1}(G_{F}; ℚ_{l}(2))
=↑  c_{2,1}↑
B(F) ⟵ H_{3}(SL(2, F); ℤ)

where the map from H_{3}(SL(2, F); ℤ) to B(F) is induced by sending the homogenous 3-simplex (g_{0}, g_{1}, g_{2}, g_{3}), g_{i} ∈ SL(2, F) onto the cross-ratio [g_{0}(∞) : g_{1}(∞) : g_{2}(∞) : g_{3}(∞)] of four points of ℙ^{1}(F) and c_{2,1} is deduced from the morphisms c_{2,1} : H_{3}(SL(2, F); ℤ/l^{n}) → H^{1}(G_{F}; ℤ/l^{n}(2)) (see [5] p.258).

The Bloch-Wigner function D(z) defines a continous cocycle on SL(2, ℂ) which is the universal class from which one gets all generators of H^{3}(SL(2, F); ℚ) for any number field F.

We shall define a cohomology class

D(F) ∈ H^{3}(SL(2, F); Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2))

by setting

D(F)(g_{0}, g_{1}, g_{2}, g_{3}) := L_{2}([g_{0}(∞) : g_{1}(∞) : g_{2}(∞) : g_{3}(∞)])

on the homogenous 3-simplex. D(F) is a cocycle because L_{2} satisfies 5-term functional equation (see [10] Theorem 11.1.14). Evaluating the cocycle D(F) on H_{3}(SL(2, F)) we get a homomorphism

H_{3}(SL(2, F); ℤ) → Hom_{Γ}(G_{F_{2}}; ℚ_{l}(2))

which factors through

D(F) : H_{3}(SL(2, F); ℤ) → H^{1}(G_{F}; ℚ_{l}(2)).

We do not know if this map is c_{2,1}.

Let F′ be a finite extension of F. We have two maps

(j_{F′/F})_{∗} : H_{3}(SL(2, F); ℤ) → H_{3}(SL(2, F′); ℤ)

and

(ι_{F′/F})^{∗} : H^{1}(G_{F}; ℚ_{l}(2)) → H^{1}(G_{F′}; ℚ_{l}(2))

induced by the inclusion F ⊂ F′. One easily sees that

(ι_{F′/F})^{∗} ∘ D(F) = D(F′) ∘ (j_{F′/F})_{∗}.

5 Action of G and Zagier conjecture for l-adic polylogarithms

We return to our study of descent properties of l-adic polylogarithms. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K).

Definition 5.1. We define P_{n}(L/K) to be the set of c_{n} := ∑_{i=1}^{N} m_{i}l_{n}(z_{i})_{γ_{i}}, where z_{1}, …, z_{N} ∈ L^{∗} \ {1}, m_{1}, …, m_{N} ∈ ℚ, the set {z_{1}, …, z_{N}} and the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] are G-invariant and c_{n} is a cocycle obtained by the way of Proposition 2.2. We define [P_{n}(L/K)] to be the image of P_{n}(L/K) in H^{1}(G_{L}; ℚ_{l}(n)).

It follows from Theorem 3.5 that

5.1.1.  [P_{n}(L/K)] ⊂ H^{1}(G_{L}; ℚ_{l}(n))^{G}.

It follows from the Lyndon spectral sequence that

5.1.2.  H^{1}(G_{L}; ℚ_{l}(n))^{G} ≃ H^{1}(G_{K}; ℚ_{l}(n)).

Hence we get that

5.1.3.  [P_{n}(L/K)] ⊂ H^{1}(G_{K}; ℚ_{l}(n)).

Definition 5.2. Let [P_{n}(K̄/K)] be the ℚ-vector subspace of H^{1}(G_{K}; ℚ_{l}(n)) generated by subspaces [P_{n}(L/K)] of H^{1}(G_{K}; ℚ_{l}(n)) for all finite Galois extensions L of K.

Conjecture 5.3. Let K be a number field. Then

dim_{ℚ}[P_{n}(K̄/K)] ≤ r_{2}(K) if n is even

and

dim_{ℚ}[P_{n}(K̄/K)] ≤ r_{1}(K) + r_{2}(K) if n is odd and n > 1.

Theorem 5.4. Let K be a number field. Then we have

dim _{ℚ}[P_{2}(K̄/K)] ≤ r_{2}(K).

Proof. Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). We have the following commutative diagram

B(L) ⟵ K_{3}(L) ⊗ ℚ
↑  ↑
B(K) ⟵ K_{3}(K) ⊗ ℚ

where the vertical maps are induced by the inclusion K ⊂ L and the horizontal maps are isomorphisms (see [7] Theorem 5.2.). The group G acts on B(L) and on K_{3}(L) ⊗ ℚ and the upper horizontal map is G-invariant. Hence we get the following diagram

B(L)^{G} ⟵ (K_{3}(L) ⊗ ℚ)^{G}
↑  ↑
B(K) ⟵ K_{3}(K) ⊗ ℚ

For the K-theory of fields we have K_{3}(K) ⊗ ℚ ≃ (K_{3}(L) ⊗ ℚ)^{G}. Hence we get

5.4.1.  B(K) ≃ B(L)^{G}.

Let c ∈ P_{2}(L/K). Then c = ∑_{i=1}^{N} m_{i}l_{2}(z_{i})_{γ_{i}} for some ∑_{i=1}^{N} m_{i}[z_{i}] ∈ B(L), such that the set {z_{1}, …, z_{N}} and the formal sum ∑_{i=1}^{N} m_{i}[z_{i}] are G-invariant. Therefore there exists ∑_{j=1}^{M} μ_{j}[ζ_{j}] ∈ B(K) corresponding to ∑_{i=1}^{N} m_{i}[z_{i}] via isomorphism 5.4.1. The cocycles c and s = ∑_{j=1}^{M} μ_{j}l_{2}(ζ_{j})_{δ_{j}} represent the same cohomology class in H^{1}(G_{L}; ℚ_{l}(2)). The cohomology class of s belongs to H^{1}(G_{K}; ℚ_{l}(2)), hence [P_{2}(L/K)] ⊂ [P_{2}(K)]. Now the theorem follows from Theorem 4.6. □

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[14] D. Zagier, Polylogarithms, Dedekind Zeta functions and the Algebraic K-theory of fields, in Arithmetic Algebraic Geometry, (ed. G. van der Geer, F.Oort, J.Steenbrick), Prog. Math. Vol 89, Birkhauser, Boston 1991, 391 – 430.

Jean-Claude DOUAI, UFR de Mathématiques, UMR AGAT CNRS, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex France E-mail address douai@agat.univ-lille1.fr Zdzislaw WOJTKOWIAK, 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 and UFR de Mathématiques, UMR AGAT CNRS, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex France