A NOTE ON FUNCTIONAL EQUATIONS OF L-ADIC POLYLOGARITHMS

ZDZISLAW WOJTKOWIAK

Contents

1. Introduction 1 2. Functional equations 2 3. l-adic polylogarithms and Soulé classes 4 References 7

Abstract. We are studying l-adic analogues of the classical polylogarithms. Using an explicit arithmetic formula we show that they satisfy the same functional equations as classical polylogarithms. We also identify l-adic polylogarithms evaluated at roots of unity with Soulé classes.

1. Introduction

In [5] we defined l-adic polylogarithms and we showed that they satisfy functional equations analogous to functional equations of classical polylogarithms. In [1], motivated by an old result of O.Gabber, we have given in a commun work with H.Nakamura an explicit formula for l-adic polylogarithms. In this note we shall use this explicit formula to get functional equations of l-adic polylogarithms. We use them in [5] to study the action of Galois groups on π_{1}(P 1 ; K̄ \ {0, μ_{n}, ∞}, 01).

Let K be a number field. We choose in K̄ a compatible family (ξ_{l^{n}})_{n∈N} of l^{n} - roots of 1 such that (ξ_{l^{n+1}})^{l} = ξ_{l^{n}}.

For any z ∈ K \ \{0\} we choose in K̄ a compatible family (z^{\frac{1}{l^{n}}})_{n∈N} of l^{n}- roots of z such that (z^{\frac{1}{l^{n+1}}} )^{l} = z^{\frac{1}{l^{n}}} .

For any z ∈ K \ {0} we define a Kummer character

κ_{z} : G_{K} → Z_{l}

setting σ(z^{\frac{1}{l^{n}}}) = (ξ_{l^{n}})^{κ_{z}(σ)} · z^{\frac{1}{l^{n}}} for all n.

Let us set G_{K} := Gal(K̄/K). For any natural numbers n and m and any i such that 0 ≤ i < l^{n} we define functions

κ_{i}^{n,m}(z) : G_{K} → Z/l^{m}Z

by the following equality

\frac{σ((1 − ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z}(σ))} · z^{\frac{1}{l^{n}}})^{\frac{1}{l^{m}}})}{(1 − ξ_{l^{n}}^{i} · z^{\frac{1}{l^{n}}})^{\frac{1}{l^{m}}}} = ξ_{l^{m}}^{κ_{i}^{n,m}(z)(σ)}.

The family of functions (κ_{i}^{n,m}(z)(σ))_{m∈N} is compatible, hence we get a function

κ_{i}^{n}(z) : G_{K} → Z_{l}.

Let us fix σ ∈ G_{K}. Then for each n we have a function

Z/l^{n} → Z_{l}, i → κ_{i}^{n}(z)(σ).

Let n > m. The identities

∏_{i≡i_{0}(l^{m})}(1 − ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z}(σ))} · z^{\frac{1}{l^{n}}}) = (1 − ξ_{l^{m}}^{χ(σ^{−1})(i_{0}−κ_{z}(σ))} · z^{\frac{1}{l^{m}}})

and

∏_{i≡i_{0}(l^{m})}(1 − ξ_{l^{n}}^{i} · z^{\frac{1}{l^{n}}}) = (1 − ξ_{l^{m}}^{i_{0}} · z^{\frac{1}{l^{m}}})

imply that

∑_{i≡i_{0}(l^{m})} κ_{i}^{n}(z)(σ) = κ_{i_{0}}^{m}(z)(σ). (1.1)

Hence we get a measure

κ(z)(σ) := (κ_{i}^{n}(z)(σ))_{n∈N,0≤i<l^{n}}

on Z_{l} = \underleftarrow{lim} Z/l^{n}. Let n > m and let p be a natural number. The equality (1.1) implies that

∑_{i=0}^{l^{n}−1} i^{p}κ_{i}^{n}(z)(σ) ≡ ∑_{i=0}^{l^{m}−1} i^{p}κ_{i}^{m}(z)(σ) mod l^{m}. (1.2)

Hence by the definition of the integration on Z_{l} we get

∫_{Z_{l}} x^{p}dκ(z)(σ) ≡ ∑_{i=0}^{l^{n}−1} i^{p}κ_{i}^{n}(z)(σ) mod l^{n}. (1.3)

It follows from (1.3) and the fact that κ_{i}^{n}(z)(σ) ≡ κ_{i}^{n,n}(z)(σ) mod l^{n} that

∫_{Z_{l}} x^{p}dκ(z)(σ) ≡ ∑_{i=0}^{l^{n}−1} i^{p}κ_{i}^{n,n}(z)(σ) mod l^{n}. (1.4)

Let us set

ℓ_{p+1}(z)(σ) := ∫_{Z_{l}} x^{p}dκ(z)(σ).

The function ℓ_{n}(z) appears naturally when studying the action of G_{K} on the π_{1}(P 1 ; K̄ \ {0, 1, ∞}, 01) - torsor of l-adic paths from 01 to z.

We embed the group π_{1}(P 1 ; K̄ \ {0, 1, ∞}, 01) into the Q_{l} -algebra Q_{l}{{X, Y }} of formal power series in two non commuting variables X and Y. The image of the loop around 0 (resp. around 1) is the power series e^{X} (resp. e^{Y}). The embedding is continous and multiplicative and we denote it by k. Let γ be a path from 01 to z. Then γ^{−1} · σ(γ) is a loop in π_{1}(P 1 ; K̄ \ {0, 1, ∞}, 01). Let x be a loop around 0 - one of the standard generators of π_{1}(P 1 ; K̄ \ {0, 1, ∞}, 01). We can write

logk(γ^{−1} · σ(γ) · x^{−κ_{z}(σ)}) = l_{1}(z)_{γ}(σ)Y + ∑_{n=2}^{∞} l_{n}(z)_{γ}(σ)[[X, Y]X^{n−2}] + ...

The main result in [1] is that the function l_{n}(z)_{γ} coinsides with ℓ_{n}(z) for some choice of a path γ from 01 to z.

2. Functional equations

In this section we show that the function ℓ_{n}(z) satisfy certain functional equations known as distribution relations. These equations were already proved in [5] using geometric methods. They were used in [5] in the study of actions of Galois

groups on π_{1}(P 1 ; K̄ \ {0, μ_{n}, ∞}, 01).

Theorem 2.1. Let r be a positive integer prime to l and let ξ_{r} be a primitive r - root of 1. We assume that ξ_{r} and z are in K. For each 0 ≤ a < r we choose a compatible family (ξr^{\frac{a}{l^{n}}})_{n∈N} of l^{n} - roots of ξ_{r}^{a} such that each ξr^{\frac{a}{l^{n}}} ∈ μ_{r}. Then we have

ℓ_{m}(z^{r}) = r^{m−1}( ∑_{ξ^{r}=1} ℓ_{m}(ξz)).

Proof. Observe that we have the identities

∏_{a=0}^{r−1} ( ∏_{i=0}^{l^{n}−1} (\frac{σ(1 − ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z}(σ))} · ξr^{\frac{a}{l^{n}}} · z^{\frac{1}{l^{n}}})}{1 − ξ_{l^{n}}^{i} · ξr^{\frac{a}{l^{n}}} · z^{\frac{1}{l^{n}}}})^{i^{m−1}} )^{r^{m−1}} =

∏_{i=0}^{l^{n}−1} (∏_{a=0}^{r−1} (\frac{σ(1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z}(σ))} · z^{\frac{1}{l^{n}}})}{1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n}}^{i} · z^{\frac{1}{l^{n}}}}))^{(ir)^{m−1}} =

∏_{i=0}^{l^{n}−1} (\frac{σ(1 − ξ_{l^{n}}^{χ(σ^{−1})(ir−rκ_{z}(σ))} · (z^{r})^{\frac{1}{l^{n}}})}{1 − ξ_{l^{n}}^{ir} · (z^{r})^{\frac{1}{l^{n}}}})^{(ir)^{m−1}} =

∏_{i=0}^{l^{n}−1} (\frac{σ(1 − ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z^{r}} (σ))} · (z^{r})^{\frac{1}{l^{n}}})}{1 − ξ_{l^{n}}^{i} · z^{r} ^{\frac{1}{l^{n}}}})^{i^{m−1}} · l^{n}−th power.

This implies

r^{m−1} ∑_{a=0}^{r−1}( ∑_{i=0}^{l^{n}−1} i^{m−1}κ_{i}^{n,n}(ξ_{r}^{a}z)) = ∑_{i=0}^{l^{n}−1} i^{m−1}κ_{i}^{n,n}(z^{r}).

The theorem follows from the congruences (1.2) and (1.4).

To consider the case when l divides r we need the following lemma.

Lemma 2.2. Let ξ_{l^{ν}} ∈ K and let z ∈ K. We have

κ_{ξ_{l^{ν}}^{j} z}(σ) = \frac{(χ(σ) − 1)j}{l^{ν}} + κ_{z}(σ).

Proof. We have \frac{σ(ξ_{l^{ν+n}}^{j} ·z^{\frac{1}{l^{n}}} )}{ξ_{l^{ν+n}}^{j} ·z^{\frac{1}{l^{n}}}} = \frac{σ(ξ_{l^{ν+n}}^{j})}{ξ_{l^{ν+n}}}·ξ_{l^{n}}^{κ_{z}(σ)} = ξ_{l^{ν+n}}^{(χ(σ)−1)j}·ξ_{l^{n}}^{κ_{z}(σ)} = ξ_{l^{n}}^{\frac{(χ(σ)−1)j}{l^{ν}} +κ_{z}(σ)}.

Theorem 2.3. Let ξ_{l^{ν}} ∈ K and let z ∈ K. We have

ℓ_{m}(z^{l^{ν}}) = l^{ν(m−1)} ∑_{j=0}^{l^{ν}−1} ℓ_{m}(ξ_{l^{ν}}^{j} · z) + ∑_{a=1}^{m−1} \binom{m − 1}{a}l^{ν(m−1−a)} ∑_{j=0}^{l^{ν}−1} j^{a}ℓ_{m−a}(ξ_{l^{ν}}^{j} · z).

Proof. We have

∏_{i=0}^{l^{n}−1} (\frac{σ(1 − ξ_{l^{n}}^{χ(σ^{−1})(i−κ_{z^{l^{ν}}} (σ))} · (z^{l^{ν}})^{\frac{1}{l^{n}}})}{(1 − ξ_{l^{n}}^{i} · (z^{l^{ν}})^{\frac{1}{l^{n}}})})^{i^{m−1}} =

∏_{0≤i<l^{n}, i=j+kl^{ν}} (\frac{σ(1 − ξ_{l^{n}}^{χ(σ^{−1})j} · ξ_{l^{n}}^{χ(σ^{−1})(kl^{ν}−l^{ν}κ_{z}(σ))} · z^{\frac{1}{l^{n−ν}}} )}{(1 − ξ_{l^{n}}^{j} · ξ_{l^{n}}^{kl^{ν}} · z^{\frac{1}{l^{n−ν}}} )})^{(j+kl^{ν})^{m−1}} =

∏_{j=0}^{l^{ν}−1} ∏_{k=0}^{l^{n−ν}−1} (\frac{σ(1 − ξ_{l^{n−ν}}^{χ(σ^{−1})(k−κ_{ξ{j \atop l^{ν}} ·z}(σ))} · (ξ_{l^{ν}}^{j} · z)^{\frac{1}{l^{n−ν}}} )}{(1 − ξ_{l^{n−ν}}^{k} · (ξ_{l^{ν}}^{j} · z)^{\frac{1}{l^{n−ν}}} )})^{(j+kl^{ν})^{m−1}} =

∏_{a=0}^{m−1} ∏_{j=0}^{l^{ν}−1} ∏_{k=0}^{l^{n−ν}−1} ((\frac{σ(1 − ξ_{l^{n−ν}}^{χ(σ^{−1})(k−κ_{ξ{j \atop l^{ν}} ·z}(σ))} · (ξ_{l^{ν}}^{j} · z)^{\frac{1}{l^{n−ν}}} )}{(1 − ξ_{l^{n−ν}}^{k} · (ξ_{l^{ν}}^{j} · z)^{\frac{1}{l^{n−ν}}} )})^{k^{m−1−a}} )^{j^{a}\binom{m−1}{a}l^{ν(m−1−a)}} .

Hence for any N we have

∑_{i=0}^{l^{n}−1} i^{m−1}κ_{i}^{n,N}(z^{l^{ν}})(σ) = l^{ν(m−1)} ∑_{0≤j<l^{ν}}( ∑_{k=0}^{l^{n−ν}−1} k^{m−1}κ_{k}^{n−ν,N}(ξ_{l^{ν}}^{j} z)(σ))+

∑_{a=1}^{m−1} \binom{m − 1}{a}l^{ν(m−1−a)}( ∑_{j=0}^{l^{ν}−1} j^{a}( ∑_{k=0}^{l^{n−ν}−1} k^{m−1−a}κ_{k}^{n−ν,N}(ξ_{l^{ν}}^{j} · z)(σ))).

