On l-adic iterated integrals, II Functional equations and l-adic polylogarithms.

Zdzisław Wojtkowiak

Abstract. We continue to study l-adic iterated integrals introduced in the first part. We shall show that the l-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying l-adic analogs of classical polylogarithms.

The plan of the paper. Part II

9. Introduction to Part II.

10. Functional equations.

11. l-adic polylogarithms.

12. Monodromy of l-adic iterated integrals and l-adic polylogarithms. Part III

13. Introduction to Part III.

14. Kummer characters.

15. Applications to Galois actions on fundamental groups.

Mathematics Subject Classification 11 Number theory §9. Introduction to Part II. 9.1. The classical complex iterated integrals satisfy functional equations (see [W1]). We shall show that l-adic iterated integrals satisfy the same functional equations as the classical complex iterated integrals.

First we introduce the following notation which we shall use in this paper. Let π (resp. L) be a group (resp. a Lie algebra). We denote by {Γ^{k}π}_{k≥1} (resp. {Γ^{k}L}_{k≥1}) the lower central series of the group π (resp. the Lie algebra L).

We set

gr_{Γ}^{k}π := Γ^{k}π/Γ^{k+1}π (resp. gr_{Γ}^{k}L := Γ^{k}L/Γ^{k+1}L).

Before we formulate our main result we make a following remark. Let Y = P_{K}^{1} \ {b_{1}, …, b_{m+1}}. Then

⨁_{k=1}^{∞} gr_{Γ}^{k}π_{1}(Y (C); x) ⊗ Q

is canonically isomorphic to a free Lie algebra over Q on m generators Y_{1}, …, Y_{m}, which we denote by Lie(Y_{1}, …, Y_{m}). Hence any linear form φ on gr_{Γ}^{k}π_{1}(Y (C); x)⊗Q corresponds to a linear form φ on Lie(Y_{1}, …, Y_{m}). Now we formulate our main result concerning functional equations. Theorem D. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let Y = P_{K}^{1} \ {b_{1}, …, b_{m+1}}. Let f_{i} : X → Y be a smooth morphism and let φ_{i} ∈ Lie(Y_{1}, …, Y_{m})^{⋄} be a linear form of degree q defined over Q for i = 1, …, N. Let n_{1}, …, n_{N} be rational numbers. If

∑_{i=1}^{N} n_{i}φ_{i} ∘ (f_{i})_{∗} = 0

in Hom(gr_{Γ}^{q}π_{1}(X(C); x); Q), where (f_{i})_{∗} : gr_{Γ}^{q}π_{1}(X(C); x) → gr_{Γ}^{q}π_{1}(Y (C); f_{i}(x)) is the map induced by f_{i} on fundamental groups for i = 1, …, N, then we have a functional equation

∑_{i=1}^{N} n_{i}L^{φ_{i}}(f_{i}(z), f_{i}(v)) = 0.

Next we generalize well known formulas

∫_{a}^{b} ω + ∫_{b}^{a} ω = 0 and ∫_{a}^{c} ω = ∫_{a}^{b} ω + ∫_{b}^{c} ω

from the elementary calculus (ω is a one-form). We show the following result.

Theorem E. Let z, y, v ∈ X̂(K) and let φ ∈ Lie(X_{1}, …, X_{n})^{⋄}. Then we have

L^{φ}(z, v) + L^{φ}(v, z) = 0

and

L^{φ}(z, v) = L^{φ}(z, y) + L^{φ}(y, v).

Let ω_{1}, ω_{2} be one-forms. The classical complex iterated integrals satisfy the following relations written here for two one-forms (see [Ch]).

i) ∫_{γ} ω_{1}, ω_{2} + ∫_{γ} ω_{2}, ω_{1} = ∫_{γ} ω_{1} · ∫_{γ} ω_{2}, ii) ∫_{αβ} ω_{1}, ω_{2} = ∫_{α} ω_{1}, ω_{2} + ∫_{α} ω_{1} · ∫_{β} ω_{2} + ∫_{β} ω_{1}, ω_{2}, iii) ∫_{γ} ω_{1}, ω_{2} = (−1)^{2} ∫_{γ^{−1}} ω_{2}, ω_{1}.

The anolog of the formula i) is satisfied by “l-adic iterated integrals” (coefficients of the power series Λ_{p}(σ)) by the very definition because the image of the inclusion map of the fundamental group into the algebra of formal non-commutative power series is of the form expL(X), where L(X) is the set of Lie elements in the algebra of formal non-commutative power series.

The formula

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

(see Part I Lemma 1.0.6), which after using suitable embeddings implies

Λ_{pq}(σ) = Λ_{p}(σ) · Λ_{q}(σ)

is the analog of the formula ii).

We do not know how to show an analog of the formula iii) for “l-adic iterated integrals” (coefficients of the power series Λ_{p}(σ)). To complete the picture we are still missing several l-adic analogs in the following table.

classical iterated integrals l-adic iterated integrals

values of Riemann zeta function Soulé classes for Q at positive integers multivalue zeta numbers values of l-adic iterated integrals at 1

and at roots of 1

multivalue zeta functions ? shuffle relations for multivalue zeta ? numbers and multivalue zeta functions ? The classical polylogarithms are the most important examples of iterated integrals.

In section 11 we introduce l-adic polylogarithms and we study their properties. We prove a theorem saying when a linear combination of l-adic polylogarithms is a cocycle. The reader can compare our result with Proposition in section 4.6 of [BD]. In section 11 we study functional equations of l-adic polylogarithms. We show that the l-adic dilogarithm satisfies the distribution relation

m( ∑_{i=0}^{m−1} l_{2}(ξ_{m}^{i}z)) = l_{2}(z^{m})

on the Galois group G_{Q(μ_{m})} and the Abel five term functional equation on G_{Q(μ_{l^{∞}})}.

These results are stronger than those in Theorem D in the sense that we get functional equations on the Galois groups G_{Q(μ_{n})} and G_{Q(μ_{l^{∞}})}, while the functional equations from Theorem D hold on the subgroup ⋂_{i=1}^{N} H_{q}(Y_{i}; f_{i}(z), f_{i}(x)) of G_{K}.

In the preprint version there is no picture. Pictures are not necessary to read and to understand this paper though they can help.

§10. Functional equations. 10.0. Let X = P_{K}^{1} \ {a_{1}, …, a_{n+1}} and let Y = P_{K}^{1} \ {b_{1}, …, b_{m+1}}. Let f : X → Y be a smooth morphism. Let z, v ∈ X̂(K). The morphism f induces

f_{∗} : π_{1}(XK̄; v) → π_{1}(YK̄; f(v))

and

f_{∗} : π(XK̄; z, v) → π(YK̄; f(z), f(v)).

Let us fix a a path p from v to z. We recall that for σ ∈ G_{K} we have defined

𝔣_{p}(σ) := p^{−1} · σ(p).

Then we have

f_{∗}(𝔣_{p}(σ)) = 𝔣_{f(p)}(σ). 10.0.1.

Let x = (x_{1}, …, x_{n+1}) (resp. y = (y_{1}, … y_{m+1})) be a sequence of geometric generators of π_{1}(X(C); v) (resp. π_{1}(Y (C); f(v)). We set X := {X_{1}, …, X_{n}} and Y := {Y_{1}, …, Y_{n}}. We recall that we have embeddings k_{x} : π_{1}(X(C); v) → Q_{l}{{X}} and k_{y} : π_{1}(Y (C); f(v)) → Q_{l}{{Y}} associated with a choice of sequences of geometric generators x of π_{1}(X(C); v) and y of π_{1}(Y (C); f(v)). There is a homomorphism of Q_{l}-algebras

f_{⋄} : Q_{l}{{X}} → Q_{l}{{Y}}

such that

f_{⋄} ∘ k_{x} = k_{y} ∘ f_{∗} and f_{⋄} ∘ k_{x,p} = k_{y,f(p)} ∘ f_{∗}. 10.0.2.

Let σ ∈ G_{K(μ_{l^{∞}})}. The equations 10.0.1 and 10.0.2 imply that
f_{⋄} ∘ σ_{x,p} = σ_{y,f(p)} ∘ f_{⋄}.

Hence we have

f_{⋄} ∘ logσ_{x,p} = logσ_{y,f(p)} ∘ f_{⋄}

and f_{⋄}((logσ_{x,p})(1)) = (logσ_{y,f(p)})(1). 10.0.3. The map f_{⋄} induces a homomorphism of Lie algebras

f_{⋄} : L(X) → L(Y).

Let

f_{•} : ⊕_{i=1}^{∞}gr_{Γ}^{i}L(X) → ⊕_{i=1}^{∞}gr_{Γ}^{i}L(Y)

be the map induced by f_{⋄} on associated graded Lie algebras. The associated graded Lie algebras are canonically isomorphic to Lie(X) and Lie(Y). Hence the map f_{⋄} induces

f_{•} : Lie(X) → Lie(Y).

Let φ ∈ Lie(Y)^{⋄} be a linear form of degree q. Let us set a_{x,p}^{φ∘f_{⋄}} := φ(f_{⋄}((logσ_{x,p})(1))). (In Part I we defined coefficients a_{x,p}^{φ} only for homogenous forms, hence we introduce this new definition.) It follows from 10.0.3 that a_{x,p}^{φ∘f_{⋄}} = a_{y,f(p)}^{φ}. 10.0.4. The map f_{⋄} is not homogenous. Therefore we have

a_{x,p}^{φ∘f_{⋄}} = a_{x,p}^{φ∘f_{•}} + ∑_{degχ<q} a_{x,p}^{χ}. 10.0.5.

It follows from 10.0.4 and 10.0.5 that L^{φ∘f_{•}}(z, v) = L^{φ}(f(z), f(v)) 10.0.6. on the subgroup H_{q}(X; z, v) of G_{K}.

Below we shall use fundamental groups of X or Y with various base points. Sequences of geometric generators and embeddings into algebras of non-commutative formal power series will be chosen as above.

Let v and v′ belong to X̂(K). If x = (x_{1}, …, x_{n+1}) is a sequence of geometric generators of π_{1}(X(C); v) and q is a path from v′ to v then q^{−1}·x·q := (q^{−1}·x_{1}·q, …, q^{−1}·x_{n+1}·q) is a sequence of geometric generators of π_{1}(X(C); v′). Then we have embeddings

k_{x} : π_{1}(X(C); v) → Q_{l}{{X}}

given by k_{x}(x_{i}) = e^{X_{i}} for i = 1, …, n and

k_{q^{−1}·x·q} : π_{1}(X(C); v′) → Q_{l}{{X}}

given by k_{q^{−1}·x·q}(q^{−1} · x_{i} · q) = e^{X_{i}} for i = 1, …, n.

Theorem 10.0.7. Let f_{i} : X → Y be a smooth morphism and let φ_{i} ∈ L(Y_{1}, …, Y_{m})^{⋄} be a linear form of degree q defined over Q for i = 1, …, N. Let n_{1}, …, n_{N} be rational numbers. If

∑_{i=1}^{N} n_{i}φ_{i} ∘ (f_{i})_{∗} = 0

in Hom(gr_{Γ}^{q}π_{1}(X(C); x); Q), where (f_{i})_{∗} : gr_{Γ}^{q}π_{1}(X(C); x) → gr_{Γ}^{q}π_{1}(Y (C), f_{i}(x)) is the map induced by f_{i} for i = 1, …, N, then we have functional equations

∑_{i=1}^{N} n_{i}L^{φ_{i}}(f_{i}(z); f_{i}(v)) = 0

on the subgroup H_{q}(X; z, v) of G_{K} and

∑_{i=1}^{N} n_{i}a_{y,f_{i}(p)}^{φ_{i}} = lower degree terms

on G_{K}, where ”lower degree terms” means a linear combination of a_{x,p}^{χ} with degree of χ strictly smaller than q.

