Specialization of the p-adic Polylogarithm to

p-th Power Roots of Unity

Dedicated to Professor Kazuya Kato

for his fiftieth birthday

Kenichi Bannai

Received: November 11, 2002 Revised: April 28, 2003

Abstract. The purpose of this paper is to calculate the restriction of the p-adic polylogarithm sheaf to p-th power torsion points.

2000 Mathematics Subject Classification: 14F30,14G20

Keywords and Phrases: p-adic polylogarithm, syntomic cohomology, rigid cohomology

1 Introduction

Fix a rational prime p. The classical polylogarithm sheaf, constructed by Beilinson and Deligne, is a variation of mixed Hodge structures on the projective line minus three points. The p-adic polylogarithm sheaf is its p-adic analogue, and is expected to be the p-adic realization of the motivic polylogarithm sheaf. In our previous paper [Ban1], we explicitly calculated the p-adic polylogarithm sheaf on the projective line minus three points, and calculated its specializations to the d-th roots of unity for d prime to p. The purpose of this paper is to extend this calculation to the d-th roots of unity for d divisible by p. In particular, we prove that the specialization of the p-adic polylogarithm sheaf to d-th roots of unity is again related to special values of the p-adic polylogarithm function defined by Coleman [Col]. Let K = ℚ_{p}(μ_{d}), with ring of integers O_{K}. Let 𝔾_{m} = Spec O_{K}[t, t^{−1}] be the multiplicative group over O_{K}. Denote by S(𝔾_{m}) the category of syntomic coefficients on 𝔾_{m}. This category is a rough p-adic analogue of the category of variation of mixed Hodge structures. Since p is in general ramified in K, we will use the definition in [Ban2], which is a generalization of the definition in [Ban1] to the case when p is ramified in K. In order to describe the polylogarithm sheaf, it is first necessary to introduce the logarithmic sheaf Log, which is a pro-object in S(𝔾_{m}). The first property we prove for this sheaf is that it satisfies the splitting principle, even at roots of unity whose order is divisible by p.

Proposition (= Proposition 5.1) Let z ≠ 1 be a d-th root of unity in K, and let i_{z} : Spec O_{K} ↪ 𝔾_{m} be the closed immersion defined by t ↦ z. Then

i_{z}^{∗}Log = ∏_{j≥0} K(j).

Let 𝕌 = 𝔾_{m} \ {1}. In our previous paper, following the method of [HW1] Definition III 2.2, we constructed the polylogarithm extension

pol ∈ Ext_{S_{syn}(𝕌)}^{1}(K(0), Log).

We first consider the case when z is a d-th root of unity, where d is an integer of the form d = Np^{r} with (N, p) = 1 and N > 1. In this case, we have a natural map i_{z} : Spec O_{K} → 𝕌. Let i_{z}^{∗} pol be the image of pol in

Ext_{S(O_{K})}^{1}(K(0), i_{z}^{∗}Log) = ∏_{j≥0} Ext_{S(O_{K})}^{1}(K(0), K(j))

with respect to the pull-back map

Ext_{S(𝕌)}^{1}(K(0), Log) ⟶^{i_{z}^{∗}} Ext_{S(O_{K})}^{1}(K(0), i_{z}^{∗}Log).

Our main result is concerned with the explicit shape of i_{z}^{∗} pol. For integers j ≥ 1, let Li_{j}(t) be the p-adic polylogarithm function defined by Coleman ([Col] VI, the function denoted ℓ_{j}(t)). It is a locally analytic function defined on ℙ^{1}(ℂ_{p}) \ {1, ∞} satisfying Li_{j}(0) = 0. On the open unit disc {z ∈ ℂ_{p} | |z|_{p} < 1}, the function is given by the usual power series

Li_{j}(t) = ∑_{n=1}^{∞} \frac{t^{n}}{n^{j}} .

To deal with the specialization at points in the open unit disc around one, we also consider the locally analytic function

Li_{j,c}(t) = Li_{j}(t) − c^{1−j} Li_{j}(t^{c}),

where c is an integer > 1. Our main theorem may be stated as follows:

Specialization to p-th Power Roots of Unity 75

Theorem 1 (= Theorem 7.3) Let z be a d-th root of unity, where d is an integer of the form d = Np^{r} with (N, p) = 1 and N > 1. Then we have

i_{z}^{∗} pol = ((−1)^{j} Li_{j}(z))_{j≥1} ∈ ∏_{j≥0} Ext_{S(O_{K})}^{1}(K(0), K(j)),

where we view (−1)^{j} Li_{j}(z) as elements of Ext_{S(O_{K})}^{1}(K(0), K(j)) through the isomorphism

Ext_{S(O_{K})}^{1}(K(0), K(j)) ≅ K.  (1)

Remark 1 The above is compatible with the results of Somekawa [So] and also Besser-de Jeu [BdJ] on the calculation of the syntomic regulator.

Remark 2 In [Ban1], we proved that when d is prime to p,

i_{z}^{∗} pol = ((−1)^{j}ℓ_{j}^{(p)}(z))_{j≥1} ,

where ℓ_{j}^{(p)}(t) is a locally analytic function on ℙ^{1}(ℂ_{p}) \ {1, ∞}, whose expansion on the open unit disc around 0 is given by

ℓ_{j}^{(p)}(t) = ∑_{n≥1,(n,p)=1} \frac{t^{n}}{n^{j}} .

The difference between this formula and the formula of the previous theorem comes from the choice of the isomorphism (1). (See Remark 7.2 for details.)

For the case when z is a p^{r}-th root of unity, let c > 1 be an integer and let [c] : 𝔾_{m} → 𝔾_{m} be the multiplication by c map induced from t ↦ t^{c}. We denote by [c]^{∗} the pull back morphism of syntomic coefficients. We define the modified polylogarithm to be

pol_{c} = pol −[c]^{∗} pol,

which we prove to be an element in Ext_{S_{syn}(𝕌_{c})}^{1}(K(0), Log) for

𝕌_{c} = Spec O_{K} [t, \frac{t − 1}{t^{c} − 1}] .

We note that this modification, which removes the singularity around one, is standard in Iwasawa theory. Our theorem in this case is:

Theorem 2 (= Theorem 8.3) Let z be a p^{r}-th root of unity. Then we have

i_{z}^{∗} pol_{c} = ((−1)^{j} Li_{j,c}(z))_{j≥1} ∈ ∏_{j≥0} Ext_{S(O_{K})}^{1}(K(0), K(j)),

where i_{z}^{∗} is the pull back of syntomic coefficient by the natural inclusion i_{z} : Spec O_{K} → 𝕌_{c}. Again, we view Li_{j,c}(z) as an element of Ext_{S(O_{K})}^{1}(K(0), K(j)) through the isomorphism (1).

Contents

1 Introduction 73

2 Review of the p-adic polylogarithm function 76

3 The Category of Syntomic Coefficients 78

4 Morphisms of Syntomic Data 82

5 The splitting principle 84

6 The specialization of pol to torsion points 88

7 The main result (Case N > 1) 91

8 The main result (Case N = 1) 94

Acknowledgement. I would like to wish Professor Kazuya Kato a happy fiftieth birthday, and to thank him for all that he has contributed to mathematics. I would also like to thank the referee for carefully reading this paper, and for giving helpful comments.

Notation Let p be a rational prime. In this paper, we let K be a finite extension of ℚ_{p} with ring of integers O_{K} and residue field k. We denote by π a generator of the maximal ideal of O_{K}. We let K_{0} the maximal unramified extension of ℚ_{p} in K, and W its ring of integers. We denote by σ the Frobenius morphism on K_{0} and W.

2 Review of the p-adic polylogarithm function

In this section, we will review the theory of p-adic polylogarithm functions due to Coleman [Col]. Since we will mainly deal with the value of the p-adic polylogarithm function at units in O_{ℂ_{p}}, we will not need the full theory of Coleman integration. As in [Col], we call any locally analytic homomorphism log : ℂ_{p}^{×} → ℂ_{p}^{+}, such that \frac{d}{dz} log(1) = 1, a branch of the logarithm. Throughout this paper, we fix once and for all a branch of the logarithm. Since we will only deal with the values of p-adic analytic functions at points outside the open unit disc where the functions have logarithmic poles, the results of this paper is independent of the choice of the branch. We define the p-adic polylogarithm function ℓ_{j}^{(p)}(t) for |t| < 1 by

ℓ_{j}^{(p)}(t) = ∑_{(n,p)=1} \frac{t^{n}}{n^{j}} (j ≥ 1).

Specialization to p-th Power Roots of Unity 77

By [Col] Proposition 6.2, this function extends to a rigid analytic function on ℂ_{p} \ {z; |z − 1|_{p} < p^{(p−1)^{−1}}}. Proposition 2.1 ([Col] Section VI) The p-adic polylogarithm function Li_{j}(t) (Denoted ℓ_{j}(t) in [Col]) is a locally analytic function on ℙ^{1}(ℂ_{p}) \ {1, ∞} satisfying

(i) Li_{0}(t) = t/(1 − t) (ii) \frac{d}{dt} Li_{j+1}(t) = \frac{1}{t} Li_{j}(t) (j ≥ 0). (iii) ℓ_{j}^{(p)}(t) = Li_{j}(t) − p^{−j} Li_{j}(t^{p}) (j ≥ 1). Definition 2.2 (i) For any integer j, we define the function u_{j}(t) by

u_{j}(t) = \begin{cases}\frac{1}{j!} log^{j}(t) & (j ≥ 0) \\ 0 & (j < 0). \end{cases}

Note that if z is a root of unity in ℂ_{p}, then u_{j}(z) = 0 (j ≠ 0). (ii) For any integer n ≥ 1, we define the function D_{n}(t) by

D_{n}(t) = ∑_{j=0}^{n−1}(−1)^{j} Li_{n−j}(t)u_{j}(t).

If z is a root of unity in ℂ_{p}, then D_{n}(z) = Li_{n}(z). To deal with the torsion points of p-th power order, we need modified versions of the above functions. Definition 2.3 Let c > 1 be an integer prime to p. We let:

(i) ℓ_{j,c}^{(p)}(z) = ℓ_{j}^{(p)}(z) − c^{1−n}ℓ_{j}^{(p)}(z^{c}) (j ≥ 1). (ii) Li_{j,c}(z) = Li_{j,c}(z) − c^{1−n} Li_{j,c}(z^{c}) (j ≥ 1). (iii)

D_{n,c}(z) = ∑_{j=0}^{n−1}(−1)^{j} Li_{n−j,c}(t)u_{j}(t).

The above functions are locally analytic on the open unit disc around one.

Documenta Mathematica · Extra Volume Kato (2003) 73–97

3 The Category of Syntomic Coefficients

In this section, we will review the construction of the category of syntomic coefficients given in [Ban2] §4. Note that since we need to deal with the case when the prime p is ramified in K, the theory of [Ban1] is not sufficient.

Definition 3.1 A syntomic datum 𝔛 = (X, \overline{X}, j, P_{X}, ϕ_{X}, ι) consists of the following:

(i) A proper smooth scheme \overline{X}, separated an of finite type over O_{K}, and an open immersion j : X ↪ \overline{X}, such that the complement D is a relative simple normal crossing divisor over O_{K}.

(ii) A formal scheme P_{X} over W.

(iii) For the formal completion \overline{X} of \overline{X} with respect to the special fiber, a closed immersion ι : \overline{X} → P_{X} ⊗_{W} O_{K}, such that both P_{X} and the morphism ι are smooth in a neighborhood of X_{k}.

(iv) A Frobenius map ϕ_{X} : P_{X} → P_{X}, which fits into the diagram

\begin{matrix}\overline{X}_{k} & ⟶^{ι} & P_{X} & ⟶ & Spf W \\ ^{F} ↓ & ^{ϕ_{X}} ↓ & & _{σ^{∗}}↓ \\ \overline{X}_{k} & ⟶^{ι} & P_{X} & ⟶ & Spf W, \end{matrix}  (2)

where F is the absolute Frobenius of \overline{X}_{k}.

We will often omit j and ι from the notation and write

𝔛 = (X, \overline{X}, P_{X}, ϕ_{X}).

Example 3.2 1. Let ℙ^{1} be the projective line over W with coordinate t, and let ℙ_{O_{K}}^{1} = ℙ^{1} ⊗ O_{K}. We let 𝔾_{m} be the syntomic datum given by

𝔾_{m} = (𝔾_{m}_{O_{K}} , ℙ_{O_{K}}^{1} , ℙ̂^{1}, ϕ) ,

where

(a) 𝔾_{m}_{O_{K}} is the multiplicative group over O_{K}, with natural inclusion j : 𝔾_{m}_{O_{K}} ↪ ℙ_{O_{K}}^{1} .

(b) ℙ̂^{1} is the p-adic formal completion of ℙ^{1}.

(c) ι : ℙ̂_{O_{K}}^{1} → ℙ̂^{1} ⊗ O_{K} is the identity.

(d) ϕ is the Frobenius given by ϕ(t) = t^{p} for the coordinate t on ℙ̂^{1}.

2. We let 𝕌 be the syntomic datum given by

𝕌 = (𝕌_{O_{K}} , ℙ_{O_{K}}^{1} , ℙ̂^{1}, ϕ) ,

where 𝕌_{O_{K}} = ℙ_{O_{K}}^{1} \{0, 1, ∞}, with the natural inclusion j : 𝕌_{O_{K}} ↪ ℙ_{O_{K}}^{1} .

3. We let O_{K} be the syntomic datum given by

O_{K} = (Spec O_{K}, Spec O_{K}, Spf W, σ),

where j and ι are the identity.

Throughout this section, we fix a syntomic datum 𝔛. We will next review the definition of the category of syntomic coefficients S(𝔛) on 𝔛. We will first define the categories S_{dR}(𝔛), S_{rig}(𝔛) and S_{vec}(𝔛). Let X_{K} = X ⊗ K and \overline{X}_{K} = \overline{X} ⊗ K.

Definition 3.3 We define the category S_{dR}(𝔛) to be the category consisting of objects the triple M_{dR} := (M_{dR}, ∇_{dR}, F^{•}), where:

(i) M_{dR} is a coherent O_{\overline{X}_{K}} module.

(ii) ∇_{dR} : M_{dR} → M_{dR} ⊗ Ω^{1}(log D_{K}) is an integrable connection on M_{dR} with logarithmic poles along D_{K} = D ⊗ K.

(iii) F^{•} is the Hodge filtration, which is a descending exhaustive separated filtration on M_{dR} by coherent sub-O_{\overline{X}_{K}} modules satisfying

∇_{dR}(F^{m}M_{dR}) ⊂ F^{m−1}M_{dR} ⊗ Ω_{\overline{X}_{K}}^{1} (log D_{K}).

Let X_{k} = X ⊗ k be the special fiber of X and X the formal completion of X with respect to the special fiber. We denote by X_{K} the rigid analytic space over K associated to X ([Ber1] Proposition (0.2.3)) and by X_{K}^{an} the rigid analytic space over K associated to X_{K} (loc. cit. Proposition (0.3.3)). We will use the same notations for \overline{X}.

Definition 3.4 We say that a set V ⊂ \overline{X}_{K} is a strict neighborhood of X_{K} in X_{K}^{an}, if V ∪ (X_{K}^{an} \ X_{K}) is a covering of X_{K}^{an} for the Grothendieck topology.

For any abelian sheaf M on X_{K}^{an}, we let

j^{†}M := \underrightarrow{lim}_{V} α_{V ∗}α_{V}^{∗} M,

where the limit is taken with respect to strict neighborhoods V of X_{K} in X_{K}^{an} with inclusion α_{V} : V ↪ \overline{X}_{K}. If M has a structure of a O_{X_{K}^{an}}-module, then j^{†}M has a structure of a j^{†}O_{X_{K}^{an}}-module.

Definition 3.5 We define the category S_{vec}(𝔛) to be the category consisting of objects the pair M_{vec} := (M_{vec}, ∇_{vec}), where:

(i) M_{vec} is a coherent j^{†}O_{X_{K}^{an}} module.

(ii) ∇_{vec} : M_{vec} → M_{vec} ⊗ Ω_{X_{K}^{an}}^{1} is an integrable connection on M_{vec}.

Let p_{dR} : X_{K}^{an} → \overline{X}_{K} be the natural map.

Definition 3.6 We define the functor

F_{dR} : S_{dR}(𝔛) → S_{vec}(𝔛)

by associating to M_{dR} := (M_{dR}, ∇_{dR}, F^{•}) the module j^{†}(p_{dR}^{∗}M_{dR}) with the connection induced from ∇_{dR}. The functor F_{dR} is exact, since it is a composition of exact functors ([Ber1] Proposition 2.1.3 (iii)).

Let P_{K_{0}} be the rigid analytic space over K_{0} associated to P_{X} ([Ber1] (0.2.2)). As in loc. cit. Définitions (1.1.2)(i), we define the tubular neighborhood of \overline{X}_{k} (resp. X_{k}) in P_{K_{0}} by

]\overline{X}_{k}[_{P}:= sp^{−1}(\overline{X}_{k}) ( resp. ]X_{k}[_{P}:= sp^{−1}(X_{k}) ) ,

where sp : P_{K_{0}} → P_{X} is the spécialization [Ber1] (0.2.2.1). The tubular neighborhoods are rigid analytic spaces over K_{0} with structures induced from that of P_{K_{0}}.

Definition 3.7 We say that a set V ⊂]\overline{X}_{k}[_{P} is a strict neighborhood of ]X_{k}[_{P} in ]\overline{X}_{k}[_{P}, if

V ∪ (]\overline{X}_{k}[_{P}\]X_{k}[_{P})

is a covering of ]\overline{X}_{k}[_{P} for the Grothendieck topology.

For any abelian sheaf M on ]\overline{X}_{k}[_{P}, we let

j^{†}M := \underrightarrow{lim}_{V} α_{V ∗}α_{V}^{∗} M,

where the limit is taken with respect to strict neighborhoods V of ]X_{k}[_{P} in ]\overline{X}_{k}[_{P} with inclusion α_{V} : V ↪]\overline{X}_{k}[_{P}. If M has a structure of a O_{]\overline{X}_{k}[_{P}} - module, then j^{†}M has a structure of a j^{†}O_{]\overline{X}_{k}[_{P}} -module. The Frobenius map ϕ_{X} : P_{X} → P_{X} induces a natural morphism of rigid analytic spaces ϕ_{X} : ]\overline{X}_{k}[_{P}→]\overline{X}_{k}[_{P}.

Definition 3.8 We define the category S_{rig}(𝔛) to be the category consisting of objects the triple M_{rig} := (M_{rig}, ∇_{rig}, Φ_{M}), where:

(i) M_{rig} is a coherent j^{†}O_{]\overline{X}_{k}[_{P}} -module.

(ii) ∇_{rig} : M_{rig} → M_{rig} ⊗ Ω_{]\overline{X}_{k}[_{P}}^{1} is an integrable connection on M_{rig}.

(iii) Φ_{M} is the Frobenius morphism, which is an isomorphism

Φ_{M} : ϕ_{X}^{∗}M_{rig} ⟶^{≅} M_{rig}

of j^{†}O_{]\overline{X}_{k}[_{P}} -modules compatible with the connection.

The map ι : \overline{X} → P_{X} ⊗_{W} O_{K} induces a map of rigid analytic spaces

p_{rig} : X_{K}^{an} →]\overline{X}_{k}[_{P}.  (3)

Definition 3.9 We define the functor

F_{rig} : S_{rig}(𝔛) → S_{vec}(𝔛)

by associating to the object M_{rig} := (M_{rig}, ∇_{rig}, Φ_{M}) the object

F_{rig}(M_{rig}) := (p_{rig}^{∗}M_{rig}, p_{rig}^{∗}∇_{rig})

in S_{vec}(𝔛). This functor is exact by definition.

Definition 3.10 We define the category of syntomic coefficients to be the category S(𝔛) such that:

(i) The objects of S(𝔛) consists of the triple M := (M_{dR}, M_{rig}, p), where:

(a) M_{typ} is an object in S_{typ}(𝔛) for typ ∈ {dR, rig}.

(b) p is an isomorphism

p : F_{dR}(M_{dR}) ⟶^{≅} F_{rig}(M_{rig})

in S_{vec}(𝔛).

(ii) A morphism f : M → N in S(𝔛) is given by a pair (f_{dR}, f_{rig}), where f_{typ} : M_{typ} → N_{typ} are morphisms in S_{typ}(𝔛) for typ ∈ {dR, rig} compatible with the comparison isomorphism p.

Example 3.11 For each integer n ∈ ℤ, we define the Tate object K(n) in S(𝔛) to be the set K(n) := (K(n)_{dR}, K(n)_{rig}, p), where:

(i) K(n)_{dR} in S_{dR}(𝔛) is given by the rank one free O_{\overline{X}_{K}} -module generated by e_{n,dR}, with connection ∇_{dR}(e_{n,dR}) = 0 and Hodge filtration

\begin{cases}F^{m}K(n)_{dR} = K(n)_{dR} & m ≤ −n \\ F^{m}K(n)_{dR} = 0 & m > −n. \end{cases}

(ii) K(n)_{rig} in S_{rig}(𝔛) is given by the rank one free j^{†}O_{]\overline{X}_{k}[_{P}} -module generated by e_{n,rig}, with connection ∇_{rig}(e_{n,rig}) = 0 and Frobenius

Φ(e_{n,rig}) := p^{−n}e_{n,rig}.

(iii) p is the isomorphism given by p(e_{n,dR}) = e_{n,rig}.

Example 3.12 (See [Ban1] Definition 5.1) We define the logarithmic sheaf

Log^{(n)} := (L_{dR}^{(n)}, L_{rig}^{(n)}, p)

in S(𝔾_{m}) by:

(i) L_{dR}^{(n)} in S_{dR}(𝔾_{m}) is given by the rank n free O_{ℙ_{K}^{1}} -module

L_{dR}^{(n)} = ∏_{j=0}^{n} O_{ℙ_{K}^{1}} e_{j,dR},

with connection ∇_{dR}(e_{j,dR}) = e_{j+1,dR} ⊗ d log t for 0 ≤ j ≤ n − 1 and ∇(e_{n,dR}) = 0, and Hodge filtration given by

F^{−m}L_{dR}^{(n)} = ∏_{j=0}^{m} O_{ℙ_{K}^{1}} e_{j,dR}.

(ii) L_{rig}^{(n)} in S_{rig}(𝔾_{m}) is given by the rank n free j^{†}O_{]ℙ_{k}^{1}[ℙ̂^{1}} -module

L_{rig}^{(n)} = ∏_{j=0}^{n} j^{†}O_{]ℙ_{k}^{1}[ℙ̂^{1}} e_{j,rig},

with connection ∇_{rig}(e_{j,rig}) = e_{j+1,rig} ⊗ d log t for 0 ≤ j ≤ n − 1 and ∇(e_{n,rig}) = 0, and Frobenius

Φ(e_{j,rig}) := p^{−j}e_{j,rig}.

(iii) p is the isomorphism given by p(e_{j,dR}) = e_{j,rig}.

4 Morphisms of Syntomic Data

Definition 4.1 Define a morphism between syntomic data u : 𝔛 → 𝔜 to be a pair (u_{dR}, u_{rig}) such that:

(i) u_{dR} : \overline{X} → \overline{Y} is a morphism of schemes over O_{K}.

(ii) u_{rig} : P_{X} → P_{Y} is a morphism of formal schemes over W compatible with the Frobenius, such that the diagram

\begin{matrix}\overline{X} ⊗ k & ⟶^{ι} & P_{X} ⊗ k \\ ^{u_{dR}}↓ & & ^{u_{rig}}↓ \\ \overline{Y} ⊗ k & ⟶^{ι} & P_{Y} ⊗ k \end{matrix}  (4)

is commutative.

Remark 4.2 Notice that in (4), contrary to [Ban2] Definition 4.2 (iii), we do not impose the commutativity of the diagram

\begin{matrix}\overline{X} & ⟶^{ι} & P_{X} \\ ^{u_{dR}}↓ & ^{u_{rig}}↓ \\ \overline{Y} & ⟶^{ι} & P_{Y} . \end{matrix}  (5)

Example 4.3 Let z be an element in O_{K}^{×}, and let 𝔾_{m} be the syntomic datum defined in Example 3.2.1. We denote by z_{0} the Teichmüller representative of z. In other words, z_{0} is a root of unity in W such that z ≡ z_{0} (mod π). Then

i_{z} = (i_{dR}, i_{rig}) : O_{K} → 𝔾_{m}

is a morphism of syntomic data, where i_{dR} : Spec O_{K} → 𝔾_{m}_{O_{K}} and i_{rig} : Spf O_{K} → ℙ̂_{W}^{1} are morphisms defined respectively by t ↦ z and t ↦ z_{0}.

Let u = (u_{dR}, u_{rig}) : 𝔛 → 𝔜 be a morphism of syntomic data. By [Ber1] (2.2.16), we have a functor u_{rig}^{∗} : S_{rig}(𝔜) → S_{rig}(𝔛).

Lemma 4.4 Let u : 𝔛 → 𝔜 be a morphism of syntomic data, and let M := (M_{dR}, M_{rig}, p) be an object in S(𝔜). Then there exists a canonical and functorial isomorphism

u^{∗}(p) : F_{dR}(u_{dR}^{∗}M_{dR}) → F_{rig}(u_{rig}^{∗}M_{rig})

in S_{vec}(𝔛).

The above lemma is trivial if we assume the commutativity of (5). Proof. Let u_{vec} : \overline{X} → \overline{Y} be the morphism of formal schemes induced from u_{dR}, and denote again by u_{vec} the map induced on the associated rigid analytic space. Then we have

F_{dR}(u_{dR}^{∗}M_{dR}) = u_{vec}^{∗}F_{dR}(M_{dR}).

Let u_{1} := ι ∘ u_{vec} and u_{2} := (u_{rig} ⊗ 1) ∘ ι be maps of formal schemes

u_{1}, u_{2} : \overline{X} → P_{Y} ⊗ O_{K}.

Then u_{vec}^{∗}F_{rig}(M_{rig}) = u_{1K}^{∗}(M_{rig} ⊗ K) and F_{rig}(u_{rig}^{∗}M_{rig}) = u_{2K}^{∗}(M_{rig} ⊗ K). Since (4) is commutative, u_{1} and u_{2} coincide on \overline{X}_{k}. Hence by [Ber1] Proposition (2.2.17), we have a canonical isomorphism

ϵ_{1,2} : u_{1K}^{∗}(M_{rig} ⊗ K) →^{≃} u_{2K}^{∗}(M_{rig} ⊗ K).  (6)

The isomorphism of the lemma is the composition of the isomorphism

F_{dR}(u_{dR}^{∗}M_{dR}) = u_{vec}^{∗}F_{dR}(M_{dR}) ⟶^{p ≅} u_{vec}^{∗}F_{rig}(M_{rig}).

with ϵ_{1,2}.

Definition 4.5 Let u : 𝔛 → 𝔜 be a morphism of syntomic data. Then

u^{∗} : S(𝔜) → S(𝔛)

is the functor defined by associating to any object M := (M_{dR}, M_{rig}, p) the object

u^{∗}M = (u_{dR}^{∗}M_{dR}, u_{rig}^{∗}M_{rig}, u^{∗}(p))

in S(𝔛).

5 The splitting principle

Let Log^{(n)} be the logarithmic sheaf defined in Example 3.12. In this section, we will extend the splitting principle of [Ban1] Proposition 5.2 to the points defined in Example 4.3.

Proposition 5.1 (splitting principle) Let d be a positive integer, and let z = ζ_{d} be a primitive d-th root of unity in K. Let

i_{z} = (i_{dR}, i_{rig}) : O_{K} → 𝔾_{m}

be the morphism of syntomic data of Example 4.3 corresponding to z. Then we have an isomorphism

i_{z}^{∗}Log^{(n)} ≅ ∏_{j=0}^{n} K(j)

in S(O_{K}).

The proof of the proposition will be given at the end of this section. In order to prove the proposition, it is necessary to explicitly calculate the map i_{z}^{∗}(p) of Lemma 4.4. For this purpose, we first review the Monsky-Washnitzer interpretation of overconvergent isocrystals and the explicit description of ϵ_{1,2} of (6) (See [Ber1] §2 and [T] §2 for details).

We assume for now that z is an arbitrary element in O_{K}^{×}. We denote by z_{0} the root of unity in W such that z ≡ z_{0} (mod π). Let A = Γ(𝔾_{m}_{O_{K}} , O_{𝔾_{m}_{O_{K}}}) = O_{K}[t, t^{−1}]. We fix a presentation

O_{K}[x_{1}, ⋯, x_{n}]/I ≅ A

over O_{K}, which defines a closed immersion

𝔾_{m}_{O_{K}} ↪ 𝔸_{O_{K}}^{n} .

Then the intersections U_{λ} of 𝔾_{mK}^{an} with the ball B(0, λ^{+}) ⊂ 𝔸_{K}^{n an} for λ → 1^{+} form a system of strict neighborhoods (Definition 3.4) of 𝔾̂_{mK} in 𝔾_{mK}^{an}. For λ > 1, we let A_{λ} = Γ(U_{λ}, O_{U_{λ}}). Then lim_{λ→1^{+}} A_{λ} = A^{†} ⊗ K, where A^{†} is the weak completion of A. Let M_{vec} = (M_{vec}, ∇_{vec}) be an object in S_{vec}(𝔾_{m}). By [Ber1] Proposition 2.2.3, M_{vec} is of the form j^{†}(M_{0}, ∇_{0}), where M_{0} is a coherent module with integrable connection ∇_{0} on a strict neighborhood U_{λ}. Let M_{λ} = Γ(U_{λ}, M_{0}). Then for λ′ < λ, the section Γ(U_{λ′}, M_{0}) is given by M_{λ′} = M_{λ} ⊗_{A_{λ}} A_{λ′}, and

M := Γ(𝔾_{mK}^{an}, M_{vec}) = \underrightarrow{lim}_{λ→1^{+}} M_{λ}.  (7)

M is a projective A^{†} ⊗ K-module with integrable connection ∇ : M → M ⊗ Ω_{A^{†}⊗K}^{1} induced from ∇_{0}. Suppose the connection ∇_{vec} is overconvergent. By [Ber1] Proposition 2.2.13, for any η < 1, there exists λ > 1 such that

∥\frac{1}{i!}∇_{λ}(∂_{t}^{i})(m)∥ η^{i} → 0 (i → ∞)  (8)

for any m ∈ M_{λ}. Here, ∇_{λ} : M_{λ} → M_{λ} ⊗ Ω_{A_{λ}/K}^{1} is the connection induced from ∇_{0}, ∂_{t} is the derivation by t, and ∥ − ∥ is a Banach norm on M_{λ}. Let M = (M_{dR}, M_{rig}, p) be an object in S(𝔾_{m}). Then

M_{vec} := F_{rig}(M_{rig}) = (M_{rig} ⊗_{K_{0}} K, ∇_{rig} ⊗_{K_{0}} K)

is an object in S_{vec}(𝔾_{m}). We have

i_{vec}^{∗}F_{rig}(M_{rig}) = M ⊗_{i_{vec}} K, F_{rig}(i_{rig}^{∗}M_{rig}) = M ⊗_{i_{rig}} K,

where M is as in (7), and i_{vec}, i_{rig} : A^{†} ⊗_{O_{K}} K → K are ring homomorphisms given respectively by t ↦ z and t ↦ z_{0}. By [Ber1] 2.2.17 Remarque,

ϵ_{1,2} : M ⊗_{i_{vec}} K ⟶^{≅} M ⊗_{i_{rig}} K

of (6) is given explicitly by the Taylor series

ϵ_{1,2}(m ⊗_{i_{vec}} 1) = ∑_{i≥0} \frac{1}{i!}∇(∂_{t}^{i})(m) ⊗_{i_{rig}} (z − z_{0})^{i}.  (9)

The existence of the Frobenius Φ_{M} on M_{rig} insures that the connection ∇_{rig} (hence ∇_{vec}) is overconvergent ([Ber1] Theorem 2.5.7). Since |z − z_{0}| < 1, the above series converges by (8). Next, let

Log^{(n)} := (L_{dR}^{(n)}, L_{rig}^{(n)}, p)

be the logarithmic sheaf of Example 3.12. As in (7), we L = Γ(𝔾_{mK}^{an}, L_{vec}^{(n)}) for L_{vec}^{(n)} = L_{rig}^{(n)} ⊗_{K_{0}} K. Then

L = ∏_{j=0}^{n}(A^{†} ⊗ K)e_{j}

for the basis e_{j} = e_{j,rig} ⊗ 1, and the connection is given by

∇(e_{j}) = e_{j+1} ⊗ \frac{dt}{t} (0 ≤ j ≤ n − 1).  (10)

Let u_{j}(t) be the function defined in Definition 2.2.

Proposition 5.2 For integers i, m ≥ 0, let a_{m}^{(i)} be elements in A_{K}^{†} such that

∇(∂_{t}^{i})(e_{0}) = ∑_{j=0}^{n} a_{j}^{(i)}e_{j}.

Then

∂_{t}^{i}(u_{m}) = ∑_{j=0}^{n} a_{j}^{(n)}u_{m−j}.

In particular, we have

a_{m}^{(i)}(z_{0}) = ∂_{t}^{i}(u_{m})(z_{0}).  (11)

Remark 5.3 The definition of a_{j}^{(i)} implies

∇(∂_{t}^{i})(e_{m}) = ∑_{j=0}^{n−m} a_{j}^{(i)}e_{m+j}.

Proof. We will give the proof by induction on i ≥ 0. Since a_{0}^{(0)} = 1, the statement is true for i = 0. Suppose for an integer i ≥ 0, we have

∂_{t}^{i}(u_{m}) = ∑_{j=0}^{n} a_{j}^{(i)}u_{m−j}.  (12)

By comparing the definition of a_{j}^{(i+1)} with the equality

∇(∂_{t}^{i+1})(e_{0}) = ∇(∂_{t}) ∘ ∇(∂_{t}^{i})(e_{0}) = ∑_{j=0}^{n} ((∂_{t}a_{j}^{(i)})e_{j} + t^{−1}a_{j}^{(i)}e_{j+1}) ,

we obtain the equality

a_{j}^{(i+1)} = ∂_{t}a_{j}^{(i)} + t^{−1}a_{j−1}^{(i)}.  (13)

Similarly, from the hypothesis (12) and ∂_{t}u_{m} = t^{−1}u_{m−1}, we have

∂_{t}^{i+1}(u_{m}) = ∂_{t} ∘ ∂_{t}^{i}(u_{m}) = ∑_{j=0}^{n} ((∂_{t}a_{j}^{(i)})u_{m−j} + t^{−1}a_{j}^{(i)}u_{m−j−1}) .

This together with (13) gives the desired result. (11) follows from the fact that since z_{0} is a root of unity, u_{m}(z_{0}) = 0 unless m = 0.

Corollary 5.4 For any integers i, m ≥ 0, we have

∇(∂_{t}^{i})(e_{m}) ⊗_{i_{rig}} 1 = ∑_{j=0}^{n−m} (e_{m+j} ⊗_{i_{rig}} ∂_{t}^{i}(u_{j})(z_{0})) .

Proof. The assertion follows immediately from Remark 5.3

Proposition 5.5 We have

ϵ_{1,2}(e_{m} ⊗_{i_{vec}} 1) = ∑_{j=0}^{n−m} (e_{m+j} ⊗_{i_{rig}} u_{j}(z))

for the map ϵ_{1,2} : L ⊗_{i_{vec}} K → L ⊗_{i_{rig}} K of (9) associated to L.

Proof. Since log(z_{0}) = 0, we have ∂_{t}^{i}(u_{j})(z_{0}) = 0 for i < j. Substituting z to the Taylor expansion of u_{j}(t) at t = z_{0} gives the equality

u_{j}(z) = ∑_{i=j}^{∞} \frac{1}{i!}∂_{t}^{i}(u_{j})(z_{0})(z − z_{0})^{i}.

The proposition now follows from the definition of ϵ_{1,2} (9) and Corollary 5.4. Let us now return to the case when z = ζ_{d} is a primitive d-th root of unity. Proof of Proposition 5.1. Since the connection is the only structure preventing L_{dR}^{(n)} and L_{rig}^{(n)} from splitting, we have

i_{dR}^{∗}L_{dR}^{(n)} = ∏_{j=0}^{n} Ke_{j,dR} i_{rig}^{∗}L_{rig}^{(n)} = ∏_{j=0}^{n} K_{0}e_{j,rig}.

It is sufficient to prove that the comparison isomorphism i_{z}^{∗}(p) respects the splitting. The isomorphism

p : i_{dR}^{∗}L_{dR}^{(n)} → L ⊗_{i_{vec}} K

is given by e_{j,dR} ↦ e_{j,rig}. Since z is a torsion point, u_{j}(z) = 0 for j ≠ 0. Hence by Proposition 5.5,

ϵ_{1,2} : L ⊗_{i_{vec}} K → L ⊗_{i_{rig}} K

maps e_{j,rig} ⊗_{i_{vec}} 1 to e_{j,rig} ⊗_{i_{rig}} 1. Hence i_{z}^{∗}(p) = ϵ_{1,2} ∘ p respects the splitting. We have

i_{z}^{∗}Log^{(n)} ≅ ∏_{j=0}^{n} K(j)

in S(O_{K}) as desired.

Remark 5.6 The calculation of Proposition 5.5 shows that if z is an arbitrary element in O_{K}^{×}, then

i_{z}^{∗}Log^{(n)} = (L_{z,dR}^{(n)}, L_{z,rig}^{(n)}, p_{z}) ∈ S(O_{K}),

where

L_{z,dR}^{(n)} = ∏_{j=0}^{n} Ke_{j,dR}, L_{z,rig}^{(n)} = ∏_{j=0}^{n} K_{0}e_{j,rig},

and

p_{z}(e_{m,dR}) = ∑_{j=0}^{n−m} e_{m+j,rig} ⊗_{K_{0}} u_{j}(z).