This implies that

ℓ_{m}(z^{l^{ν}}) = l^{ν(m−1)} ∑_{j=0}^{l^{ν}−1} ℓ_{m}(ξ_{ν}^{j}z) + ∑_{a=1}^{m−1} \binom{m − 1}{a}l^{ν(m−1−a)} ∑_{j=0}^{l^{ν}−1} j^{a}ℓ_{m−a}(ξ_{ν}^{j}z).

Now we consider a general case.

Theorem 2.4. Let p be a positive integer. Assume that p = r · l^{ν} where r is a prime to l. Assume that ξ_{r} and ξ_{l^{ν}} are in K. Then we have

ℓ_{m}(z^{p}) = p^{m−1} ∑_{ξ^{p}=1} ℓ_{m}(ξz)+r^{m−1} ∑_{a=1}^{m−1} \binom{m − 1}{a}l^{ν(m−1−a)} ∑_{k=0}^{r−1} ∑_{0≤j<l^{ν}} j^{a}ℓ_{m−a}(ξ_{l^{ν}}^{j} ξ_{r}^{k}z).

Proof. The theorem follows from Theorems 2.1 and 2.3.

Corollary 2.5. Let G_{n} ⊂ G_{K} be a subgroup on which functions ℓ_{k}(z) vanish for all z ∈ K and all k < n. Then

ℓ_{m}(z^{p}) = p^{m−1} ∑_{ξ^{p}=1} ℓ_{m}(ξz)

on G_{n}.

3. l-adic polylogarithms and Soulé classes

3.1. We recall briefly the construction of the Soulé classes. Let K be a number field. Let us set K_{n} := K(μ_{l^{n}}) and K_{∞} := K(μ_{l^{∞}}). The extension K ⊂ K_{n} is unramified outside l. Let M be a maximal Galois extension of K unramified outside l.

Let us set G := Gal(M/K), G_{n} := Gal(M/K_{n}), G_{∞} := Gal(M/K_{∞}), Γ_{n} := Gal(K_{n}/K) and Γ_{∞} := Gal(K_{∞}/K). The groups G_{n} and G_{∞} are normal subgroups of G and we have G/G_{n} ≃ Γ_{n} and G/G_{∞} ≃ Γ_{∞}. Let us set R_{n} := O_{K_{n}}[\frac{1}{l}] and R := O_{K}[\frac{1}{l}]. Let us set X_{n} := SpecR_{n}, X_{∞} := \underleftarrow{lim}_{n}X_{n} and X := SpecR.

The exact sequence of sheaves on X_{n}

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

induces a long exact sequence of cohomology

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

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

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

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

x_{n}^{i} := N_{n}(α_{n} ∪ δ(u_{n})).

The next lemma one finds in [3].

Lemma 3.1. Let (u_{n})_{n∈N}, u_{n} ∈ R_{n}^{∗} be a family of l - units such that N_{n+1,n}(u_{n+1}) = u_{n}, where N_{n+1,n} : R_{n+1}^{∗} → R_{n}^{∗} is a norm. Then r_{n+1,n}(x_{n+1}^{i}) = x_{n}^{i}, where r_{n+1,n} : H^{1}(X, μ_{l^{n+1}}^{⊗i}) → H^{1}(X, μ_{l^{n}}^{⊗i}) is induced by the projection μ_{l^{n+1}} → μ_{l^{n}}. Hence the family (x_{n}^{i})_{n∈N} defines an element x^{i} = x^{i}((u_{n})_{n∈N}) = (x_{n}^{i})_{n∈N} ∈ H^{1}(X, Z_{l}(i)).

The covering p_{n} : X_{n} → X is Galois. It follows from [2] Lemma 6 that we have a commutative diagram

\xymatrix{(R_{n}^{∗}/(R_{n}^{∗})^{l^{n}})⊗μ_{l^{n}}^{⊗(i−1)} \ar[r]_{δ⊗id_{μ{⊗(i−1) \atop l^{n}}}} \ar[d]_{N_{Γn}} & H^{1}(X_{n}; μ_{l^{n}}^{⊗i}) \ar[d]_{p_{n}^{∗}∘N_{n}} \\ (R_{n}^{∗}/(R_{n}^{∗})^{l^{n}})⊗μ_{l^{n}}^{⊗(i−1)} \ar[r]_{δ⊗id_{μ{⊗(i−1) \atop l^{n}}}} & H^{1}(X_{n}; μ_{l^{n}}^{⊗i})}

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

Observe that

N_{Γ_{n}}(u_{n}⊗ξ_{l^{n}}^{⊗(i−1)}) = ∏_{σ∈Γ_{n}}(σ(u_{n}))⊗(ξ_{l^{n}}^{χ(σ)})^{⊗(i−1)} =

∏_{σ∈Γ_{n}}(σ(u_{n}))^{χ̄(σ)^{i−1}}⊗ξ_{l^{n}}^{⊗(i−1)},

where χ̄(σ) is a positive integer such that 0 ≤ χ̄(σ) < |Γ_{n}| and χ̄(σ) ≡ χ(σ) mod |Γ_{n}|. Hence we have proved the following result.

Lemma 3.2. The element

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

is given by the Kummer character associated to ∏_{σ∈Γ_{n}}(σ(u_{n}))^{χ̄(σ)^{i−1}}.

The element x^{i} = x^{i}((u_{n})_{n∈N}) ∈ H^{1}(X, Z_{l}(i)) is called a Soulé class. Further we shall identify Soulé classes with ℓ-adic polylogarithms.

3.2. Let r be a positive integer prime to l. Let K = Q(μ_{r}). Let 0 ≤ q < r. Observe that

∑_{i=0}^{l^{n}−1} i^{m−1}κ_{i}^{n,M}(ξ_{r}^{q}) = ∑_{i=0,(i,l)=1}^{l^{n}−1} i^{m−1}κ_{i}^{n,M}(ξ_{r}^{q}) + ∑_{i=0,(i,l)≥l}^{l^{n}−1} i^{m−1}κ_{i}^{n,M}(ξ_{r}^{q}) (3.1)

and

κ_{li}^{n,M}(ξ_{r}^{q}) = κ_{j}^{n−1,M}(ξ_{r}^{ql^{−1}}). (3.2)

It follows from 3.1. and 3.2 and from section 1 that

∫_{Z_{l}} x^{m−1}dκ(ξ_{r}^{q})(σ) = ∫_{Z_{l}\lZ_{l}} x^{m−1}dκ(ξ_{r}^{q})(σ) + l^{m−1} ∫_{Z_{l}} x^{m−1}dκ(ξ_{r}^{ql^{−1}})(σ).

6 ZDZISLAW WOJTKOWIAK

Hence by the definition of l-adic polylogarithms we have ℓ_{m}(ξ_{r}^{q})(σ) = ∫_{Z_{l}\lZ_{l}} x^{m−1}dκ(ξ_{r}^{q})(σ) + l^{m−1}ℓ_{m}(ξ_{r}^{ql^{−1}})(σ). (3.3)

The function

G_{K} ∋ σ → ∫_{Z_{l}\lZ_{l}} x^{m−1}dκ(ξ_{r}^{q})(σ) ∈ Z_{l}

we shall identify with a Soulé class.

The family of l-units u_{n}^{(q)} := (1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}}) (n ∈ N) is compatible i.e. N_{n+1,n}(1 − ξr^{\frac{q}{l^{n+1}}} · ξ_{l^{n+1}}) = (1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}}) for all n ∈ N. It follows from Lemma 3.2 that the corresponding Soulé classe x^{m}((1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}})_{n∈N}) restricted to G_{Q(μ_{r·l^{∞}})} := Gal(Q̄/Q(μ_{r·l^{∞}})) is given by the Kummer character associated to ∏_{i=0, (i,l)=1}(1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}})^{i^{m−1}}. This implies that

x^{m}((1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}})_{n∈N})(σ) = ∫_{Z_{l}\lZ_{l}} x^{m−1}dκ(ξ_{r}^{q})(σ) (3.4)

for any σ ∈ G_{Q(μ_{r·l^{∞}})}. Proposition 3.3. (see also [5]) We have

x^{m}((1 − ξr^{\frac{q}{l^{n}}} · ξ_{l^{n}})_{n∈N}) = ℓ_{m}(ξ_{r}^{q}) − l^{m−1}ℓ_{m}(ξ_{r}^{ql^{−1}})

on G_{Q(μ_{r·l^{∞}})}. Proof. The proposition follows from (3.3) and (3.4). Corollary 3.4. Let p be the order of l in (Z/r)^{∗}. Then

