A remark on nilpotent polylogarithmic extensions of the field of rational functions of

one variable over ℂ ^{∗}

Z. Wojtkowiak

Laboratoire Paul Painlevé CNRS UMR 8524 Université des Sciences et Technologies de Lille, F–59665 Villeneuve d’Ascq Cedex et Université de Nice Mathématiques, Parc Valrose F – 06108 Nice Cedex 2 wojtkow@math.unice.fr

Abstract

In this note we construct explicitly a family of nilpotent polylogarithmic extensions unramified outside three points of the field ℂ(z) of rational functions of one variable over ℂ. We show how to use these extensions to calculate explicitly l-adic polylogarithms introduced in [5].

Résumé

Dans cette note, nous construisons une famille des extensions polygarithmiques nonramifiées en dehors de trois points du corps ℂ(z) de fonctions rationnelles d’une variable sur ℂ. Nous montrons comment calculer explicitement des polylogarithmes l-adiques introduits en [5].

Mots clé : polylogarithms, field extensions, Galois groups, free groups, monodromy transformations.

∗preprint

0. Classical complex polylogarithms Li_{n}(z) = ∑_{k=1}^{∞} \frac{z^{k}}{k^{n}} are related to certain nilpotent quotients of π_{1}(ℙ^{1}(ℂ)\\{0, 1, ∞\} ; \overrightarrow{01}). More precisely, the monodromy representation of the matrice function (see [2])

( 1 ; log(1 − z) 1  O ; − Li_{2}(z)  logz 1 ; − Li_{3}(z)  \frac{(logz)^{2}}{2!} logz 1 ; ⋮ ; − Li_{n}(z)  …)

factors throught the polylogarithmic quotient π_{1}(ℙ^{1}(ℂ) \ \{0, 1, ∞\} ; \overrightarrow{01})/ < Γ^{n+1}, y^{2} >, where < Γ^{n+1}, y^{2} > is a normal subgroup of π_{1}(ℙ^{1}(ℂ)\\{0, 1, ∞\}; \overrightarrow{01}) generated by commutators of length n + 1 and by commutators which contain at least two y’s. (We recall that x - loop around 0 and y - loop around 1 are standard generators of π_{1}(ℙ^{1}(ℂ) \ \{0, 1, ∞\} ; \overrightarrow{01}).

Let K be a finite extension of ℚ. In [5], we have defined l-adic polylogarithms l_{n}(z), which are coefficients of some big Galois representations of the group Gal(\overline{ℚ}/K) for z ∈ K and which non-normalized versions appear in “the same place” as the classical polylogarithms Li_{n}(z). The l-adic polylogarithms l_{n}(z) are calculated explicitly in [3].

In this note, we shall recover the explicit formula from [3] studying nilpotent polylogarithmic quotients of Galois groups of Ihara elementary extensions of ℂ(z) - the field of rational functions of one variable over ℂ. The approach is very elementary, though it is related to the one presented in [3]. It is different from the Gabber construction of the Heisenberg cover of ℙ^{1} \ {0, 1, ∞}.

1. Let ℂ(z) be a field of rational functions of one variable over ℂ, i.e., the field of rational functions on ℙ_{ℂ}^{1}. We start with the study of the Galois group of the two-stage Ihara elementary extension of ℂ(z) (see [1]).

Let l be a given prime number. Let a and b be two points of ℙ^{1}(ℂ) \ {∞}. We define extensions K_{1}^{(n)} and K_{2}^{(n,m)} of the field ℂ(z) setting

K_{1}^{(n)} := ℂ(z)((z − a)^{1/l^{n}})

and

K_{2}^{(n,m)} := K_{1}^{(n)}(((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i}(z − a)^{1/l^{n}})^{1/l^{m}} ; 0 ≤ i < l^{n}).

Observe that these extensions of ℂ(z) are algebraic, unramified outside a, b and ∞.

Below we shall calculate the Galois group of K_{2}^{(n,m)} over ℂ(z). This group is a quotient of π_{1}(ℙ^{1}(ℂ) \ {a, b, ∞} ; v) - a free group on two generators, x - loop around a and y - loop around b.

The composition of loops is from right to left. Actions of groups are left actions, i.e., (α · β)(u) = α(β(u)).

Let us denote by

φ_{n,m} : π_{1}(ℙ^{1}(ℂ) \ {a, b, ∞} ; v) → Gal(K_{2}^{(n,m)}/ℂ(z))

the natural epimorphism. Lemma 1.1 The group Gal(K_{2}^{(n,m)}/ℂ(z)) has the following presentation

Gal(K_{2}^{(n,m)}/ℂ(z)) =< α, β_{i} ; i ∈ ℤ/l^{n} | α^{l^{n}} = 1 , β_{i}^{l^{m}} = 1 , β_{i}β_{j} = β_{j}β_{i} , αβ_{i}α^{−1} = β_{i+1} >,

where α := φ_{n,m}(x) and β_{i} := φ_{n,m}(x^{i}yx^{−i}). Proof : Let us define automorphisms α and β_{i} (i ∈ ℤ/l^{n}) of the field K_{2}^{(n,m)} by setting

α((z − a)^{1/l^{n}}) := ξ_{l^{n}}^{1}(z − a)^{1/l^{n}},

α(((b − a)^{1/l^{n}} − ξ_{l^{n}}^{k}(z − a)^{1/l^{n}})^{1/l^{m}}):= ((b − a)^{1/l^{n}} − ξ_{l^{n}}^{k+1}(z − a)^{1/l^{n}})^{1/l^{m}}

for k = 0, 1, …, l^{n} − 1 and

β_{i}((z − a)^{1/l^{n}}) := (z − a)^{1/l^{n}},

β_{i}(((b − a)^{1/l^{n}} − ξ_{l^{n}}^{k}(z − a)^{1/l^{n}})^{1/l^{m}}):= ξ_{l^{m}}^{δ_{i}^{k}}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{k}(z − a)^{1/l^{n}})^{1/l^{m}}

for k = 0, 1, …, l^{n} − 1 and i ∈ ℤ/l^{n}. One easily checks that α and β_{i} (i ∈ ℤ/l^{n}) satisfy the relations of the Lemma.

Let G be a subgroup of Gal(K_{2}^{(n,m)}/ℂ(z)) generated by α and β_{i} (i ∈ ℤ/l^{n}). Observe that G has l^{n} · (l^{m})^{l^{n}} elements and that G has a presentation as in the Lemma. The field extension K_{2}^{(n,m)} of ℂ(z) is finite of degree l^{n} · (l^{m})^{l^{n}}, hence K_{2}^{(n,m)} is a Galois extension of ℂ(z) and Gal(K_{2}^{(n,m)}/ℂ(z)) = G.

Studying monodromy transformations of functions (z − a)^{1/l^{n}} and ((b − a)^{1/l^{n}} − ξ_{l^{n}}^{k}(z − a)^{1/l^{n}})^{1/l^{m}} for 0 ≤ k < l^{n} along elements x and x^{i}yx^{−i} for 0 ≤ i < l^{n} we show that α = φ_{n,m}(x) and β_{i} = φ_{n,m}(x^{i}yx^{−i}) for 0 ≤ i < l^{n}. □ Corollary 1.2 In the group Gal(K_{2}^{(n,m)}/ℂ(z)) we have

(α^{−1} · β_{0})^{l^{n}} = β_{l^{n}−1} · β_{l^{n}−2}, …, β_{1} · β_{0},

hence α^{−1} · β_{0} is an element of order l^{n+m}. □ Corollary 1.3 The group Gal(K_{2}^{(n,m)}/ℂ(z)) is isomorphic to the semi-direct product

ℤ/l^{m}[ℤ/l^{n}] ˜×_{φ}ℤ/l^{n},

where φ : ℤ/l^{n} → Aut(ℤ/l^{m}[ℤ/l^{n}]) is given by φ(˜1)(∑_{i=0}^{l^{n}−1} α_{i}[\overline{i}]) = ∑_{i=0}^{l^{n}−1} α_{i}[\overline{i + 1}]. □

2. In this section, we shall study nilpotent polylogarithmic quotients of a free group on two generators.

For any group F we denote by {Γ^{n}F }_{n∈ℕ} the lower central series of F. We denote by (Γ^{2}F, Γ^{2}F) the double commutator of F.