Proof. It follows from 10.0.6 that ∑_{i=1}^{N} n_{i}L^{φ_{i}}(f_{i}(z); f_{i}(v)) = ∑_{i=1}^{N} n_{i}L^{φ_{i}∘(f_{i})_{•}}(z, v) = L^{∑_{i=1}^{N} n_{i}φ_{i}∘(f_{i})_{•}}(z, v) = 0. It follows from 10.0.4 and 10.0.5 that ∑_{i=1}^{N} n_{i}a_{y,f_{i}(p)}^{φ_{i}} = ∑_{i=1}^{N} n_{i}a_{x,p}^{φ_{i}∘(f_{i})_{⋄}} = ∑_{i=1}^{N} n_{i}a_{x,p}^{φ_{i}∘(f_{i})_{•}} + lower degree terms = a_{x,p}^{∑_{i=1}^{N} n_{i}φ_{i}∘(f_{i})_{•}}+ lower degree terms = lower degree terms.

10.1. Let p be a path from v to z. Let x = (x_{1}, …, x_{n+1}) be a sequence of geometric generators of π_{1}(X(C); v). Then x′ := (p · x_{1} · p^{−1}, …, p · x_{n+1} · p^{−1}) is a sequence of geometric generators of π_{1}(X(C); z) . The action of σ_{p^{−1}} on π_{1}(XK̄; z) can be expressed in the following way by the action of σ_{p} on π_{1}(XK̄; v). Let ω ∈ π_{1}(XK̄; z). Then σ_{p^{−1}}(ω) = p · σ(p^{−1} · ω · p) · 𝔣_{p}(σ)^{−1} · p^{−1}. This implies that on Q_{l}{{X}} we have

σ_{x,p} = L_{Λ_{p}(σ)} ∘ σ_{x} and σ_{x′,p^{−1}} = R_{Λ_{p}(σ)^{−1}} ∘ σ_{x}. 10.1.0.

Lemma 10.1.1. Let D be a derivation of the algebra Q_{l}{{X}} and let ω ∈ L(X). Then

L_{ω} ◯ D = L_{ζ} + D and R_{−ω} ◯ D = R_{−ζ} + D

for some ζ ∈ L(X). Proof. The lemma follows from the identities

[L_{ω}, D] = L_{−D(ω)}, [R_{−ω}, D] = R_{D(ω)}

and

[L_{ω}, L_{−D(ω)}] = L_{−[ω,D(ω)]}, [R_{−ω}, R_{D(ω)}] = R_{[ω,D(ω)]}.

Theorem 10.1.2. Let z, v ∈ X̂(K) and let p be a path from v to z. Then we have

i) L^{e}(z, v) + L^{e}(v, z) = 0,
ii) a_{x,p}^{e} + a_{pxp^{−1},p^{−1}}^{e} = 0.

Proof. It follows from 10.1.0 that

(logσ_{x′,p^{−1}})(1) = (R_{−logΛ_{p}(σ)} ◯ logσ_{x})(1).

It follows from Lemma 10.1.1 that

(R_{−logΛ_{p}(σ)} ◯ logσ_{x})(1) = −(L_{logΛ_{p}(σ)} ◯ logσ_{x})(1).

Hence we get that

(logσ_{x′,p^{−1}})(1) = −(logσ_{x,p})(1).

Evaluating a linear form on both sides of the equation we get the theorem.

Theorem 10.1.3. Let z, y, v ∈ X̂(K). Then we have

L^{e}(z, v) = L^{e}(z, y) + L^{e}(y, v).

Proof. Let p be a path from v to y, let r be a path from y to z and let q = r · p. We have σ_{p} = L_{𝔣_{p}(σ)} ∘ σ and σ_{q} = L_{𝔣_{q}(σ)} ∘ σ on π_{1}(XK̄; v) and σ_{r} = L_{𝔣_{r}(σ)} ∘ σ on π_{1}(XK̄, y). It follows from Lemma 1.0.6 that σ_{q} = L_{p^{−1}𝔣_{r}(σ)p}∘σ_{p}. Let us choose a sequence x of geometric generators of π_{1}(XK̄, y). Then x′ = p^{−1} · x · p is a sequence of geometric generators of π_{1}(XK̄; v). Observe that

σ_{x′,q} = σ_{x,r} ∘ σ_{x}^{−1} ∘ σ_{x′,p}.

Hence we get

logσ_{x′,q} = logσ_{x,r} ◯ logσ_{x}^{−1} ◯ logσ_{x′,p}.

Let σ belongs to the degree m step of the filtration defined in section 3, i.e., σ ∈ K_{m}^{T}(X) for some finite subset T ⊂ X̂(K)^{2}. Then

(logσ_{x′,q})(1) ≡ (logσ_{x,r})(1) + (logσ_{x′,p})(1) mod Γ^{m+1}L(X).

Evaluating a linear form of degree m on both sides of the congruence we get the theorem. 10.2. It follows from Proposition 7.1.10 that relations between functions L^{e}(z, v) imply relations between symbols {z, v}_{e}. Hence we get the following result

Corollary 10.2.1. Assume that Conjectures D_{n} are true for all n. Assume that for all n the maps realization : Ext_{MM_{K}}^{1} (Q(0), Q(n)) ⊗ Q → H^{1}(G_{K}, Q_{l}(n)) are injective. Then we have

{z, v}_{e} + {v, z}_{e} = 0

and

{z, v}_{e} = {z, y}_{e} + {y, v}_{e}

in L^{K}(X).

Proof. The corollary follows from Theorems 10.1.2 and 10.1.3 and Proposition 7.1.10.

10.3. Let π be a group. If π is nilpotent then we denote by π ⊗ Q its rationalization. For an arbitrary group π, π ⊗ Q := \underleftarrow{lim}_{n}((π/Γ^{n}π) ⊗ Q) is a rational completion of π. The group π_{1}(XK̄; v) is equipped with a pro-finite topology. Hence every quotient π_{1}(XK̄; v)/Γ^{n}π_{1}(XK̄; v) is equipped with a pro-finite topology. Therefore rationalization (π_{1}(XK̄; v)/Γ^{n}π_{1}(XK̄; v)) ⊗ Q is a Q_{l}-Lie group. Hence π_{1}(XK̄; v) ⊗ Q = \underleftarrow{lim}_{n}((π_{1}(XK̄; v)/Γ^{n}π_{1}(XK̄; v)) ⊗ Q) is equipped with a topology of the inverse limit of Q_{l}-Lie groups. The action of G_{K} on π_{1}(XK̄; v) extends uniquely to a continous action of G_{K} on π_{1}(XK̄; v) ⊗ Q.

Now we shall define a rational completion of π_{1}(XK̄; v)-torsor π(XK̄; z, v). We introduce an equivalence relation on the product π(XK̄; z, v) × π_{1}(XK̄; v) ⊗ Q. We say that a pair (p, S) is equivalent to a pair (q, T) and we write (p, S) ∼ (q, T) if S = (p^{−1} · q) · T in π_{1}(XK̄; v) ⊗ Q.

We set

π(XK̄; z, v) ⊗ Q := (π(XK̄; z, v) × π_{1}(XK̄; v) ⊗ Q)/ ∼ .

The Galois group G_{K} acts on the product π(XK̄; z, v) × π_{1}(XK̄; v) ⊗ Q component wise. The group π_{1}(XK̄; v) ⊗ Q acts on the product π(XK̄; z, v) × π_{1}(XK̄; v) ⊗ Q by the right multiplication on the second factor. Both actions are compatible with the equivalence relation ∼ and continous. Hence G_{K} acts on the set of equivalence classes π(XK̄; z, v)⊗Q. The action of π_{1}(XK̄; v)⊗Q on the product π(XK̄; z, v)×π_{1}(XK̄; v)⊗Q induces a structure of π_{1}(XK̄; v) ⊗ Q-torsor on the set of equivalence classes π(XK̄; z, v) ⊗ Q. Elements of π(XK̄; z, v) ⊗ Q have the form p · S, where p is in π(XK̄; z, v) and S ∈ π_{1}(XK̄; v) ⊗ Q. We shall call them Q_{l}-paths.

Lemma 10.3.1. The embedding k_{x} : π_{1}(XK̄; v) → Q_{l}{{X}} extends uniquely to a continous multiplicative embedding k̄_{x} : π_{1}(XK̄; v) ⊗ Q → Q_{l}{{X}}.

Proof. The image of k_{x} is contained in Q_{l}{{X}}^{∗}. The group Q_{l}{{X}}^{∗} is a pro-unipotent group with exponents in Q_{l}. Hence k_{x} extends to k̄_{x} : π_{1}(XK̄; v) ⊗ Q → Q_{l}{{X}}.

Further we shall denote the embedding k̄_{x} by k_{x}. One shows that the formulas

𝔣_{p·q}(σ) = q^{−1} · 𝔣_{p}(σ) · q · 𝔣_{q}(σ), Λ_{p·q}(σ) = Λ_{p}(σ) · Λ_{q}(σ), Λ_{p}(τ · σ) = Λ_{p}(τ) · τ(Λ_{p}(σ)) and g_{∗}(𝔣_{p}) = 𝔣_{g(p)}, where g : X_{K} → X_{K} is a regular map, are valid also for Q_{l}-paths p and q.

§11. l-adic polylogarithms. 11.0. In this subsection we introduce l-adic polylogarithms. We give sufficient conditions when a linear combination of l-adic polylogarithms is a cocycle. Next we are studying a relative version of l-adic polylogarithms. We also show that l-adic polylogarithms are special case of l-adic iterated integrals introduced in section 5.

Let K be a number field . Let V = P_{K}^{1} \{0, 1, ∞}. Let x and y be standard generators of π_{1}(VK̄; \overrightarrow{01}) - loops around 0 and 1 respectively (see the Picture 1).

Picture 1

Let k : π_{1}(VK̄; \overrightarrow{01}) ⊗ Q → Q_{l}\{\{X, Y \}\} be a multiplicative continous embedding given by k(x) = e^{X} and k(y) = e^{Y} . We denote by Lie(X, Y) a free Lie algebra over Q_{l} on X and Y and by L(X, Y) a completion of Lie(X, Y) with respect to the lower central series. We identify L(X, Y) with the Lie algebra of Lie elements in Q_{l}{{X, Y }}.

Let us set E_{1} := Y, E_{k+1} := [E_{k}, X]. Let B be a base of Lie(X, Y) given by basic Lie elements. We assume that E_{k} ∈ B for k = 1, 2, ….

Let z ∈ V̂ (K) and let p be a Q_{l}-path from \overrightarrow{01} to z. We recall that 𝔣_{p}(σ) = p^{−1} · σ(p) ∈ π_{1}(VK̄; \overrightarrow{01}) ⊗ Q and Λ_{p}(σ) := k(𝔣_{p}(σ)) ∈ Q_{l}\{\{X, Y \}\} for any σ ∈ G_{K}.

If e ∈ B we denote by e^{∗} the dual linear form to e with respect to B. Definition 11.0.1. Let σ ∈ G_{K}. We set

l_{n}(z)(σ) := E_{n}^{∗}(logΛ_{p}(σ)) and l(z)(σ) := X^{∗}(logΛ_{p}(σ)).

The coefficient l_{n}(z) is an l-adic polylogarithm (n-th order l-adic polylogarithm) evaluated at z. It is a function from G_{K} to Q_{l}(n). It depends on a choice of p in π(VK̄; z, \overrightarrow{01})⊗Q. The coefficient l(z) is an l-adic logarithm evaluated at z. If we are using various paths and it is important to indicate the dependence of l_{n}(z) (resp. l(z)) on a path p we shall write l_{n}(z)_{p} (resp. l(z)_{p}).

Definition 11.0.2. We set

L_{n}(z) := l_{n}(z)_{|H_{n}(V;z,\overrightarrow{01})}.

Observe that L_{n}(z) depends only on z.

Let us set e_{1} := y and e_{k+1} := (e_{k}, x). Observe that any element g ∈ π_{1}(VK̄; \overrightarrow{01}) ⊗ Q can be written in the following form

g = x^{α^{0}(g)} · y^{α^{1}(g)} · e_{2}^{α^{2}(g)} · e_{3}^{α^{3}(g)} · ((y, x)y)^{β(g)} · e_{4}^{α^{4}(g)} · f_{4} · . . . · e_{n}^{α^{n}(g)} · f_{n} · . . . ,

where the exponents are in Q_{l} and each f_{n} is a product of powers of commutators of length n, which contain y at least twice. Definition 11.0.3. Let σ ∈ G_{K}. We define functions κ_{z}^{n} : G_{K} → Q_{l} by the identity

𝔣_{p}(σ) = x^{κ_{z}^{0}(σ)} · y^{κ_{z}^{1}(σ)} · e_{2}^{κ_{z}^{2}(σ)} · e_{3}^{κ_{z}^{3}(σ)} · f_{3} · e_{4}^{κ_{z}^{4}(σ)} · f_{4} · . . . · e_{n}^{κ_{z}^{n}(σ)} · f_{n} · . . . .

Let n ≥ 1. Then κ_{z}^{n} we view as a function from G_{K} to Q_{l}(n). κ_{z}^{0} we view as a function from G_{K} to Q_{l}(1). We shall also use the notation κ_{0}(z) := κ_{z}^{0} and κ_{1}(z) := κ_{z}^{1}. If we are using various paths and it is important to indicate the dependence of κ_{z}^{n}(σ) on a path p we shall write κ_{z}^{n}(σ)_{p}.

We shall express l-adic polylogarithms in terms of functions κ_{z}^{n}.

Let f ∈ L(X, Y). We define a derivation ad f of L(X, Y) setting (ad f)(g) = [f, g] for any g ∈ L(X, Y).

Let I_{k} be a Lie ideal of L(X, Y) generated topologically by Lie brackets which contain Y at least k-times.

Lemma 11.0.4. We have

log(k(e_{n+1})) = (−1)^{n} ∑_{k_{1},...,k_{n}=1}^{∞} \frac{1}{k_{1}! · . . . · k_{n}!}(ad X)^{k_{1}+...+k_{n}}(Y) mod I_{2}.

Lemma 11.0.5. (see [B] chapitre II). We have

log(e^{X} · e^{Y}) = X + Y + \frac{1}{2}[X, Y] + ∑_{n=1}^{∞} \frac{1}{(2n)!}B_{2n}(ad X)^{2n}(Y) mod I_{2}.

Proposition 11.0.6. Let σ ∈ G_{K}. We have

logΛ_{p}(σ) = κ_{z}^{0}(σ)X + ∑_{i=1}^{∞}(−1)^{i−1}κ_{z}^{i}(σ)( ∑_{k_{1},...,k_{i−1}=1}^{∞} \frac{1}{k_{1}! · . . . · k_{i−1}!}(ad X)^{k_{1}+...+k_{i−1}}(Y))

+\frac{1}{2}[κ_{z}^{0}(σ)X, ∑_{i=1}^{∞}(−1)^{i−1}κ_{z}^{i}(σ)( ∑_{k_{1},...,k_{i−1}=1}^{∞} \frac{1}{k_{1}! · . . . · k_{i−1}!}(ad X)^{k_{1}+...+k_{i−1}}(Y))]+

∑_{n=1}^{∞} \frac{(κ_{z}^{0}(σ))^{2n}}{(2n)!} B_{2n}(ad X)^{2n}( ∑_{i=1}^{∞}(−1)^{i−1}κ_{z}^{i}(σ)( ∑_{k_{1},...,k_{i−1}=1}^{∞} \frac{1}{k_{1}! · . . . · k_{i−1}!}(ad X)^{k_{1}+...+k_{i−1}}(Y)))

mod I_{2}. Proof. The proposition follows from Lemmas 11.0.4 and 11.0.5.

Using Proposition 11.0.6 we can easily calculate l-adic polylogarithms in terms of functions κ_{z}^{n}. For example in small degrees we get the following result. Corollary 11.0.7. We have

l(z) = κ_{z}^{0}, l_{1}(z) = κ_{z}^{1}, l_{2}(z) = κ_{z}^{2} − \frac{1}{2}κ_{z}^{0} · κ_{z}^{1}

and

l_{3}(z) = κ_{z}^{3} − \frac{1}{2}κ_{z}^{0} · κ_{z}^{2} + \frac{1}{12}(κ_{z}^{0})^{2} · κ_{z}^{1} − \frac{1}{2}κ_{z}^{2}.

Proposition 11.0.8. Let ζ ∈ V̂ (K) and let p be a Q_{l}-path from \overrightarrow{01} to ζ. Let q be the standard path from \overrightarrow{01} to \overrightarrow{10} (an interval [0, 1]). Let g : V_{K} → V_{K} be given by g(z) = 1−z. Then we have

l_{1}(ζ)_{p} = l(1 − ζ)_{g(p)·q}.

Proof. It follows from Corollary 11.0.7 that

𝔣_{p} ≡ x^{l(ζ)_{p}} · y^{l_{1}(ζ)_{p}} mod Γ^{2}(π_{1}(VK̄; \overrightarrow{01}) ⊗ Q).

Observe that g(p) · q is a Q_{l}-path from \overrightarrow{01} to 1 − ζ. Hence we have

𝔣_{g(p)·q} ≡ x^{l(1−ζ)_{g(p)·q}} · y^{l_{1}(1−ζ)_{g(p)·q}} mod Γ^{2}(π_{1}(VK̄; \overrightarrow{01}) ⊗ Q).

On the other side 𝔣_{g(p)·q} = q^{−1} · 𝔣_{g(p)} · q · 𝔣_{q} = q^{−1} · g_{∗}(𝔣_{p}) · q · 𝔣_{q} ≡ x^{l_{1}(ζ)_{p}} · y^{l(ζ)_{p}} mod Γ^{2}(π_{1}(VK̄; \overrightarrow{01}) ⊗ Q) because q^{−1} · g_{∗}(x) · q = y, q^{−1} · g_{∗}(y) · q = x and 𝔣_{q} ≡ 1 mod Γ^{2}(π_{1}(VK̄; \overrightarrow{01}) ⊗ Q). The proposition follows from the last two congruences.

Theorem 11.0.9. Let z_{i} ∈ V̂ (K), let p_{i} ∈ π(VQ̄; z_{i}, \overrightarrow{01}) ⊗ Q and let n_{i} ∈ Q_{l} for i = 1, …, N. Let us assume that l-adic polylogarithms l_{k}(z_{i}) calculated along the Q_{l}-paths p_{i} for i = 1, …, N satisfy the following conditions

i) ∑_{i=1}^{N} n_{i}(l(z_{i})(τ))^{α} · (l(z_{i})(σ))^{β} · (l(z_{i})(τ) · l_{1}(z_{i})(σ) − l(z_{i})(σ) · l_{1}(z_{i})(τ)) = 0 for any τ, σ ∈ G_{K} and for any α and β such that α + β = n − 2,

ii) ∑_{i=1}^{N} n_{i}(l(z_{i})(τ))^{α} · (l(z_{i})(σ))^{β} · l_{k}(z_{i})(σ) = 0 for any τ, σ ∈ G_{K}, for k = 2, …, n − 1 and for any α and β such that α + β = n − k.

Then ∑_{i=1}^{N} n_{i}l_{n}(z_{i}) is a cocycle on G_{K} with values in Q_{l}(n).

Proof. The equality Λ_{p}(τσ) = Λ_{p}(τ) · τ(Λ_{p}(σ)) implies

logΛ_{p}(τσ) = logΛ_{p}(τ) + logτ(Λ_{p}(σ)) + \frac{1}{2}[logΛ_{p}(τ), log(τ(Λ_{p}(σ)))]−

\frac{1}{12}[[logΛ_{p}(τ), log(τ(Λ_{p}(σ)))], logΛ_{p}(τ)] + \frac{1}{12}[[logΛ_{p}(τ), log(τ(Λ_{p}(σ)))], log(τ(Λ_{p}(σ)))]

−\frac{1}{24}[[[logΛ_{p}(τ), log(τ(Λ_{p}(σ)))], log(τ(Λ_{p}(σ)))]logΛ_{p}(τ)] + ….

Comparing coefficients at E_{n} we get

l_{n}(z)(τσ) = l_{n}(z)(τ)+χ(τ)^{n}l_{n}(z)(σ)+\frac{1}{2}(l_{n−1}(z)(τ)χ(τ)l(z)(σ)−χ(τ)^{n−1}l_{n−1}(z)(σ)l(z)(τ))−

\frac{1}{12}(l_{n−2}(z)(τ)χ(τ)l(z)(τ)l(z)(σ) − χ(τ)^{n−2}l_{n−2}(z)(σ)(l(z)(τ))^{2})+

\frac{1}{12}(l_{n−2}(z)(τ)χ(τ)^{2}(l(z)(σ))^{2} − χ(τ)^{n−1}l_{n−2}(z)(σ)l(z)(τ)l(z)(σ))−

\frac{1}{24}(l_{n−3}(z)(τ)χ(τ)^{2}l(z)(τ)(l(z)(σ))^{2} − χ(τ)^{n−2}l_{n−3}(z)(σ)(l(z)(τ))^{2}l(z)(σ)) + ….

The assumptions of the theorem imply that

∑_{i=1}^{N} n_{i}l_{n}(z_{i})(τσ) = ∑_{i=1}^{N} n_{i}l_{n}(z_{i})(τ) + χ(τ)^{n} ∑_{i=1}^{N} n_{i}l_{n}(z_{i})(σ).

The l-adic polylogarithm l_{n}(z)_{p} depends on a choice of a Q_{l}-path from \overrightarrow{01} to z. We have the following elementary result.

Lemma 11.0.10. Let p be a Q_{l}-path from \overrightarrow{01} to z and let S ∈ π_{1}(VK̄, \overrightarrow{01})⊗Q. If S = x^{α}·y^{β} mod Γ^{2}(π_{1}(VK̄; \overrightarrow{01}) ⊗ Q) then l(z)_{pS} = l(z)_{p} + α(χ − 1) and l_{1}(z)_{pS} = l_{1}(z)_{p} + β(χ − 1).

