CRYSTALLINE SHEAVES, SYNTOMIC COHOMOLOGY AND

p-ADIC POLYLOGARITHMS

(NOTES OF THE SEMINAR AT CAL TECH ON FEB. 20, 2001)

TAKESHI TSUJI

In [BD92] (see also [HW98]), A. A. Beilinson and P. Deligne constructed the motivic polylogarithmic sheaf on ℙ_{ℚ}^{1}\{0, 1, ∞}. Its specializations at primitive d-th roots of unity give the Beilinson’s elements of H_{M}^{1}(ℚ(μ_{d}), ℚ(m)) = K_{2m−1}(ℚ(μ_{d}))⊗ ℚ (m ≥ 1), whose images under the regulator maps to Deligne cohomology are the values of m-th polylogarithmic functions at primitive d-th roots of unity. The polylogarithmic functions appear as the (complex) period functions. In these notes, we will show that the following two p-adic realizations correspond to each other via the theory of crystalline sheaves by G. Faltings [Fal89] V f).

One is the realization in the category of smooth ℚ_{p}-sheaves on (ℙ_{ℚ_{p}}^{1}\{0, 1, ∞})_{ét}. Its specializations at primitive d-th roots of unity give the images of the Beilinson’s elements in H^{1}(ℚ_{p}(μ_{d}), ℚ_{p}(m)) = Ext_{Rep_{ℚp} (G_{K})}^{1}(ℚ_{p}, ℚ_{p}(m)) (m ≥ 1) under the regulator maps. It is known that they also coincide with the Soulé’s cyclotomic elements.

The other is the realization in the category of (log) filtered convergent F -isocrystals on ℙ_{ℤ_{p}}^{1} endowed with the log structure associated to the divisor {0, 1, ∞}. In [Ban00a], K. Bannai constructed the realization in the category of filtered overconvergent F -isocrystals on ℙ_{ℤ_{p}}^{1}\{0, 1, ∞} using rigid syntomic cohomology, gave an explicit description of it in terms of p-adic polylogarithmic functions, and then proved that the specializations of the crystalline polylogarithmic sheaf at primitive d-th roots of unity for d prime to p give the values of p-adic polylogarithmic functions at primitive d-th roots of unity. His construction and calculation also work for log filtered convergent F -isocrystals and log syntomic cohomology, and we prefer the log version to the overconvergent one because we have the theory of crystalline sheaves by G. Faltings for the former.

We will see that some functions which live in a big ring B_{crys} and satisfy the same differential equations as the complex polylogarithmic functions, appear as the p-adic period functions of these p-adic realizations (Proposition 6.8).

By combining the result of K. Bannai explained above with our comparison theorem, we immediately obtain a new proof of the following fact: The images of the Beilinson’s elements in H^{1}(ℚ_{p}(μ_{d}), ℚ_{p}(m)) (m ≥ 1) under the regulator maps for d prime to p coincide with the images of the values of m-th p-adic polylogarithmic functions at primitive d-th roots of unity under the map:

K(m) = K(m)/(1 − φ)(F il^{0}K(m)) ⟶^{≅} Ext_{MF_{K}(φ)}^{1}(K, K(m))

⟶^{⊂} Ext_{Rep_{ℚp} (G_{K})}^{1}(ℚ_{p}, ℚ_{p}(m)) = H^{1}(K, ℚ_{p}(m)).

Here K = ℚ_{p}(μ_{d}).

In [Ban00c], K. Bannai also studied the crystalline realization of the motivic elliptic polylogarithm on a CM elliptic curve minus 0 and its relation with the values of the p-adic L-function at positive integers (a p-adic analogue of the Beilinson conjecture). Our argument also works in this case.

I would like to thank K. Bannai for kindly explaining me some basic facts on the motivic polylogarithm and for useful discussions.

1. Filtered convergent F -isocrystals

Let K be a complete discrete valuation field of mixed characteristic (0, p) with perfect residue field k, and let O_{K} denote the ring of integers. let K_{0} be the field of fractions of the ring of Witt vectors W (k). For a scheme X_{0} of finite type over Spec(k), P. Berthelot and A. Ogus ([Ber86], [Ogu84]) defined the notion of convergent isocrystals on X_{0}/K, which can be regarded as p-adic local systems in characteristic p at least in the case X_{0} is proper over k; they are defined in an abstract way using the notion of enlargements of X_{0} in general, but if a global embedding of X_{0} into a p-adic smooth formal scheme Y over Spf(O_{K}) is given, then they can be described more explicitly as modules with connections on the tubular neighborhood ]X_{0}[_{Y_{K}} of X_{0} in the rigid analytic space Y_{K} associated to Y.

Suppose that X_{0} is the special fiber of a proper smooth scheme X over Spec(O_{K}). In this case, G. Faltings ([Fal89] V f)) considered filtered convergent F -isocrystals on X, which are just convergent isocrystals on X_{0}/K_{0} endowed with Frobenius automorphisms Φ and some kind of descending filtrations after the base change to K, and generalized the theory of Fontaine on crystalline p-adic representations and filtered φ-modules ([Fon82], [Fon94a], [Fon94b]) to crystalline p-adic sheaves on (X_{K})_{ét} and filtered convergent F -isocrystals. See §3 for more details.