(1 − l^{p(m−1)})ℓ_{m}(ξ_{r}^{q}) = ∑_{i=0}^{p−1} l^{i(m−1)}x^{m}((1 − ξr^{\frac{q·l^{−i}}{l^{n}}} · ξ_{l^{n}})_{n∈N})

on G_{Q(μ_{r·l^{∞}})}. 3.3. Let r be an integer prime to l and let ν be a positive integer. Let K = Q(μ_{l^{ν}·r}). Then K_{n+ν} = Q(μ_{l^{n+ν}·r}) and Γ_{n+ν} = Z/l^{n}. Let α ∈ Γ_{n+ν} = Z/l^{n}. Then α(ξ_{l^{n+ν}}) = ξ_{l^{n+ν}}^{1+l^{ν}α}. Let q be an integer prime to l and let 0 ≤ a < r. The family of l-

units (1−ξr^{\frac{a}{l^{n−ν}}} ·ξ_{l^{n}}^{q})_{n≥ν} is compatible i.e. N_{n+1,n}(1−ξr^{\frac{a}{l^{n−ν+1}}} ·ξ_{l^{n+1}}^{q}) = (1−ξr^{\frac{a}{l^{n−ν}}} ·ξ_{l^{n}}^{q})

for all n ≥ ν. We have

N_{Γ_{n+ν}} ((1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n+ν}}^{q})⊗ξ_{l^{n+ν}}^{⊗(m−1)}) = ∏_{α∈Z/l^{n}}(1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n+ν}}^{(1+l^{ν}α)q})⊗(ξ_{l^{n+ν}}^{1+l^{ν}α})^{⊗(m−1)} =

∏_{α=0}^{l^{n}−1} (1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n+ν}}^{(1+l^{ν}α)q})^{(1+l^{ν}α)^{(m−1)}} ⊗ξ_{l^{n+ν}}^{⊗(m−1)} = ∏_{k=0}^{m−1} ∏_{α=0}^{l^{n}−1} (1 − ξr^{\frac{a}{l^{n}}} · ξ_{l^{n}}^{αq} · ξ_{l^{n+ν}}^{q} )^{\binom{m−1}{k}l^{νk}α^{k}} ⊗ξ_{l^{n+ν}}^{⊗(m−1)}

= ∏_{k=0}^{m−1} ( ∏_{α=0}^{l^{n}−1}(1 − ξ_{l^{n}}^{αq} · (ξ_{r}^{a} · ξ_{l^{ν}}^{q})^{\frac{1}{l^{n}}})^{(αq)^{k}})^{\binom{m−1}{k}l^{νk}q^{−k}} ⊗ξ_{l^{n+ν}}^{⊗(m−1)}.

The Kummer character associated to the last term is

∑_{k=0}^{m−1} \binom{m − 1}{k}l^{νk} · q^{−k}( ∑_{j=0}^{l^{n}−1} j^{k}κ_{j}^{n,n}(ξ_{l^{ν}}^{q} · ξ_{r}^{a})).

Hence we get the following result.

Proposition 3.5. We have

x^{m}((1−ξr^{\frac{a}{l^{n−ν}}} ·ξ_{l^{n}}^{q})_{n≥ν}) = l^{ν(m−1)}q^{−(m−1)}ℓ_{m}(ξ_{r}^{a}·ξ_{l^{ν}}^{q})+ ∑_{k=0}^{m−2} \binom{m − 1}{k}·l^{νk}·q^{−k}·ℓ_{k+1}(ξ_{r}^{a}·ξ_{l^{ν}}^{q})

on Gal(Q̄/Q(μ_{l^{∞}}).

In the next proposition we generalize Corollary 8.4.2 from [5].

Theorem 3.6. The functions ℓ_{m}(ξ_{r}^{q} · ξ_{l^{ν}}^{q}) for 0 < a ≤ \frac{r−1}{2}, (a, r) = 1 and 0 < q ≤ \frac{l^{ν}}{2} , (q, l) = 1 are linearly independent over Z_{l}.

Proof. It is sufficient to show that the Soulé classes x^{m}((1 − ξr^{\frac{a}{l^{n−ν}}} · ξ_{l^{n}}^{q})_{n≥ν}) for 0 < a ≤ \frac{r−1}{2}, (a, r) = 1 and 0 < q ≤ \frac{l^{ν}}{2} , (q, l) = 1 are linearly independent over Z_{l}. This follows from the fact that l-units (1 − ξ_{r}^{a} · ξ_{l^{ν}}^{q}) for 0 < a ≤ \frac{r−1}{2}, (a, r) = 1 and 0 < q ≤ \frac{l^{ν}}{2} , (q, l) = 1 are linearly independent in the group of cyclotomic l-units of Q(μ_{rl^{ν}}) tensored by μ_{l^{n}}^{⊗(m−1)}.

References

[1] Nakamura, H. and Wojtkowiak, Z., On the explicit formulae for l-adic polylogarithms , to appear.

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

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

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

[5] Wojtkowiak, Z., On l-adic iterated integrals.