Proof. We have 𝔣_{pS}(σ) = S^{−1} · 𝔣_{p}(σ) · σ(S). Hence Λ_{pS}(σ) = k(S)^{−1} · Λ_{p}(σ) · k(σ(S)). Let S = x^{α} · y^{β} · e_{2}^{β_{2}} · e_{3}^{β_{3}} · f_{3} · e_{4}^{β_{4}} · . . . . Therefore logΛ_{pS}(σ) = ( − log(e^{αX} · e^{βY} · (e^{X} · e^{Y} · e^{−X} · e^{−Y})^{β_{2}} ·. . .))◯logΛ_{p}(σ)◯log(e^{αχ(σ)X} ·e^{βχ(σ)Y} ·(e^{χ(σ)X} ·e^{χ(σ)Y} ·e^{−χ(σ)X} ·e^{−χ(σ)Y})^{β_{2}} ·. . .) ≡ −αX − βY + l(z)_{p}X + l_{1}(z)_{p}Y + αχ(σ)X + βχ(σ)Y mod Γ^{2}L(X, Y). The lemma follows from the congruence.

Theorem 11.0.11. Let z_{i} ∈ V (K), let p_{i} ∈ π(VK̄; z_{i}, \overrightarrow{01}) ⊗ Q and let n_{i} ∈ Q for i = 1, …, N. Let S be a subgroup of K^{∗} ⊗ Q generated by z_{i} and 1 − z_{i} for i = 1, …, N. Assume that

i) the map φ : S → Z^{1}(G_{K}; Q_{l}(1)) given by φ(z_{i}) = l(z_{i})_{p_{i}} and φ(1 − z_{i}) = l_{1}(z_{i})_{p_{i}} is well defined and it is a homomorphism;

ii) ∑_{i=1}^{N} n_{i}ν_{1}(z_{i}) · . . . · ν_{n−2}(z_{i})(z_{i}) ∧ (1 − z_{i}) = 0 in (S ∧ S) ⊗ Q_{l} for any homomorphisms

ν_{1}, …, ν_{n−2} from S to Q_{l}; iii) ∑_{i=1}^{N} n_{i} · ν_{1}(z_{i})^{α} · ν_{2}(z_{i})^{β} · l_{k}(z_{i})(σ) = 0 for any homomorphisms ν_{1} and ν_{2} from S to

Q_{l}, for any σ ∈ G_{K}, for k = 2, …,n − 1 and for any α and β such that α + β = n − k.

Then ∑_{i=1}^{N} n_{i}l_{n}(z_{i})_{p_{i}} is a cocycle on G_{K} with values in Q_{l}(n).

Proof. Let us fix τ ∈ G_{K}. The map S → Q_{l}(1) given by s → φ(s)(τ) (z_{i} → l(z_{i})(τ)) is a homomorphism. Let us fix τ, σ ∈ G_{K}. The map S ⊗ S → Q_{l}(2), x ⊗ y → φ(x)(τ) · φ(y)(σ) − φ(x)(σ) · φ(y)(τ) (z_{i} ⊗ (1 − z_{i}) → l(z_{i})(τ) · l_{1}(z_{i})(σ) − l(z_{i})(σ) · l_{1}(z_{i})(τ)) factors through S ∧ S. Hence the theorem follows from Theorem 11.0.9.

Corollary 11.0.12. Let ξ_{m} be a m-th root of 1 different from 1. There is a Q_{l}-path p from \overrightarrow{01} to ξ_{m} such that l_{n}(ξ_{m})_{p} is a cocycle on G_{Q(μ_{m})}. If l does not divide m then one can choose the path p in π(VQ̄; ξ_{m}, \overrightarrow{01}).

Proof. Let m = l^{k_{0}} · r, where l does not divide r. Let q be a path from \overrightarrow{01} to ξ_{m}. There are α, β and γ in Z_{l} such that (ξ_{l^{k_{0}+n}}^{α} · ξr^{\frac{β}{l^{n}}} · ξ_{l^{n}}^{γ})_{n∈N} is a compatible family of l^{n}-th roots of ξ_{m} determined by the path q. Hence l(ξ_{m})_{q} = (\frac{α}{k_{0}} + γ)(χ − 1). Lemma 11.0.10 implies that there is a Q_{l}-path p from \overrightarrow{01} to ξ_{m} such that l(ξ_{m})_{p} = 0. Theorem 11.0.9 implies that l_{n}(ξ_{m})_{p} is a cocycle. Observe that if k_{0} = 0 then one can choose p in π(VQ̄; ξ_{m}, \overrightarrow{01}).

The classical polylogarithms are iterated integrals defined by ∫_{0}^{z} \frac{dz}{1−z} , \frac{dz}{z} , …, \frac{dz}{z} . The iterated integral ∫_{a}^{b} \frac{dz}{1−z} , \frac{dz}{z} , …, \frac{dz}{z} can be express by classical polylogarithms. Now we shall define a normalized analog of the iterated integral ∫_{a}^{b} \frac{dz}{1−z} , \frac{dz}{z} , …, \frac{dz}{z} .

Let z, v ∈ V̂ (K). Let q be a path from \overrightarrow{01} to v and let p be a path from v to z. We shall define relative polylogarithms l_{n}(z, v). Let us set x_{1} := q · x · q^{−1}, y_{1} := q · y · q^{−1}. Observe that x_{1}, y_{1} are generators of π_{1}(VK̄; v). Let G_{n+1} ⊂ π_{1}(VK̄; \overrightarrow{01}) (resp. G′_{n+1} ⊂ π_{1}(VK̄; v)) be a closed normal subgroup generated by Γ^{n+1}π_{1}(VK̄; \overrightarrow{01}) (resp. Γ^{n+1}π_{1}(VK̄; v)) and all commutators which contain y (resp. y_{1} ) at least twice. Let π := π_{1}(VK̄; \overrightarrow{01})/G_{n+1} and π′ := π_{1}(VK̄; v)/G′_{n+1}.

It follows from Proposition 2.2.1 that the action of G_{K} on π′ is given by

σ(x_{1}) = (q · 𝔣_{q}(σ) · q^{−1}) · x_{1}^{χ(σ)} · (q · (𝔣_{q}(σ))^{−1} · q^{−1}) mod G′_{m+1}

and

σ(y_{1}) = (q · 𝔣_{q}(σ) · q^{−1}) · y_{1}^{χ(σ)} · (q · (𝔣_{q}(σ))^{−1} · q^{−1}) mod G′_{m+1}.

Lemma 11.0.13. The action of G_{K} on π_{1}(VK̄; v) induced from the action on the torsor π(VQ̄; z, v) by the isomorphism t_{p} (see Part I Section 1) is given by

σ_{p}(w) = (q · 𝔣_{pq}(σ) · q^{−1}) · σ̄(w) · (q · (𝔣_{q}(σ))^{−1} · q^{−1}) mod G′_{n+1},

where σ̄(x_{1}) = x_{1}^{χ(σ)}, σ̄(y_{1}) = y_{1}^{χ(σ)} and σ̄ is continous and multiplicative.

Proof. The formula for σ_{p}(w) follows from Lemma 1.0.2 and Lemma 1.0.6.

Let I be the augmentation ideal of Q_{l}{{X, Y }} and let J_{n+1} be a closed ideal of Q_{l}{{X, Y }} generated by I^{n+1} and all monomials which contain Y at least twice. We define two maps

k : π_{1}(VK̄; \overrightarrow{01}) → Q_{l}\{\{X, Y \}\}/J_{n+1} and k′ : π_{1}(VK̄; v) → Q_{l}\{\{X, Y \}\}/J_{n+1}

by k(x) = e^{X}, k(y) = e^{Y} and k′(x_{1}) = e^{X}, k′(y_{1}) = e^{Y} .

Let ( )_{p} : G_{K} → GL (Q_{l}{{X, Y }}/J_{n+1}) be the action of G_{K} induced from the action of G_{K} on the torsor π(VQ̄; z, v) by the isomorphism t_{p} and the embedding k′.

Let us set

ψ_{p}(σ) := σ_{p} ∘ ρ(χ(σ)^{−1}).

We recall that E_{1} := Y and E_{k+1} := [E_{k}, X] for k = 1, …, n − 1. Then any Lie element of Q_{l}{{X, Y }}/J_{n+1} is a linear combination with Q_{l} coefficients of X, E_{1}, …, E_{n}. If g ∈ π′ then logk′(g) is a Lie element of Q_{l}{{X, Y }}/J_{n+1}. Definition 11.0.14. Let σ ∈ G_{K}. We set

(logψ_{p}(σ))(1) = l(z, v)_{p}(σ)X + ∑_{k=1}^{n} l_{k}(z, v)_{p}(σ)E_{k}.

Proposition 11.0.15. We have

l_{n}(z, v)_{p} = l_{n}(z)_{pq} − l_{n}(v)_{q}.

Proof. Observe that k′(q · 𝔣_{pq}(σ) · q^{−1}) = k(𝔣_{pq}(σ)) = Λ_{pq}(σ) and k′(q · (𝔣_{q}(σ))^{−1} · q^{−1}) = k((𝔣_{q}(σ))^{−1}) = (Λ_{q}(σ))^{−1}. Let σ ∈ G_{K}. It follows from Lemma 11.0.13 that

ψ_{p}(σ) = L_{Λ_{pq}(σ)} ∘ R_{(Λ_{q}(σ))^{−1}}.

This implies that

logψ_{p}(σ) = L_{logΛ_{pq}(σ)} ◯ R_{−logΛ_{q}(σ)}.

The operators L_{logΛ_{pq}(σ)} and R_{−logΛ_{q}(σ)} commute. Hence logψ_{p}(σ) = L_{logΛ_{pq}(σ)}+R_{−logΛ_{q}(σ)}. This implies the proposition. Corollary 11.0.16. We have

l_{n}(z, \overrightarrow{01})_{p} = l_{n}(z)_{p}.

Proof. It follows from Proposition 11.0.15 that l_{n}(z, \overrightarrow{01})_{p} = l_{n}(z)_{p} − l_{n}(\overrightarrow{01})_{c}, where c is a constant path. For such a path l_{n}(\overrightarrow{01})_{c} = 0. Remark. The relative polylogarithm l_{n}(z, v) is the function a_{p}^{E_{n}} from section 5. Hence the l-adic polylogarithm l_{n}(z)_{p} is also a special case of l-adic iterated integrals defined in section 5.

We finish this subsection with a result expressing coefficiets of 𝔣_{p} in degree one for an arbitrary X by l-adic logarithms.

Proposition 11.0.17. 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. Then

𝔣_{p} ≡ x_{1}^{l(z−a_{1})_{g_{1}(p)·q_{1}} −l(v−a_{1})_{q_{1}}} · . . . · x_{n}^{l(z−a_{n})_{gn(p)·qn} −l(v−a_{n})_{qn}} mod Γ^{2}π_{1}(XK̄; v),

where q_{i} is any path from \overrightarrow{01} to v − a_{i} on P1 ; K̄ \ \{0, ∞\} for i = 1, …, n.

Proof. Without loss of generality we can suppose that X = P_{K}^{1} \ {a, ∞} and g : X → P_{K}^{1} \ {0, ∞} is given by g(z) = z − a. Let p be a path from v to z on XK̄. Then g(p) is a path from v − a to z − a on P_{K}^{1} \ \{0, ∞\}. Let q be any path from \overrightarrow{01} to v − a on

P1 ; K̄ \ {0, ∞}. We have

𝔣_{g(p)·q} = q^{−1} · 𝔣_{g(p)} · q · 𝔣_{q} = q^{−1} · g_{∗}(𝔣_{p}) · q · 𝔣_{q}.