6 The specialization of pol to torsion points

In this section, we will first introduce the p-adic polylogarithmic extension pol calculated in [Ban1]. Then we will calculate its restriction to d-th roots of unity, where d is an integer of the form d = Np^{r} with (N, p) = 1 and N > 1. The case N = 1 will be treated in Section 8. Let 𝕌 be the syntomic datum correspoinding to the projective line minus three points, as defined in Definition 3.2. The p-adic polylogarithm sheaf is an extension in S(𝕌) of the trivial object K(0) by the logarithmic sheaf Log having a certain residue. In our previous paper, we determined the explicit shape of this sheaf.

Theorem 6.1 ([Ban1] Theorem 2) The p-adic polylogarithmic extension pol^{(n)} is the extension

0 → Log^{(n)} → pol^{(n)} → K(0) → 0

in S(𝕌), given explicitly by pol^{(n)} := (P_{dR}^{(n)}, P_{rig}^{(n)}, p), where:

(i) P_{dR}^{(n)} in S_{dR}(𝕌) is given by

P_{dR}^{(n)} = O_{ℙ_{K}^{1}} e_{dR} ⨁ L_{dR}^{(n)},

with connection ∇_{dR}(e_{dR}) = e_{1,dR} ⊗d log(t−1) and Hodge filtration given by the direct sum.

(ii) P_{rig}^{(n)} in S_{rig}(𝕌) is given by

P_{rig}^{(n)} = j^{†}O_{]𝕌_{k}[ℙ̂^{1}} e_{rig} ⨁ L_{rig}^{(n)},

with connection ∇_{rig}(e_{rig}) = e_{1,rig} ⊗ d log(t − 1) and Frobenius

Φ(e_{rig}) := e_{rig} + ∑_{j=1}^{n}(−1)^{j+1}ℓ_{j}^{(p)}(t)e_{j,rig}.  (14)

(iii) p is the isomorphism given by p(e_{dR}) = e_{rig} ⊗ 1.

Remark 6.2 In [Ban1] Theorem 2, the Frobenius is written as

Φ(e_{rig}) := e_{rig} + ∑_{j=1}^{n}(−1)^{j}ℓ_{j}^{(p)}(t)e_{j,rig}.

This is due to an error in the calculation of the proof. The correct Frobenius is the one given in (14).

Let z be a d-th root of unity, where d is an integer of the form d = Np^{r} with (N, p) = 1 and N > 1, and let z_{0} ∈ W such that z ≡ z_{0} (mod π). The purpose of this section is to prove the following theorem.

Theorem 6.3 The specialization of the polylogarithm at z is explicitly given as follows:

(i) i_{z}^{∗}P_{dR}^{(n)} = Ke_{dR} ⊕ ⨁_{j=0}^{n} Ke_{j,dR} with the natural Hodge filtration.

(ii) i_{z}^{∗}P_{rig}^{(n)} = K_{0}e_{rig} ⊕ ⨁_{j=0}^{n} K_{0}e_{j,rig} with Frobenius

Φ(e_{rig}) := e_{rig} + ∑_{j=1}^{n}(−1)^{j+1}ℓ_{j}^{(p)}(z_{0})e_{j,rig}.

(iii) p is the isomorphism given by

p(e_{dR}) = e_{rig} ⊗ 1 + ∑_{j=1}^{n} e_{j,rig} ⊗ (−1)^{j}(D_{j}(z) − D_{j}(z_{0})),

where D_{j}(t) is the function defined in Definition 2.2.

The proof of the theorem will be given at the end of this section. As in the case of Log, we first consider the Monsky-Washnitzer interpretation of pol^{(n)}. Let B_{K}^{†} = Γ(𝕌_{K}^{an}, j^{†}O_{𝕌_{K}^{an}}),

P_{vec}^{(n)} := F_{rig}(M_{rig}) = (M_{rig} ⊗_{K_{0}} K, ∇_{rig} ⊗_{K_{0}} K),

and P^{(n)} = Γ(𝕌_{K}^{an}, P_{vec}^{(n)}). Then we have

P^{(n)} = B_{K}^{†}e ⨁ ∏_{j=0}^{n} B_{K}^{†}e_{j}

where e = e_{rig} ⊗ 1 and e_{j} = e_{j,rig} ⊗ 1, with connection ∇(e) = e ⊗ d log(1 − t) and ∇(e_{j}) = e_{j+1} ⊗ d log t.

Proposition 6.4 For integers i, m > 0, let b_{m}^{(i)} be elements in B_{K}^{†} such that

∇(∂_{t}^{i})(e) = ∑_{j=1}^{n}(−1)^{j}b_{j}^{(i)}e_{j}.

Then

∂_{t}^{i}(D_{m}) = ∑_{j=1}^{n}(−1)^{m−j}b_{j}^{(i)}u_{m−j}.

In particular, we have

b_{m}^{(i)}(z_{0}) = ∂_{t}^{i}(D_{m})(z_{0}).  (15)

Proof. The proof is again by induction on i > 0. We first consider the case when i = 1. In this case, b_{1}^{(1)} = (1 − t)^{−1}. Since Li_{m−j}(t) and u_{j}(t) satisfy the differential equations

∂_{t}(Li_{j}(t)) = \frac{1}{t} Li_{j−1}(t) (j ≥ 1) ∂_{t}(u_{j}(t)) = \frac{u_{j−1}}{t} (∀j),

the definition of D_{m}(t) (Definition 2.2) and the fact that u_{j}(t) = 0 for j < 0 implies that:

∂_{t}(D_{m}) = ∑_{j=0}^{m−1}(−1)^{j}∂_{t}(Li_{m−j}(t)u_{j}(t))

= ∑_{j=0}^{m−1} \frac{(−1)^{j}}{t} (Li_{m−j−1}(t)u_{j}(t) + Li_{m−j}(t)u_{j−1}(t))

= \frac{(−1)^{m−1}}{t} Li_{0}(t)u_{m−1}(t) = (−1)^{m−1} \frac{u_{m−1}(t)}{1 − t}

= (−1)^{m−1}b_{1}^{(1)}(t)u_{m−1}(t).

Hence the statement is true for i = 1. Suppose for an integer i ≥ 1, we have

∂_{t}^{i}(D_{m}) = ∑_{j=1}^{n}(−1)^{m−j}b_{j}^{(i)}u_{m−j}.  (16)

Specialization to p-th Power Roots of Unity 91

By comparing the definition of b_{j}^{(i+1)} with the equality

∇(∂_{t}^{i+1})(e_{0}) = ∇(∂_{t}) ∘ ∇(∂_{t}^{i})(e_{0}) = ∑_{j=1}^{n}(−1)^{j} ((∂_{t}b_{j}^{(i)})e_{j} + t^{−1}b_{j}^{(i)}e_{j+1}) ,

we obtain the equality

b_{j}^{(i+1)} = ∂_{t}b_{j}^{(i)} − t^{−1}b_{j−1}^{(i)} (i ≥ 1, j > 1). (17)

Similarly, from the hypothesis (16) and ∂_{t}u_{m} = t^{−1}u_{m−1}, we have

\begin{aligned}∂_{t}^{i+1}(D_{m}) & = ∂_{t} (∑_{j=1}^{i}(−1)^{m−j}b_{j}^{(i)}u_{m−j}) \\ & = ∑_{j=1}^{n}(−1)^{m−j} ((∂_{t}b_{j}^{(i)})u_{m−j} + t^{−1}b_{j}^{(i)}u_{m−j−1}) . \end{aligned}

This together with (17) gives the desired result. (15) follows from the fact that since z_{0} is a root of unity, u_{m}(z_{0}) = 0 unless m = 0. Proposition 6.5 We have

ϵ_{1,2}(e ⊗_{i_{vec}} 1) = e ⊗_{i_{rig}} 1 + ∑_{j=1}^{n} (e_{j} ⊗_{i_{rig}} (−1)^{j}(D_{j}(z) − D_{j}(z_{0})))

for the map ϵ_{1,2} : P ⊗_{i_{vec}} K → P ⊗_{i_{rig}} K of (9) associated to P. Proof. Substituting z to the Taylor expansion of D_{j}(t) at t = z_{0} gives the equality

D_{j}(z) = ∑_{i=0}^{∞} \frac{1}{i!}∂_{t}^{i}(D_{j})(z_{0})(z − z_{0})^{i}.

The proposition now follows from the definition of ϵ_{1,2} and Proposition 6.4. 7 The main result (Case N > 1) The following lemma is well-known. Lemma 7.1 There is a canonical isomorphism

Ext_{S(O_{K})}^{1}(K(0), K(j)) = K(j)_{dR}  (18)

for j > 0.

Documenta Mathematica · Extra Volume Kato (2003) 73–97

Proof. Suppose M̃ = (M̃_{dR}, M̃_{rig}, p̃) is an extension of K(0) by K(j) in S(O_{K}). We have exact sequences

0 → K(j)_{dR} → M̃_{dR} → K(0)_{dR} → 0
0 → K(j)_{rig} → M̃_{rig} → K(0)_{rig} → 0.

Denote by e_{j,dR} and e_{j,rig} the basis of K(j)_{dR} and K(j)_{rig}, and let ẽ_{0,dR} and ẽ_{0,rig} respectively be the liftings of e_{0,dR} and e_{0,rig} in M̃_{dR} and M̃_{rig}. If we map ẽ_{0,dR} to e_{0,dR}, then we have an isomorphism

M̃_{dR} ≅ K(0)_{dR} ⨁ K(j)_{dR}

in S_{dR}(O_{K}). Next, since the quotient of M by K(j) is isomorphic to K(0), the Frobenius and p̃ is given by

p̃(ẽ_{0,dR}) = ẽ_{0,rig} ⊗ 1 + e_{j,rig} ⊗ a
ϕ^{∗}(ẽ_{0,rig}) = ẽ_{0,rig} + ce_{j,rig}

for some a ∈ K and c ∈ K_{0}. If we take b ∈ K_{0} such that (1 − σ/p^{j})b = c, then we have an isomorphism

M̃_{rig} ≅ K(0)_{rig} ⨁ K(j)_{rig}

in S_{rig}(O_{K}) given by ẽ_{0,rig} ↦ e_{0,rig} − be_{j,rig}. The above shows that we have an isomorphism

M̃ ≅ (K(0)_{dR} ⨁ K(j)_{dR}, K(0)_{rig} ⨁ K(j)_{rig}, p)

of extensions of K(0) by K(j) in S(O_{K}), where p is the isomorphism given by

p(e_{0,dR}) = ẽ_{0,rig} ⊗ 1 + e_{j,rig} ⊗ a
= e_{0,rig} ⊗ 1 + e_{j,rig} ⊗ (a + b).

The canonical map of the lemma is given by associating to M̃ the element (a + b)e_{j,dR} in K(j)_{dR}. The inverse of this canonical map is constructed by associating to we_{j,dR} in K(j)_{dR} the extension

(K(0)_{dR} ⨁ K(j)_{dR}, K(0)_{rig} ⨁ K(j)_{rig}, p) ,

where

p(e_{0,dR}) = e_{0,rig} ⊗ 1 + e_{j,rig} ⊗ w.

This construction shows that the canonical map is in fact an isomorphism.

Remark 7.2 Suppose K = K_{0}. Then by [Ban1] Theorem 1 and Example 2.8, we have an isomorphism

Ext_{S(O_{K})}^{1}(K(0), K(j)) ⟶^{≅} H_{syn}^{1}(O_{K}, K(j)) = K(j)_{rig}.  (19)

If M is an extension in S(O_{K}) corresponding to ae_{j,dR} in Lemma 7.1, then M maps by (19) to ((1 − p^{−j}σ)a)e_{j,rig} in K(j)_{rig} .

The following theorem is Theorem 1 of the introduction.

Theorem 7.3 Let z be a torsion point of order d = Np^{r}, where (N, p) = 1 and N > 1. Then

i_{z}^{∗} pol^{(n)} = ((−1)^{j} Li_{j}(z)e_{j,dR})_{j≥1}

in

Ext_{S(O_{K})}^{1}(K(0), i_{z}^{∗}Log(1)) = ∏_{j=0}^{n} Ext_{S(O_{K})}^{1}(K(0), K(j)),

where we view (−1)^{j} Li_{j}(z)e_{j,dR} as an element in Ext_{S(O_{K})}^{1}(K(0), K(j)) through the isomorphism of lemma 7.1

Proof. By Theorem 6.3, the image of i_{z}^{∗} pol^{(n)} in Ext_{S(O_{K})}^{1}(K(0), K(j)) is the extension M̃ = (M_{dR}, M̃_{rig}, p̃) given as follows: M_{dR} is the direct sum

M_{dR} = K(0)_{dR} ⨁ K(j)_{dR},

M̃_{rig} is the extension of K(0)_{rig} by K(j)_{rig} with the Frobenius given by

Φ(ẽ_{0,rig}) = ẽ_{0,rig} + (−1)^{j+1}ℓ_{j}^{(p)}(z_{0})e_{j,rig}

for the lifting ẽ_{0,rig} of e_{0,rig} in M̃_{rig}, and p̃ is the isomorphism given by

p̃(e_{0,dR}) = ẽ_{0,rig} ⊗ 1 + e_{j,rig} ⊗ (−1)^{j}(Li_{j}(z) − Li_{j}(z_{0})).

This implies that, in the notation of Lemma 7.1, we have

\begin{aligned}a & = (−1)^{j}(Li_{j}(z) − Li_{j}(z_{0})) \\ c & = (−1)^{j+1}ℓ_{j}^{(p)}(z_{0}). \end{aligned}

Since z_{0} is a root of unity prime to p, the Frobenius acts by σ(z_{0}) = z_{0}^{p}. Hence the Formula of Propisition 2.1 (iii) gives

ℓ_{j}^{(p)}(z_{0}) = (1 − \frac{σ}{p^{j}} ) Li_{j}(z_{0}).

Again, in the notation of Lemma 7.1, we have

c = (−1)^{j+1} Li_{j}(z_{0}).

Since a + b = (−1)^{j} Li_{j}(z), the construction of the canonical map shows that the image of i_{z}^{∗} pol^{(n)} in Ext_{S(O_{K})}^{1}(K(0), K(j)) maps to (−1)^{j} Li_{j}(z)e_{j,dR} in K(j)_{dR}.

8 The main result (Case N = 1)

In this section, we will consider the specialization of the polylogarithm sheaf to p-th power roots of unity. As mentioned in the introduction, we will consider a slightly modified version of the polylogarithm. Let c > 1 be an integer prime to p, and let 𝕌_{c,O_{K}}^{0} = Spec O_{K}[t, (1 − t^{c})^{−1}]. We denote by 𝕌_{c}^{0} the syntomic data

𝕌_{c}^{0} = (𝕌_{c,O_{K}}^{0} , ℙ_{O_{K}}^{1} , ℙ̂^{1}, ϕ).

The multiplication by [c] map on 𝔾_{m}_{O_{K}} defines a morphism of syntomic datum

[c] : 𝕌_{c}^{0} → 𝕌.

Definition 8.1 We define the modified p-adic polylogarithmic pol_{c}^{(n)} by

pol_{c}^{(n)} = pol^{(n)} −[c]^{∗} pol^{(n)} ∈ Ext_{S(𝕌_{c}^{0})}^{1}(K(0), Log^{(n)}).

The explicit shape of pol^{(n)} given in Theorem 6.1 and the definition of the pull-back [c]^{∗} gives the following proposition. Let

θ_{c}(t) = \frac{1 − t^{c}}{1 − t} .

Proposition 8.2 The modified p-adic polylogarithmic pol_{c}^{(n)} is the extension in S(𝕌_{c}^{0}), given explicitly by pol_{c}^{(n)} := (P_{dR}^{(n)}, P_{rig}^{(n)}, p), where:

(i) P_{dR}^{(n)} in S_{dR}(𝕌_{c}^{0}) is given by

P_{dR}^{(n)} = O_{ℙ_{K}^{1}} e_{dR} ⨁ L_{dR}^{(n)},

with connection ∇_{c,dR}(e_{dR}) = e_{1,dR}⊗d log θ_{c}(t) and Hodge filtration given by the direct sum.

(ii) P_{rig}^{(n)} in S_{rig}(𝕌_{c}^{0}) is given by

P_{rig}^{(n)} = j^{†}O_{]𝕌_{c,k}^{0}[ℙ̂^{1}} e_{rig} ⨁ L_{rig}^{(n)},

with connection ∇_{c,rig}(e_{rig}) = e_{1,rig} ⊗ d log θ_{c}(t) and Frobenius

Φ(e_{rig}) := e_{rig} + ∑_{j=1}^{n}(−1)^{j+1}ℓ_{j,c}^{(p)}(t)e_{j,rig},

(iii) p is the isomorphism given by p(e_{dR}) = e_{rig} ⊗ 1.

Let 𝕌_{c,O_{K}} = Spec O_{K}[t, θ_{c}(t)^{−1}], and denote by 𝕌_{c} the syntomic data

𝕌_{c} = (𝕌_{c,O_{K}} , ℙ_{O_{K}}^{1} , ℙ̂^{1}, ϕ).

The explicit shape of pol_{c}^{(n)} given in the previous proposition shows that pol_{c}^{(n)} is in fact an object in S(𝕌_{c}). In particular, we can specialize pol_{c}^{(n)} at points on the open unit disc around one. Similar caluclations as that of Theorem 6.3 with ℓ_{j}^{(p)}, D_{j}^{(p)} and D_{j} replaced by ℓ_{j,c}^{(p)}, D_{j,c}^{(p)} and D_{j,c} gives the following theorem, which is Theorem 2 of the introduction.

Theorem 8.3 Let z be a p^{r}-th root of unity, and let z_{0} = 1. Then the specialization of the modified polylogarithm at z is explicitly given as follows:

(i) i_{z}^{∗}P_{dR}^{(n)} = Ke_{dR} ⊕ ⨁_{j=0}^{n} Ke_{j,dR} with the natural Hodge filtration.

(ii) i_{z}^{∗}P_{rig}^{(n)} = Ke_{rig} ⊕ ⨁_{j=0}^{n} Ke_{j,rig} with Frobenius

Φ(e_{rig}) := e_{rig} + ∑_{j=1}^{n}(−1)^{j+1}ℓ_{j,c}^{(p)}(z_{0})e_{j,rig}.

(iii) p_{c} is the isomorphism given by

p_{c}(e_{dR}) = e_{rig} ⊗ 1 + ∑_{j=1}^{n} e_{j,rig} ⊗ (−1)^{j}(D_{j,c}(z) − D_{j,c}(z_{0})).

As a corollary, we obtain the following result.

Corollary 8.4 Let z be a torsion point of order p^{r}. Then

i_{z}^{∗} pol_{c}^{(n)} = ((−1)^{j} Li_{j}(z)e_{j,dR})_{j≥1}

in

Ext_{S(O_{K})}^{1}(K(0), i_{z}^{∗}Log(1)) = ∏_{j=0}^{n} Ext_{S(O_{K})}^{1}(K(0), K(j)),

where we view (−1)^{j} Li_{j.c}(z)e_{j,dR} as an element in Ext_{S(O_{K})}^{1}(K(0), K(j)) through the isomorphism of lemma 7.1

References

[BC] F. Baldassarri and B. Chiarellotto, Algebraic versus rigid cohomology with logarithmic coefficients. In: V. Cristante, W. Messing (eds.), Barsotti Symposium in Algebraic Geometry, Perspectives in Math. 15, Academic Press (1994).

[Ban1] K. Bannai, Rigid syntomic cohomology and p-adic polylogarithms, J. Reine Angew. Math. 529 (2000), 205-237.

[Ban2] K. Bannai, Syntomic cohomology as a p-adic absolute Hodge cohomology, Math. Z. 242 (2002) No. 3, 443-480.

[Ber1] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, Première partie, Prépublication IRMAR 96-03, Université de Rennes (1996).

[Ber2] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. math. 128 (1997), 329-377.

[Bei] A. A. Beilinson, Polylogarithm and Cyclotomic Elements, typewritten preprint, MIT 1989 or 1990.

[BD] A. A. Beilinson and P. Deligne, Motivic Polylogarithm and Zagier’s conjecture, preprint, 1992.

[Bes1] A. Besser, Syntomic Regulators and p-adic Integration I: Rigid Syntomic Regulators, Israel. J. Math. 120 (2000), 291-334.

[Bes2] A. Besser, Syntomic Regulators and p-adic Integration II: K_{2} of curves, Israel. J. Math. 120 (2000), 335-359.

[BdJ] A. Besser and R. de Jeu, The syntomic regulator for K-theory of fields, preprint 15 December 2001, arXiv:math.AG/0110334v2.

[BGR] S. Bosch, U. Güntzer, and R. Remmert, Non-archimedean analysis, Grundlehren der math. Wissenschaften 261, Springer-Verlag (1984).

[Ch] B. Chiarellotto, Weights in rigid cohomology Applications to unipotent F -isocrystals, Ann. Scient. École Norm. Sup. 31 (1998), 683-715.

[Col] R. Coleman, Dilogarithms, Regulators and p-adic L-functions, Inv. math. 69 (1982), 171-208.

[C-LS] J. Y. Etesse and B. Le Stum, Fonctions L associées aux F -isocristaux surconvergents. I. Interprétation cohomologique, Math. Ann. 296 (1993), 557-576.

[D] P. Deligne, Le Groups Fondamental de la Droite Projective Moins Trois Points, In:Galois Groups over ℚ, Proceedings of a workshop held March 23-27, 1987 at the MSRI, Berkeley, California, Springer 1989, 79-297.

[Fon1] J.-M. Fontaine, Modules Galoisiens, modules filtrés et anneaux de Barsotti-Tate, Astérisque 65 (1979), 3-80.

[Fon2] J.-M. Fontaine, Exposé III: Représentations p-adiques semi-stables, Astérisque 223 (1994), 113-184.

[GK] M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques I (avec un appendice par Masato Kurihara), Invent. Math. 99 (1990), 293-320.

[G] M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques II, Invent. Math. 115 (1994), 61-79.

[HW1] A. Huber and J. Wildeshaus, The Classical Polylogarithm, Abstract of a series of lectures given at a workshop on polylogs in Essen, May 1-4 (1997).

[HW2] A. Huber and J. Wildeshaus, Classical Motivic Polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27-133.

[MW] P. Monsky and G. Washnitzer, Formal Cohomology: I, Annals of Math 88 (1968), 181-217.

[Ni] W. Nizioł, Cohomology of crystalline smooth sheaves, Comp. Math. 129 (2001), 123-147.

[So] M. Somekawa, Log syntomic regulators and p-adic polylogarithms, K-Theory 17 (1999), 256-294.

[Su] S. Sugimoto, Filtered modules and p-adic polylogarithms, Ph. D. thesis, University of Tokyo (1992).

[T] N. Tsuzuki, On the Gysin isomorphism of rigid cohomology, Hiroshima Math. J. 29 no. 3 (1999), 479-527.

[W] J. Wildeshaus, Realization of Polylogarithms, SLN 1650 (1997).

Kenichi Bannai

Graduate School of Mathematics

Nagoya University

Furo-cho, Chikusa-ku, Nagoya 464-8602 Japan bannai@math.nagoya-u.ac.jp 98

Documenta Mathematica · Extra Volume Kato (2003)