Let F (x, y) be a free group on two generators x and y. Let us introduce the following notation :

(x^{(0)}, y) := y , (x^{(1)}, y) := x · y · x^{−1} · y^{−1}

and

(x^{(m+1)}, y) := (x, (x^{(m)}, y)) for m > 0.

Lemma 2.1 For any natural numbers i and j, we have

x^{i} · y · x^{−i} ≡ ∏_{k=0}^{i}(x^{(k)}, y)^{\binom{i}{k}} mod (Γ^{2}F (x, y), Γ^{2}F (x, y))

and

x^{i} · (x^{(j)}, y) · x^{−i} ≡ ∏_{k=0}^{i}(x^{(k+j)}, y)^{\binom{i}{k}} mod (Γ^{2}F (x, y), Γ^{2}F (x, y)).

Proof : Observe that x · y · x^{−1} = x · y · x^{−1} · y^{−1} · y = (x^{(1)}, y) · y. Hence the first congruence is shown for i = 1. Assume that it is true for i ≤ m. Observe that x · (x^{(a)}, y) · x^{−1} = (x^{(a+1)}, y) · (x^{(a)}, y). Hence we get

x^{m+1} · y · x^{−m−1} ≡ x · ∏_{k=0}^{m}(x^{(k)}, y)^{\binom{m}{k}}) · x^{−1} ≡

∏_{k=0}^{m} ((x^{k+1}), y)^{\binom{m}{k}}·(x^{(k)}, y)^{\binom{m}{k}}) ≡ ∏_{k=0}^{m+1}(x^{(k)}, y)^{\binom{m+1}{k}} mod (Γ^{2}F (x, y), Γ^{2}F (x, y)).

The proof of the second congruence is the same. □

3. Now we shall construct nilpotent polylogarithmic quotients of F (x, y) and in consequence finite nilpotent polylogarithmic coverings of P 1 ; \overline{ℚ} \ \{a, b, ∞\} unramified outside a, b and ∞.

In a finite polylogarithmic quotient of F (x, y), we want x^{l^{n}} = 1. On the other side by Lemma 2.1 we have

x^{l^{n}} · y · x^{−l^{n}} ≡ ∏_{k=0}^{l^{n}}(x^{(k)}, y)^{\binom{l^{n}}{k}} mod (Γ^{2}F (x, y), Γ^{2}F (x, y)).

Hence in a finite polylogarithmic quotient of F (x, y) we must require that ∏_{k=0}^{l^{n}}(x^{(k)}, y)^{\binom{l^{n}}{k}} ≡ y.

We start with the discussion of arithmetic properties of numbers \binom{l^{n}}{k}. If m is a natural number we denote by v_{l}(m) the l-adic valuation of m. Lemma 3.0 Let k be a natural number such that 0 < k < l^{n}. Then we have

v_{l}(\binom{l^{n}}{k}) = n − v_{l}(k).

Proof : Let m = ∑_{i=0}^{M} a_{i} · l^{i} be the l-adic developement of a natural number m. Then the l-adic valuation of m! is given by v_{l}(m!) = \frac{1}{l−1}(m − ∑_{i=0}^{M} a_{i}).

Let k = ∑_{i=0}^{n−1} a_{i}l^{i} be the l-adic developpement of k. Using the formula for l-adic valuation of m! we get

v_{l}(l^{n}!) = \frac{1}{l − 1}(l^{n} − 1) and v_{l}(k!) = \frac{1}{l − 1}(k − ∑_{i=0}^{n−1} a_{i}).

Let us set ι := v_{l}(k). Then (l−a_{ι})l^{ι}+∑_{i=ι+1}^{n−1}(l−a_{i}−1)l^{i} is the l-adic developpement of l^{n} −k. Hence v_{l}((l^{n} −k)!) = \frac{1}{l−1}(l^{n} −k−(n−ι)l+∑_{i=0}^{n−1} a_{i} +(n−ι−1)). The lemma follows immediately from the equality v_{l}(\binom{l^{n}}{k}) = v_{l}(l^{n}!) − v_{l}(k!) − v_{l}((l^{n} − k)!). □

Let N be a positive integer such that N ≤ l^{n}. We define a natural number q(n, N) by

q(n, N) := min(\{v_{l}(\binom{l^{n}}{k}) | 0 < k < N\}).

It follows from Lemma 3.0 that i) q(n, N) ≤ n; ii) if N is fixed and n → ∞ then q(n, N) → ∞.

Lemma 3.1 Let N be a positive integer such that N ≤ l^{n}. Then for any i and any k such that i > 0 and 0 < k < N we have

\binom{i + l^{n}}{k} ≡ \binom{i}{k} mod l^{q(n,N)}.

Proof : The lemma follows from the identity (1 + T)^{i+l^{n}} = (1 + T)^{i} · (1 + T)^{l^{n}} and from Lemma 3.0. □

Now we introduce some notations and definitions.

We denote by < y^{(2)} > a normal subgroup of F (x, y) generated by commutators in x and y which contain two or more y’s.

We define a polylogarithmic quotient of F (x, y) by

P(n; N) := F (x, y)/ϕ(n, N),

where

ϕ(n, N) := ⟨Γ^{N+1}F (x, y), < y^{(2)} >, x^{l^{n}}, (x^{(k)}, y)^{l^{q(n,N)}} | 0 ≤ k ≤ N − 1⟩

is a normal subgroup of F (x, y) generated by Γ^{N+1}F (x, y), < y^{(2)} >, x^{l^{n}} and (x^{(k)}, y)^{l^{q(n,N)}} for 0 ≤ k ≤ N − 1.

It follows from Lemmas 3.0 and 3.1 that

P(n; N) ≈ (⨁_{k=0}^{N−1} ℤ/l^{q(n,N)}(x^{(k)}, y)) ˜×ℤ/l^{n}x,

i.e., P(n; N) is semi-direct product of N copies of ℤ/l^{q(n,N)} by ℤ/l^{n} and that the action of a generator x of ℤ/l^{n} on (x^{(k)}, y) is given by

x((x^{(k)}, y)) = (x^{(k)}, y) + (x^{(k+1)}, y)

if k < N − 1 and

x((x^{(N−1)}, y)) = (x^{(N−1)}, y).

Proposition 3.2 Let N ≤ l^{n} and m ≥ q(n, N). There is an epimorphism

ψ : Gal(K_{2}^{(n,m)}/ℂ(z)) → P(n, N)

such that ψ(α) ≡ x mod ϕ(n, N) and ψ(β_{i}) ≡ x^{i} · y · x^{−i} mod ϕ(n, N) for 0 ≤ i < l^{n}. Proof : We must show that ker φ_{n,m} ⊂ ϕ(n, N). Let ε_{n} : F (x, y) → ℤ/l^{n} be given by ε_{n}(x) = 1 and ε_{n}(y) = 0. Then

ker ε_{n} = F (x^{l^{n}}, x^{i}yx^{−i}; i ∈ {0, 1, …, l^{n} − 1})

is a free group on x^{l^{n}} and x^{i}yx^{−i} for i ∈ {0, 1, …, l^{n} − 1}. Hence any element g of F (x, y) can be written as a product

g = x^{a} · ∏_{i=0}^{l^{n}−1}(x^{i}yx^{−i})^{a_{i}} · g_{1},

where g_{1} ∈ Γ^{2}ker ε_{n}. If g ∈ ker φ_{n,m} then a ≡ 0 mod l^{n} and a_{i} ≡ 0 mod l^{m} for i = 0, 1, …, l^{n} − 1. Notice that the commutator subgroup of a free group is the smalllest normal subgroup containing commutators of all pairs of free generators of the group. The last two observations imply that ker φ_{n,m} ⊂ ϕ(n, N). □

We shall study ker ψ. Let us observe that any element of Gal(K_{2}^{(n,m)}/ℂ(z)) can be written uniquely in the form α^{a} · ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}}, where 0 ≤ a < l^{n} and 0 ≤ a_{i} < l^{n}. Lemma 3.3 The element α^{a} · ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} ∈ Gal(K_{2}^{(n,m)}/ℂ(z)) belongs to ker ψ if and only if a = 0 and ∑_{i=k}^{l^{n}−1} \binom{i}{k}a_{i} ≡ 0 mod l^{q(n,N)} for k < N. Proof : It follows from Lemma 2.1 that we have