Using the theory of logarithmic structures in the sense of Fontaine and Illusie ([Kat89]), one can generalize, in an almost straightforward way, the notion of convergent isocrystals to fine log schemes of finite type over Spec(k), and the notion of filtered convergent F -isocrystals to fine log formal schemes smooth over Spf(O_{K}). Here we endow Spec(k) and Spf(O_{K}) with the trivial log structures. These generalizations for a proper smooth scheme X over Spec(O_{K}) with the log structure defined by a divisor D with relative normal crossings, were also studied by G. Faltings ([Fal89], [Fal90]). See also [Shi00] 5.1.

In this section, we assume K = K_{0}, consider a p-adic smooth formal scheme X over Spf(O_{K}) endowed with the log structure M defined by a divisor D with relative normal crossings and explain the local description of filtered convergent F -isocrystals on (X, M) in terms of modules with connections. To understand this description, we do not need any knowledge on log structures.

First assume X is affine and set A = Γ(X, O_{X}). We further assume that there exist t_{1}, t_{2}, …, t_{d} ∈ A such that D is defined by the equation t_{1}t_{2} ⋯ t_{d} = 0, the divisor defined by t_{i} = 0 is empty or smooth over Spf(O_{K}) for each i, and {dt_{i}|1 ≤ i ≤ d} is a basis of Ω_{X/O_{K}}^{1} . We call such set of elements t_{1}, t_{2}, …, t_{d} a log coordinate of (X, M).

Set

Ω_{A}(log) := Γ(X, Ω_{X/O_{K}} (log D)) ≅ ⨁_{1≤i≤d} A · \frac{dt_{i}}{t_{i}}

and

Ω_{A}^{i}(log) := ∧_{A}^{i}Ω_{A}(log) (i ∈ ℤ, i ≥ 0).

p-ADIC POLYLOGARITHMS 3

Choose a lifting of the absolute Frobenius of (X, M) ⊗_{O_{K}} k to (X, M) which is equivalent to giving a lifting F_{X}: X → X of the absolute Frobenius of X ⊗_{O_{K}} k such that F_{X}(t_{i}) = t_{i}^{p}·(unit) for each i such that t_{i} is not invertible. We denote by φ the endomorphism of A induced by F_{X}.

Giving a filtered convergent F -isocrystal on (X, M) is equivalent to giving the following data (E, F il^{i}, ∇, Φ):

• a finite projective K ⊗ A-module E.

• a decreasing filtration F il^{i}E (i ∈ ℤ) by direct factors of E as K ⊗ A-modules such that F il^{i}E = 0 (i >> 0) and F il^{i}E = E (i << 0).

• an integrable connection ∇: E → E ⊗_{A} Ω_{A}(log) satisfying:

– Griffiths transversality: ∇(F il^{i}E) ⊂ F il^{i−1}E ⊗_{A} Ω_{A}(log) (i ∈ ℤ)

– Convergence: For any x ∈ E and any δ ∈ ℝ_{>0},

p^{[(n_{1}+n_{2}+···+n_{d})δ]} \frac{1}{n_{1}!n_{2}! ⋯ n_{d}!}( ∏_{1≤i≤d} ∏_{0≤j<n_{i}}(∇_{i}^{log} − j))(x),

(n_{1}, n_{2}, …, n_{d}) ∈ ℕ^{d},

converges to 0 as n_{1} + n_{2} + ⋯ + n_{d} → ∞. Here, for a ∈ ℝ, [a] denotes the maximal integer ≤ a, the endomorphisms ∇_{i}^{log} on E are defined by the formula ∇(y) = ∑_{1≤i≤d} ∇_{i}^{log}(y)d log(t_{i}) (y ∈ E), and E is endowed with the p-adic topology induced by a finitely generated A-submodule E^{0} of E such that K ⊗ E^{0} = E, which is independent of the choice of E^{0}.

• an isomorphism Φ: A_{φ} ⊗_{A} E → E which is horizontal with respect to the connection ∇.

If you choose another lifting of Frobenius F′_{X}, then the integrable connection induces a canonical isomorphism:

A_{φ} ⊗_{A} E ≅ A_{φ′} ⊗_{A} E, (1.1)

which sends 1 ⊗ x to

∑_{(n_{1},... ,n_{d})∈ℕ^{d}}(n_{1}! ⋯ n_{d}!)^{−1}( ∏_{1≤i≤d}(φ(t_{i})φ′(t_{i})^{−1}−1)^{n_{i}})⊗( ∏_{1≤i≤d} ∏_{0≤j<n_{i}}(∇_{i}^{log}−j))(x).

Here we need the convergence of ∇. Hence the definition of Φ is independent of the choice of F_{X}. The convergence condition on the connection ∇ is independent of the choice of t_{1}, t_{2}, …, t_{d}, but the author does not know a direct proof without using enlargements which involves the theory of log structures.

For a general (X, M), there exists an étale covering {X_{α} → X} such that each X_{α} satisfies the above conditions. If we choose such an étale covering, then giving a filtered convergent F -isocrystal on (X, M) is equivalent to giving a data (E_{α}, F il_{α}^{i}, ∇_{α}, Φ_{α}) for each (X_{α}, M_{α}) and a descent data in the obvious sense.

In the case that K is ramified i.e. K ≠ K_{0}, to describe filtered convergent F - isocrystals similarly, we need to choose a lifting (X′, D′) of (X ⊗ k, D ⊗ k) over O_{K_{0}} and an isomorphism (X, D) ≅ (X′, D′) ⊗_{O_{K_{0}}} O_{K}. Once we choose them, the description is almost the same as in the case K = K_{0}. We denote by

MF_{(X,M)/K}^{∇}(Φ)

the category of filtered convergent F -isocrystals on X with the log structure M defined by D.

2. Syntomic cohomology

The syntomic cohomology with constant p-torsion coefficients was introduced by J.-M. Fontaine and W. Messing ([FM87]) as the cohomology groups of some sheaves called S_{n}^{r} on the syntomic sites in their proof of the crystalline conjecture, which compares the p-adic étale cohomology and the crystalline cohomology for a proper smooth scheme over O_{K}. The projections of S_{n}^{r} on the étale site, which are sometimes called the syntomic complexes, can be described explicitly in terms of the de Rham complexes with coefficients in some PD-envelopes and the Frobenius endomorphisms on them; the syntomic cohomology is given as the étale cohomology of the syntomic complex. This construction using de Rham complexes can be generalized to non-constant coefficients. Such a generalization is studied by W. Niziol ([Niz97], [Niz]) and K. Bannai ([Ban00b]).

Assume K = K_{0} and let (X, M) and D be as in §1. First assume X is affine and (X, M) has a log coordinate globally as in §1. Choose a lifting of Frobenius F_{X} on (X, M) and denote F_{X}^{∗} on A and Ω_{A}^{q}(log) by φ. In this case, for a filtered convergent F -isocrystal E corresponding to (E, F il^{i}, ∇, Φ), we define the de Rham complex DR_{X,F_{X}} ((X, M), E) to be E ⊗_{A} Ω_{A}^{•}(log). We define the filtration F il^{i} (i ∈ ℤ) on it by F il^{i}(E ⊗_{A} Ω_{A}^{q}(log)) = F il^{i−q}E ⊗_{A} Ω_{A}^{q}(log). The isomorphism Φ induces a semi-linear endomorphism on E, which we denote by φ. Note that φ depends on the choice of the lifting of Frobenius F_{X}. Since Φ is horizontal, φ ⊗ φ defines an endomorphism on DR_{X,F_{X}} ((X, M), E), which we again denote by φ.

We define the complex S_{X,F_{X}} ((X, M), E) to be the mapping fiber of

1 − φ: F il^{0}(DR_{X,F_{X}} ((X, M), E)) ⟶ DR_{X,F_{X}} ((X, M), E).

The degree q-part of the complex S_{X,F_{X}} ((X, M), E) is the direct sum:

(F il^{−q}E) ⊗_{A} Ω_{A}^{q}(log) ⨁ E ⊗_{A} Ω_{A}^{q−1}(log)

and the differential map is given by

d(x, y) = (∇(x), (1 − φ)(x) − ∇(y)).

We define the crystalline cohomology and the syntomic cohomology of E by

H_{crys}^{m}((X, M), E) := H^{m}(DR_{X,F_{X}} ((X, M), E)),
H_{syn}^{m}((X, M), E) := H^{m}(S_{X,F_{X}} ((X, M), E)).

The endomorphism φ on the de Rham complex induces a semi-linear endomorphism φ of the crystalline cohomology. A priori, the definitions of the syntomic cohomology and φ on the crystalline cohomology depend on the choice of the lifting of Frobenius F_{X}. To prove the independence, we need a more general construction which will be explained later.

If we define F il^{i}H_{crys}^{m}((X, M), E) to be H^{m}(F il^{i}(DR_{X}((X, M), E))), which may not be a submodule of H_{crys}^{m}((X, M), E), we have a long exact sequence:

(2.1) → H_{syn}^{m}((X, M), E) → F il^{0}H_{crys}^{m}((X, M), E) ⟶^{1−φ} H_{crys}^{m}((X, M), E) → · · · .

For m = 1, we have the following interpretation of syntomic cohomology, which generalize [BK90] Lemma 4.4.

Proposition 2.2 (cf. [Ban00a] §4). There exists a canonical isomorphism:

H_{syn}^{1}((X, M), E) ≅ Ext_{MF_{(X,M)/K}^{∇}(Φ)}^{1}(O, E),

where O denotes the constant filtered convergent F -isocrystal, which corresponds to K ⊗ A with F il^{i}(K ⊗ A) = K ⊗ A (i ≤ 0), 0 (i > 0) and the obvious ∇ and Φ.

Proof. We only give a construction of the map from the LHS to the RHS. The proof of the isomorphism is straightforward and is left to the reader. Let x ∈ (F il^{−1}E) ⊗_{A} Ω_{A}(log) and y ∈ E and assume (1 − φ)(x) = ∇(y) and ∇(x) = 0. Then the extension corresponding to the class of the cocycle (x, y) is defined to be F := (K ⊗ A) ⊕ E with F il^{i} = F il^{i} ⊕ F il^{i}, ∇(1, 0) = (0, x) and Φ(1, 0) = (1, −y). The integrability of ∇ on F is equivalent to ∇(x) = 0 and the condition that Φ on F is horizontal is equivalent to (1 − φ)(x) = ∇(y).

For a general (X, M), the construction becomes more complicated because a lifting of Frobenius does not exist globally in general. We choose an étale hypercovering X_{•} → X, a closed immersion of (X_{•}, M|_{X_{•}}) into a simplicial smooth fine log formal scheme (Y_{•}, N_{•}) over Spf(O_{K}) and a lifting of Frobenius F_{Y_{•}} of (Y_{•}, N_{•}) such that, for each i ∈ ℕ, X_{i} and Y_{i} are affine. Such a hypercovering, a closed immersion and a lifting of Frobenius always exist. We consider the “closed tube of radius p^{− \frac{1}{n}} ” (Z_{•}^{(n)}, L_{•}^{(n)}) of (X_{•}, M|_{X_{•}}) in (Y_{•}, N_{•}). Let A_{i}, B_{i} and C_{i}^{(n)} denote the rings of coordinates of X_{i}, Y_{i} and Z_{i}^{(n)} respectively. If M|_{X_{i}} is the pull-back of N_{i}, then C_{i}^{(n)} is the p-adic completion of B_{i}[\frac{(I_{i})^{n}}{p} ], where I_{i} denotes the kernel

of B_{i} → A_{i}, and L_{i}^{(n)} is the pull-back of N_{i}. The log formal scheme (Z_{i}^{(n)}, L_{i}^{(n)}) can be naturally regarded as an enlargement of (X, M) over Spf(O_{K}). Hence one can evaluate a filtered F -isocrystal E on it and obtains a K ⊗ C_{i}^{(n)}-module E_{i}^{(n)} endowed with an integrable connection, a filtration and a Frobenius. We define the de Rham complex DR_{X_{•},Y_{•},F_{Y•}} ((X, M), E) to be the projective limit with respect to n of the complex associated to the double complex:

E_{0}^{(n)} ⊗_{B_{0}^{(n)}} Ω_{B_{0}^{(n)}}^{•}(log) ⟶ E_{1}^{(n)} ⊗_{B_{1}^{(n)}} Ω_{B_{1}^{(n)}}^{•}(log) ⟶ E_{2}^{(n)} ⊗_{B_{2}^{(n)}} Ω_{B_{2}^{(n)}}^{•}(log) ⟶ ⋯,

which is filtered and has a Frobenius endomorphism φ. The differential maps are, as usual, defined to be alternating sums of the pull-backs by the projections. We define the syntomic complex S_{X_{•},Y_{•},F_{Y•}} ((X, M), E) to be the mapping fiber of

1 − φ: F il^{0}(DR_{X_{•},Y_{•},F_{Y•}} ((X, M), E)) ⟶ DR_{X_{•},Y_{•},F_{Y•}} ((X, M), E).

The image of the syntomic complex in the derived category D^{+}(ℚ_{p}-Vect) of the category ℚ_{p}-Vect of ℚ_{p}-vector spaces is unique up to canonical isomorphisms: For another (X′_{•}, M|_{X_{•}}) ↪ (Y′_{•}, N′_{•}), F_{Y′_{•}}, we take the fiber product X′′_{•} of X_{•} and X′_{•} over X and the fiber product (Y′′_{•}, N′′_{•}) of (Y_{•}, N_{•}) and (Y′_{•}, N′_{•}) over Spf(O_{K}). Then the natural morphisms from the syntomic complexes for (X_{•}, Y_{•}) and (X′_{•}, Y′_{•}) to that for (X′′_{•}, Y′′_{•}) are both quasi-isomorphisms. Here we use the Poincaré lemma and the vanishing of R^{1} \underleftarrow{lim}_{n}E_{i}^{(n)}. For the Poincaré lemma holds, we need to work for an “open tube” and that is the reason why we take the projective limit with respect to n in the definition of the de Rham and syntomic complexes. The above argument also works for the de Rham complex with F il^{•} and φ.

We define the crystalline cohomology and the syntomic cohomology by

H_{crys}^{m}((X, M), E) := H^{m}(DR_{X_{•},Y_{•},F_{Y•}} ((X, M), E))
H_{syn}^{m}((X, M), E) := H^{m}(S_{X_{•},Y_{•},F_{Y•}} ((X, M), E)).

Proposition 2.2 is still valid for a general (X, M). If we define F il^{i}H_{crys}^{m}((X, M), E) to be the cohomology of F il^{i} of the de Rham complex, then we have the exact sequence (2.1).

Assume (X, M) is the p-adic formal completion of a proper smooth scheme X over Spec(O_{K}) with the log structure defined by a divisor with relative normal crossings. Set X_{K} := X ×_{Spec(O_{K})} Spec(K). Then a filtered convergent F -isocrystal on (X, M) defines a locally free O_{X_{K}} -module E endowed with an integrable connection with log poles ∇ and a decreasing filtration F il^{•} by direct factors, and we have canonical isomorphisms:

H_{crys}^{m}((X, M), E) ≅ H^{m}(X_{K}, E ⊗_{O_{X_{K}}} Ω_{X_{K}}^{•} (log)),
F il^{i}H_{crys}^{m}((X, M), E) ≅ H^{m}(X_{K}, F il^{i−•}E ⊗_{O_{X_{K}}} Ω_{X_{K}}^{•} (log)).

Hence these are finite dimensional K-vector spaces and F il^{i} = 0 (i >> 0). Furthermore, if the spectral sequence with respect to the filtration F il^{i−•}E ⊗ Ω_{X}^{•} (log) degenerates at E_{1}, then the natural map F il^{i}H_{crys}^{m}((X, M), E) → H_{crys}^{m}((X, M), E) (i ∈ ℤ) is injective and the crystalline cohomology can be regarded as a filtered φ-module (= a filtered convergent F -isocrystal on Spec(O_{K}) over K). In this case, the long exact sequence (2.1) is rewritten as short exact sequences:

(2.3) 0 ⟶ H_{syn}^{1}(O_{K}, H_{crys}^{m−1}((X, M), E)) ⟶ H_{syn}^{m}((X, M), E)

⟶ H_{syn}^{0}(O_{K}, H_{crys}^{m}((X, M), E)) → 0.

We can define a syntomic cohomology even over a ramified base similarly noting that giving a filtration on a convergent crystal E on X/K_{0} is equivalent to giving a filtration on the inverse image of E to X/K. In [Niz], W. Niziol gave a different construction in the case D = ∅ generalizing the method of of Bloch-Kato in [BK90] §3, where they treat the case X = Spec(O_{K}).

3. Crystalline sheaves

Let X be a proper smooth scheme over Spec(O_{K}) and let M be the log structure on X defined by a divisor D with relative normal crossings. In [Fal89] V f), G. Faltings defined the notion “crystalline” for smooth ℚ_{p}-sheaves on the étale site of X_{triv} := X_{K}\D_{K} generalizing the Fontaine’s theory of crystalline p-adic representations of Gal(\overline{K}/K) ([Fon82], [Fon94a], [Fon94b]). Similarly as crystalline p-adic representations, we have a fully-faithful functor from the category of crystalline sheaves to the category of filtered convergent F -isocrystals on (X, M) over K. Note that, in the case X = Spec(O_{K}), the latter category coincides with the category MF_{K}(φ) of filtered φ-modules over K considered by Fontaine.

In this section, we will explain crystalline sheaves in the case K = K_{0}. Our approach is slightly different from Faltings’; we use the description of filtered convergent F -isocrystals as modules with connections, which is reviewed in §1, while Faltings consider their values on every enlargement.

Assume K = K_{0}. Let X be a smooth affine formal scheme over Spf(O_{K}) endowed with the fine log structure M defined by a divisor D with relative normal crossings. We set A = Γ(X, O_{X}). We assume that (X, M) has a log coordinate t_{1}, t_{2}, …, t_{d} ∈ A globally as in §1. We choose a lifting of Frobenius F_{X} of the log formal scheme (X, M). We further assume that X is connected and hence A is a regular domain. Choose an algebraic closure \overline{Frac(A)} of the field of fractions Frac(A) of A, and define \overline{A} to be the union of the normalization of A in L, where L ranges over all finite extensions of Frac(A) contained in \overline{Frac(A)} such that the normalizations of A[(pt_{1}t_{2} ⋯ t_{d})^{−1}] in L are étale. We set G := Gal(Frac(\overline{A})/Frac(A)), which is canonically isomorphic to the fundamental group of Spec(A[(pt_{1}t_{2} ⋯ t_{d})^{−1}]) for the base point Spec(\overline{Frac(A)}).

Associated to the ring \overline{A} with the natural action of G and (X, M), F_{X}, there exists a K ⊗ A-algebra B_{crys}(\overline{A}) endowed with the following structures:

• an action of G.

• a decreasing filtration F il^{i}B_{crys}(\overline{A}) (i ∈ ℤ) by K ⊗ A-submodules such that ∪_{i}F il^{i}B_{crys}(\overline{A}) = B_{crys}(\overline{A}) and ∩_{i}F il^{i}B_{crys}(\overline{A}) = 0.

• an integrable connection: ∇: B_{crys}(\overline{A}) → B_{crys}(\overline{A})⊗_{A} Ω_{A}(log) satisfying Griffiths transversality: ∇(F il^{i}) ⊂ F il^{i−1} ⊗_{A} Ω_{A}(log) (i ∈ ℤ).

• an injective endomorphism (called the Frobenius) φ: B_{crys}(\overline{A}) → B_{crys}(\overline{A}) horizontal with respect to the connection and semi-linear with respect to F_{X}.

The filtration is stable under the action of G and the connection and the Frobenius commute with the action of G. If we set B_{crys}(\overline{A}) := B_{crys}(\overline{A})^{∇=0} and endow it with the induced filtration, then ℚ_{p}(i) (i ∈ ℤ) is contained in F il^{i}B_{crys}(\overline{A}) and

gr^{i}B_{crys}(\overline{A}) is canonically embedded into (K ⊗_{O_{K}} ˆ \overline{A})(i). If we use the surjectivity of the absolute Frobenius of \overline{A}/p\overline{A}, which follows from of the theory of almost étale extensions by Faltings, then we can prove that this embedding is an isomorphism.

The ring B_{crys}(\overline{A}) is defined along the following lines. We first define the ring R\overline{A} to be the perfection of \overline{A}/p\overline{A} i.e. the projective limit of \overline{A}/p\overline{A} ← \overline{A}/p\overline{A} ← \overline{A}/p\overline{A} ← ⋯ where the transition maps are the absolute Frobenius. We have a canonical embedding ℤ_{p}(1)(\overline{A}) ↪ R∗ ; \overline{A} sending ε = (ε_{n})_{n≥0}, ε_{0} = 1, ε_{n+1}^{p} = ε_{n} to \underline{ε} := (ε_{n} mod p)_{n≥0}. We define the ring A_{crys}(\overline{A}) to be the p-adic completion of some subring of ℚ_{p} ⊗ (A ⊗_{O_{K}} W (R\overline{A})) containing A ⊗ W (R\overline{A}). For any ε ∈ ℤ_{p}(1), log(1 ⊗ [\underline{ε}] − 1) converges in A_{crys} p-adically and we obtain a canonical embedding ℤ_{p}(1) ↪ A_{crys}. We regard ℤ_{p}(1) as a submodule of A_{crys}. The ring B_{crys}(\overline{A}) is defined to be A_{crys}[p^{−1}, ε^{−1}], where ε is any non-zero element of ℤ_{p}(1).

The following two properties are important.

Proposition 3.1. (1) The G-invariant part of B_{crys}(\overline{A}) is K ⊗_{O_{K}} A.

(2) ℚ_{p} with the trivial action of G is quasi-isomorphic to the mapping fiber of the morphism of complexes of G-modules:

1 − φ ⊗ φ: F il^{0}(B_{crys}(\overline{A}) ⊗_{A} Ω_{A}^{•}(log)) → B_{crys}(\overline{A}) ⊗_{A} Ω_{A}^{•}(log).

Here the degree q-part of F il^{0} is defined to be F il^{−q}B_{crys}(\overline{A}) ⊗_{A} Ω_{A}^{q}(log).

Let V be a p-adic representation of G, i.e. a finite dimensional ℚ_{p}-vector space endowed with a continuous and linear action of G, which corresponds to a smooth ℚ_{p}-sheaf on the étale site of Spec(A[(pt_{1}t_{2} ⋯ t_{d})^{−1}]). Similarly as the Fontaine’s theory, we define D_{crys}(V) and a filtration on it by

D_{crys}(V) := (B_{crys}(\overline{A}) ⊗_{ℚ_{p}} V)^{G},
F il^{i}D_{crys}(V) := (F il^{i}B_{crys}(\overline{A}) ⊗_{ℚ_{p}} V)^{G} (i ∈ ℤ).

Here an element g ∈ G acts on B_{crys}(\overline{A}) ⊗_{ℚ_{p}} V by g ⊗ g. The connection and the Frobenius on B_{crys}(\overline{A}) induce those on D_{crys}(V) and they satisfy ∇(F il^{i}) ⊂ F il^{i−1} ⊗_{A} Ω_{A}(log) and ∇ ∘ φ = (φ ⊗ φ) ∘ ∇.

Definition 3.2. We say that V is crystalline if it satisfies the following two conditions:

(1) The canonical map B_{crys}(\overline{A}) ⊗_{K⊗A} D_{crys}(V) → B_{crys}(\overline{A}) ⊗_{ℚ_{p}} V is a filtered isomorphism.

(2) D_{crys}(V) and gr^{i}D_{crys}(V) (i ∈ ℤ) are finite projective K ⊗_{O_{K}} A-modules.

If we use the flatness of K ⊗_{O_{K}} ˆ \overline{A} (where ˆ \overline{A} denotes the p-adic completion of \overline{A}) over K ⊗_{O_{K}} A, which follows from the theory of almost étale extensions by Faltings, then we can prove (1) implies (2). However we avoid to use this fact here because it is very hard to understand all the details of the Faltings’ theory.

If V is crystalline, then (D_{crys}(V), F il^{i}D_{crys}(V), ∇, φ) defines a filtered convergent F -isocrystal on (X, M) over K, which we also denote by D_{crys}(V).

Theorem 3.3 (G. Faltings [Fal89] Lemma 5.5). The functor

D_{crys} : Rep_{crys}(G) ⟶ MF_{(X,M)/K}^{∇}(Φ)

from the category of crystalline representations of G to the category of filtered convergent F -isocrystals is fully faithful and exact. Furthermore the essential image is stable under extensions.

Proof of fully faithfulness. By Proposition 3.1 (2), we have

V = F il^{0}(B_{crys}(\overline{A}) ⊗_{K⊗A} D_{crys}(V))^{∇=0,φ=1},

which immediately implies the fully faithfulness.

Let V be a crystalline p-adic representation of G and let E be the corresponding filtered convergent F -isocrystal on (X, M)/K. Then we can construct canonical homomorphisms:

H_{syn}^{m}((X, M), E) ⟶ H_{cont}^{m}(G, V) (m ∈ ℤ) (3.4)

functorial on V as follows. See [Niz] for the case D = ∅. By tensoring V to the resolution of ℚ_{p} of Proposition 3.1 (2) and using the filtered isomorphism in Definition 3.2 (1), we obtain the resolution of V by the mapping fiber \overline{S}_{X,F_{X}} ((X, M), E) of the morphism of complexes:

1 − φ: F il^{0}(B_{crys}(\overline{A}) ⊗_{K⊗A} E ⊗_{A} Ω_{A}^{•}(log)) ⟶ B_{crys}(\overline{A}) ⊗_{K⊗A} E ⊗_{A} Ω_{A}^{•}(log).

We define the homomorphisms (3.4) by taking the cohomology of the following maps:

S_{X,F_{X}} ((X, M), E) → Γ(G, \overline{S}_{X,F_{X}} ((X, M), E))

→ “RΓ_{cont}(G, \overline{S}_{X,F_{X}} ((X, M), E))” ←^{∼} RΓ_{cont}(G, V)

The ring B_{crys}(\overline{A}) can be described as a union ∪_{m≥0}t^{−m}A_{crys}(\overline{A}) of p-adically complete and separated G-stable submodules t^{−m}A_{crys}(\overline{A}) (where 0 ≠ t ∈ ℤ_{p}(1) ⊂ A_{crys}(\overline{A})) and it induces a description of \overline{S}_{X,F_{X}} ((X, M), E) as a union of complexes of G-modules S_{m} (m ∈ ℕ) p-adically complete and separated. The third term “RΓ_{cont} … ” is defined to be the inductive limit of the inhomogeneous continuous cochain complexes of S_{m} with the p-adic topology.

Lemma 3.5. The homomorphism (3.4) for m = 1 coincides with the map:

H_{syn}^{1}((X, M), E)) ≅ Ext_{MF_{(X,M)/K}^{∇}(Φ)}^{1}(O, E) → Ext_{Rep_{ℚp} (G)}^{1}(ℚ_{p}, V) ≅ H_{cont}^{1}(G, V)

induced by D_{crys}.

Proof. Choose x ∈ F il^{−1}E ⊗_{A}Ω_{A}(log) and y ∈ E and assume (1−φ)(x) = ∇(y) and ∇(y) = 0. Then there exists a ∈ F il^{0}(B_{crys} ⊗_{K⊗A} E) ≅ F il^{0}(B_{crys} ⊗_{ℚ_{p}} V) unique up to addition by an element of V such that y = (1−φ)(a) and x = ∇(a) and the map (3.4) sends the class of (x, y) to the class of the cocycle G → V; g ↦ −(g(a)−a). On the other hand, if we define the extension F of O by E as in the proof of Proposition 2.2, then the element (1, −a) ∈ B_{crys} ⊗ F = B_{crys} ⊕ (B_{crys} ⊗ E) is contained in the extension of ℚ_{p} by V corresponding to F.

We still assume K = K_{0} and let X be a proper smooth scheme over Spec(O_{K}) endowed with the fine log structure M defined by a divisor D with relative normal crossings. Set X_{triv} := X_{K}\D_{K}. For a ℚ_{p}-smooth sheaf V on (X_{triv})_{ét}, we say that V is crystalline if, for each affine étale scheme U = Spec(A) of X having a global log coordinate t_{1}, t_{2}, …, t_{d}, the pull-back of V on Spec(Â[(pt_{1}t_{2} ⋯ t_{d})^{−1}]) is crystalline. Here Â denotes the p-adic completion of A. By gluing the functor D_{crys} for Spf(Â) with the inverse image of M, we obtain a functor D_{crys} from the category of crystalline sheaves on (X_{triv})_{ét} to the category MF_{(X,M)/K}^{∇}(Φ) of filtered convergent F -isocrystals on (X, M)/K and Theorem 3.3 holds. By taking X_{•}, (X_{•}, M|_{X_{•}}) ↪ (Y_{•}, N_{•}), F_{Y_{•}} as in the definition of the syntomic cohomology in the general case and using B_{crys} associated to these data, one can construct canonical homomorphisms:

(3.6) H_{syn}^{m}((X, M), D_{crys}(V)) → H_{ét}^{m}(X_{triv}, V)

similarly as (3.4). Here the RHS is the continuous étale cohomology ([Jan88]). See [Niz] in the case D = ∅. Lemma 3.5 is still valid.

4. Log

In this section, we review the realizations of the motivic pro-sheaf Log = (Log^{(n)})_{n≥1} on 𝔾_{m,ℚ} = ℙ_{ℚ}^{1}\{0, ∞} in the category of smooth ℚ_{p}-sheaves on (𝔾_{m,ℚ_{p}})_{ét} and the category of filtered convergent F -isocrystals on ℙ_{ℤ_{p}}^{1} with the log structure defined by {0, ∞}. The realization in filtered overconvergent F -isocrystals on 𝔾_{m,ℤ_{p}} was studied by K. Bannai in [Ban00a], but from his explicit description, it is clear that it actually comes from log filtered convergent F -isocrystals.

Let 𝔾_{m,ℤ_{p}}^{log} be ℙ_{ℤ_{p}}^{1} endowed with the fine log structure define by {0, ∞}. We denote by K(r) the object of MF_{ℚ_{p}}(φ) i.e. the filtered φ-module over ℚ_{p} corresponding to the crystalline p-adic representation ℚ_{p}(r) (r ∈ ℤ) and by O(r) its inverse image in MF_{𝔾_{m,ℤp}^{log} /ℚ_{p}}^{∇}(Φ). Then we have the following exact sequences. See

(2.3) for the syntomic case.

(4.1) 0 ⟶ H_{ét}^{1}(ℚ_{p}, ℚ_{p}(1)) ⟶^{(∗)} H_{ét}^{1}(𝔾_{m,ℚ_{p}}, ℚ_{p}(1)) ⟶ H^{0}(ℚ_{p}, H_{ét}^{1}(𝔾_{m,\overline{ℚ_{p}}}, ℚ_{p}(1)))

⟶ H_{ét}^{2}(ℚ_{p}, ℚ_{p}(1)) ⟶^{(∗)} H_{ét}^{2}(𝔾_{m,ℚ_{p}}, ℚ_{p}(1)),

0 → H_{syn}^{1}(ℤ_{p}, K(1)) ⟶^{(∗)} H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, O(1)) → H_{syn}^{0}(ℤ_{p}, H_{crys}^{1}(𝔾_{m,ℤ_{p}}^{log}, O(1))) → 0. (4.2)

Here H_{ét}^{∗} denotes the continuous étale cohomology ([Jan88]). The pull-backs by the point 1 give the splittings of the homomorphisms (∗). On the other hand the third terms of the above exact sequences are both canonically isomorphic to ℚ_{p}. Thus we obtain isomorphisms:

(4.3)  H_{ét}^{1}(𝔾_{m,ℚ_{p}}, ℚ_{p}(1)) ≅ H_{ét}^{1}(ℚ_{p}, ℚ_{p}(1)) ⊕ ℚ_{p},
(4.4)  H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, O(1)) ≅ H_{syn}^{1}(ℤ_{p}, K(1)) ⊕ ℚ_{p}.

We define the smooth ℚ_{p}-sheaf Log_{ét}^{(1)} on (𝔾_{m,ℚ_{p}})_{ét} and the filtered convergent F -isocrystal Log_{crys}^{(1)} on 𝔾_{m,ℤ_{p}}^{log} to be the extensions:

0 ⟶ ℚ_{p}(1) ⟶ Log_{ét}^{(1)} ⟶ ℚ_{p} ⟶ 0,
0 ⟶ O(1) ⟶ Log_{crys}^{(1)} ⟶ O ⟶ 0

corresponding to the elements (0, 1) via the above isomorphisms. See Proposition 2.2 for the interpretation of the first syntomic cohomology as Ext^{1}. The specializations of these extensions at the point 1 split.

We define Log_{ét}^{(n)} and Log_{crys}^{(n)} to be the n symmetric tensors of Log_{ét}^{(1)} and Log_{crys}^{(1)} respectively. Let pr denote the projections from Log_{ét}^{(1)} to ℚ_{p} and Log_{crys}^{(1)} to O and let pr_{i}: (Log_{•}^{(1)})^{⊗n} → (Log_{•}^{(1)})^{⊗(n−1)} (1 ≤ i ≤ n) be the tensor product of pr on the i-th term and id on the other terms. Then the sum ∑_{i=1}^{n} pr_{i} induces Log_{•}^{(n)} → Log_{•}^{(n−1)}. We have short exact sequences:

(4.5) 0 ⟶ ℚ_{p}(n) ⟶ Log_{ét}^{(n)} ⟶ Log_{ét}^{(n−1)} ⟶ 0,

(4.6) 0 ⟶ O(n) ⟶ Log_{crys}^{(n)} ⟶ Log_{crys}^{(n−1)} ⟶ 0.

Proposition 4.7. Let d be a positive integer. Then the pull-back of Log_{ét}^{(n)} by Spec(ℚ_{p}(μ_{d})) = Spec(ℚ_{p}[t]/Φ_{d}(t)) ↪ 𝔾_{m,ℚ_{p}} is canonically isomorphic to ⊕_{0≤r≤n}ℚ_{p}(r), and the pull-back of Log_{crys}^{(n)} by Spec(ℤ_{p}[μ_{d}]) = Spec(ℤ_{p}[t]/Φ_{d}(t)) ↪ 𝔾_{m,ℤ_{p}}^{log} is canonically isomorphic to ⊕_{0≤r≤n}ℚ_{p}(μ_{d})(r).

Proof. It suffices to prove the claim in the case n = 1. Let [d] denote the multiplication by d on 𝔾_{m,ℚ_{p}}. Then the pull-back by [d] preserves the decomposition (4.3) and it induces a multiplication by d on the factor ℚ_{p}. Hence [d]^{∗}(Log_{ét}^{(1)}) is isomorphic to the fiber product of Log_{ét}^{(1)} → ℚ_{p} ←^{d}_{∼} ℚ_{p}. Since [d]^{∗}(Log_{ét}^{(1)}) splits at

Spec(ℚ_{p}(μ_{d})) ↪ 𝔾_{m,ℚ_{p}}, so does Log_{ét}^{(1)}. If d is prime to p, [d] induces a morphism of log schemes 𝔾_{m,ℤ_{p}}^{log} → 𝔾_{m,ℤ_{p}}^{log} and we can prove the second claim similarly.

Proposition 4.8. Log_{ét}^{(n)} is crystalline and D_{crys}(Log_{ét}^{(n)}) is canonically isomorphic to Log_{crys}^{(n)}.

Proof. It suffices to prove in the case n = 1. We have a commutative diagram:

Γ(𝔾_{m,ℤ_{p}}, O^{∗}) ⟶ H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, O(1))
∥  ↓^{∩}
Γ(𝔾_{m,ℤ_{p}}, O^{∗}) ