Bounds for the Dimensions

of p-Adic Multiple L-Value Spaces

Dedicated to Professor Andrei Suslin

on the occasion of his 60th birthday

Go Yamashita^{1}

Received: January 13, 2009 Revised: October 12, 2009

Abstract. First, we will define p-adic multiple L-values (p-adic MLV’s), which are generalizations of Furusho’s p-adic multiple zeta values (p-adic MZV’s) in Section 2.

Next, we prove bounds for the dimensions of p-adic MLV-spaces in Section 3, assuming results in Section 4, and make a conjecture about a special element in the motivic Galois group of the category of mixed Tate motives, which is a p-adic analogue of Grothendieck’s conjecture about a special element in the motivic Galois group. The bounds come from the rank of K-groups of ring of S-integers of cyclotomic fields, and these are p-adic analogues of Goncharov-Terasoma’s bounds for the dimensions of (complex) MZV-spaces and Deligne-Goncharov’s bounds for the dimensions of (complex) MLV-spaces. In the case of p-adic MLV-spaces, the gap between the dimensions and the bounds is related to spaces of modular forms similarly as the complex case.

In Section 4, we define the crystalline realization of mixed Tate motives and show a comparison isomorphism, by using p-adic Hodge theory.

2010 Mathematics Subject Classification: Primary 11R42; Secondary 11G55, 14F42, 14F30. Keywords and Phrases: p-adic multiple zeta values, mixed Tate motives, algebraic K-theory, p-adic Hodge theory.

1Partially supported by JSPS Research Fellowships for Young Scientists. The author also thanks the financial support by the 21st Century COE Program in Kyoto University “Formation of an international center of excellence in the frontiers of mathematics and fostering of researchers in future generations”, and EPSRC grant EP/E049109/1.

Supported by JSPS Research Fellowships for Young Scientists.

Documenta Mathematica · Extra Volume Suslin (2010) 687–723

Contents

1 Introduction. 688

2 p-adic Multiple L-values. 691 2.1 The Twisted p-adic Multiple Polylogarithm. . . . . . . . . . . . 692 2.2 The p-adic Drinfel’d Associator for Twisted p-adic Multiple

Polylogarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

3 Bounds for Dimensions of p-adic Multiple L-value spaces. 696 3.1 The Motivic Fundamental Groupoids of 𝕌_{N}. . . . . . . . . . . 696 3.2 The p-adic MLV-space in the Sense of Deligne. . . . . . . . . . 699 3.3 The Tannakian Interpretations of Two p-adic MLV’s. . . . . . . 703

4 Crystalline Realization of Mixed Tate Motives. 708 4.1 Crystalline Realization. . . . . . . . . . . . . . . . . . . . . . . 708 4.2 Comparison Isomorphism. . . . . . . . . . . . . . . . . . . . . . 711 4.3 Some Remarks and Questions. . . . . . . . . . . . . . . . . . . 715

1 Introduction.

For the multiple zeta values (MZV’s)

ζ(k_{1}, …, k_{d}) := ∑_{n_{1}<···<n_{d}} \frac{1}{n_{1}^{k_{1}} ⋯ n_{d}^{k_{d}}} (= lim_{ℂ∋z→1} Li_{k_{1},...,k_{d}}(z))

(k_{1}, …, k_{d−1} ≥ 1, k_{d} ≥ 2), Zagier conjectures the dimension of the space of MZV’s

Z_{w} := ⟨ζ(k_{1}, …, k_{d}) | d ≥ 1, k_{1} + ⋯ + k_{d} = w, k_{1}, …, k_{d−1} ≥ 1, k_{d} ≥ 2⟩_{ℚ} ⊂ ℝ,

and Z_{0} := ℚ (Here, ⟨⋯ ⟩_{ℚ} means the ℚ-vector space spanned by ⋯) as follows.

Conjecture 1 (Zagier) Let D_{n+3} = D_{n+1} + D_{n}, D_{0} = 1, D_{1} = 0, D_{2} = 1

(that is, the generating function ∑_{n=0}^{∞} D_{n}t^{n} is \frac{1}{1 − t^{2} − t^{3}} ). Then, for w ≥ 0 we have

dim_{ℚ} Z_{w} = D_{w}.

Terasoma, Goncharov, and Deligne-Goncharov proved the upper bound:

Theorem 1.1 (Terasoma [T], Goncharov [G1], Deligne-Goncharov [DG]) For w ≥ 0, we have

dim_{ℚ} Z_{w} ≤ D_{w}.

Bounds for the Dimensions of p-Adic MLV-Spaces 689

Deligne-Goncharov also proved an upper bound for dimensions of multiple Lvalue (MLV) spaces. ([DG]) On the other hand, Furusho defined p-adic MZV’s [Fu1] by using Coleman’s iterated integral theory:

ζ_{p}(k_{1}, …, k_{d}) := lim_{ℂ_{p}∋z→1}′Li_{k_{1},...,k_{d}}^{a}(z).

where Li^{a} is the p-adic multiple polylogarithm defined by Coleman’s iterated integral, and a is a branching parameter (For the notations lim′, see [Fu1, Notation 2.12]). For k_{d} ≥ 2, RHS converges, and the limit value is independent of a and lands in ℚ_{p} ([Fu1, Theorem 2.13, 2.18, 2.25]). Put

Z_{w}^{p} := ⟨ζ_{p}(k_{1}, …, k_{d}) | d ≥ 1, k_{1}+⋯+k_{d} = w, k_{1}, …, k_{d−1} ≥ 1, k_{d} ≥ 2⟩_{ℚ} ⊂ ℚ_{p},

and Z_{0}^{p} := ℚ. Note that for k_{d} = 1, p-adic MZV’s may converge, however, these are ℚ-linear combinations of p-adic MZV’s corresponding to the same weight indices with k_{d} ≥ 2 (See, [Fu1, Theorem 2.22]). The following conjecture is proposed.

Conjecture 2 (Furusho-Y.) Let d_{n+3} = d_{n+1} + d_{n}, d_{0} = 1, d_{1} = 0, d_{2} = 0

(that is, the generating function ∑_{n=0}^{∞} d_{n}t^{n} is \frac{1 − t^{2}}{1 − t^{2} − t^{3}} ). Then, for w ≥ 0

we have

dim_{ℚ} Z_{w}^{p} = d_{w}.

From the fact ζ_{p}(2) = 0 and the motivic point of views (see, Remark 3.7, p-adic analogue of Grothendieck’s conjecture about an element of a motivic Galois group (Conjecture 4), and Proposition 3.12), it seems natural to conjecture as above.

Remark 1.2 The conjecture implies that dim_{ℚ} Z_{w}^{p} is independent of p. On the other hand, ζ_{p}(2k + 1) ≠ 0 is equivalent to the higher Leopoldt conjecture in the Iwasawa theory. For a regular prime p, or a prime p satisfying (p − 1) | 2k, we have ζ_{p}(2k + 1) ≠ 0. However, it is not known if ζ_{p}(2k + 1) is zero or not in general. Thus, it is non-trivial that dim_{ℚ} Z_{w}^{p} is independent of p (See also [Fu1, Example 2.19 (b)]). It seems that the above conjecture contains the “Leopoldt conjecture for higher depth”.

For Conjecture 2, we will prove the following result.

Theorem 1.3 For w ≥ 0, we have

dim_{ℚ} Z_{w}^{p} ≤ d_{w}.

We can also define p-adic multiple L-values for N-th roots of unity ζ_{1}, …, ζ_{d} and k_{1}, …, k_{d} ≥ 1, (k_{d}, ζ_{d}) ≠ (1, 1) and a prime ideal 𝔭 ∤ N above p in the cyclotomoic field ℚ(μ_{N}),

L_{𝔭}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) ∈ ℚ(μ_{N})_{𝔭},

by Coleman’s iterated integral as Furusho did for MZV’s (See, Section 2.1). Here, ℚ(μ_{N})_{𝔭} is the completion of ℚ(μ_{N}) at the finite place 𝔭. Put

Z_{w}^{𝔭}[N] :=⟨L_{𝔭}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) | d ≥ 1, k_{1} + ⋯ + k_{d} = w, k_{1}, …, k_{d} ≥ 1,
ζ_{1}^{N} = ⋯ = ζ_{d}^{N} = 1, (k_{d}, ζ_{d}) ≠ (1, 1)⟩_{ℚ} ⊂ ℚ(μ_{N})_{𝔭},
and Z_{0}^{𝔭}[N] := ℚ.

This Z_{w}^{p}[1] is equal to the above Z_{w}^{p}. We will also prove bounds for the dimensions of p-adic MLV’s.

Theorem 1.4 For w ≥ 0, we have

dim_{ℚ} Z_{w}^{𝔭}[N] ≤ d[N]_{w}.

Here, d[N]_{w} is defined as follows:

1. For N = 1, d[1]_{n+3} = d[1]_{n+1} + d[1]_{n} (n ≥ 0), d[1]_{0} = 1, d[1]_{1} = 0,

d[1]_{2} = 0, that is, the generating function is \frac{1 − t^{2}}{1 − t^{2} − t^{3}} (This d[1]_{n} is

equal to the above d_{n}).

2. For N = 2, d[2]_{n+2} = d[2]_{n+1} + d[2]_{n} (n ≥ 1), d[2]_{0} = 1, d[2]_{1} = 1,

d[2]_{2} = 1, that is, the generating function is \frac{1 − t^{2}}{1 − t − t^{2}} .

3. For N ≥ 3, d[N]_{n+2} = (\frac{φ(N)}{2} + ν) d[N]_{n+1} − (ν − 1)d[N]_{n} (n ≥ 0),

d[N]_{0} = 1, d[N]_{1} = \frac{φ(N)}{2} + ν − 1, that is, the generating function is

\frac{1 − t}{1 − (\frac{φ(N)}{2} + ν) t + (ν − 1)t^{2}} . Here, φ(N) := #(ℤ/Nℤ)^{×}, and ν is the

number of prime divisors of N.

Remark 1.5 It is not known that dim_{ℚ} Z_{w}^{𝔭}[N] is independent of p.

Remark 1.6 In the proof of the above bounds, we use some kinds of (pro- )varieties, which are related to the algebraic K-theory. For N > 4, the above bounds are not best possible in general, because in the proof, we use smaller varieties in general than varieties, which give the above bounds. The gap of dimensions is related to the space of cusp forms of weight 2 on X_{1}(N) if N is a prime. See also [DG, 5.27][G2].

In the proof of the above theorem, we use a special element in motivic Galois group of the category of mixed Tate motives like in the complex case ([DG]). We also propose a p-adic analogue of Grothendieck’s conjecture on this special element (see Section 3 for the details):

Conjecture 3 (= Conjecture 4 in Section 3, p-adic analogue of Grothendieck’s conjecture) The element φ_{𝔭} ∈ U_{ω}(ℚ(μ_{N})_{𝔭}) is ℚ-Zariski dense. That means that if a subvariety X of U_{ω} over ℚ satisfies φ_{𝔭} ∈ X(ℚ(μ_{N})_{𝔭}), then X = U_{ω}.

Finally, we will give the plan of this paper. First, we define the p-adic MLV’s, twisted p-adic multiple polylogarithms (twisted p-adic MPL’s), and p-adic Drinfel’d associator for twisted p-adic MPL’s in Section 2. Next, assuming results of Section 4, we will show bounds for dimensions of p-adic MLV-spaces in the sense of Deligne [D1][DG], by using the motivic fundamental groupoid constructed in [DG] in Section 3.2. Lastly, we show bounds for dimensions of Furusho’s p-adic MLV-spaces, by comparing the two p-adic MLV-spaces in the Tannakian interpretation in Section 3.3. In Section 4, we construct the crystalline realization of mixed Tate motives, and prove a comparison isomorphism, by using p-adic Hodge theory. In the end of this article, we propose some questions. We fix conventions. We use the notation γ′γ for a composition of paths, which means that γ followed by γ′. Similarly, we use the notation g′g for a product of elements in a motivic Galois group, which means that the action of g followed by the one of g′. Acknowledgement. In the proof of the main theorem, a crucial ingredient is the algebraic K-theory, the area of mathematics to which Professor Andrei A. Suslin greatly contributed. It is great pleasure for the author to dedicate this paper to Professor Andrei A. Suslin. He sincerely thanks to Hidekazu Furusho for introducing to the author the theory of multiple zeta values and the theory of Grothendieck-Teichmüller group, and for helpful discussions. He also expresses his gratitude to Professor Pierre Deligne for helpful discussions for the crystalline realization. The last chapter of this paper is written during the author’s staying at IHES from January/2006 to July/2006. He also thanks to the hospitality of IHES. Finally, he thanks the referee for kind comments.

2 p-adic Multiple L-values.

In this section, we define twisted p-adic multiple polylogarithms (twisted p-adic MPL), p-adic multiple L-values (p-adic MLV), p-adic KZ-equation for twisted p-adic MPL, and p-adic Drinfel’d associator for twisted p-adic MPL, similarly as Furusho’s definitions in [Fu1]. We discuss the fundamental properties of them. Fix a prime ideal 𝔭 in ℚ(μ_{N}), and an embedding ι_{𝔭} : ℚ(μ_{N}) ↪ ℂ_{p}. Put S := \{0, ∞\} ∪ μ_{N}, 𝕌_{N} := ℙ_{ℚ(μ_{N})}^{1} \ S, and \overline{𝕌_{N}} := 𝕌_{N} ⊗_{ℚ(μ_{N})} ℂ_{p} (The variety 𝕌_{N} is defined over ℚ, however, we use 𝕌_{N} over ℚ(μ_{N}) for the purpose of bounding dimensions in the next section).

2.1 The Twisted p-adic Multiple Polylogarithm.

We use the same notations as in [Fu1]: the tube ]x[⊂ ℙ_{ℂ_{p}}^{1} of x ∈ (𝕌_{N})_{𝔽_{p}}(\overline{𝔽_{p}}), the algebra A(U) of rigid analytic functions on U, and the algebra A_{Col}^{a} of Coleman functions on \overline{𝕌_{N}} with a branching parameter a.

Definition 2.1 For 𝔭 ∤ N, k_{1}, …, k_{d} ≥ 1, and ζ_{1} …, ζ_{d} ∈ μ_{N}, we define the (one variable) twisted p-adic multiple polylogarithm (twisted p-adic MPL) Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) ∈ A_{Col}^{a} attached to a ∈ ℂ_{p} by the following integrals inductively:

Li_{(1;ζ_{1})}^{a}(z) := − log^{a}(ι_{𝔭}(ζ_{1}) − z) := ∫_{0}^{z} \frac{dt}{ι_{𝔭}(ζ_{1}) − t},

Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) := \begin{cases}∫_{0}^{z} \frac{1}{t}Li_{(k_{1},...,(k_{d}−1);ζ_{1},...,ζ_{d})}^{a}(t)dt & k_{d} ≠ 1, \\ ∫_{0}^{z} \frac{1}{ι_{𝔭}(ζ_{d}) − t}Li_{(k_{1},...,k_{(d−1)};ζ_{1},...,ζ_{d−1})}^{a}(t)dt & k_{d} = 1. \end{cases}

Here, log^{a} is the logarithm with a branching parameter a, which means log^{a}(p) = a.