ψ(α^{a} · ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}}) ≡ x^{a} · ∏_{i=0}^{l^{n}−1}(∏_{k=0}^{i}(x^{(k)}, y)^{\binom{i}{k}a_{i}}) ≡ x^{a} · ∏_{k=0}^{l^{n}−1}(x^{(k)}, y)^{∑_{i=k}^{l^{n}−1} \binom{i}{k}a_{i}} ≡

≡ x^{a} · ∏_{k=0}^{N−1}(x^{(k)}, y)^{∑_{i=k}^{l^{n}−1} \binom{i}{k}a_{i}} mod ϕ(n, N).

This finishes the proof of the lemma. □

Now we shall look for a subfield of K_{2}^{(n,m)} fixed by ker ψ. The base of K_{2}^{(n,m)} over ℂ(z) is given by

(z − a)^{k/l^{n}} ∏_{i=0}^{l^{n}−1}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i}(z − a)^{1/l^{n}})^{k_{i}/l^{m}},

where 0 ≤ k < l^{n} and 0 ≤ k_{0}, …, k_{l^{n}−1} < l^{m}. The element ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} acts on K_{2}^{(n,m)} in the following way :

( ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}})( ∑_{0≤k<l^{n}, 0≤k_{0},...,k_{l^{n}−1}<l^{m}} f_{k,k_{0},...,k_{l^{n}−1}}(z) · (z − a)^{k/l^{n}}·

∏_{i=0}^{l^{n}−1}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i}(z − a)^{1/l^{n}})^{k_{i}/l^{m}}) =

= ∑_{0≤k<l^{n}, 0≤k_{0},...,k_{l^{n}−1}<l^{m}} ξ_{l^{m}}^{∑_{i=0}^{l^{n}−1} a_{i}k_{i}} · f_{k,k_{0},...,k_{l^{n}−1}}(z) · (z − a)^{k/l^{n}}·

∏_{i=0}^{l^{n}−1}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i}(z − a)^{1/l^{n}})^{k_{i}/l^{m}}.

We assume that ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} ∈ ker ψ and we look for elements of K_{2}^{(n,m)} fixed by ker ψ.

We recall that \binom{i}{j} = 0 if i < j. Lemma 3.4 The product ∏_{i=0}^{l^{n}−1}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i}(z − a)^{1/l^{n}})^{k_{i}/l^{m}} is fixed by ker ψ if and only if (k_{0}, k_{1}, …, k_{l^{n}−1}) ∈ (ℤ/l^{m})^{l^{n}} is a linear combination of vectors

e_{j} = l^{m−q(n,N)} (\binom{0}{j}, \binom{1}{j}, …, \binom{l^{n} − 1}{j}) , 0 ≤ j < N.

Proof : Assume that (k_{0}, k_{1}, …, k_{l^{n}−1}) is a linear combination of vectors e_{0}, e_{1},. …, e_{N−1}. Then it follows from Lemma 3.3 that ∑_{i=0}^{l^{n}−1} k_{i}a_{i} ≡ 0 mod l^{m} for any sequence (a_{0}, a_{1}, …, a_{l^{n}−1}) such that ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} ∈ kerψ. Therefore it follows from the formula expressing action of Gal(K_{2}^{(n,m)}/ℂ(z)) on elements of K_{2}^{(n,m)} that ∏_{i=0}^{l^{n}−1}((b − a)^{\frac{1}{l^{n}}} − ξ_{l^{n}}^{i}(z − a)^{\frac{1}{l^{n}}})^{\frac{k_{i}}{l^{m}}} is fixed by kerψ.

Now we assume that ∏_{i=0}^{l^{n}−1}((b − a)^{\frac{1}{l^{n}}} − ξ_{l^{n}}^{i}(z − a)^{\frac{1}{l^{n}}})^{\frac{k_{i}}{l^{m}}} is fixed by kerψ. This means that ∑_{i=0}^{l^{n}−1} k_{i}a_{i} ≡ 0 mod l^{m} for any vector (a_{0}, a_{1}, …, a_{l^{n}−1}) such that ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} ∈ kerψ.

Let κ := ∑_{i=0}^{l^{n}−1} k_{i}X_{i} be a linear form from (ℤ/l^{m})^{l^{n}} to ℤ/l^{m} vanishing on any vector (a_{0}, a_{1}, …, a_{l^{n}−1}) such that ∏_{i=0}^{l^{n}−1} β_{i}^{a_{i}} ∈ kerψ. Let us consider linear forms d_{k} := l^{m−q(n,N)} ∑_{i=k}^{l^{n}−1} \binom{i}{k}X_{i} from (ℤ/l^{m})^{l^{n}} to ℤ/l^{m} for 0 ≤ k < N.

It follows from Lemma 3.3 that ⋂_{k=0}^{N−1} kerd_{k} ⊂ kerκ. Hence κ is a linear combination of forms d_{k} for 0 ≤ k < N. This finishes the proof of the lemma. □ Corollary 3.5 The subfield of K_{2}^{(n,m)} fixed by ker ψ is a field

ℙ(n, N) := ℂ(z)((z−a)^{1/l^{n}}, ∏_{i=0}^{l^{n}−1}((b−a)^{1/l^{n}}−ξ_{l^{n}}^{i}(z−a)^{1/l^{n}})^{\binom{i}{k}/l^{q(n,N)}} ; 0 ≤ k < N).

The Galois group Gal(ℙ(n, N)/ℂ(z)) is equal P(n, N) = F (x, y)/ϕ(n, N). □

4. We shall study monodromy transformations of functions appearing in Corollary 3.5.

Let us set

f_{k}^{(−s)} := ∏_{i=0}^{l^{n}−1}((b − a)^{1/l^{n}} − ξ_{l^{n}}^{i−s}(z − a)^{1/l^{n}})^{\binom{i}{k}/l^{q(n,N)}}

for 0 ≤ s < l^{n}. Lemma 4.1 The monodromy transformation of f_{k}^{(−s)} is given by

(x^{(p)}, y) : f_{k}^{(−s)} → ξ\binom{s}{k−p}_{l^{q(n,N)}} · f_{k}^{(−s)}.

Proof : First, we notice that

y : f_{k}^{(−s)} → ξ\binom{s}{k}_{l^{q(n,N)}} · f_{k}^{(−s)}.

We calculate monodromy transformations of functions f_{k}^{(−s)} by induction. Observe that

(x^{(p+1)}, y) = (x, (x^{(p)}, y)) : f_{k}^{(−s)} ⟶^{(x^{(p)},y)^{−1}} ξ_{l^{q(n,N)}}^{−\binom{s}{k−p}} · f_{k}^{(−s)} ⟶^{x^{−1}}

ξ_{l^{q(n,N)}}^{−\binom{s}{k−p}} · f_{k}^{(−s−1)} ⟶^{(x^{(p)},y)} ξ_{l^{q(n,N)}}^{−\binom{s}{k−p}} · ξ\binom{s+1}{k−p}_{l^{q(n,N)}} · f_{k}^{(−s−1)} →^{x} ξ_{l^{q(n,N)}}^{−\binom{s}{k−p}+\binom{s+1}{k−p}} · f_{k}^{(−s)}.

The lemma follows from the identity \binom{s+1}{k−p} − \binom{s}{k−p} = \binom{s}{k−(p+1)}. □ Corollary 4.2 Let ϵ ∈< y^{(2)} >. Then

ϵ : f_{k}^{(−s)} → f_{k}^{(−s)}.

Proof : Observe that monodromy transformations of f_{k}^{(−s)} along (x^{(p)}, y) and (x^{(q)}, y) commute. Hence the commutator of (x^{(p)}, y) and (x^{(q)}, y) acts trivially on f_{k}^{(−s)}. □ Corollary 4.3 The monodromy transformation of f_{k}^{(0)} along (x^{(k)}, y) is given by :

(x^{(k)}, y) : f_{k}^{(0)} → ξ_{l^{q(n,N)}} · f_{k}^{(0)}.

□

We shall show how to use the functions ∏_{i=0}^{l^{n}−1}((b−a)^{1/l^{n}}−ξ_{l^{n}}^{i}(z−a)^{1/l^{n}})^{\binom{i}{k}/l^{q(n,N)}} to calculate l-adic polylogarithms introduced in [5].

Let K be a number field. Let

V := ℙ_{K}^{1} \ {a, b, ∞},

where a, b ∈ K. Let v and z_{0} be K-points or tangential K-points of V. Let γ be a path from v to z_{0}. Let π_{1}(VK̄; v) be a pro-l completion of the etale fundamental group of VK̄ based at v.

Let δ ∈ G_{K}. The element 𝔤_{γ}(δ) := γ^{−1} · δ · γ · δ^{−1} ∈ π_{1}(VK̄; v) (see [4] Definition 1.0.1) we write as an infinite convergent product

…, ϵ_{n+1} · (x^{(n)}, y)^{α_{n+1}(δ)} … ϵ_{3} · (x^{(2)}, y)^{α_{3}(δ)} · (x, y)^{α_{2}(δ)} · y^{α_{1}(δ)} · x^{α(δ)},

where ϵ_{n} is a product of commutators in x and y of length n belonging to < y^{(2)} >. Proposition 4.4 Let δ ∈ G_{K} be such that 𝔤_{γ}(δ) ≡ 1 mod Γ^{k}π_{1}(VK̄; v), i.e., 𝔤_{γ}(δ) ≡ (x^{(k−1)}, y)^{α_{k}(δ)} · ϵ_{k} mod Γ^{k+1}π_{1}(VK̄; v). Assume that k ≥ 2. Then the exponent α_{k}(δ) is given by the formula

\frac{δ^{−1}(f_{k−1}^{(0)}(v))}{f_{k−1}^{(0)}(v)} · \frac{δ(f_{k−1}^{(0)}(z_{0}))}{f_{k−1}^{(0)}(z_{0})} = ξ_{l^{q(n,N)}}^{α_{k}(δ)}.

Proof : Observe that (x^{(k−1)}, y) : f_{k−1}^{(0)} → ξ_{l^{q(n,N)}} · f_{k−1}^{(0)} by Corollary 4.3. The elements of Γ^{k+1}π_{1}(VK̄; v) and ϵ_{k} fix f_{k−1}^{(0)}.

On the other side 𝔤_{γ}(δ) transforms f_{k−1}^{(0)} into δ^{−1}(f_{k−1}^{(0)}(v)) · (f_{k−1}^{(0)}(v))^{−1} · δ(f_{k−1}^{(0)}(z_{0})) · (f_{k−1}^{(0)}(z_{0}))^{−1} · f_{k−1}^{(0)}. This implies the proposition. □ Corollary 4.5 Let a = 0, b = 1 and v = \overrightarrow{01}. Let γ be a path from \overrightarrow{01} to a

point z_{0}. Let δ ∈ G_{K} be such that 𝔤_{γ}(δ) ≡ (x^{(k−1)}, y)^{α_{k}(δ)} · ϵ_{k} mod Γ^{k+1}π_{1}(ℙ1 ; K̄ \

\{0, 1, ∞\} ; \overrightarrow{01}). Then

(−1)^{k−1}α_{k}(δ) = l_{k}(z_{0})(δ).

Proof : The formula follows immediately from the definition of l-adic polylogarithms given in [5] .□

References

[1] G. Anderson, Y. Ihara, Pro-l branched coverings of P^{1} and higher circular l-units, Annals of Math. 128 (1988), 271–293.

[2] 2 A. Beilinson, P. Deligne, Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, in Motives (U. Jannsen, S.L. Kleiman and J.-P. Serre, eds.), Proc. of Symp. in Pure Math. 55, Part II, AMS (1994), 97–121.

[3] H. Nakamura, Z. Wojtkowiak, On explicit formulae for l-adic polylogarithms in Arithmetic Fundamental Groups and Noncommutative Algebra, (M. Fried, Y. Ihara, eds.), Proc. of Symp. in Pure Math. Vol. 70, 2002, 285–294.

[4] Z. Wojtkowiak, On l-adic iterated integrals, I Analog of Zagier conjecture, Nagoya Math. J. Vol. 176, (2004), 113–158.

[5] Z. Wojtkowiak, On l-adic iterated integrals, II Functional equations and l-adic polylogarithms, Nagoya Math. J. Vol. 177, (2005), 117-153.