It follows from Corollary 11.0.7 that

x^{l(z−a)_{g(p)·q}} = q^{−1} · g_{∗}(𝔣_{p}) · q · x^{l(v−a)_{q}},

where x is a loop around 0. Hence we get that 𝔣_{p} = x_{a}^{l(z−a)_{g(p)·q}−l(v−a)_{q}}, where g_{∗}(x_{a}) = q · x · q^{−1}. 11.1. In this subsection we shall study functional equations of l-adic polylogarithms. We shall prove the distribution relation and the Abel five term equation for l-adic dilogarithms. We shall show that l-adic dilogarithms satisfy these functional equations without lower degree terms.

We start with the discussion of the l-adic analog of the functional equation

log (x · y) = log x + log y

of the classical logarithm.

Proposition 11.1.0. Let ζ, y ∈ P^{1}(K) \ \{0, ∞\}. Then there exist paths γ from \overrightarrow{01} to ζ, δ from \overrightarrow{01} to y and φ from \overrightarrow{01} to y · ζ such that

l(y · ζ)_{φ} = l(y)_{δ} + l(ζ)_{γ}

on G_{K}.

Proof. Let g : P_{K}^{1} \ {0, ∞} → P_{K}^{1} \ {0, ∞} be given by g(z) = y · z. Let p be a path from \overrightarrow{01} to ζ. Then g(p) is a path from \overrightarrow{0y} to y · ζ. We recall that x is a standard generator of π_{1}(P1 ; K̄ \ \{0, ∞\}, \overrightarrow{01}). Let us fix a path q from \overrightarrow{01} to \overrightarrow{0y}. Let us set x′ = q · x · q^{−1}. Then x′ is a generator of π_{1}(P1 ; K̄ \ \{0, ∞\}, \overrightarrow{0y}). Observe that g_{∗}(x) = x′. It follows from Corollary 11.0.7 that

𝔣_{p}(σ) = x^{l(ζ)_{p}(σ)} and 𝔣_{g(p)·q}(σ) = x^{l(y·ζ)_{g(p)·q}(σ)}.

On the other side we have

𝔣_{g(p)·q}(σ) = q^{−1}·𝔣_{g(p)}(σ)·q·𝔣_{q}(σ) = q^{−1}·g_{∗}(𝔣_{p}(σ))·q·𝔣_{q}(σ) = x^{l(ζ)_{p}(σ)}·x^{l(\overrightarrow{0y})_{q}(σ)} = x^{l(ζ)_{p}(σ)+l(\overrightarrow{0y})_{q}(σ)}.

Comparing exponents we get

l(y · ζ)_{g(p)·q} = l(ζ)_{p} + l(\overrightarrow{0y})_{q}.

Let t be the canonical path from \overrightarrow{0y} to y. Then t · q is a path from \overrightarrow{01} to y (see Picture 2).

Picture 2

We have x^{l(y)_{t·q}(σ)} = 𝔣_{t·q}(σ) = q^{−1} · 𝔣_{t}(σ) · q · 𝔣_{q}(σ) = q^{−1} · 𝔣_{t}(σ) · q · x^{l(\overrightarrow{0y})_{q}(σ)}. Without loss of generality we can suppose that y = 1 and t is the canonical path from \overrightarrow{01} to 1. Then it is clear that 𝔣_{t}(σ) = 1. Hence l(\overrightarrow{0y})_{q} = l(y)_{t·q}. This finishes the proof of the proposition.

Now we shall discuss the l-adic analog of the functional equation

Li_{2}(z^{m}) = m(∑_{i=0}^{m} Li_{2}(ξ_{m}^{i}z))

of the classical dilogarithm. Let Y = P_{Q(μ_{m})}^{1} \ {0, μ_{m}, ∞} and V = P_{Q(μ_{m})}^{1} \ {0, 1, ∞}. We choose generators x′, y_{0}, …, y_{m−1} of π_{1}(YQ̄, \overrightarrow{01}) as on the picture.

Picture 3

Let f : Y → V be given by f(z) = z^{m}. We have f_{∗}(x′) = x^{m}, f_{∗}(y_{0}) = y and f_{∗}(y_{i}) = x^{−i} · y · x^{i}. Let z ∈ Ŷ (Q(μ_{m})) and let p be a path from \overrightarrow{01} to z. We define functions λ(z), μ_{0}(z), …, μ_{m−1}(z), ν_{0}(z), …, ν_{m−1}(z) from G_{Q(μ_{m})} to Z_{l} by the following congruence

𝔣_{p} ≡ x^{′λ(z)} · y_{0}^{μ_{0}(z)} · y_{1}^{μ_{1}(z)} · . . . · y_{m−1}^{μ_{m−1}(z)} · (y_{0}, x′)^{ν_{0}(z)} · . . . · (y_{m−1}, x′)^{ν_{m−1}(z)}· 11.1.1.

∏_{i<j}(y_{i}, y_{j})^{α_{ij}(z)} mod Γ^{3}π_{1}(YQ̄; \overrightarrow{01}).

Observe that f(p) is a path from \overrightarrow{01} to z^{m}. Hence we have

𝔣_{f(p)} ≡ x^{κ_{z^{m}}^{0}} · y^{κ_{z^{m}}^{1}} · (y, x)^{κ_{z^{m}}^{2}} mod Γ^{3}π_{1}(VQ̄; \overrightarrow{01}),

(see Definition 11.0.3.)

Lemma 11.1.2. We have κ_{z^{m}}^{0} = mλ(z), κ_{z^{m}}^{1} = μ_{0}(z) + μ_{1}(z) + … + μ_{m−1}(z) and κ_{z^{m}}^{2} = m(ν_{0}(z) + … + ν_{m−1}(z)) + μ_{1}(z) + … + iμ_{i}(z) + … + (m − 1)μ_{m−1}(z).

Proof. We have

f_{∗}𝔣_{p} ≡ x^{mλ(z)}·y^{μ_{0}(z)}·x^{−1}·y^{μ_{1}(z)}·x·. . .·x^{−(m−1)}·y^{μ_{m−1}(z)}·x^{m−1}·(y, x)^{m(ν_{0}(z)+...+ν_{m−1}(z))} ≡

x^{mλ(z)} · y^{μ_{0}(z)+...+μ_{m−1}(z)} · (y, x)^{m(ν_{0}(z)+...+ν_{m−1}(z))+∑_{i=0}^{m−1} iμ_{i}(z)} mod Γ^{3}π_{1}(VQ̄; \overrightarrow{01}).

Observe that f_{∗}𝔣_{p} = 𝔣_{f(p)}. Comparing exponents of f_{∗}𝔣_{p} and 𝔣_{f(p)} we get the equalities of the lemma.

Let q_{i} be a path from \overrightarrow{0ξ_{m}^{i}} to \overrightarrow{01} as on Picture 4.

Picture 4

Let us set x_{i} := q_{i}^{−1} · x′ · q_{i} and y_{k}^{(i)} := q_{i}^{−1} · y_{k} · q_{i}. Let f_{i} : Y → V be given by f_{i}(z) = ξ_{m}^{−i}·z. Observe that (f_{i})_{∗}\overrightarrow{0ξ_{m}^{i}} = \overrightarrow{01}, (f_{i})_{∗}(x_{i}) = x, (f_{i})_{∗}(y_{i}^{(i)}) = y and (f_{i})_{∗}(y_{k}^{(i)}) = 1 for k ≠ i.

Lemma 11.1.3. We have

κ_{ξ_{m}^{−i}z}^{0} = λ(z) + \frac{i}{m}(1 − χ), κ_{ξ_{m}^{−i}z}^{1} = μ_{i}(z) and κ_{ξ_{m}^{−i}z}^{2} = ν_{i}(z) + \frac{i}{m}(1 − χ)μ_{i}(z).

Proof. f_{i}(pq_{i}) is a path from \overrightarrow{01} to ξ_{m}^{−i}z. Hence we have

𝔣_{f_{i}(pq_{i})} ≡ x^{κ_{ξ{−i \atop m}z}^{0}} · y^{κ_{ξ{−i \atop m}z}^{1}} · (y, x)^{κ_{ξ{−i \atop m}z}^{2}} mod Γ^{3}π_{1}(VQ̄; \overrightarrow{01}),

by the Definition 11.0.3. On the other side

(f_{i})_{∗}𝔣_{pq_{i}} = (f_{i})_{∗}(q_{i}^{−1} · 𝔣_{p} · q_{i}) · (f_{i})_{∗}(𝔣_{q_{i}}).

Hence it follows from 11.1.1 that

(f_{i})_{∗}𝔣_{pq_{i}} ≡ x^{λ(z)} ·y^{μ_{i}(z)} ·(y, x)^{ν_{i}(z)} ·x^{\frac{i}{m} (1−χ)} ≡ x^{λ(z)+ \frac{i}{m} (1−χ)} ·y^{μ_{i}(z)} ·(y, x)^{ν_{i}(z)+ \frac{i}{m} (1−χ)μ_{i}(z)}

mod Γ^{3}π_{1}(VQ̄; \overrightarrow{01}). We have the identity

(f_{i})_{∗}𝔣_{pq_{i}} = 𝔣_{f_{i}(pq_{i})}.

Hence comparing exponents of (f_{i})_{∗}𝔣_{pq_{i}} and 𝔣_{f_{i}(pq_{i})} we get the equalities of the lemma.

Proposition 11.1.4. Let l_{2}(z^{m}) be calculated along the path f(p) and let l_{2}(ξ_{m}^{−i}z) be calculated along the Q_{l}-path f_{i}(pq_{i}) · x^{\frac{i}{m}} for i = 0, 1, …, m − 1. Then we have

l_{2}(z^{m}) = m( ∑_{i=0}^{m−1} l_{2}(ξ_{m}^{−i}z)).

Proof. It follows from Corollary 11.0.7 that

l_{2}(z)_{p}(σ) = κ_{z}^{2}(σ)_{p} − \frac{1}{2}κ_{z}^{0}(σ)_{p} · κ_{z}^{1}(σ)_{p}.

Hence it follows from Lemma 11.1.2 that

l_{2}(z^{m})_{f(p)} = m( ∑_{i=0}^{m−1} ν_{i}(z)) + ∑_{i=0}^{m−1} iμ_{i}(z) − \frac{1}{2}mλ(z)( ∑_{i=0}^{m−1} μ_{i}(z)).

Let us calculate l_{2}(ξ_{m}^{−i}z)_{f_{i}(pq_{i})·x^{\frac{i}{m}}} . We have

𝔣_{f_{i}(pq_{i})·x^{\frac{i}{m}}} (σ) = x^{− \frac{i}{m}} · 𝔣_{f_{i}(pq_{i})}(σ) · x^{\frac{i}{m} χ(σ)}.

Hence it follows from Lemma 11.1.3 that

l_{2}(ξ_{m}^{−i}z)_{f_{i}(pq_{i})x^{\frac{i}{m}}} = ν_{i}(z) + \frac{i}{m}μ_{i}(z) − \frac{1}{2}λ(z)μ_{i}(z).

Comparing formulas for l_{2}(z^{m})_{f(p)} and l_{2}(ξ_{m}^{−i}z)_{f_{i}(pq_{i})·x^{\frac{i}{m}}} we get

l_{2}(z^{m})_{f(p)} = m( ∑_{i=0}^{m−1} l_{2}(ξ_{m}^{−i}z)_{f_{i}(pq_{i})·x^{\frac{i}{m}}}).

The classical dilogarithm satisfy the functional equation

Li_{2}(\frac{(1 − y)z}{z − 1} ) − Li_{2}(yz) + Li_{2}(\frac{(z − 1)y}{1 − y}) − Li_{2}(\frac{y}{y − 1}) + Li_{2}(z) = lower degree terms.