Remark 2.2 For |z|_{p} < 1, it is easy to see that

Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) = ∑_{0<n_{1}<···<n_{d}} \frac{ι_{𝔭}(ζ_{1}^{−n_{1}}ζ_{2}^{n_{1}−n_{2}} ⋯ ζ_{d}^{n_{d−1}−n_{d}})z^{n_{d}}}{n_{1}^{k_{1}} ⋯ n_{d}^{k_{d}}} .

Inductively, we can easily verify that Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z)|_{]0[} ∈ A(]0[), Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z)|_{]∞[} ∈ A(]∞[)[log^{a} t^{−1}], and Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z)|_{]ι_{𝔭}(ζ)[} ∈ A(]ι_{𝔭}(ζ)[)[log^{a}(z − ι_{𝔭}(ζ))] for ζ ∈ μ_{N}.

Proposition 2.3 Fix k_{1}, …, k_{d} ≥ 1, and N-th roots of unity ζ_{1}, …, ζ_{d} ∈ μ_{N}. Then the convergence of lim_{ℂ_{p}∋z→1}′Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) is independent of branches

a ∈ ℂ_{p}. Moreover, if it converges in ℂ_{p}, the limit value is independent of branches a ∈ ℂ_{p} and lands in ℚ(μ_{N})_{𝔭} (For the notation lim′, see [Fu1, Notation 2.12]).

Proof The same as [Fu1, Theorem 2.13, Theorem 2.25].

Definition 2.4 When the limit lim′_{ℂ_{p}∋z→1} Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) converges, we define the corresponding p-adic multiple L-value to be its limit value:

L_{𝔭}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) := lim_{ℂ_{p}∋z→1}′Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z)

For example, L_{𝔭}(1; ζ) = − log^{a}(ι_{𝔭}(ζ) − 1) (1 ≠ ζ ∈ μ_{N}) is independent of a, since log^{a}(z) does not depend on a for |z| = 1. (Recall that we assume 𝔭 ∤ N.)

2.2 The p-adic Drinfel’d Associator for Twisted p-adic Multiple Polylogarithms.

Let A_{ℂ_{p}}^{∧} := ℂ_{p}⟨⟨A, B_{ζ} | ζ ∈ μ_{N}⟩⟩ be the non-commutative formal power series ring with ℂ_{p} coefficients generated by variables A and B_{ζ} for ζ ∈ μ_{N}. For a word W consisting of A and {B_{ζ}}_{ζ∈μ_{N}}, we call the sum of all exponents of A and {B_{ζ}}_{ζ∈μ_{N}} the weight of W, and the sum of all exponents of {B_{ζ}}_{ζ∈μ_{N}} the depth of W.

Definition 2.5 Fix a prime ideal 𝔭 above p in ℚ(μ_{N}) and an embedding ι_{𝔭} : ℚ(μ_{N}) ↪ ℂ_{p}. The p-adic Knizhnik-Zamolodchikov equation (p-adic KZequation) is the differential equation

\frac{dG}{dz}(z) = (\frac{A}{z} + ∑_{ζ∈μ_{N}} \frac{B_{ζ}}{z − ι_{𝔭}(ζ)}) G(z),

where G(z) is an analytic function in variable z ∈ \overline{𝕌_{N}} with values in A_{ℂ_{p}}^{∧}. Here, G = ∑_{W} G_{W} (z)W is ‘analytic’ means each of whose coefficient G_{W} (z) is locally p-adically analytic.

Proposition 2.6 Fix a ∈ ℂ_{p}. Then, there exist unique solutions G_{0}^{a}(z), G_{1}^{a}(z) ∈ A_{Col}^{a} ˆ⊗A_{ℂ_{p}}^{∧}, which are locally analytic on ℙ^{1}(ℂ_{p}) \ S and satisfy G_{0}^{a}(z) ≈ z^{A} (z → 0), and G_{1}^{a}(z) ≈ (1 − z)^{B_{1}} (z → 1).

Here, the notations u^{A} means ∑_{n=0}^{∞} \frac{1}{n!}(A log^{a} u)^{n}. Note that it depends on a. For the notations G_{0}^{a}(z) ≈ z^{A} (z → 0), see [Fu1, Theorem 3.4].

Remark 2.7 We do not have the symmetry z ↦ 1−z on \overline{𝕌_{N}}. Thus, we do not have a simple relation between G_{0}^{a}(z) and G_{1}^{a}(z) as in [Fu1, Proposition 3.8]. On the other hand, we have the symmetry z ↦ z^{−1} on \overline{𝕌_{N}}. Thus, we have a unique locally analytic solution G_{∞}^{a}(z) with G_{∞}^{a}(z) ≈ (z^{−1})^{−A−∑_{ζ∈μ_{N}} B_{ζ}} (z → ∞), and have a relation

G_{∞}^{a}(A, {B_{ζ}}_{ζ∈μ_{N}})(z) = G_{0}^{a}(−A − ∑_{ζ∈μ_{N}} B_{ζ}, {B_{ζ^{−1}}}_{ζ∈μ_{N}})(z^{−1}).

However, when we define a Drinfel’d associator by using G_{0}^{a} and G_{∞}^{a} similarly as below (Definition 2.8), there appears

lim_{ℂ_{p}∈z→∞}′Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z)

in the coefficient of that Drinfel’d associator. What we want is lim_{ℂ_{p}∈z→1}′. Thus, we use the boundary condition at z = 1.

Proof The uniqueness is easy. In [Fu1], he cites Drinfel’d’s paper [Dr] for the existence of a solution of the KZ-equation. Here, we give an alternative proof of the existence without using the quasi-triangular quasi-Hopf algebra theory and the quasi-tensor category theory. In fact, we put G_{0}^{a}(z) to be ∑_{W} (−1)^{depth(W)}Li_{W}^{a} (z)W. Here, for a word W, we define Li_{W}^{a} (z) inductively as following: Li_{A^{n}}^{a}(z) := \frac{1}{n!}(log^{a} z)^{n}, Li_{AW}^{a} (z) := ∫_{0}^{z} \frac{1}{t}Li_{W}^{a} (t)dt, for W ≠ A^{n} (n ≥ 0), Li_{B_{ζ}W}^{a} (z) := ∫_{0}^{z} \frac{1}{ι_{𝔭}(ζ)−t}Li_{W}^{a} (t)dt, for ζ ∈ μ_{N}. It is easy to verify that ∑_{W} (−1)^{depth(W)}Li_{W}^{a} (z)W satisfies the p-adic KZ-equation. As for the boundary condition G_{0}^{a}(z) ≈ z^{A} (z → 0), it is easy to show that

∑_{W:W ≠W′A,W′≠∅}(−1)^{depth(W)}Li_{W}^{a} (z)W

satisfies the above boundary condition. Thus, it remains to show that Li_{W′A^{n}}^{a}(z) → 0 (z → 0) for n > 0, W′ ≠ ∅. For Li_{B_{ζ}A^{n}}^{a},

Li_{B_{ζ}A^{n}}^{a}(z) = ∫_{0}^{z} \frac{1}{ι_{𝔭}(ζ) − t}Li_{A^{n}}^{a}(t)dt = \frac{1}{n!} ∫_{0}^{z} ζ^{−1} ∑_{k=0}^{∞}(ζ^{−1}t)^{k}(log^{a} t)^{n}dt,

in |z| < 1. Since ∫_{0}^{z} t^{k} log^{a} tdt = \frac{z^{k+1}}{k+1} log^{a} z − \frac{z^{k+1}}{(k+1)^{2}} , we have ∫_{0}^{z} t^{k} log^{a} tdt → 0 (z → 0). Inductively, we have ∫_{0}^{z} t^{k}(log^{a} t)^{n}dt → 0 (z → 0). Thus, we showed Li_{B_{ζ}A^{n}}^{a}(z) → 0 (z → 0). For general Li_{W′A}^{a}(z)’s, we can inductively show Li_{W′A}^{a}(z) → 0 (z → 0) by using the following fact for f(z) = Li_{∗∗}^{a}(z): For a locally analytic function f(z) satisfying f(0) = 0, we have ∫_{0}^{z} \frac{1}{t}f(t)dt → 0 (z → 0), ∫_{0}^{z} \frac{1}{ι_{𝔭}(ζ)−t}f(t)dt → 0 (z → 0). As for G_{1}^{a}(z), the same argument works, by replacing Li_{A^{n}}^{a}(z) := \frac{1}{n!}(log^{a} z)^{n} by Li_{B_{1}^{n}}^{a}(z) := \frac{1}{n!}(log^{a}(1 − z))^{n}, and ∫_{0}^{z} by ∫_{1}^{z}.

Definition 2.8 We define the p-adic Drinfel’d associator for twisted p-adic multiple polylogarithms to be Φ_{KZ}^{𝔭}(A, {B_{ζ}}_{ζ∈μ_{N}}) := G_{1}^{a}(z)^{−1}G_{0}^{a}(z). It is in A_{ℂ_{p}}^{∧} = ℂ_{p}⟨⟨A, {B_{ζ}}_{ζ∈μ_{N}} ⟩⟩, and independent of a by the same argument in [Fu1, Remark 3.9, Theorem 3.10].

By the same arguments as in [Fu1], we can show the following propositions.

Proposition 2.9 lim_{ℂ_{p}∈z→1}′Li_{(k_{1},...,k_{d};ζ_{1},...,ζ_{d})}^{a}(z) converges when (k_{d}, ζ_{d}) ≠ (1, 1).

Proof See, [Fu1, Theorem 2.18] for the case where N = 1.

For W in A · A_{ℂ_{p}}^{∧} · B_{ζ} or B_{ζ′} · A_{ℂ_{p}}^{∧} · B_{ζ} (ζ′ ≠ 1), we define L_{𝔭}(W) to be lim_{ℂ_{p}∈z→1}′Li_{W}^{a} (z).

Proposition 2.10 (Explicit Formulae) The coefficient I_{𝔭}(W) of W in the padic Drinfel’d associator for twisted p-adic MPL’s is the following: When W is written as B_{1}^{r}V A^{s} for (r, s ≥ 0), V is in A · A_{ℂ_{p}}^{∧} · B_{ζ} or B_{ζ′} · A_{ℂ_{p}}^{∧} · B_{ζ} (ζ′ ≠ 1),

I_{𝔭}(W) = (−1)^{depth(W)}(−1)^{a+b} ∑_{0≤a≤r,0≤b≤s} L_{𝔭}(f(B_{1}^{a} ∘ B_{1}^{r−a}V A^{s−b} ∘ A^{b})).

In particular, when W is in A · A_{ℂ_{p}}^{∧} · B_{ζ} or B_{ζ′} · A_{ℂ_{p}}^{∧} · B_{ζ} (ζ′ ≠ 1), I_{𝔭}(W) = (−1)^{depth(W)}L_{𝔭}(W). Here, f : A_{ℂ_{p}}^{∧} → A_{ℂ_{p}}^{∧} is the composition of

A_{ℂ_{p}}^{∧} ↠ A_{ℂ_{p}}^{∧}/(B_{1} ·A_{ℂ_{p}}^{∧} +A_{ℂ_{p}}^{∧} ·A), A_{ℂ_{p}}^{∧}/(B_{1} ·A_{ℂ_{p}}^{∧} +A_{ℂ_{p}}^{∧} ·A) →^{∼} ℂ_{p} ·1+A·A_{ℂ_{p}}^{∧} ·B_{1},

and ℂ_{p} · 1 + A · A_{ℂ_{p}}^{∧} · B_{1} ↪ A_{ℂ_{p}}^{∧}.

For the definition of the shuffle product ∘, see [Fu0, Definition 3.2.2].

Proof See, [Fu1, Theorem 3.28] for the case where N = 1. Note we use G_{i}^{a}(A − α, B_{1} − β, {B_{ζ}}_{ζ∈μ_{N} ,ζ≠1})(z) = z^{−α}(1 − z)^{−β}G_{i}^{a}(A, {B_{ζ}}_{ζ∈μ_{N}})(z) for i = 0, 1.

Proposition 2.11 Suppose lim_{ℂ_{p}∈z→1}′Li_{(k_{1},...,k_{d−1},1;ζ_{1},...,ζ_{d−1},1)}^{a}(z) converges. Then, the limit value is a p-adic regularized MLV, that is, L_{𝔭}(k_{1}, …, k_{d−1}, 1; ζ_{1}, …, ζ_{d−1}, 1) = (−1)^{depth(W)}I_{𝔭}(W). In particular, L_{𝔭}(k_{1}, …, k_{d−1}, 1; ζ_{1}, …, ζ_{d−1}, 1) can be written as a ℚ-linear combination of p-adic MLV’s corresponding to the same weight indices with (k_{d}, ζ_{d}) ≠ (1, 1).

Proof See, [Fu1, Theorem 2.22] for the case where N = 1.

Definition 2.12 We define the p-adic multiple L-value space of weight w Z_{w}^{𝔭}[N] to be the finite dimensional ℚ-linear subspace of ℚ(μ_{N})_{𝔭} generated by the all p-adic MLV’s of indices of weight w, ζ_{1}^{N} = ⋯ = ζ_{d}^{N} = 1. Put Z_{0}^{𝔭}[N] := ℚ. We define Z_{•}^{𝔭}[N] to be the formal direct sum of Z_{w}^{𝔭}[N] for w ≥ 0.

Remark 2.13 By Proposition 2.11, we see that

Z_{w}^{𝔭}[N] :=⟨L_{𝔭}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) | d ≥ 1, k_{1} + ⋯ + k_{d} = w, k_{1}, …, k_{d} ≥ 1,
ζ_{1}^{N} = ⋯ = ζ_{d}^{N} = 1, (k_{d}, ζ_{d}) ≠ (1, 1)⟩_{ℚ}
= ⟨I_{𝔭}(W) | the weight of W is w⟩_{ℚ} ⊂ ℚ(μ_{N})_{𝔭}.

Proposition 2.14 We have Δ(Φ_{KZ}^{𝔭}) = Φ_{KZ}^{𝔭} ˆ⊗Φ_{KZ}^{𝔭}. In particular, the graded ℚ-vector space Z_{•}^{𝔭}[N] has a ℚ-algebra structure, that is, Z_{a}^{𝔭}[N] · Z_{b}^{𝔭}[N] ⊂ Z_{a+b}^{𝔭}[N] for a, b ≥ 0.

Proof See, [Fu1, Proposition 3.39, Theorem 2.28] for the case where N = 1.

Proposition 2.15 (Shuffle Product Formulae) For W, W′ ∈ (A · A_{ℂ_{p}}^{∧} · B_{ζ}) ∪ ∪_{ζ′≠1}(B_{ζ′} · A_{ℂ_{p}}^{∧} · B_{ζ}), we have

L_{𝔭}(W ∘ W′) = L_{𝔭}(W)L_{𝔭}(W′).

Proof This follows from Proposition 2.10 and Proposition 2.14. See, [Fu1, Corollary 3.42] for the case where N = 1.

3 Bounds for Dimensions of p-adic Multiple L-value spaces.

In this section, we show Theorem 1.4, by the method of Deligne-Goncharov [DG], assuming results of Section 4. First, we recall some facts about the motivic fundamental groupoids in [DG]. Next, we show that bounds for dimensions of p-adic MLV-spaces in the sense of Deligne [D1][DG]. Lastly, we show that p-adic MLV-spaces in the previous section is equal to p-adic MLV-spaces in the sense of Deligne by the Tannakian interpretations.

3.1 The Motivic Fundamental Groupoids of 𝕌_{N}.

Deligne-Goncharov constructed the category MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]) of mixed Tate motives over ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}], the fundamental MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])- group π_{1}^{M}(𝕌_{N}, x) and the fundamental MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-groupoid P_{y,x}^{M} for 𝕌_{N} not only for rational base points x, y, but also for tangential base points x, y [DG, Theorem 4.4, Proposition 5.11]. Here, w | N runs through primes w dividing N, and ζ_{w} is a w-th root of unity (Since 𝕌_{N} is defined over ℚ, π_{1}^{M}(𝕌_{N}, x), P_{y,x}^{M} are also MAT(ℚ(μ_{N})/ℚ)-schemes. However, we do not use this fact. Here, MAT(ℚ(μ_{N})/ℚ) is the category of mixed Artin-Tate motives for ℚ(μ_{N})/ℚ). For T -schemes, T -group schemes, and T -groupoids for a Tannakian category T , see [D1, §5, §6], [D2, 7.8], and [DG, 2.6]. First, we recall some facts about them. Let

G := π_{1}(MT(ℤ[μ_{N}, \{\frac{1}{1 − ζ_{w}} \}_{w|N}])) ∈ pro-MT(ℤ[μ_{N}, \{\frac{1}{1 − ζ_{w}} \}_{w|N}])

be the fundamental MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-group [D1, §6][D2, Definition 8.13]. Then, by its action on ℚ(1), we have a surjection G ↠ 𝔾_{m} (Here, we regard 𝔾_{m} as an MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-group). The kernel U of the map G → 𝔾_{m} is a pro-unipotent group. Then, we have an isomorphism [DG, 2.8.2]:

Lie(U^{ab}) ≅ ∏_{n} Ext_{MT(ℤ[μ_{N} ,\{ \frac{1}{1−ζw} \}_{w|N}])}^{1}(ℚ(0), ℚ(n))^{∨} ⊗ ℚ(n)

∈ pro-MT(ℤ[μ_{N}, \{\frac{1}{1 − ζ_{w}} \}_{w|N}]).

The extension group is related to the algebraic K-theory [DG, 2.1.3]:

Ext_{MT(ℤ[μ_{N} ,\{ \frac{1}{1−ζw} \}_{w|N}])}^{1}(ℚ(0), ℚ(n)) = \begin{cases}0 & n ≤ 0, \\ ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]^{×} ⊗_{ℤ} ℚ & n = 1, \\ K_{2n−1}(ℚ(μ_{N})) ⊗_{ℤ} ℚ & n ≥ 2. \end{cases}

Let ω be the canonical fiber functor ω : MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]) → Vect_{ℚ}, which sends a motive M to ⊕_{n}Hom(ℚ(n), Gr_{−2n}^{W}(M)). Here, W_{m}(M) is the weight filtration of M. Let G_{ω} := ω(G) = \underline{Aut}^{⊗}(MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]), ω) be the motivic Galois gruop of MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]) with respect to the canonical fiber functor ω (For the de Rham realization M_{dR} of a motive M ∈ MT(ℚ(μ_{N})), we have M_{dR} = ω(M) ⊗_{ℚ} ℚ(μ_{N}) [DG, Proposition 2.10]). Then, the ω-realization of the exact sequence 0 → U → G → 𝔾_{m} → 0 is split by the action of 𝔾_{m}, which gives the grading by weights,

G_{ω} = 𝔾_{m} ⋉ U_{ω}.

Here, U_{ω} := ω(U). Let τ denote the splitting 𝔾_{m} → G_{ω}. The pro-unipotent group U_{ω} is equipped with the grading {(U_{ω})_{n}}_{n}. Put (LieU_{ω})^{gr} := ⊕_{n}(LieU_{ω})_{n}. Then, (LieU_{ω})^{gr} is a free Lie algebra, since we have Ext_{MT(ℤ[μ_{N} ,\{ \frac{1}{1−ζw} \}_{w|N}])}^{2}(ℚ(0), ℚ(n)) = K_{2n−2}(ℚ(μ_{N})) ⊗_{ℤ} ℚ = 0 [DG, Proposition 2.3]. Thus, the generating function of the universal envelopping algebra of (LieU_{ω})^{gr} is ∑_{n=0}^{∞} f(t)^{n}, where

f(t)

= \begin{cases}t^{3} + t^{5} + t^{7} + ⋯ = \frac{t^{3}}{1−t^{2}} & N = 1, \\ t + t^{3} + t^{5} + ⋯ = \frac{t}{1−t^{2}} & N = 2, \\ (\frac{φ(N)}{2} + ν − 1) t + \frac{φ(N)}{2}t^{2} + \frac{φ(N)}{2}t^{3} + ⋯ = \frac{φ(N)}{2} \frac{t}{1−t} + (ν − 1)t & N ≥ 3. \end{cases}

Therefore, we have

∑_{n=0}^{∞} f(t)^{n} = \frac{1}{1 − f(t)} = \begin{cases}\frac{1 − t^{2}}{1 − t^{2} − t^{3}} & N = 1, \\ \frac{1 − t^{2}}{1 − t − t^{2}} & N = 2, \\ \frac{1 − t}{1 − (\frac{φ(N)}{2} + ν) t + (ν − 1)t^{2}} & N ≥ 3. \end{cases}

That is the generating function of d[N]_{n}’s in Section 1. Let P_{y,x}^{M} be the fundamental MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-groupoid of 𝕌_{N} at (tangential) base points x and y. We consider only tangential base points λ_{x} at x ∈ S := {0, ∞} ∪ μ_{N} with tangent vectors λ in roots of unity under the identification the tangent space at x with 𝔾_{a}. Then, P_{λ′_{y},λ_{x}}^{M} depends only on x and y, by the triviality of a Kummer ℚ(1)-torsor [DG, 5.4]. Let P_{y,x}^{M} denote P_{λ′_{y},λ_{x}}^{M}. We have the following structures of the system of MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])- schemes {P_{y,x}^{M}}_{x,y∈S} [DG, 5.5, 5.7]: [The system of groupoids in the level of motives]

(1)^{M} The Tate object ℚ(1),

(2)^{M} For x, y ∈ S, the fundamental MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-groupoid P_{y,x}^{M},

(3)^{M} The composition of paths,

(4)^{M} For x ∈ S, a morphism of MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-group scheme (the local monodromy around x):

ℚ(1) → P_{x,x}^{M},

(5)^{M} An equivariance under the dihedral group ℤ/2ℤ ⋉ μ_{N}.

By applying a fiber functor F to the category of K-vector spaces, where K is a field of characteristic 0, we get the following structure [DG, 5.8]: [The system of groupoids under the fiber functor F]

(1)^{F} A vector space K(1) of dimension 1,

(2)^{F} For x, y ∈ S, a scheme P_{y,x}^{F} over K,

(3)^{F} a system of morphisms of schemes P_{z,y}^{F} × P_{y,x}^{F} → P_{z,x}^{F} making P_{y,x}^{F}’s a groupoid. The group schemes P_{x,x}^{F} are pro-unipotent,

(4)^{F} For x ∈ S, a morphism

(additive group K(1)) → P_{x,x}^{F}.

That is equivalent to giving K(1) → LieP_{x,x}^{F},

(5)^{F} An ℤ/2ℤ ⋉ μ_{N}-equivariance.

In particular, we take the canonical fiber functor ω as F, and we consider the following weakened structure (forgetting the conditions at infinity) [DG, 5.8]. Note that in the realization ω, the weight filtrations split and give the grading, and that all π_{1}^{ω}(𝕌_{N}, x)-groupoids are trivial since H^{1}(𝕌_{N}, O_{𝕌_{N}}) = 0. Let L be the Lie algebra freely generated by symbols A, and {B_{ζ}}_{ζ∈μ_{N}}. Let Π be the pro-unipotent group

Π := \underleftarrow{lim}_{n} exp(L/degree ≥ n).

Then, we have the following structure [DG, 5.8]:

[The (weakened) system of groupoids under the canonical fiber functor ω] (1)^{ω} The vector space ℚ,

(2)^{ω} A copy Π_{0,0} of Π, and the trivial Π_{0,0}-torsor Π_{1,0}. The twist of Π_{0,0} by this torsor is a new copy of Π, denoted by Π_{1,1},

(3)^{ω} The group law of Π,

(4)^{ω} The morphism

ℚ → L^{∧} : 1 ↦ A, ℚ → L^{∧} : 1 ↦ B_{1}.
for x = 0, 1 respectively. Here, L^{∧} := \underleftarrow{lim}_{n} L/(degree ≥ n),

(5)^{ω} The action μ_{N} on Π_{0,0}, which induces on the Lie algebra B_{ζ} ↦ B_{σζ}.

Let H_{ω} be the group scheme of automorphisms of ℚ and Π preserving the above structure (1)^{ω}-(5)^{ω}. The action of H_{ω} on the one dimensional vector space (1)^{ω} gives a morphism H_{ω} ↠ 𝔾_{m}. Let V_{ω} be the kernel. The grading gives a splitting,

H_{ω} = 𝔾_{m} ⋉ V_{ω}.

Also let τ denote the splitting 𝔾_{m} → V_{ω}. The action G_{ω} on the above structure factors through H_{ω}, which sends U_{ω} to V_{ω}.

\xymatrix{1 \ar[r] & U_{ω} \ar[r] \ar[d] & G_{ω} \ar[r] \ar[d] & 𝔾_{m} \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & V_{ω} \ar[r] & H_{ω} \ar[r] & 𝔾_{m} \ar[r] & 1.}

Let ι denote both of G_{ω} → H_{ω}, and U_{ω} → V_{ω}. The above diagram comes from MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-schemes (splitting does not come from MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}])-schemes), however we do not use this fact (see, [DG, 5.12.1]). For the details of affine T -schemes, where T is a Tannakian category, see [D1, §5, §6], [D2, 7.8], and [DG, 2.6]. By the Proposition 5.9 in [DG], the map

η : V_{ω} → Π_{1,0} (v ↦ v(γ_{dR}))

is bijective. Here, γ_{dR} is the neutral element of Π_{1,0}, that is, γ_{dR} is the canonical path from 0 to 1 in the realization of ω.

3.2 The p-adic MLV-space in the Sense of Deligne.

We will discuss the crystalline realization of mixed Tate motives, and now we assume the results of Section 4 (See, Remark 4.8). We use the word “crystalline”, not “rigid” for the purpose of fixing terminologies. In [D1], Deligne has found the p-adic zeta values (i.e., the p-adic MZV’s of depth 1), and the p-adic differential equation of p-adic polylogarithms in the study of crystalline aspects of the fundamental group of 𝕌_{N} modulo depth ≥ 2 [D1, 19.6]. Deligne-Goncharov proposed that the coefficients of the image of

φ_{𝔭} := F_{𝔭}^{−1}τ(q)^{−1} ∈ U_{ω}(ℚ(μ_{N})_{𝔭})

by the map

η · ι : U_{ω}(ℚ(μ_{N})_{𝔭}) → V_{ω}(ℚ(μ_{N})_{𝔭}) →^{∼} Π(ℚ(μ_{N})_{𝔭}) ⊂ ℚ(μ_{N})_{𝔭}⟨⟨A, {B_{ζ}}_{ζ∈μ_{N}} ⟩⟩

“seem” to be p-adic analogies of MZV’s [DG, 5.28]. Here, τ is the splitting 𝔾_{m} → G_{ω}, F_{𝔭} is the Frobenius endomorphism at 𝔭, q is the cardinality of the residue field at 𝔭, and Π(ℚ(μ_{N})_{𝔭}) is the ℚ(μ_{N})_{𝔭}-valued points of Π in the previous subsection. Note that we have the Frobenius endomorphism on M_{ω} ⊗ ℚ(μ_{N})_{𝔭} ≅ M_{crys} for M ∈ MT(ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}]) by Remark 4.8. Here, M_{crys} is the crystalline realization of M.

Definition 3.1 We define the p-adic multiple L-values in the sense of Deligne of weight w to be the coefficients I_{𝔭}^{D}(W) of words W of weight w in ηι(φ_{𝔭}) ∈ Π(ℚ(μ_{N})_{𝔭}) ⊂ ℚ(μ_{N})_{𝔭}⟨⟨A, {B_{ζ}}_{ζ∈μ_{N}} ⟩⟩. We define the p-adic L-value spaces in the sense of Deligne of weight w Z_{w}^{𝔭,D}[N] to be the finite dimensional ℚ-linear subspace of ℚ(μ_{N})_{𝔭} generated by all p-adic MLV’s in the sense of Deligne of indices of weight w. By the definition, we have Z_{0}^{𝔭,D}[N] = ℚ. We define Z_{•}^{𝔭,D}[N] to be the formal direct sum of Z_{w}^{𝔭,D}[N] for w ≥ 0.

On the othe hand, we call p-adic MLV’s defined in Section 2.1 p-adic MLV’s in the sense of Furusho.

Remark 3.2 If we calculate the action of Frobenius F_{𝔭}^{−1} on (P_{1,0})_{ω}, we get the following KZ-like p-adic differential equation by the same arguments as in [D1, 19.6]:

dG(t) =

− qG(t) (\frac{dt}{t}A + ∑_{ζ∈μ_{N}} \frac{dt}{t − ι_{𝔭}(ζ)}ζ(Φ_{D}^{𝔭})^{−1}B_{ζ}ζ(Φ_{D}^{𝔭}))

+ (\frac{d(t^{q})}{t^{q}}A + ∑_{ζ∈μ_{N}} \frac{d(t^{q})}{t^{q} − ι_{𝔭}(ζ)}B_{ζ} ) G(t).

Here, ζ(Φ_{D}^{𝔭}) means the action of ζ on Φ_{D}^{𝔭} determined by ζ(A) = A and ζ(B_{ζ′}) = B_{ζζ′}. Here, Φ_{D}^{𝔭} is the Deligne associator (See, the subsection of Tannakian interpretions, and Proposition 3.10). The coefficient of a word W in the solution of the above p-adic differential equation is q^{w(W)}I_{𝔭}^{D}(W) in the limit t → 1, that is, p-adic MLV’s in the sense of Deligne (multiplied by q^{w(W)}). (More precisely, we have to consider the effect (1 − t)^{−B_{1}} of the tangential base point in taking the limit). The first term in RHS is multiplied by G from the left, and the second term in RHS is multiplied by G from the right. Thus, the inductive procedure of determining coefficients is more complicated. In [D1, 19.6], Deligne calculated the Frobenius action on π_{1}^{ω}(𝕌_{N}, 1_{0}) = (P_{1,0})_{ω} modulo depth ≥ 2, however, we get the above p-adic differential equation by the same arguments. Here we give a sketch. We use some notations in [D1]. The above equation arises from the horizontality of Frobenius ([D1, 19.6.2]):

F_{𝔭}^{−1}(e^{−1}∇e) = G^{−1}∇G.

Here, e is the identity element. The above F_{𝔭}^{−1} and G are F_{∗} and v in [D1] respectively. On the LHS, we have [D1, 12.5, 12.12, 12.15]

e^{−1}∇e = −α = − (\frac{dt}{t}A + ∑_{ζ∈μ_{N}} \frac{dt}{t − ι_{𝔭}(ζ)}B_{ζ} ) .

Here, α is the Maurer-Cartan form ([D1, 12.5.5]). On the RHS, since the connection is the one of F̃^{∗}(P_{1,0})_{ω}, we have ∇e = −F̃^{∗}α, where F̃^{∗} means the Frobenius lift t ↦ t^{q}. Combining these and ∇G = dG + (∇e)G, we get

− qG (\frac{dt}{t}A + ∑_{ζ∈μ_{N}} \frac{dt}{t − ι_{𝔭}(ζ)}F_{𝔭}^{−1}(B_{ζ})) =

dG − (\frac{d(t^{q})}{t^{q}}A + ∑_{ζ∈μ_{N}} \frac{d(t^{q})}{t^{q} − ι_{𝔭}(ζ)}B_{ζ} ) G.

This gives the equation (For F_{𝔭}^{−1}(B_{ζ}), see the proof of Proposition 3.10).

Example 1 From the p-adic differential equation in the above Remark 3.2, the coefficient of A^{k−1}B in ηι(F_{p}^{−1}τ(p)^{−1}) in the case where N = 1 is the limit value at z = 1 of the p-adic analytic continuation of the following analytic function on |z|_{p} < 1 [D1, 19.6]:

∑_{p∤n} \frac{z^{n}}{n^{k}} .

That limit value is (1 − p^{−k})ζ_{p}(k). From the condition p ∤ n in the summation, we lose the Euler factor at p for p-adic MZV’s of depth 1 in the sense of Deligne.

Proposition 3.3 For a, b ≥ 0, we have

Z_{a}^{𝔭,D}[N] · Z_{b}^{𝔭,D}[N] ⊂ Z_{a+b}^{𝔭,D}[N].

Proof The group Π(ℚ(μ_{N})_{𝔭}) is the subgroup of group-like elements in ℚ(μ_{N})_{𝔭}⟨⟨A, {B_{ζ}}_{ζ∈μ_{N}} ⟩⟩, and ηι(φ_{𝔭}) is an element of Π(ℚ(μ_{N})_{𝔭}) by the definition. Thus, we have Δ(ηι(φ_{𝔭})) = ηι(φ_{𝔭})ˆ⊗ηι(φ_{𝔭}). This implies the proposition.

Proposition 3.4 For w ≥ 0, we have

dim_{ℚ} Z_{w}^{𝔭,D}[N] ≤ d[N]_{w}.

Proof Let U_{ω} = SpecR and ηι(U_{ω}) = SpecS. The algebras R = ∏_{n} R^{n} and S = ∏_{n} S^{n} are graded algebras over ℚ. Here, the grading of R and S come from the grading of U_{ω}. Then, ηι(φ_{𝔭}) ∈ ηι(U_{ω})(ℚ(μ_{N})_{𝔭}) gives a homomorphism ψ_{𝔭} : S → ℚ(μ_{N})_{𝔭}. The coefficients of ηι(φ_{𝔭}) of weight w are contained in ψ_{𝔭}(S^{w}). Thus, we have Z_{w}^{𝔭,D}[N] ⊂ ψ_{𝔭}(S^{w}). By the surjection

ι : U_{ω} ↠ ι(U_{ω})(⊂ V_{ω} ≅^{η} Π), the dimension of S^{w} is at most the one of the w-th graded part of the universal envelopping algebra of (LieU_{ω})^{gr}. That dimension is d[N]_{w}. We are done.

Remark 3.5 As remarked in [DG, 5.27], ι : LieU_{ω} → LieV_{ω} is not injective for N > 4 in general. Thus, the above bounds are not best possible for N > 4 in general. The kernel is related to the space of cusp forms of weight 2 on X_{1}(N) if N is a prime. See also [G2].

Remark 3.6 In the complex case [DG], dch(σ) is in (P_{1,0})_{ω} ⊗ ℂ = Π(ℂ) ←^{∼} V_{ω}(ℂ). (Here, dch(σ) is the “droit chemin” from 0 to 1 in the Betti realization with respect to σ : ℚ(μ_{N}) ↪ ℂ.) Thus, Deligne-Goncharov relate dch(σ) to the motivic Galois group U_{ω} for the purpose of bounds for the dimensions in [DG, Proposition 5.18, 5.19, 5.20, 5.21, 5.22]. (The point is that V_{ω} is too big, and U_{ω} is small enough.) However, in the p-adic situation, φ_{𝔭} is contained a priori in a small enough variety, i.e., we have φ_{𝔭} ∈ U_{ω}(ℚ(μ_{N})_{𝔭}) by the definition. Thus, the bounds from K-theory of p-adic MLV’s in the sense of Deligne are almost trivial.

We give remarks on ζ_{p}(2).

Remark 3.7 By Proposition 3.4 and Example 1, we have ζ_{p}(2) = 0, since dim_{ℚ} Z_{2}^{p,D}[1] = 0. It is another proof of that well-known fact. To bound dimensions, Deligne-Goncharov used ι(U_{ω})×𝔸^{1} in the complex case [DG, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25]. This affine line corresponds to “π^{2}”, and we need this affine line simply because π^{2} is not in ℚ. In the p-adic case, we do not need such an affine line, simply because the image of F_{p}^{−1} in (𝔾_{m})_{ω} (i.e., p) is in ℚ. This gives a motivic interpretation of ζ_{p}(2) = 0.

Remark 3.8 It is well-known that ζ_{p}(2m) = 0. However, it is non-trivial because we do not know how to show directly

“ ∑_{ℂ_{p}∋z→1} \frac{z^{n}}{n^{2m}} = 0”

(We add a double quotation in the above, since we have to take p-adic analytic continuation). The well-known proof of ζ_{p}(2m) = 0 is following (also see, [Fu1, Example 2.19(a)]): By the Coleman’s comparison [C], we have lim_{ℂ_{p}∋z→1} Li_{k}^{a}(z) = (1 − p^{−k})^{−1}L_{p}(k, ω^{1−k}) for k ≥ 2. Here, L_{p} is the padic L-function of Kubota-Leopoldt, ω is the Teichmüller character. This is the values of the p-adic L-function at positive integers. On the other hand, the p-adic L-function interpolates the values of usual L-functions at negative integers, thus, L_{p}(z, ω^{1−k}) is constantly zero for even k. Therefore, we have ζ_{p}(2m) = 0. That proof is indirect. Furusho informed to the author that 2-, and 3-cycle relations induce ζ_{p}(2m) = 0 similarly as in [D1, §18] (In the notations in [D1, §18], we can take γ =(the unique Frobenius invariant path from 0 to 1) (see, the next subsection,) and x = 0). These relations come from the geometry of ℙ^{1} \ {0, 1, ∞}. Thus, it seems that it comes from “the same origin” that ‘ζ_{p}(2) = 0 from cycle relations’ and ‘ζ_{p}(2) = 0 from the bounds by K-theory’. Furusho also comments that we may translate ‘ζ_{p}(2m) = 0 from cycle relations’ into ‘ζ_{p}(2m) = 0 from padic differential equation’, i.e., we may show that ζ_{p}(2k) = 0 directly from the p-adic analytic function ∑_{n≥1} \frac{z^{n}}{n^{2m}}.

3.3 The Tannakian Interpretations of Two p-adic MLV’s.

Besser proved that there exists a unique Frobenius invariant path in the fundamental groupoids of certain p-adic analytic spaces [B, Corollary 3.2]. Furthermore, Besser showed the existence of Frobenius invariant path on p-adic analytic spaces is equivalent to the Coleman’s integral theory [B, §5]. Let γ_{crys} be the unique Frobenius invariant path in (P_{1,0})_{crys}. To a differential form ω, the path γ_{crys} associates the Colman integration ∫_{0}^{1} ω. Let γ_{dR} ∈ (P_{1,0})_{ω} be the canonical path from 0 to 1 under the realization ω. Furusho proved the path α_{F} := γ_{dR}^{−1}γ_{crys} ∈ π_{1}^{crys}(𝕌_{N}, 1_{0}) is equal to the p-adic Drinfel’d associator Φ_{KZ}^{p} for p-adic MZV’s, that is, for N = 1 in [Fu2]. By the same argument, we can verify that α_{F} = Φ_{KZ}^{𝔭} for p-adic MLV’s. Briefly, we review the argument. For details, see [Fu2] (See also [Ki, Proposition 4]). The coefficient of a word A^{k_{d}−1}B_{ζ_{d}} ⋯ A^{k_{1}−1}B_{ζ_{1}} in α_{F} = γ_{dR}^{−1}γ_{crys} ∈ π_{1}^{crys}(𝕌_{N}, 1_{0}) ⊂ ℚ(μ_{N})_{𝔭}⟨⟨A, {B_{ζ}}_{ζ∈μ_{N}} ⟩⟩ for (k_{d}, ζ_{d}) ≠ (1, 1) is an iterated integral

∫_{0}^{1} \frac{dt}{t} ⋯ ∫_{0}^{t} \frac{dt}{t} ∫_{0}^{t} \frac{dt}{t − ι_{𝔭}(ζ_{d})} ∫_{0}^{t} \frac{dt}{t} ⋯ ∫_{0}^{t} \frac{dt}{t} ∫_{0}^{t} \frac{dt}{t − ι_{𝔭}(ζ_{1})}

by the characterization of γ_{crys} with respect to Coleman’s integration theory (Here, the succesive numbers of dt/t are k_{d} −1, k_{d−1} −1, ⋯, k_{2} −1 and k_{1} −1). For words beginning from A or ending B_{1}, the coefficients are regularized p-adic MLV’s, because the coefficients in α_{F} are the one in lim′_{ℂ_{p}∋z→1}(1−z)^{−B_{1}}G_{0}(z) by using the tangential base point. Thus, α_{F} is the p-adic Drinfel’d associator Φ_{KZ}^{𝔭} for twisted p-adic MPL’s in Section 2.2:

α_{F} := γ_{dR}^{−1}γ_{crys} = Φ_{KZ}^{𝔭} = ∑_{W} I_{𝔭}(W)W.

On the other hand, ηι(φ_{𝔭}) ∈ Π_{0,0}(ℚ(μ_{N})_{𝔭}) = π_{1}^{crys}(𝕌_{N}, 1_{0}) is γ_{dR}^{−1}φ_{𝔭}(γ_{dR}) by

the definition (Recall that V_{ω} ≅^{η} Π_{1,0} and Π_{0,0} ≅ Π_{1,0} : 1 ↦ γ_{dR}). Briefly, padic MLV’s in the sense of Furusho come from α_{F} = γ_{dR}^{−1}γ_{crys}, and p-adic MLV’s in the sense of Deligne come from α_{D} := γ_{dR}^{−1}φ_{𝔭}(γ_{dR}). That is the Tannakian interpretations of p-adic MLV’s. In [Fu2], he calls Φ_{D}^{𝔭} := γ_{dR}^{−1}F_{𝔭}^{−1}(γ_{dR}) the Deligne associator.

Remark 3.9 In both of complex and p-adic cases, the iterated integrals appear in the theory of MZV’s. However, the iterated integrals come from different origins in the complex case and the p-adic case. In the complex case, the iterated integrals appear in the comparison map between the Betti fundamental group π_{1}^{B} ⊗_{ℚ} ℂ tensored by ℂ of ℙ^{1} \ {0, 1, ∞} and the de Rham fundamental group π_{1}^{dR} ⊗_{ℚ} ℂ tensored by ℂ of ℙ^{1} \ {0, 1, ∞}. The difference between the ℚ-structure π_{1}^{B} and the ℚ-structure π_{1}^{dR} under the comparison π_{1}^{B} ⊗_{ℚ} ℂ ≅ π_{1}^{dR} ⊗_{ℚ} ℂ is expressed by iterated integrals. In the p-adic case, iterated integrals do not appear in the comparison map between the de Rham fundamental group π_{1}^{dR} ⊗_{ℚ} ℚ_{p} tensored by ℚ_{p} and the crystalline fundamental group π_{1}^{crys}. Furthermore, there is no ℚ-structure on π_{1}^{crys}. For p-adic MZV’s in the sense of Deligne, iterated integrals appear in the difference between the ℚ-structure π_{1}^{dR} and the ℚ-structure F_{𝔭}^{−1}(π_{1}^{dR}) in P_{1,0}^{crys} under the comparison P_{1,0}^{crys} ≅ P_{1,0}^{dR} ⊗_{ℚ} ℚ_{p} = π_{1}^{dR} ⊗_{ℚ} ℚ_{p}. For p-adic MZV’s in the sense of Furusho, they appear in the difference between ℚ-structure π_{1}^{dR} and the ℚ-structure απ_{1}^{dR} in π_{1}^{crys} under the comparison π_{1}^{crys} ≅ π_{1}^{dR} ⊗_{ℚ} ℚ_{p}. Here, α is a unique element in π_{1}^{crys} such that γ_{dR} · α ∈ P_{1,0}^{crys} is invariant under the Frobenius (Thus, α is equal to α_{F}). From this, it seems difficult to find a “motivic Drinfel’d associator”, which is an origin of both complex and p-adic MZV’s, and a motivic element, which is an origin of linear relations of both complex and p-adic MZV’s. Note also that roughly speaking, the complex Drinfel’d associator is the differenc between Betti and de Rham realizations ([DG, 5.19]), and the p-adic Drinfel’d associator is the Frobenius element at p.

Example 2 1. (Kummer torsor) Let K(x)_{ω} be the fundamental groupoid from 1 to x on 𝔾_{m} with respect to the realization ω. Deligne calculated in [D1, 2.10] the action of F_{p}^{−1} on K(x)_{ω} ⊂ K(x)_{crys}:

F_{p}^{−1}(γ_{dR}) = γ_{dR} + log^{a} x^{1−p}.

Here, γ_{dR} is the canonical de Rham path from 1 to x, and + means the right action of π_{1}^{crys}(𝔾_{m}, 1) = ℚ(1)_{crys} = ℚ_{p}(1) on K(x)_{crys}. From this, we have

F_{p}^{−1}(γ_{dR} + log^{a} x) = γ_{dR} + log^{a} x^{1−p} + p log^{a} x = γ_{dR} + log^{a} x.

Thus, γ_{dR} + log^{a} x is Frobenius invariant, that is, the unique crystalline path γ_{crys} from 1 to x.

2. (Polylogarithm torsor) Let P_{1,k}(ζ)_{ω} be the k-th polylogarithm torsor with respect to the realization ω for ζ ∈ μ_{N} (see, [D1, Definition 16.18]). The polylogarithm torsors are not fundamental groupoids, but quotients of fundamental groupoids. However, we use the terminology “ℤ(k)-torsor of ℤ(k)-paths from 0 to ζ” in [D1, 13.15]. Here, we consider as ℚ(k)_{ω}- torsor not as ℤ(k)_{ω}-torsor, and we do not multiply \frac{1}{(k−1)!} on the integral structure unlike as [D1]. Deligne calculated in [D1, 19.6, 19.7] the action of F_{p}^{−1} on P_{1,k}(ζ)_{ω} ⊂ P_{1,k}(ζ)_{crys}:

F_{p}^{−1}(γ_{dR}) = γ_{dR} + p^{k}(1 − p^{−k})N^{k−1}Li_{k}^{a}(ζ)

(That is, F_{p}^{−1}τ(p)^{−1}(γ_{dR}) = γ_{dR} +(1−p^{−k})N^{k−1}Li_{k}^{a}(ζ)). Here, + means the right action of ℚ(k)_{crys} = ℚ_{p}(k) on P_{1,k}(ζ)_{crys}. From this, we have

F_{p}^{−1}(γ_{dR} − N^{k−1}Li_{k}^{a}(ζ)) = γ_{dR} + p^{k}(1 − p^{−k})N^{k−1}Li_{k}^{a}(ζ) − p^{k}N^{k−1}Li_{k}^{a}(ζ)
= γ_{dR} − N^{k−1}Li_{k}^{a}(ζ).

Thus, γ_{dR} − N^{k−1}Li_{k}^{a}(ζ) is Frobenius invariant, that is, the unique crystalline path γ_{crys} from 0 to ζ.

3. In the case where N = 1, the coefficient of A^{k−1}B in Φ_{KZ}^{𝔭} is −ζ_{p}(k) and the one of A^{k−1}B in ηι(F_{p}^{−1}τ(p)^{−1}) is (1 − p^{−k})ζ_{p}(k), from the above example.

4. (Furusho) The coefficient of A^{b−1}BA^{a−1}B in F_{p}^{−1}τ(p)^{−1} in the case where N = 1 is

(\frac{1}{p^{a+b}} − 1) ζ_{p}(a, b) − (\frac{1}{p^{a}} − 1) ζ_{p}(a)ζ_{p}(b)

+ ∑_{r=0}^{a−1}(−1)^{r} (\frac{1}{p^{b+r}} − 1) \binom{b − 1 + r}{b − 1}ζ_{p}(a − r)ζ_{p}(b + r)

+ (−1)^{a+1} ∑_{s=0}^{b−1} (\frac{1}{p^{a+s}} − 1) \binom{a − 1 + s}{a − 1}ζ_{p}(a + s)ζ_{p}(b − s),

for b > 1.

The following proposition combined with Proposition 3.4 gives a proof of Theorem 1.4. The author learned the following proposition from Furusho’s caluculation Example 2(4).

Proposition 3.10 For w ≥ 0, we have

Z_{w}^{𝔭}[N] = Z_{w}^{𝔭,D}[N].

Proof The effect of τ(q) is the multiplication by q^{w} on p-adic MLV’s of weight w in the sense of Deligne. Thus, Z_{w}^{𝔭,D}[N] is not changed when we replace F_{𝔭}^{−1} ∈ G_{ω}(ℚ(μ_{N})_{𝔭}) by φ_{𝔭} = F_{𝔭}^{−1}τ(q)^{−1} ∈ G_{ω}(ℚ(μ_{N})_{𝔭}) in α_{D} = γ_{dR}^{−1}φ_{𝔭}(γ_{dR}). Let J_{𝔭}^{D}(W) be the coefficient of a word W in Φ_{D}^{𝔭} := γ_{dR}^{−1}F_{𝔭}^{−1}(γ_{dR}). We have

Z_{w}^{𝔭,D}[N] = ⟨J_{𝔭}^{D}(W) | the weight of W is w⟩_{ℚ} ⊂ ℚ(μ_{N})_{𝔭}

(We recall that the coefficient of a word W in α_{F} is I_{𝔭}(W)). We have

α_{F} = γ_{dR}^{−1}γ_{crys} = γ_{dR}^{−1}F_{𝔭}^{−1}(γ_{dR}) · (F_{𝔭}^{−1}(γ_{dR}))^{−1}F_{𝔭}^{−1}(γ_{crys}) = Φ_{D}^{𝔭}F_{𝔭}^{−1}(α_{F})

= (∑_{W} J_{𝔭}^{D}(W)W ) (∑_{W} I_{𝔭}(W)F_{𝔭}^{−1}(W))

(By a theorem of Besser [B, Theorem 3.1], we see that α_{F} and α_{D} determine each other from the above formula). We compute the action F_{𝔭}^{−1} on a word W. Let γ_{dR,ζ} be the canonical path from 0 to ζ under the realization ω, that is, γ_{dR,1} = γ_{dR}, γ_{dR,ζ} = ζ(γ_{dR,1}). Here, ζ(γ_{dR,1}) is the action of ζ ∈ μ_{N} on Π. Then, B_{ζ} = (γ_{dR,ζ})^{−1}A · γ_{dR,ζ} ([DG, (5.11.3)]). Thus, we have F_{𝔭}^{−1}(A) = qA and

F_{𝔭}^{−1}(B_{ζ}) = (F_{𝔭}^{−1}(γ_{dR,ζ}))^{−1}qAF_{𝔭}^{−1}(γ_{dR,ζ}) = qζ(Φ_{D}^{𝔭})^{−1}B_{ζ}ζ(Φ_{D}^{𝔭})

= q (∑_{W} J_{𝔭}^{D}(ζ^{−1}(W))W )^{−1} B_{ζ} (∑_{W} J_{𝔭}^{D}(ζ^{−1}(W))W ) .

Here, the action of ζ ∈ μ_{N} on words is given by ζ(A) = A, and ζ(B_{ζ′}) = B_{ζζ′}. From the above formula about α_{F}, we have

α_{F} = Φ_{D}^{𝔭}F_{𝔭}^{−1}(α_{F}) = (∑_{W} J_{𝔭}^{D}(W)W ) (∑_{W} I_{𝔭}(W)F_{𝔭}^{−1}(W))

= (∑_{W} J_{𝔭}^{D}(W)W ) [ ∑_{W =A^{k_{d}} B_{ζ_{d}} ···A^{k_{1}} B_{ζ_{1}} A^{k_{0}}} q^{k_{0}+···+k_{d}+d}I_{𝔭}(W)A^{k_{d}}

· (∑_{W} J_{𝔭}^{D}(ζ_{d}^{−1}(W))W )^{−1} B_{ζ_{d}} (∑_{W} J_{𝔭}^{D}(ζ_{d}^{−1}(W))W ) ⋯

· (∑_{W} J_{𝔭}^{D}(ζ_{1}^{−1}(W))W )^{−1} B_{ζ_{1}} (∑_{W} J_{𝔭}^{D}(ζ_{1}^{−1}(W))W ) A^{k_{0}} ] ,

There, by using Proposition 2.14 and Proposition 3.3, for a word W of weight w we have

(1 − q^{w})I_{𝔭}(W) − J_{𝔭}^{D}(W) ∈ ∑_{w=w′+w′′:w′<w,w′′<w} Z_{w′}^{𝔭} · Z_{w′′}^{𝔭,D}.

By induction, we have Z_{w}^{𝔭} = Z_{w}^{𝔭,D}.

Finally, we remark on some conjectures. The following conjecture is a p-adic analogue of Grothendieck’s conjecture [DG, 5.20], which says that a_{σ} ∈ G_{ω}(ℂ) is ℚ-Zariski dense (weakly, a_{σ}^{0} := a_{σ}τ(2π\sqrt{−1})^{−1} ∈ U_{ω}(ℂ) is ℚ-Zariski dense). Here, a_{σ} is the “difference” between the Betti realization with respect to σ and the de Rham realization (For elements a_{σ} and a_{σ}^{0}, see [DG, Proposition 2.12] and [D1, 8.10 Proposition]).

Conjecture 4 The element φ_{𝔭} ∈ U_{ω}(ℚ(μ_{N})_{𝔭}) is ℚ-Zariski dense. That means that if a subvariety X of U_{ω} over ℚ satisfies φ_{𝔭} ∈ X(ℚ(μ_{N})_{𝔭}), then X = U_{ω}.

Remark 3.11 We have the Chebotarev density theorem for usual Galois groups. So, the author expects that there may be “Chebotarev density like” theorem for the Frobenius element in the motivic Galois group varying the prime number p. It will be interesting to study for this “Chebotarev density like” theorem varying p, adèle valued points of the motivic Galois group, and possible relations among “Chebotarev density like” theorem varying p, Grothendieck’s conjecture about the motivic element, and the above p-adic analogue of Grothendieck’s conjecture about the Frobenius element.

The following conjecture in the case N = 1 (i.e. p-adic MZV’s) is proposed by Furusho (non published).

Conjecture 5 All linear relations among p-adic MLV’s are linear combinations of linear relations among p-adic MLV’s with same weights.

The following proposition is obvious (cf. [DG, 5.27]).

Proposition 3.12 We consider the following statements:

1. The inequality in Theorem 1.4 is an equality (For N = 1, this is Conjecture 2).

2. The map ι : U_{ω} → V_{ω} is injective.

3. Conjecture 4.

4. Conjecture 5.

Then, (1) is equivalent to the combination of (2) and (3), and implies (4).

Remark 3.13 The statement (2) is true for N = 2, 3, 4. For N > 4, the statement (2) is false in general. The kernel is related to the space of cusp forms of weight 2 on X_{1}(N) if N is a prime. See, [DG, 5.27][G2].

4 Crystalline Realization of Mixed Tate Motives.

In this section, we consider the construction of the crystalline realization of mixed Tate motives, and Berthelot-Ogus isomorphism for the de Rham and crystalline realizations of mixed Tate motives.

4.1 Crystalline Realization.

Let k be a number field, v be a finite place of k, and G_{k} be the absolute Galois group of k. First, we define the crystalline inertia group at v. Let p be a prime divided by v. Let \underline{Rep}_{ℚ_{p}}(G_{k}), and \underline{Rep}_{ℚ_{p}}^{crys,v}(G_{k}) be the category of finite dimensional representations of G_{k} over ℚ_{p}, and the subcategory of crystaline representations of G_{k} at v.

Definition 4.1 (crystalline inertia group) The inclusion \underline{Rep}_{ℚ_{p}}^{crys,v}(G_{k}) ↪ \underline{Rep}_{ℚ_{p}}(G_{k}) induces the map of Tannaka dual groups with respect to the forgetful fiber functor. We define a crystalline inertia group I_{v}^{crys}(⊂ G_{k,p} := \underline{Aut}^{⊗}(\underline{Rep}_{ℚ_{p}}(G_{k}))) at v to be its kernel.

Here, G_{k,p} is the (algebraic group over ℚ_{p})-closure of G_{k}. The group I_{v}^{crys} is a pro-algebraic group over ℚ_{p}. Note that by the definition, the action of G_{k} on M_{p} is crystalline at v if and only if the action of I_{v}^{crys} on M_{p} is trivial. We recall Bloch-Kato’s group H_{f}^{1}. Let O_{(v)} be the localization at v of the ring of integers of k, and k_{v} be the completion of k with respect to v. For a finite dimensional representation V of G_{k_{v}} over ℚ_{ℓ}, they defined [BK, §3]

H_{f}^{1}(k_{v}, V) := \begin{cases}ker(H^{1}(k_{v}, V) → H^{1}(k_{v}^{ur}, V)) & v ∤ ℓ, \\ ker(H^{1}(k_{v}, V) → H^{1}(k_{v}, B_{crys} ⊗ V)) & v | ℓ. \end{cases}

Here, k_{v}^{ur} is the maximal unramified extension of k_{v}, and B_{crys} is the Fontaine’s p-adic period ring (See, [Fo1]). For a prime ℓ not divided by v, Hom_{Gal(\overline{k_{v}}/k_{v}^{ur})}(ℚ_{ℓ}(m), ℚ_{ℓ}(m + n)) is trivial for n ≥ 2. Thus, we have

H_{f}^{1}(k_{v}, ℚ_{ℓ}(n)) = \begin{cases}O_{(v)}^{×} ⊗ ℚ_{ℓ} & n = 1, \\ H^{1}(k_{v}, ℚ_{ℓ}(n)) & n ≥ 2. \end{cases}

In the crystalline case, we have from the calculations

H_{f}^{1}(k_{v}, ℚ_{p}(n)) = \begin{cases}O_{(v)}^{×} ⊗ ℚ_{p} & n = 1, \\ H^{1}(k_{v}, ℚ_{p}(n)) & n ≥ 2, \end{cases}  (4.1)

(See, [BK, Example 3.9]) monodromy informaions of I_{v}^{crys} on mixed Tate motives. We recall that the fact H_{f}^{1}(k_{v}, ℚ_{p}(n)) = H^{1}(k_{v}, ℚ_{p}(n)) for n ≥ 2, v | p follows from

dim_{ℚ_{p}} H_{f}^{1}(k_{v}, ℚ_{p}(n))
= dim_{ℚ_{p}} D_{dR}(ℚ_{p}(n))/Fil^{0}D_{dR}(ℚ_{p}(n)) + dim_{ℚ_{p}} H^{0}(k_{v}, ℚ_{p}(n))
= [k_{v} : ℚ_{p}] + 0 = −χ(ℚ_{p}(n)) = dim_{ℚ_{p}} H^{1}(k_{v}, ℚ_{p}(n))

(See, [BK, Corollary 3.8.4, Example 3.9]). Here, D_{dR} is the Fontaine’s functor ([Fo2]), and χ(V) is the Euler characteristic of V for a Galois representation V. Thus, it holds without assuming that k_{v} is unramified over ℚ_{p}. Let H_{f}^{1}(k, V) be the inverse image of H_{f}^{1}(k_{v}, V) via the restriction map H^{1}(k, V) → H^{1}(k_{v}, V).

Theorem 4.2 (cf. [DG, Proposition 1.8]) Let k be a number field, and v be a finite place of k. Take a mixed Tate motive M in MT(k). Then, the following statements are equivalent.

1. The motive M is unramified at v, that is, M ∈ MT(O_{(v)}).

2. For a prime ℓ not divided by v, the ℓ-adic realization M_{ℓ} of M is an unramified representation at v.

3. For all prime ℓ not divided by v, the ℓ-adic realization M_{ℓ} of M is an unramified representation at v.

4. For the prime p divided by v, the p-adic realization M_{p} of M is a crystalline representation at v.

Proof The equivalence of (1), (2), and (3) is proved in [DG, Proposition 1.8]. We show that (1) is equivalent to (4). The proof is a crystalline analogue of [DG, Proposition 1.8]. The Kummer torsor K(a) for a ∈ k^{×} ⊗ℚ is crystalline at v, if and only if a ∈ O_{(v)}^{×}⊗ℚ (See, the isomorphism (4.1) H_{f}^{1}(k_{v}, ℚ_{p}(1)) ≅ O_{(v)}^{×}⊗ℚ_{p}). Since Kummer torsors generate Ext_{MT(k)}^{1}(ℚ(0), ℚ(1)), it suffices to show that the following statement: For a mixed Tate motive M ∈ MT(k), the action of I_{v}^{crys} on M_{p} is trivial if the action of I_{v}^{crys} on W_{−2n}M_{p}/W_{−2(n+2)}M_{p} is trivial for each n ∈ ℤ. Assume that the action of I_{v}^{crys} on W_{−2n}M_{p}/W_{−2(n+2)}M_{p} is trivial for each n ∈ ℤ. We show that the action of I_{v}^{crys} on W_{−2n}M_{p}/W_{−2(n+r)}M_{p} is trivial by the induction on r. For r = 2, it is the hypothesis. For r > 2, the induction hypothesis assure that the action of I_{v}^{crys} is trivial on W_{−2n}/W_{−2(n+r−1)} and W_{−2(n+1)}/W_{−2(n+r)}. Thus, the action of σ ∈ I_{v}^{crys} is of the form 1 + ν(σ), where ν(σ) is the composite:

W_{−2n}/W_{−2(n+r)} ↠ Gr_{−2n}^{W} ⟶^{μ(σ)} Gr_{−2(n+r−1)}^{W} ↪ W_{−2n}/W_{−2(n+r)}.

We have μ(σ_{1}σ_{2}) = μ(σ_{1}) + μ(σ_{2}). This μ is compatible with the action of G_{k,p}. It suffices to show that the map μ(σ) : Gr_{−2n}^{W} → Gr_{−2(n+r−1)}^{W} is trivial.

This follows from

Hom_{G_{k,p}}(I_{v}^{crys}, Hom(ℚ_{p}(n), ℚ_{p}(n + r − 1)))
≅ Ext_{\underline{Rep}_{ℚp} (I_{v}^{crys})}^{1}(ℚ_{p}(n), ℚ_{p}(n + r − 1))^{G_{k,p}/I_{v}^{crys}}
≅ Ext_{\underline{Rep}_{ℚp} (G_{k,p})}^{1}(ℚ_{p}(n), ℚ_{p}(n + r − 1))/Ext_{\underline{Rep}_{ℚp} (G_{k,p}/I_{v}^{crys})}^{1}
≅ Ext_{\underline{Rep}_{ℚp} (G_{k})}^{1}(ℚ_{p}(n), ℚ_{p}(n + r − 1))/Ext_{\underline{Rep}_{ℚp}^{crys,v}(G_{k})}^{1}
≅ H^{1}(k, ℚ_{p}(r − 1))/H_{f}^{1}(k, ℚ_{p}(r − 1)) = 0,

where we abbreviate Ext_{\underline{Rep}_{ℚp} (G_{k,p}/I_{v}^{crys})}^{1}(ℚ_{p}(n), ℚ_{p}(n + r − 1)) and

Ext_{\underline{Rep}_{ℚp}^{crys,v}(G_{k})}^{1}(ℚ_{p}(n), ℚ_{p}(n+r−1)) as Ext_{\underline{Rep}_{ℚp} (G_{k,p}/I_{v}^{crys})}^{1} and Ext_{\underline{Rep}_{ℚp}^{crys,v}(G_{k})}^{1}

respectively by a typesetting reason. The second isomorphism follows from the fact that Ext_{\underline{Rep}_{ℚp}^{crys,v}(G_{k})}^{2} = 0, and the action of I_{v}^{crys} on ℚ_{p}(r − 1) is

trivial, and the last equality follows from the isomorphism (4.1). (We have Ext_{\underline{Rep}_{ℚp}^{crys,v}(G_{k})}^{2} = 0 from the elemental theory of the category of filtered

φ-modules. In fact, RHom is calculated by a complex, which is concentrated only in degree 0 and 1.)

Remark 4.3 If we have a full sub-Tannakian category MT(O_{(v)})^{good} of MT(k) satisfying

Ext_{MT(O_{(v)})^{good}}^{1}(ℚ(0), ℚ(1)) ≅ \begin{cases}O_{(v)}^{×} ⊗ ℚ, & n = 1, \\ Ext_{MT(k)}^{1}(ℚ(0), ℚ(n)), & n ≥ 2, \end{cases}

and

Ext_{MT(O_{(v)})^{good}}^{2}(ℚ(0), ℚ(n)) = 0 for any n,

then by introducing the “motivic inertia group” at v

I_{v}^{M} := ker\{\underline{Aut}^{⊗}(ω_{MT(k)}) → \underline{Aut}^{⊗}(ω_{MT(O_{(v)})^{good}})\},

we can prove the similar result for MT(O_{(v)})^{good}, that is, M is in MT(O_{(v)}) if and only if M is in MT(O_{(v)})^{good} by the “motivic analogue” of the above proof. In a naive way, we cannot define “M ⊗_{O_{(v)}} k(v)” the reduction at v of an object M in MT(O_{(v)}), since MT(O_{(v)}) is not defined by a “geometrical way”. So, the author hopes that this remark will be useful to construct “the reduction at v” of object in MT(O_{(v)}). If we “geometrically” construct a full sub-Tannakian category MT(O_{(v)})^{good} of MT(k) satisfying the above conditions, then we can get a good definition of “the reduction at v”. Here, the word “geometrically” means that returning the definition of Voevodsky’s category DM(k). See also the proof of Theorem 4.6.

Definition 4.4 For a mixed Tate motive M ∈ MT(O_{(v)}) unramified at v, we define the crystalline realization M_{crys,v} to be D_{crys}(M_{p}). Here D_{crys} is the Fontaine’s functor (B_{crys} ⊗_{ℚ_{p}} −)^{G_{kv}}, and M_{p} is the p-adic realization of M.

Note that M_{p} is a crystalline representaion of G_{k_{v}} by Theorem 4.2, so we have dim_{k_{0,v}} M_{crys,v} = dim_{ℚ_{p}} M_{p} = dim_{ℚ} M_{ω}. Here, k_{0,v} is the fraction field of the ring of Witt vectors with coefficients in the residue field k(v) of O_{(v)}. Note also that the pair (M_{crys,v}, M_{crys,v} ⊗_{k_{0.v}} k_{v}) gives an admissible filtered φ-module in the sense of Fontaine ([Fo1], [Fo2]). The crystalline realization is functorial, and defines a fiber functor MT(O_{(v)}) → Vect_{k_{0,v}}, which factors through the category of admissible filtered φ-modules MF_{k_{0,v}}^{ad}(φ).

Remark 4.5 By using the fact that H_{st}^{1}(k_{v}, ℚ_{p}(1)) = H^{1}(k_{v}, ℚ_{p}(1)) and introducing “semistable inertia group” at v, we can show that M_{p} is a semistable representation of G_{k_{v}} for any mixed Tate motive M in MT(k), similarly as the proof of Theorem 4.2. Thus, we can define the crystalline realization (or semistable realization) M_{crys,v} to be D_{st}(M_{p}) = (B_{st} ⊗_{ℚ_{p}} M_{p})^{G_{kv}} for all M ∈ MT(k), and get a functor MT(k) → MF_{k_{0,v}}^{ad}(φ, N) to the category of admissible filtered (φ, N)-modules.

4.2 Comparison Isomorphism.

In this subsection, we prove a “Berthelot-Ogus like” comparison isomorphism between the crystalline realization and the de Rham realization. We defined the crystalline realization by using Fontaine’s functor, so we need another “geometrical” construction of the crystalline realization to compare it with the de Rham realization (it is not obvious that the other construction is functorial). For preparing the following theorem, we briefly recall that Voevodsky’s category DM(k) (see, [V]), Levine’s category MT(k) (see, [L]), and Deligne-Goncharov’s category MT(O_{(v)}) (see, [DG]). Let k be a field. First, let SmCor(k) be the additive category whose objects are smooth separated scheme over k, and morphisms Hom(X, Y) are free abelian group generated by reduced irreducible closed subschemes Z of X × Y, which are finite over X and dominate a connected component of X. Then, Voevodsky’s tensor triangulated category DM(k) is constructed from the category of bounded complexes K^{b}(SmCor(k)) of SmCor(k) by localizing the thick subcategory generated by [X × 𝔸^{1}] → [X] (homotopy invariance), and [U ∩ V] → [U] ⊕ [V] → [X] for X = U ∪ V (Mayer-Vietoris), adding images of direct factors of idempotents, and inverting formally ℤ(1). Let k be a number field. Then, the vanishing conjecture of Beilinson-Soulé holds for k. From the vanishing conjecture of Beilinson-Soulé, Levine constructed the Tannakian category of mixed Tate motives MT(k) from DMT(k) by taking a heart with respect to a t-structure. Here, DMT(k) is the sub-tensor triangulated category of DM(k)_{ℚ} generated by ℚ(n)’s.

For a finite place v of k, let O_{(v)} denote the localization of k at v. Deligne-Goncharov defined the full subcategory MT(O_{(v)}) of mixed Tate motives unramified at v in MT(k), whose objects are mixed Tate motives M in MT(k) such that for each subquotient E of M, which is an extension of ℚ(n) by ℚ(n + 1), the extension class of E in

Ext_{MT(k)}^{1}(ℚ(n), ℚ(n + 1)) ⟵^{≅} Ext_{MT(k)}^{1}(ℚ(0), ℚ(1)) ≅ k^{×} ⊗ ℚ

is in O_{(v)}^{×}⊗ℚ(⊂ k^{×}⊗ℚ). The following theorem is the comparison isomorphism between crystalline realization and de Rham realization. However, we defined the crystalline realization by using p-adic étale realization. So, the content of the following theorem is the comparison isomorphism between p-adic étale realization and the pair of crystalline and de Rham realizations.

Theorem 4.6 (Berthelot-Ogus isomorphism) For any mixed Tate motive M in MT(O_{(v)}), we have a canonical isomorphism

k_{v} ⊗_{k_{0,v}} M_{crys,v} ≅ k_{v} ⊗_{k} M_{dR}.

Remark 4.7 (Hyodo-Kato isomorphism) After choosing a uniformizer π of k_{v}, we can prove a canonical isomorphism

k_{v} ⊗_{k_{0,v}} M_{crys,v} ≅ k_{v} ⊗_{k} M_{dR}

for any mixed Tate motive M in MT(k) by the same way (cf. Remark 4.5).

Remark 4.8 From the functorial isomorphism M_{crys,v} ⊗_{k_{0,v}} k_{v} ≅ M_{dR} ⊗_{k} k_{v}, we have G_{ω} ⊗_{ℚ} k_{v} ≅ G_{crys} ⊗_{k_{0,v}} k_{v}. Here, G := π_{1}(MT(O_{(v)})) ∈ pro-MT(O_{(v)}) is the fundamental MT(O_{(v)})-group (See, [D1, §6][D2, Definition 8.13]). Thus, we can consider the Frobenius element F_{𝔭}^{−1} ∈ G_{ω}(k_{v}) if k_{0,v} = k_{v} (For example, in the case where k is ℚ(μ_{N}) and v is a prime ideal not dividing (N)).

Proof First, we observe the following thing. Let X and Y be smooth schemes over k, and Γ be an integral closed subschemes of X × Y, which is finite surjective over a component of X. Then, by using de Jong’s alterations, there exists a finite extension k′ of k, a prime ideal w over v, semistable pairs (cf. [dJ]) (X , D) and (Y, E) over O_{(w)}, such that f_{X} : (X \ D)_{k′} → X and f_{Y} : (Y \ E)_{k′} → Y are generically étale alterations of X, and Y, respectively. Put [Γ̃′_{k′}] := (f_{X} × f_{Y})^{!}[Γ ⊗_{k} k′]. Here, (f_{X} × f_{Y})^{!} : CH^{∗}(Γ ⊗_{k} k′) → CH^{∗}(Γ ⊗_{k} k′ ×_{(X×_{k}Y)⊗_{k}k′} ((X \ D)_{k′} × (Y \ E)_{k′})) is the Fulton-MacPherson’s refined Gysin map. Let Γ̃_{k′} denote the closure of Γ̃′_{k′} in X_{k′} ×Y_{k′}, Then, we have Γ̃_{k′} ∩ (X_{k′} × E_{k′}) ⊂ Γ̃_{k′} ∩ (D_{k′} × Y_{k′}). After choosing a uniformizer π′ ∈ k′_{w}, we have the comparison isomorphisms B_{st} ⊗_{k′_{0,w}} H_{log-crys}^{m}((X \ D)_{k′(w)}) ≅ B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}((X \ D)\overline{k}), and B_{st} ⊗_{k′_{0,w}} H_{log-crys}^{m}((Y \ E)_{k′(w)}) ≅ B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}((Y \ E)\overline{k}) proved in [Y]. By Γ̃_{k′} ∩ (X_{k′} × E_{k′}) ⊂ Γ̃_{k′} ∩ (D_{k′} × Y_{k′}), we can define the cycle classes (cf. [Y])

cl(Γ̃\overline{k}) ∈ H_{ét}^{2 dim Y} (X\overline{k} × Y\overline{k}, (X\overline{k} × E\overline{k})_{!}, (D\overline{k} × Y\overline{k})_{∗}),

and

cl(Γ̃_{k′}) ∈ H_{dR}^{2 dim Y} (X_{k′} × Y_{k′}, (X_{k′} × E_{k′})_{!}, (D_{k′} × Y_{k′})_{∗}).

Then, by using these cycle classes, we get a commutative diagram ([Y])

\xymatrix{k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}((Y \ E)\overline{k})) \ar[r]^{≅} \ar[d]^{[Γ̃\overline{k}]^{∗}} & k′_{w} ⊗_{k′} H_{dR}^{m}((Y \ E)_{k′}) \ar[d]^{[Γ̃_{k′}]^{∗}} \\ k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}((X \ D)\overline{k})) \ar[r]^{≅} & k′_{w} ⊗_{k′} H_{dR}^{m}((X \ D)_{k′}),}

where we used Hyodo-Kato isomorphism [Y]. Let [Ξ_{X}] ∈ CH(X_{k′} ×_{(X_{k′} ×X_{k′})} (X × X)) be (f_{X} × f_{X})^{!}([Δ_{X_{k′}}]), where f_{X} is the morphism X_{k′} → X_{k′}, (f × f)^{!} means Fulton-MacPherson’s refined Gysin homomorphism, and Δ_{X_{k′}} is the diagonal class of X_{k′}. We define [Ξ_{Y}] by the same way, then by using these cycle classes and the compatibility of the comparison isomorphism with cycle classes, we get commutative diagrams

[figure: D_{st,k′_{w}} (H{m \atop ét}((X \ D)\overline{k}))_{k′_{w}}; f_{X∗}; ≅; D_{st,k′_{w}} (H{m \atop ét}(X\overline{k}))_{k′_{w}}; f^{∗}_{X}; D_{st,k′_{w}} (H{m \atop ét}((X \ D)\overline{k}))_{k′_{w}}; ≅; H{m \atop dR}((X \ D)_{k′})_{k′_{w}}; f_{X∗}; H{m \atop dR}(X_{k′})_{k′_{w}}; f^{∗}_{X}; H{m \atop dR}((X \ D)_{k′})_{k′_{w}} ,]

[figure: D_{st,k′_{w}} (H{m \atop ét}((Y \ E)\overline{k}))_{k′_{w}}; f_{Y ∗}; ≅; D_{st,k′_{w}} (H{m \atop ét}(Y\overline{k}))_{k′_{w}}; f^{∗}_{Y}; D_{st,k′_{w}} (H{m \atop ét}((Y \ E)\overline{k}))_{k′_{w}}; ≅; H{m \atop dR}((Y \ E)_{k′})_{k′_{w}}; f_{Y ∗}; H{m \atop dR}(Y_{k′})_{k′_{w}}; f^{∗}_{Y}; H{m \atop dR}((Y \ E)_{k′})_{k′_{w}} k,]

where we abbreviate k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(−) and k′_{w} ⊗_{k′} H_{dR}^{m}(−) as D_{st,k′_{w}}(−)_{k′_{w}} and H_{dR}^{m}(−)_{k′_{w}} respectively by a typesetting reason. So, we get isomorphisms

k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}(X\overline{k})) ≅ k′_{w} ⊗_{k′} H_{dR}^{m}(X_{k′}),

and

k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}(Y\overline{k})) ≅ k′_{w} ⊗_{k′} H_{dR}^{m}(Y_{k′}).

By using the following commutative diagrams

\xymatrix{H_{ét}^{m}((Y \ E)\overline{k}) \ar@{->>}[r]^{f_{Y ∗}} \ar[d]^{[Γ̃\overline{k}]^{∗}} & H_{ét}^{m}(Y\overline{k}) \ar@{^{(}->}[r]^{f_{Y}^{∗}} & H_{ét}^{m}((Y \ E)\overline{k}) \ar[d]^{[Γ̃\overline{k}]^{∗}} \\ H_{ét}^{m}((X \ D)\overline{k}) \ar@{->>}[r]^{f_{X∗}} & H_{ét}^{m}(X\overline{k}) \ar@{^{(}->}[r]^{f_{X}^{∗}} & H_{ét}^{m}((X \ D)\overline{k}),}

and

\xymatrix{H_{dR}^{m}((Y \ E)_{k′}) \ar@{->>}[r]^{f_{Y ∗}} \ar[d]^{[Γ̃_{k′}]^{∗}} & H_{dR}^{m}(Y_{k′}) \ar@{^{(}->}[r]^{f_{Y}^{∗}} & H_{dR}^{m}((Y \ E)_{k′}) \ar[d]^{[Γ̃_{k′}]^{∗}} \\ H_{dR}^{m}((X \ D)_{k′}) \ar@{->>}[r]^{f_{X∗}} & H_{dR}^{m}(X_{k′}) \ar@{^{(}->}[r]^{f_{X}^{∗}} & H_{dR}^{m}((X \ D)_{k′}),}

we finally get a commutative diagram

\xymatrix{k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}(Y\overline{k})) \ar[r]^{≅} \ar[d]_{restriction of [Γ̃\overline{k}]^{∗}} & k′_{w} ⊗_{k′} H_{dR}^{m}(Y_{k′}) \ar[d]^{restriction of [Γ̃_{k′}]^{∗}} \\ k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(H_{ét}^{m}(X\overline{k})) \ar[r]^{≅} & k′_{w} ⊗_{k′} H_{dR}^{m}(X_{k′}).}

Now, take a triple (X^{•}, f, n) for the given motive M in MT(O_{(v)}), such that f(X^{•})(n) represents M, where X^{•} ∈ K^{b}(SmCor(k)), n ∈ ℤ, and f is an idempotent in K^{b}(SmCor(k)). We will proceed the above construction successively for the complex X^{•} in SmCor(k), by replacing the finite extension k′ one by one (Here, X^{•} is bounded. So, we can start from the first non-empty place and make the above construction and the above commutative diagram. Next, we make the above construction and commutative diagram in the next place after a finite base extension. We replace the first place by the finite base extension...). By using ((X^{•}, D^{•}), {Γ_{j_{•}}^{•}}_{j_{•}}), we can define sequences ((C^{•})_{ét}^{•}, d_{ét}^{•}), and ((C^{•})_{dR}^{•}, d_{dR}^{•}) of cohomological complexes, where (C^{•})_{ét}^{i} and (C^{•})_{dR}^{i} calculate the étale cohomology and de Rham cohomology of X i ; \overline{k} and X_{k}^{i} respectively, and d_{ét}^{i} and d_{dR}^{i} are defined by \{Γ_{\overline{k},j_{•}}^{i}\}_{j_{•}}, and \{Γ_{k,j_{•}}^{i}\}_{j_{•}} respectively. Note that we do not define the crystalline version ((C^{•})_{crys}^{i}, d_{crys}^{•}). Even if we define it by taking integral models of Γ_{j}^{•}’s, we do not have d_{crys}^{i+1} ∘ d_{crys}^{i} = 0 for the sequence of complexes (C^{•})_{crys}^{•} in general, because of the lack of the uniqueness of the extensions Γ_{j}^{•}’s (cf. [DG, Lemma 1.5.1]). So, we cannot define a crystalline realization by using (C^{•})_{crys}^{•} at least in the present situation (Note that we do not need to get d_{crys}^{•} by integral models of Γ_{j}^{•}’s in this proof). On the other hand, we have d_{ét/dR}^{i+1} ∘ d_{ét/dR}^{i} = 0 for the sequence of complexes (C^{•})_{ét/dR}^{•}, because they live on the generic fiber (cf. [DG, Lemma 1.5.1]) and we have the uniqueness (Note that the above construction and the above commutative diagram work after replacing H^{m} by RΓ, because we do not use integral models of Γ_{j}^{•}’s, but only use the generic fiber of them. See [DG, 1.5] for the de Rham part (C^{•})_{dR}^{•}). Therefore, we get an isomorphism k′_{w} ⊗_{k′_{0,w}} M_{crys,w} ≅ k′_{w} ⊗_{k′_{0,w}} D_{st,k′_{w}}(M_{p}) ≅ k′_{w} ⊗_{k′} M_{dR,k′} ≅ k′_{w} ⊗_{k} M_{dR} . Now, we use the condition that M is in MT(O_{(v)}). The p-adic realization M_{p} is crystalline at v by Thoerem 4.2. So, we have M_{crys,w} ≅ k′_{0,w} ⊗_{k_{0,v}} M_{crys,v}. Therefore, we have an isomorphism k′_{w} ⊗_{k_{0,v}} M_{crys,v} ≅ k′_{w} ⊗_{k} M_{dR}. In general, for any element τ ∈ Gal(k′_{w}/k_{v}), we have an isomorphism k_{w}^{′,τ} ⊗_{k_{0,v}} M_{crys,v} ≅ k_{w}^{′,τ} ⊗_{k} M_{dR} by using the triple {(X^{•,τ} , D^{•,τ}), f^{τ} , n}. Thus, we have an isomorphism k_{v} ⊗_{k_{0,v}} M_{crys,v} ≅ k_{v} ⊗_{k} M_{dR} by the descent. Since M_{p} is crystalline at v, this isomorphism does not depend on the choice of π′, and we can show that this isomorphism does not depend on the choice of good reduction models and this isomorphism is functorial by using the standard product argument.

4.3 Some Remarks and Questions.

The crystalline realization to the category of φ-modules (not to the category of admissible filtered φ-modules) is split, because we have

Ext_{MT(O_{(v)})}^{1}(ℚ(0), ℚ(n)) = 0

for n ≤ 0 and Ext_{Mod_{k_{0,v}} (φ)}^{1}(k_{0,v}(0), k_{0,v}(n)) = 0 for n > 0. So, we can expect that the crystalline realization MT(O_{(v)}) → Vect_{k_{0,v}} factors through MT(k(v)). Note that the weight filtration of mixed Tate motives over a finite field is split by Quillen’s calculations of K-groups of finite fields ([Q]). Thus, they are sums of ℚ(n)’s. The weight filtration is motivic, and both of the de Rham realization and the crystalline realization are split. However, the splittings do not coincide, that is, the splitting of the crystalline realization does not coincide to the splitting of the de Rham realization via the Berthelot-Ogus isomorphism of Theorem 4.6. The iterated integrals and p-adic MLV’s appear in the difference of these splittings. See also Remark 3.9.

Remark 4.9 We have Ext_{Mod_{k_{0,v}} (φ)}^{1}(k_{0,v}(0), k_{0,v}(0)) ≅ ℚ_{p} ≠ 0, and this gap corresponds to the “near critical strip case” of Beilinson’s conjecture and Bloch-Kato’s Tamagawa number conjecture, that is, we need not only regulator maps, but also Chow groups to formulate these conjectures near the critical strip case (that is, the case where the weight of motive is 0 or −2). In this case, this corresponds to the “dual” of the fact that the image of the Dirichlet regulator is not a lattice of ℝ^{r_{1}+r_{2}}, but a lattice of a hyperplane of ℝ^{r_{1}+r_{2}}. The author does not know a direct proof of the fact that the non-trivial extension in Ext_{Mod_{k_{0,v}} (φ)}^{1}(k_{0,v}(0), k_{0,v}(0)) = ℚ_{p} does not occur in the crystalline realization.

Example 3 (Kummer torsor) Let K be a finite extension of ℚ_{p}, K_{0} be the fraction field of the ring of Witt vectors with coefficient in the residue field of K. Let z ∈ 1 + πO_{K}. Let

0 → ℚ_{p}(1) → V (z)_{p} → ℚ_{p}(0) → 0

be the extension of p-adic realization corresponding to z. Fix e_{0} a generator of ℚ_{p}(1) corresponding {ζ_{n}}_{n}, and e_{1} the generator of ℚ_{p}(0) corresponding 1.

Then, the action of Galois group is the following:

\begin{cases}ge_{0} = χ(g)e_{0}, \\ ge_{1} = e_{1} + ψ_{z}(g)e_{0}. \end{cases}

Here, χ is the p-adic cyclotomic character, and ψ_{z} is characterized by g(z^{1/p^{n}}) = ζ_{n}^{ψ_{z}(g)}z^{1/p^{n}}. Then, V (z)_{crys} ≅ (B_{crys} ⊗_{ℚ_{p}} V (z)_{p})^{G_{K}} has the following basis:

\begin{cases}t^{−1} ⊗ e_{0} =: x_{0}, \\ e_{1} − t^{−1} log[\underline{z}] ⊗ e_{0} =: x_{1}. \end{cases}

Here, t := log[\underline{ζ}], log[\underline{z}] ∈ B_{crys}. Thus, the Frobenius action is the following:

\begin{cases}ϕ(x_{0}) = \frac{1}{p}x_{0}, \\ ϕ(x_{1}) = x_{1}. \end{cases}

The filtration after K⊗_{K_{0}} is the following:

Fil^{−1}V (z)_{dR} = V (z)_{dR} = ⟨x_{0}, x_{1}⟩_{K},
Fil^{0}V (z)_{dR} = ⟨x_{1} + (log z)x_{0}⟩_{K},
Fil^{1}V (z)_{dR} = 0

(In B_{dR}, we have t^{−1} log \frac{z}{[\underline{z}]} ∈ Fil^{0}B_{dR}). Thus, we have splittings:

V (z)_{crys} = ⟨x_{0}⟩_{K_{0}} ⊕ ⟨x_{1}⟩_{K_{0}} = K_{0}(1) ⊕ K_{0}(0),
V (z)_{dR} = ⟨x_{0}⟩_{K} ⊕ ⟨x_{1} + (log z)x_{0}⟩_{K} = K(1) ⊕ K(0).

These splittings do not coincide in general. We will recover the calculation ϕ^{−1}(0) = log z^{1−p} in [D1, 2.9, 2.10]. In this case, we assume K = K_{0}. By the above calculation, the Kummer torsor K(z)_{dR} is

K(z)_{dR} = −(x_{1} + (log z)x_{0}) + Kx_{0}

(For the purpose of making satisfy ∇(u) = du − \frac{dz}{z} in [D1, 2.10], we use the above sign convention). Then, we have

ϕ^{−1}(0) ↔ ϕ^{−1}(−(x_{1} + (log z)x_{0}) + 0) = −(x_{1} + p(log z)x_{0})
= −(x_{1} + (log z)x_{0}) + (1 − p)(log z)x_{0}
= −(x_{1} + (log z)x_{0}) + (log z^{1−p})x_{0}
↔ log z^{1−p}.

This coincides the calculation in [D1, 2.10]. Here, ↔ is the identification via K(z)_{dR} = −(x_{1} + (log z)x_{0}) + Kx_{0} ≅ K.

Next, we define polylogarithm extensions. In the following, we consider the case where k is a cyclotimic field ℚ(μ_{N}) for N ≥ 1. For ζ ∈ μ_{N}, let U_{ζ} ∈ pro-MT(ℚ(μ_{N})) be the kernel of π_{1}^{M}(ℙ^{1} \{0, 1∞}, ζ) → π_{1}^{M}(𝔾_{m}, ζ). We define Log_{ζ} to be the abelianization of U_{ζ} Tate-twisted by (−1). We define Pol_{ζ} with Tate twist (1) to be the push-out in the following diagram (see also, [D1, §16]):

\xymatrix{1 \ar[r] & U_{ζ} \ar[r] \ar@{->>}[d] & π_{1}^{M}(ℙ^{1} \ \{0, 1, ∞\}, ζ) \ar[r] \ar@{->>}[d] & π_{1}^{M}(𝔾_{m}, ζ) \ar[r] \ar[d]^{=} & 1 \\ 0 \ar[r] & Log_{ζ}(1) \ar[r] & Pol_{ζ}(1) \ar[r] & ℚ(1) \ar[r] & 0.}

(4.2)

For n ≥ 1, we also define Pol_{n,ζ} to be the push-out under Log_{ζ} = Π_{n≥0}ℚ(n) → ℚ(n) (see also, [D1, §16]):

\xymatrix{0 \ar[r] & Log_{ζ} \ar[r] \ar@{->>}[d] & Pol_{ζ} \ar[r] \ar@{->>}[d] & ℚ(0) \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & ℚ(n) \ar[r] & Pol_{n,ζ} \ar[r] & ℚ(0) \ar[r] & 0.}

The extension class [Pol_{n,ζ}] lives in Ext_{MT(ℚ(μ_{N}))}^{1}(ℚ(0), ℚ(n)) ≅ K_{2n−1}(ℚ(μ_{N}))_{ℚ}. Let μ_{N}^{0} be the group of primitive N-th roots of unity. Recall that Huber-Wildeshaus constructed motivic polylogarithm classes pol_{ζ} ∈ ∏_{n≥2} K_{2n−1}(ℚ(μ_{N}))_{ℚ} (not extensions of motives) in [HW].

Proposition 4.10 Let n be an integer greater than or equal to 2, and ζ be an N-th root of unity. Then, the n-th component of Huber-Wildeshaus’ motivic polylogarithm class pol_{ζ} (see, [HW, Definition 9.4]) is equal to (−1)^{n−1} \frac{n!}{N^{n−1}} [Pol_{n,ζ}] under the identification

K_{2n−1}(ℚ(μ_{N}))_{ℚ} ≅ Ext_{MT(ℚ(μ_{N}))}^{1}(ℚ(0), ℚ(n)).

In particular, the extension classes {[Pol_{n,ζ}]}_{ζ∈μ_{N}^{0}} generate K_{2n−1}(ℚ(μ_{N}))_{ℚ}.

Proof It is sufficient to show the equality after taking the Hodge realization. This follows from [D1, §3, §16, §19] and [HW, Theorem 9.5, Corolary 9.6]. Note that we consider as ℚ(n)_{ω}-torsor not as ℤ(n)_{ω}-torsor, and we do not multiply \frac{1}{(n−1)!} on the integral structure unlike as [D1] (See also Example (2, 2)).

Fix a place v ∤ N of ℚ(μ_{N}). Put K := ℚ(μ_{N})_{v}. Let p be the prime devied by v. Note that K is unramified over ℚ_{p}. Let σ denote the Frobenius endomorphism on K. For a mixed Tate motive [0 → ℚ(n) → M → ℚ(0) → 0] ∈ Ext_{MT(O_{(v)})}^{1}(ℚ(0), ℚ(n)), the pair M_{syn} := (M_{crys,v}, M_{dR} ⊗_{ℚ(μ_{N})} K) defines a extension of filtered φ-modules:

0 → K(n) → M_{syn} → K(0) → 0.

Here, K(i) is the Tate object in the category of filtered φ-modules over K. Thus, we have a map

r_{n} : K_{2n−1}(O_{(v)})_{ℚ} ≅ Ext_{MT(O_{(v)})}^{1}(ℚ(0), ℚ(n))
→ Ext_{MF_{K}^{f}}^{1} (K(0), K(n)) ≅ H_{syn}^{1}(K, K(n)).

See, [Ba] for the last isomorphism. We call r_{n} the n-th syntomic regulator map. Recall that H_{syn}^{1} is a finite dimensional ℚ_{p}-vector space, not a K-vector space. We fix an isomorphism H_{syn}^{1}(K, K(n)) ≅ K as ℚ_{p}-vector spaces for n ≥ 1 as follows.

H_{syn}^{1}(K, K(n))

≅ coker(K(n)_{crys} ⟶^{a↦(ā,(1−φ)(a))} (K(n)_{dR}/Fil^{0}K(n)_{dR}) ⊕ K(n)_{crys})

≅ coker(K ⟶^{a↦(a,(1−p^{−n}σ)(a))} K ⊕ K)

≅^{[(a,b)]↦b−(1−p^{−n}σ)(a)} K.

In general, note that for a filtered φ-module D and for

[(x, y)] ∈ coker(D ⟶^{a↦(ā,(1−φ_{D})(a))} (D/Fil^{0}D) ⊕ D) ≅ Ext_{MF_{K}^{f}}^{1} (K(0), D),

the corresponding extension E of K(0) by D is the following: E = D ⊕ Ke_{0}

\begin{cases}Fil^{i}E = Fil^{i}D + ⟨x + e_{0}⟩_{K} & for i ≤ 0, \\ Fil^{i}E = Fil^{i}D & for i > 0, \end{cases}

\begin{cases}φ_{E}(a) = φ_{D}(a) & for a ∈ D, \\ φ_{E}(e_{0}) = e_{0} + y. \end{cases}

Proposition 4.11 The syntomic regulator map

r_{1} : K_{1}(O_{(v)})_{ℚ} ≅ O_{(v)}^{×} ⊗ ℚ → H_{syn}^{1}(K, K(1)) ≅ K

is given by z ↦ −(1 − \frac{1}{p}) log z. For n ≥ 2, the syntomic regulator map

r_{n} : K_{2n−1}(ℚ(μ_{N}))_{ℚ} → H_{syn}^{1}(K, K(n)) ≅ K

sends [Pol_{n,ζ}] to −N^{n−1}(1 − \frac{1}{p^{n}})Li_{n}^{a}(ζ).

Note that Coleman’s p-adic polylogarithm (1 − \frac{1}{p^{n}})Li_{n}^{a}(ζ) is often written by

ℓ_{n}^{(p)}(ζ), and does not depend on the chice of a.

Remark 4.12 In the above proposition, we used the homomorphism induced by crystalline realizations and the isomorphism between K-theory and Ext_{MT}^{1} as a regulator. For a purely K-theoretic definition of a regulator and its calculation, see [BdJ].

Remark 4.13 If we use an identification

coker(K ⟶^{a↦(a,(1−p^{−n}σ)(a))} K ⊕ K) ≅^{[(a,b)]↦a−(1−p^{−n}σ)^{−1}(b)} K

(note that 1 − p^{−n}σ is a bijection on K for n ≥ 1), then the above formula changes as the following: the map

r_{1} : K_{1}(O_{(v)})_{ℚ} ≅ O_{(v)}^{×} ⊗ ℚ → H_{syn}^{1}(K, K(1)) ≅ K

is given by z ↦ log z. For n ≥ 2, the map

r_{n} : K_{2n−1}(ℚ(μ_{N}))_{ℚ} → H_{syn}^{1}(K, K(n)) ≅ K

sends [Pol_{n,ζ}] to N^{n−1}Li_{n}^{a}(ζ).

Proof The first assertion follows from Example (3). The second assertion follows from the following structure of (Pol_{n,ζ})_{syn} = ((Pol_{n,ζ})_{crys}, (Pol_{n,ζ})_{dR}): (Pol_{n,ζ})_{crys} = ⟨x_{0}, x_{1}⟩_{K}

\begin{cases}φ(x_{0}) = \frac{1}{p^{n}} x_{0}, \\ φ(x_{1}) = x_{1} − N^{n−1}(1 − p^{−n})Li_{n}^{a}(ζ), \end{cases}

Fil^{−n}(Pol_{n,ζ})_{dR} = ⟨x_{0}, x_{1}⟩_{K},
Fil^{i}(Pol_{n,ζ})_{dR} = ⟨x_{1}⟩_{K} for − n < i ≤ 0,
Fil^{1}(Pol_{n,ζ})_{dR} = 0.

This structure follows from Example (2).

Remark 4.14 We have an isomorphism

B_{crys} ⊗_{ℚ_{p}} (P_{y,x}^{M})_{p} ≅ B_{crys} ⊗_{K_{0}} (P_{y,x}^{M})_{crys}.

Here, P_{y,x}^{M} is a fundamental groupoid of ℙ^{1} \ {0, ∞} ∪ μ_{N}. This induces an isomorphism

B_{crys} ⊗_{ℚ_{p}} (Pol_{ζ})_{p} ≅ B_{crys} ⊗_{K_{0}} (Pol_{ζ})_{crys}.

Thus, we have the following commutative diagram for n ≥ 2:

[figure: K_{2n−1}(ℚ(μ_{N}))_{ℚ}; H^{1}(K, ℚ_{p}(n)); ≅; [Pol_{n,ζ}]; [(Pol_{n,ζ})_{p}]; H_{syn}^{1}(K, K(n)); [(Pol_{n,ζ})_{syn}].]

Here, K denotes ℚ_{p}(μ_{N}), ζ is in μ_{N}, and p does not divide N. The horizontal map sends the extension class [Pol_{n,ζ}] to the one [(Pol_{n,ζ})_{p}], and the oblique map sends the extension class [Pol_{n,ζ}] to the one [(Pol_{n,ζ})_{syn}]. The fact that [(Pol_{n,ζ})_{p}] is sent to [(Pol_{n,ζ})_{syn}] was first shown by T. Tsuji. Unfortunately, no preprint is available yet. His method is totally different. He does not use motivic theory or motivic π_{1}. He used the classical characterization (or the definition) of p-adic and syntomic polylogarithm sheaves as a specified extension (via residue isomorphisms etc.) of the constant sheaf by Log, and checked the characterization coincides via the p-adic Hodge comparison isomorphism.

Finally, we’d like to propose some very vague questions. If we take the Hodge (resp. ℓ-adic) realization of the lower line of (4.2), and specialize it to the roots of unity, then we get the special values of polylogarithms (resp. the Soulé elements). This fact is important of Bloch-Kato’s Tamagawa number conjecture ([BK]) for Tata motives. Furthermore, the Soulé elements form an Euler system, which has a power to show a half of Iwasawa main conjecture. The Soulé elements are sent to the Kubota-Leopoldt’s p-adic L-function via Bloch-Kato’s dual exponentioal map.

Question 1 Can we “suitably lift” this theory to the upper line of (4.2)?

More concretely:

Question 2 This will give a theory between non-commutative extensions of cyclotomic fields and multiple zeta values?

(It seems that Massey products play some roles instead of Ext^{1}.)

Question 3 This is related with Ihara’s higher cyclotomic fields, Anderson-Ihara’s higher circular units ([AI]), and Ozaki’s non-commutative Iwasawa theory?

(Ozaki considered the maximal pro-p extensions unramified outside p of the cyclotmic fields, and its graded quotients of the lower central series, and he showed that Iwasawa class number formura for each graded quotient.) Wojtkoviak studied ([W]) ℓ-adic iterated integrals, which specialize to the Soulé elements at the roots of unity in the case where the depth is one.

Question 4 What are the properties and axioms of “iterated integrals of Euler system”?

There are many difficulties to establish the above theory. It seems that the origin of the difficulties is that there are no good analytic properties for the zeta function in the higher depth cases. The above things are questions above “non-commutative Iwasawa theory in the mixed Tate sense”.

Next, we propose some very vague questions about “non-commutative class field theory in the mixed Tate sense”. We have the universal mixed Tate representation

Gal(\overline{k}/k) ⟶ G_{𝔸_{f}} (𝔸_{f})

for any number field k and ring of S-integers O_{S}, where G_{𝔸_{f}} is the motivic Galois group of MT(O_{S}) with respect to the finite adele realization.

Question 5 Can we relate this with an automorphic representation of G_{𝔸}?

(We also note that X^{∗}(U_{ω}) ≅ X^{∗}(U_{ω}^{ab}) ≅ X^{∗}(LieU_{ω}^{ab}) ≅ ⊕_{n≥1}Ext_{MT(O_{S})}^{1}(ℚ(0), ℚ(n)) ≅ K_{2n−1}(O_{S})_{ℚ}.) It seems that the concept of the automorphic representation is not good for unipotent groups. So, the author thinks that it will not be successful to consider automorphic representations. He also thinks that this corresponds that we cannot consider the L-factors and the functional equations in the higher depth cases. We modify the above question as follows (it becomes more vague):

Question 6 Can we find some “automorphy” in the lattice G_{ω}(ℚ) ↪ G_{ω}(𝔸)?

Manin studied the iterated integrals of modular forms ([M]). However, the analytic properties of them (e.g. “automorphy in the higher depth cases”) are not clarified.

Question 7 Are there some kinds of relations among a ∈ G_{ω}(ℂ), F_{𝔭}^{−1} ∈ G_{ω}(k_{𝔭,0}), and Frob_{𝔭} ∈ G_{𝔸_{f}^{p}} (𝔸_{f}^{p}) for 𝔭 ∉ S?

The lower bounds of (p-adic) multiple zeta value spaces are (p-adic) transcendental number theoritic problem. The author thinks that we cannot show the lower bounds by using only algebraic arithmetic geometry, and that we need (p-adic) transcendental number theory (or ergodic theory) to show them. However, we might be able to attack the following weaker statement by using only algebraic arithmetic geometry.

Question 8 By finding some kinds of “automorphy” in the case where k = ℚ, and O_{S} = ℤ, can we show that the lower bounds of the dimensions of the p-adic multiple zeta value spaces for p ≤ ∞ except p_{0} are equivalent to the lower bounds of the dimensions of the p-adic multiple zeta value spaces for all p ≤ ∞?

Take a 2-step unipotent quotient of U_{ω}. Then, we can consider the adelic theta theory for this group.

Question 9 Can we describe explicitly the theta theory for (p-adic) multiple L-values?

Question 10 By studying this, can we formulate a conjecture about the precise dimensions of (p-adic) multiple L-value spaces?

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Go Yamashita

School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD, United Kingdom Go.Yamashita@nottingham.ac.uk 724

Documenta Mathematica · Extra Volume Suslin (2010)