We shall prove its l-adic analog.

Let V = P_{K}^{1} \ \{0, 1, ∞\} and let Y = P_{K}^{1} \ \{0, 1, \frac{1}{y} , ∞\}, where y ∈ K \ \{0, 1\}. Let ξ, ζ ∈ V̂ (K) and let π be a path from ξ to ζ and let ρ be a path from \overrightarrow{01} to ξ (see Picture 5).

Picture 5

Let us set

x′ = ρ · x · ρ^{−1} and y′ = ρ · y · ρ^{−1}

where x, y are generators of π_{1}(VK̄; \overrightarrow{01}) as in 11.0. We define functions 𝔨(π), 𝔨_{1}(π) and 𝔨_{2}(π) from G_{K} to Z_{l} by the following congruence

𝔣_{π} ≡ x^{′𝔨(π)} · y^{′𝔨_{1}(π)} · (y′, x′)^{𝔨_{2}(π)} mod Γ^{3}π_{1}(VK̄; ξ).

Lemma 11.1.5. i). We have

l_{2}(ζ)_{πρ} − l_{2}(ξ)_{ρ} = 𝔨_{2}(π) − \frac{1}{2}𝔨(π)𝔨_{1}(π) − \frac{1}{2}κ_{ζ}^{0}κ_{ξ}^{1} + \frac{1}{2}κ_{ξ}^{0}κ_{ζ}^{1}.

ii). If we replace ρ by ρ_{1} = ρ·x^{a} then in terms of new generators x′′ = ρ_{1} ·x·ρ_{1}^{−1}, y′′ = ρ_{1} · y ·ρ_{1}^{−1} the triple 𝔨(π), 𝔨_{1}(π), 𝔨_{2}(π) is replaced by the triple 𝔨(π), 𝔨_{1}(π), 𝔨_{2}(π)+a𝔨_{1}(π).

Proof. It follows from the formula 𝔣_{πρ} = ρ^{−1}𝔣_{π}ρ · 𝔣_{ρ} (see Lemma 1.0.6) that Λ_{πρ}(σ) = Λ_{π}(σ)·Λ_{ρ}(σ), where Λ_{π}(σ) is the image of 𝔣_{π} by the embedding of π_{1}(VK̄; ξ) into Q_{l}{{X, Y }} sending x′ to e^{X} and y′ to e^{Y} . Applying logarithm we get

logΛ_{πρ}(σ) ◯ (−logΛ_{ρ}(σ)) = logΛ_{π}(σ).

Comparing coefficient at [Y, X] we get

l_{2}(ζ)_{πρ} − l_{2}(ξ)_{ρ} = 𝔨_{2}(π) − \frac{1}{2}𝔨(π)𝔨_{1}(π) − \frac{1}{2}κ_{ζ}^{0}κ_{ξ}^{1} + \frac{1}{2}κ_{ξ}^{0}κ_{ζ}^{1}.

The second part of the lemma follows from the congruence y′′ = x′^{a}·y′·x′^{−a} ≡ y′·(y′, x′)^{−a} mod Γ^{3}π_{1}(VK̄; ξ). Definition 11.1.6. Let us set

K(ζ, ξ) := −κ_{ζ}^{0}κ_{ξ}^{1} + κ_{ξ}^{0}κ_{ζ}^{1}.

Observe that K(ζ, ξ) is a function from G_{K} to Q_{l}. After the restriction to G_{K(μ_{l^{∞}})} the function K(ζ, ξ) does not depend on a choice of paths from \overrightarrow{01} to ξ and ζ.

Now we start to look for l-adic analog of the 5-term functional equation of the classical dilogarithm. Let f(z) = \frac{(1−y)z}{z−1} , g(z) = yz, h(z) = \frac{(z−1)y}{1−y} and k(z) = z. Observe that f, g, h and k define regular maps from Y to V. Let v be a tangential base point at 0 corresponding to the local parameter yz at 0. Let x_{0}, x_{1}, x_{y}, x_{∞} be geometric generators of π_{1}(YQ̄; v) - loops around 0, 1, \frac{1}{y} and ∞ respectively (see Picture 6).

Picture 6

We assume that

x_{∞} · x_{y} · x_{1} · x_{0} = 1.

Let z ∈ Ŷ (K) and let p ∈ π(YK̄; z, v). We introduce functions λ(z), μ(z), ν(z), α(z), β(z) and γ(z) from G_{K} to Z_{l} by the following congruence 𝔣_{p} ≡ x_{0}^{λ(z)} · x_{1}^{μ(z)} · x_{y}^{ν(z)} · (x_{1}, x_{0})^{α(z)} · (x_{y}, x_{0})^{β(z)} · (x_{y}, x_{1})^{γ(z)} mod Γ^{3}π_{1}(YK̄; v). 11.1.7.

We recall that f : Y → V is given by f(z) = \frac{(1−y)z}{z−1} . Observe that f_{∗}(v) = w, where w is a tangential base point at 0 corresponding to the local parameter \frac{y}{y−1} · z at 0. Let us set x′ := f_{∗}(x_{0}) and y′ := f_{∗}(x_{y}). Observe that f(∞) = 1 − y. This implies that f_{∗}(x_{∞}) = 1. Therefore f_{∗}(x_{1}) = y^{′−1} · x^{′−1}. Let q be a path from \overrightarrow{01} to f_{∗}(v) such that q · x · q^{−1} = x′ and q · y · q^{−1} = y′ (see Picture 7).

Picture 7

By the definition of functions 𝔨 and 𝔨_{i} we have

𝔣_{f(p)} ≡ x^{′𝔨(f(p))} · y^{′𝔨_{1}(f(p))} · (y′, x′)^{𝔨_{2}(f(p))} mod Γ^{3}π_{1}(VK̄, f_{∗}(v)).

Applying f_{∗} to 11.1.7 we get f_{∗}𝔣_{p} ≡ x^{′λ(z)−μ(z)} · y^{′ν(z)−μ(z)} · (y′, x′)^{−α(z)+β(z)−γ(z)+ \frac{1}{2} μ(z)^{2}+ \frac{1}{2} μ(z)} mod Γ^{3}π_{1}(VK̄, f_{∗}(v)). The equality f_{∗}𝔣_{p} = 𝔣_{f(p)} implies 𝔨(f(p)) = λ(z) − μ(z), 𝔨_{1}(f(p)) = ν(z) − μ(z) 11.1.8. and

𝔨_{2}(f(p)) = −α(z) + β(z) − γ(z) + \frac{1}{2}μ(z)^{2} + \frac{1}{2}μ(z). 11.1.8.

We recall that g : Y → V is given by g(z) = yz. Observe that g_{∗}(v) = \overrightarrow{01}, g_{∗}(x_{0}) = x, g_{∗}(x_{1}) = 1 and g_{∗}(x_{y}) = y. Comparing coefficients of 𝔣_{g(p)} and g_{∗}𝔣_{p} we get

𝔨(g(p)) = λ(z), 𝔨_{1}(g(p)) = ν(z), 𝔨_{2}(g(p)) = β(z). 11.1.9.

We recall that h : Y → V is given by h(z) = \frac{(z−1)y}{1−y} . Observe that h_{∗}(v) = \frac{y}{y−1}. Let us set x_{h} := h_{∗}(x_{1}) and y_{h} := h_{∗}(x_{y}). Notice that h_{∗}(x_{0}) = 1. Let r be a path from \overrightarrow{01} to \frac{y}{y−1} such that r · x · r^{−1} = x_{h} and r · y · r^{−1} = y_{h} (see Picture 8).

Picture 8

Comparing coefficients of 𝔣_{h(p)} and h_{∗}𝔣_{p} we get

𝔨(h(p)) = μ(z), 𝔨_{1}(h(p)) = ν(z), 𝔨_{2}(h(p)) = γ(z). 11.1.10.

We recall that k : Y → V is given by k(z) = z. Observe that k_{∗}(v) = v. Let us set x_{k} := k_{∗}(x_{0}) and y_{k} := k_{∗}(x_{1}). We have k_{∗}(x_{y}) = 1. Let s be a path from \overrightarrow{01} to v such that

s · x · s^{−1} = x_{k} and s · y · s^{−1} = y_{k}

(see Picture 9).

Picture 9

Comparing coefficients of 𝔣_{k(p)} and k_{∗}𝔣_{p} we get

𝔨(k(p)) = λ(z), 𝔨_{1}(k(p)) = μ(z), 𝔨_{2}(k(p)) = α(z). 11.1.11.

It follows from the equalities 11.1.8 - 11.1.11 that

𝔨_{2}(f(p)) − \frac{1}{2}𝔨(f(p))𝔨_{1}(f(p)) − 𝔨_{2}(g(p)) + \frac{1}{2}𝔨(g(p))𝔨_{1}(g(p))+ 11.1.12.

𝔨_{2}(h(p)) − \frac{1}{2}𝔨(h(p))𝔨_{1}(h(p)) + 𝔨_{2}(k(p)) − \frac{1}{2}𝔨(k(p))𝔨_{1}(k(p)) = \frac{1}{2}μ(z).

Lemma 11.1.13. On G_{K(μ_{l^{∞}})} we have the following equality

K(f(z), f_{∗}(v)) − K(g(z), g_{∗}(v)) + K(h(z), h_{∗}(v)) + K(k(z), k_{∗}(v)) = 0.

Proof. We recall that κ_{0}(z) = κ_{z}^{0} and κ_{1}(z) = κ_{z}^{1}. Hence we have

K(f(z), f_{∗}(v))−K(g(z), g_{∗}(v))+K(h(z), h_{∗}(v))+K(k(z), k_{∗}(v)) = K(f(z), w)−K(g(z), \overrightarrow{01})+

K(h(z), \frac{y}{y − 1}) + K(k(z), v) = −κ_{0}(\frac{(1 − y)z}{z − 1} )κ_{1}(w) + κ_{0}(w)κ_{1}(\frac{(1 − y)z}{z − 1} ) + κ_{0}(yz)κ_{1}(\overrightarrow{01})

−κ_{0}(\overrightarrow{01})κ_{1}(yz)−κ_{0}(\frac{(z − 1)y}{1 − y})κ_{1}(\frac{y}{y − 1})+κ_{0}(\frac{y}{y − 1})κ_{1}(\frac{(z − 1)y}{1 − y})−κ_{0}(z)κ_{1}(v)+κ_{0}(v)κ_{1}(z).

One checks that κ_{0}(\overrightarrow{0a}) = κ_{0}(a) and κ_{1}(\overrightarrow{0a}) = 0. The lemma follows from the fact that κ_{0}(x · y) = κ_{0}(x) + κ_{0}(y) and κ_{1}(z) = κ_{0}(1 − z) on G_{K(μ_{l^{∞}})}.

Theorem 11.1.14. There are paths (Q_{2}-paths if l = 2) from \overrightarrow{01} to points \frac{(1−y)z}{z−1} , yz, \frac{(z−1)y}{1−y} , \frac{y}{y−1} and z such that on G_{K(μ_{l^{∞}})} for l-adic dilogarithms calculated along these paths we have

l_{2}(\frac{(1 − y)z}{z − 1} ) − l_{2}(yz) + l_{2}(\frac{(z − 1)y}{1 − y}) − l_{2}(\frac{y}{y − 1}) + l_{2}(z) = 0.

Proof. It follows from Lemma 11.1.5, the equality 11.1.12 and Lemma 11.1.13 that

l_{2}(f(z))−l_{2}(f_{∗}(v))−l_{2}(g(z))+l_{2}(g_{∗}(v))+l_{2}(h(z))−l_{2}(h_{∗}(v))+l_{2}(k(z))−l_{2}(k_{∗}(v)) = \frac{1}{2}μ(z).

To eliminate \frac{1}{2}μ(z) we replace the path s by s′ = s · x^{− \frac{1}{2}} . Then x′_{k} = s′ · x · s^{′−1} = x_{k} and y′_{k} = (s · x^{− \frac{1}{2}} ) · y · (s · x^{− \frac{1}{2}} )^{−1} = s · y · (y, x)^{\frac{1}{2}} · s^{−1} = y_{k} · (y_{k}, x_{k})^{\frac{1}{2}} . In terms of generators x′_{k} and y′_{k} of π_{1}(VK̄; v) we have

𝔨_{2}(k(p)) = α(z) − \frac{1}{2}μ(z).

Observe that l_{2}(\overrightarrow{0a}) = 0. Hence we get

l_{2}(\frac{(1 − y)z}{z − 1} ) − l_{2}(yz) + l_{2}(\frac{(z − 1)y}{1 − y}) − l_{2}(\frac{y}{y − 1}) + l_{2}(z) = 0

for l-adic dilogarithms calculated along the paths f(p) · q, g(p), h(r) · r, r and k(p) · s · x^{− \frac{1}{2}} respectively.

It would be interesting to choose paths in such a way that we get the Abel equation on G_{K} without lower degree terms. 11.2. Now we shall discuss functional equations of arbitrary l-adic polylogarithms. The next result is a corollary of Theorem 10.0.7. We recall that a subgroup G_{n+1} of π_{1}(VQ̄; \overrightarrow{01}) was defined at the end of subsection 11.0.

We are not able to show that after a suitable choice of paths l-adic polylogarithms satisfy functional equations without lower degree terms. We have only the following result.

Theorem 11.2.1. Let K be a number field and let V = P_{K}^{1} \ {0, 1, ∞}. Let a_{1}, …, a_{m+1} be K-points of P_{K}^{1} and let Y = P_{K}^{1} \ {a_{1}, …, a_{m+1}}. Let n_{i} ∈ Z for i = 1 …, N and let f_{i} : Y → V be regular maps defined over K for i = 1, …, N. Let z, v ∈ Ŷ (K). Let us assume that ∑_{i=1}^{N} n_{i}(f_{i})_{∗} = 0 in

Hom(Γ^{n}π_{1}(YK̄; ∗)/Γ^{n+1}π_{1}(YK̄; ∗); Γ^{n}π_{1}(VQ̄; \overrightarrow{01})/G_{n+1}).

Then we have a functional equation

∑_{i=1}^{N} n_{i}(L_{n}(f_{i}(z)) − L_{n}(f_{i}(v)) = 0

on the subgroup H_{n}(Y; z, v) of G_{K}.

Proof. The theorem follows from Theorem 10.0.7 and Proposition 11.0.15.

Corollary 11.2.2. Let ξ_{m} be a primitive m-th root of 1. Then we have

m^{n−1}( ∑_{k=0}^{m−1} L_{n}(ξ_{m}^{k}z)) = L_{n}(z^{m})

on the subgroup H_{n}(P_{Q(μ_{m})}^{1} \ \{0, μ_{m}, ∞\}; z, \overrightarrow{01}) of G_{Q(μ_{m})}.

In Part III we shall need a special case of the equality from Corollary 11.2.2.

Corollary 11.2.3. Let ξ_{m} be a primitive m-th root of 1. Then we have

m^{n−1}( ∑_{k=0}^{m−1} L_{n}(ξ_{m}^{k})) = L_{n}(1)

on the subgroup H_{n}(P_{Q(μ_{m})} \ \{0, μ_{m}, ∞\}; \overrightarrow{10}, \overrightarrow{01}) of G_{Q(μ_{m})}, where L_{n}(1) := L_{n}(\overrightarrow{10}).

Both corollaries follow immediately from Theorem 11.2.1. We give however a detailed proof of Corollary 11.2.3 because of it importance in Part III. Proof of Corollary 11.2.3.

We shall use the notation of section 11.1, where we discussed the l-adic analog of the functional equation Li_{2}(z^{m}) = m(∑_{i=0}^{m} Li_{2}(ξ_{m}^{i}z)). We shall use also the following notation. If a and b are elements of a group then (a, b^{1}) := (a, b) = a · b · a^{−1} · b^{−1} and (a, b^{n}) := ((a, b^{n−1}), b) for n > 1.

We recall that Y = P_{Q(μ_{m})}^{1} \ {0, μ_{m}, ∞} and f : Y → V is given by f(z) = z^{m}. Let p be a path from \overrightarrow{01} to \overrightarrow{10}, the interval [0, 1]. Let σ ∈ H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}). Then we have

𝔣_{p}(σ) ≡ (y_{0}, x^{′n−1})^{ν_{0}^{n}(\overrightarrow{10})(σ)} · . . . · (y_{m−1}, x^{′n−1})^{ν_{m−1}^{n}(\overrightarrow{10})(σ)}

modulo a subgroup generated by Γ^{n+1}π_{1}(YK̄; \overrightarrow{01}) and commutators which contain at least two y’s. Observe that f(p) is a path from \overrightarrow{01} to m · \overrightarrow{10}. Then for any σ ∈ H_{n}(V; \overrightarrow{10}, \overrightarrow{01}), and therefore also for any σ ∈ H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}) we have

𝔣_{f(p)}(σ) ≡ (y, x^{n−1})^{κ\overrightarrow{10}^{n}(σ)} mod G_{n+1}.

It follows from the equality f_{∗}𝔣_{p} = 𝔣_{f(p)} that

m^{n−1}(ν_{0}^{n}(\overrightarrow{10}) + … + ν_{m−1}^{n}(\overrightarrow{10})) = κ\overrightarrow{10}^{n} 11.2.4.

on H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}). We recall that f_{i} : Y → V is given by f_{i}(z) = ξ_{m}^{−i} · z. Observe that (f_{i})_{∗}𝔣_{pq_{i}}(σ) ≡ (y, x^{n−1})^{ν_{i}(\overrightarrow{10})(σ)} mod G_{n+1} for σ ∈ H_{n}(Y; \overrightarrow{10}, \overrightarrow{0ξ_{m}^{i}}) = H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}) and 𝔣_{f_{i}(pq_{i})}(σ) ≡ (y, x^{n−1})^{κ_{ξ{−i \atop m}}^{n} (σ)} mod G_{n+1} for σ ∈ H_{n}(Y; \overrightarrow{ξ_{n}^{−i}0}, \overrightarrow{01}) = H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}), where q_{i} is a path from \overrightarrow{0ξ_{m}^{i}} to \overrightarrow{01} as on Picture 4. Hence we get

ν_{i}^{n}(\overrightarrow{10}) = κ_{ξ_{m}^{−i}}^{n} 11.2.5. on H_{n}(Y; \overrightarrow{ξ_{n}^{−i}0}, \overrightarrow{01}) = H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}). It follows from 11.2.4 and 11.2.5 that

m^{n−1}( ∑_{i=0}^{m−1} κ_{ξ_{m}^{−i}}^{n}) = κ\overrightarrow{10}^{n}

on H_{n}(Y; \overrightarrow{ξ_{n}^{−i}0}, \overrightarrow{01}) = H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}). For σ ∈ H_{n}(Y; \overrightarrow{10}, \overrightarrow{01}) we have κ_{ξ_{m}^{−i}}^{n}(σ) = L_{n}(ξ_{m}^{−i})(σ) and κ\overrightarrow{10}^{n}(σ) = L_{n}(\overrightarrow{10})(σ). This finishes the proof of Corollary 11.2.3.

One of the most useful functional equations of classical polylogarithms is the relation between Li_{n}(z) and Li_{n}(\frac{1}{z}). For l-adic polylogarithms we have the following result.

Corollary 11.2.6. For any z ∈ V (K), we have

L_{n}(z) + (−1)^{n}L_{n}(\frac{1}{z}) = 0

on the subgroup H_{n}(VQ̄; z, \overrightarrow{01}).

Proof. It follows from Theorem 11.2.1 that

L_{n}(z) − L_{n}(\overrightarrow{01}) + (−1)^{n}(L_{n}(\frac{1}{z}) − L_{n}(\overrightarrow{∞1})) = 0.

L_{n}(\overrightarrow{01}) vanishes. Hence we have to calculate L_{n}(\overrightarrow{∞1}). Let p a path from \overrightarrow{01} to \overrightarrow{10} and let s a path from \overrightarrow{10} to \overrightarrow{1∞} as on the picture.

Picture 10

Let g : V → V be given by g(z) = \frac{1}{z}. Let us set q := g(p)^{−1} · s · p. We denote by π′′ the subgroup [Γ^{2}π_{1}(VK̄; \overrightarrow{01}), Γ^{2}π_{1}(VK̄; \overrightarrow{01})] of π_{1}(VK̄; \overrightarrow{01}). Let (Γ^{n+1}π_{1}(VK̄; \overrightarrow{01}), π′′) be a normal subgroup of π_{1}(VK̄; \overrightarrow{01}) generated by Γ^{n+1}π_{1}(VK̄; \overrightarrow{01}) and π′′.

Let σ ∈ H_{n}(VQ̄; z, \overrightarrow{01}). Then we have

𝔣_{q}(σ) = ∏_{i+j=n,i≥1,j≥1}(((y, x)x^{i−1})y^{j−1})^{κ_{i,j}(\overrightarrow{∞1})(σ)} mod (Γ^{n+1}π_{1}(VK̄, \overrightarrow{01}), π′′) 11.2.7.

for some κ_{i,j}(\overrightarrow{∞1})(σ) ∈ Z_{l}. It follows from Lemma 1.0.6 and from equality 10.0.1 that

𝔣_{q} = q^{−1} · g_{∗}(𝔣_{p})^{−1} · q · p^{−1} · 𝔣_{s} · p · 𝔣_{p}. 11.2.8.

Observe that q^{−1} · g_{∗}(y) · q = y and q^{−1} · g_{∗}(x) · q = x^{−1} · y^{−1}. 11.2.9. Let σ ∈ H_{n}(VQ̄; z, \overrightarrow{01}). Then we have

𝔣_{p}(σ) = ∏_{i+j=n,i≥1,j≥1}(((y, x)x^{i−1})y^{j−1})^{κ_{i,j}(\overrightarrow{10})(σ)} mod (Γ^{n+1}π_{1}(VK̄, \overrightarrow{01}), π′′) 11.2.10.

for some κ_{i,j}(\overrightarrow{10})(σ) ∈ Z_{l}. It follows from 11.2.7 - 11.2.10 that

κ_{n−1,1}(\overrightarrow{∞1}) = (−1)^{n}κ_{n−1,1}(\overrightarrow{10})(σ) + κ_{n−1,1}(\overrightarrow{10})(σ).

Hence κ_{n−1,1}(\overrightarrow{∞1}) = 0 if n is odd.

We shall show that κ_{n−1,1}(\overrightarrow{10}) vanishes for n even. Let x, y and z be generators of π_{1}(VK̄; \overrightarrow{01}) as on the picture.

Picture 11

Then we have z · y · x = 1. It follows from Proposition 2.2.1 that

(𝔣_{q}(σ)(x, y))^{−1} · z^{χ(σ)} · (𝔣_{q}(σ)(x, y)) · (𝔣_{p}(σ)(x, y))^{−1} · y^{χ(σ)} · (𝔣_{p}(σ)(x, y)) · x^{χ(σ)} = 1. Let σ ∈ H_{n}(VK̄; z, \overrightarrow{01}). It follows from 11.2.8 and 11.2.9 that

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

Hence we get

(𝔣_{p}(σ)(x, y))^{−1} · (𝔣_{p}(σ)(x^{−1}y^{−1}, y)) · x^{−1} · y^{−1} · (𝔣_{p}(σ)(x^{−1}y^{−1}, y))^{−1} · (𝔣_{p}(σ)(x, y))·

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

Comparing exponents at (y, x^{n}) we get (1 + (−1)^{n})κ_{n−1,1}(\overrightarrow{10}) = 0. Hence κ_{n−1,1}(\overrightarrow{10}) = 0 for n even (see also [I1], [I2] and [D], where the element 𝔣_{p}(σ) is studied). Therefore κ_{n−1,1}(\overrightarrow{∞1}) = 0 for any n. The equality κ_{n−1,1}(\overrightarrow{∞1}) = L_{n}(\overrightarrow{∞1}) implies the corollary.

The fact that κ_{n−1,1}(\overrightarrow{10}) vanishes for n even implies the following well known result. Corollary 11.2.11.

L_{2n}(\overrightarrow{10}) = 0.

§12. Monodromy of l-adic iterated integrals and l-adic polylogarithms. 12.0. We shall show here that suitable defined l-adic polylogarithms form a local system with the similar shape of the monodromy representation as the local system of classical polylogarithms given in [BD]. We start with the discussion of the monodromy of arbitrary l-adic iterated integrals. The notation is the same as in section 10.

Let p be a path from v to z on XK̄ and let S ∈ π_{1}(XK̄; v). Then we have

𝔣_{pS}(σ) = S^{−1} · 𝔣_{p}(σ) · S · 𝔣_{S}(σ). 12.0.0.

Let Map(G_{K}; π_{1}(XK̄; v)) be the set of all maps from G_{K} to π_{1}(XK̄; v). We define a map

r_{z,v;p} : π_{1}(XK̄; v) → Aut_{set}(Map (G_{K}; π_{1}(XK̄; v)))

setting

r_{z,v;p}(S)(w)(σ) := S^{−1} · w(σ) · S · 𝔣_{S}(σ),

for S ∈ π_{1}(XK̄; v), w ∈ Map (G_{K}; π_{1}(XK̄; v)) and σ ∈ G_{K}.

Further we drop the indices _{z,v;p} to simplify the notation.

Lemma 12.0.1. The map r_{z,v;p} is a representation of π_{1}(XK̄; v).

Proof. Let S, T ∈ π_{1}(XK̄; v). We have r(T)(r(S)w)(σ) = T^{−1}(S^{−1} · w(σ) · S · 𝔣_{S}(σ)) · T · 𝔣_{T} (σ) = (S · T)^{−1} · w(σ) · (S · T) · (T^{−1} · 𝔣_{S}(σ) · T · 𝔣_{T} (σ)) = (S · T)^{−1} · w(σ) · (S · T) · 𝔣_{ST} (σ) = r(S · T)(w)(σ). We recall that in our notation S · T means that first we go along T and then along S. Therefore r is a representation of π_{1}(XK̄; v).

We recall that k_{x} : π_{1}(XK̄; v) → Q_{l}{{X}} is a continous multiplicative embedding given by k_{x}(x_{i}) = e^{X_{i}} for i = 1, …, n and that for a path p from v to z we set Λ_{p}(σ) := k_{x}(𝔣_{p}(σ)).

Let Map(G_{K}; Q_{l}{{X}}) be the set of all maps from G_{K} to Q_{l}{{X}}. Observe that Map(G_{K}; Q_{l}{{X}}) is a vector space over Q_{l}. We denote by GL(Map (G_{K}; Q_{l}{{X}})) the group of linear automorphisms of the vector space Map(G_{K}; Q_{l}{{X}}).

Let us define a map

R_{z,v;p} : π_{1}(XK̄; v) → GL(Map (G_{K}; Q_{l}{{X}}))

setting

R_{z,v;p}(S)(W)(σ) := k_{x}(S)^{−1} · W (σ) · k_{x}(S) · Λ_{S}(σ).

Proposition 12.0.2. The map R_{z,v;p} is a representation of π_{1}(XK̄; v).

Proof. To simplify the notation let us set R = R_{z,v;p}. Let S, T ∈ π_{1}(XK̄; v). We have R(T)(R(S)(W))(σ) = k_{x}(T)^{−1} ·(R(S)(W)(σ))·k_{x}(T)·Λ_{T} (σ) = k_{x}(T)^{−1} ·(k_{x}(S)^{−1} ·W (σ)· k_{x}(S)·Λ_{S}(σ))·k_{x}(T)·Λ_{T} (σ) = k_{x}(S·T)^{−1}·W (σ)·k_{x}(S·T)·k_{x}(T)^{−1}·Λ_{S}(σ)·k_{x}(T)·Λ_{T} (σ) = R(S · T)(W)(σ).

It follows from Lemma 10.3.1 that the embedding k_{x} : π_{1}(XK̄; v) → Q_{l}{{X}} extends uniquely to a continous multiplicative embedding k̄_{x} : π_{1}(XK̄; v) ⊗ Q → Q_{l}{{X}}.

Proposition 12.0.3. The representation R_{z,v;p} extends to the representation

R̄_{z,v;p} : π_{1}(XK̄; v) ⊗ Q → GL(Map (G_{K}; Q_{l}{{X}})).

Let S ∈ π_{1}(XK̄; v) ⊗ Q. Then we have

R̄_{z,v;p}(S)(W)(σ) = k̄_{x}(S)^{−1} · W (σ) · k̄_{x}(S) · R̄_{z,v;p}(S)(1)(σ).

Proof. We define an increasing filtration {W_{−i}}_{i∈N} of the Q_{l}-vector space Map (G_{K}; Q_{l}{{X}}) setting

W_{−2k} = W_{−2k−1} to be a set of all maps from G_{K} to I^{k},

where I^{k} is a k-th power of the augmentation ideal of Q_{l}{{X}}. Let S ∈ π_{1}(XK̄; v) and let W ∈ W_{−2k}. Then we have

R_{z,v;p}(S)(W) ≡ W mod W_{−2(k+1)}.

Hence the image of R_{z,v;p} is in the subgroup of pro-unipotent automorphisms of the vector space Map (G_{K}; Q_{l}{{X}}). This implies that the representation R_{z,v;p} extends to the representation

R̄_{z,v;p} : π_{1}(XK̄; v) ⊗ Q → GL(Map (G_{K}; Q_{l}{{X}})).

Let S ∈ π_{1}(XK̄; v) ⊗ Q be such that S^{l^{m}} ∈ π_{1}(XK̄; v). Then we have

R_{z,v;p}(S^{l^{m}})(W)(σ) = k_{x}(S^{l^{m}})^{−1} · W (σ) · k_{x}(S^{l^{m}}) · Λ_{S^{l^{m}}} (σ),

where Λ_{S^{l^{m}}} (σ) = R_{z,v;p}(S^{l^{m}})(1)(σ). This implies that

R̄_{z,v;p}(S)(W)(σ) = k̄_{x}(S)^{−1} · W (σ) · k̄_{x}(S) · R̄_{z,v;p}(S)(1)(σ).

The elements S ∈ π_{1}(XK̄; v) ⊗ Q such that S^{l^{m}} ∈ π_{1}(XK̄; v) for some m are dense in π_{1}(XK̄; v) ⊗ Q hence the last formula holds for any S ∈ π_{1}(XK̄; v) ⊗ Q. This finishes the proof of the proposition. 12.1. Now we shall study monodromy of l-adic polylogarithms, more exactly, we shall study monodromy of coefficients at X^{n−1}Y of the power series Λ_{p}(σ). Let V = P_{K}^{1} \ \{0, 1, ∞\} and let p be a path from \overrightarrow{01} to z. From now on the notation is the same as in section 11.0.

We define functions λ_{i}(z)_{p}, μ_{j}(z)_{p} and ν_{i,j}(z)_{p} from G_{K} to Q_{l} by the congruence

Λ_{p}(σ) ≡ 1+∑_{k=1}^{∞} \frac{(l(z)_{p}(σ))^{k}}{k!} X^{k}+∑_{i=1}^{∞} λ_{i}(z)_{p}(σ)X^{i−1}Y +∑_{j=2}^{∞} μ_{j}(z)_{p}(σ)Y X^{j−1}+ ∑_{i,j=1}^{∞} ν_{i,j}(z)_{p}(σ)X^{i}Y X^{j}

modulo the ideal generated by monomials with at least two Y ’s.

The function λ_{1}(z)_{p} = l_{1}(z)_{p} and the l-adic polylogarithms l_{k}(z)_{p} can be expressed by the function λ_{k}(z)_{p} and the functions l(z)_{p} and λ_{i}(z)_{p} with i < k.

Proposition 12.1.1. The monodromy transformation of functions l(z)_{p} and λ_{n}(z)_{p} is as follows:

x : l(z)_{p} → l(z)_{p} + (χ − 1), λ_{n}(z)_{p} → λ_{n}(z)_{p} + ∑_{i=1}^{n−1} \frac{(−1)^{n−i}}{(n − i)!} λ_{i}(z)_{p},

μ_{n}(z)_{p} → μ_{n}(z)_{p} + ∑_{i=2}^{n−1} \frac{χ^{n−i}}{(n − i)!}μ_{i}(z)_{p} + \frac{χ^{n−1}}{(n − 1)!}λ_{1}(z)_{p}

and

y : l(z)_{p} → l(z)_{p}, λ_{1}(z)_{p} → λ_{1}(z)_{p} + (χ − 1), λ_{n}(z)_{p} → λ_{n}(z)_{p} + χ\frac{(l(z)_{p})^{n−1}}{(n − 1)!}

for n > 1 and μ_{n}(z)_{p} → μ_{n}(z)_{p} − \frac{(l(z)_{p})^{n−1}}{(n−1)!} .

Proof. The proposition follows from the formula

Λ_{p·S}(σ) = k(S)^{−1} · Λ_{p}(σ) · k(S) · Λ_{S}(σ),

which for S = x gives

Λ_{p·x}(σ) = e^{−X} · Λ_{p}(σ) · e^{χ(σ)X}.

For S = y the formula is more complicated, however when we restrict our attention to coefficients with only one Y then the formula have the same simple form.

References

[B] N.Bourbaki, Eléments de mathématiques, Groupes et algèbres de Lie, Diffusion C.C.L.S. Paris 1972.

[BD] A.A.Beilinson, P.Deligne, Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulareurs, in U.Jannsen, S.L.Kleiman, J.-P.Serre, Motives Proc. of Symp. in Pure Math. 55, Part II AMS 1994 pp. 97-121.

[Ch] K.T.Chen, Iterated integrals, fundamental groups and covering spaces, Trans. of the Amer. Math. Soc. Vol. 206, 1975, 83-98.

[D] P.Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois Groups over Q (ed. Y.Ihara, K.Ribet and J.-P. Serre), Mathematical Sciences Research Institute Publications, no 16 (1989), pp. 79-297.

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

[I2] Y.Ihara, Braids, Galois Groups and Some Arithmetic Functions, Proc.of the Int. Cong. of Math. Kyoto 1990, 99-119.

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

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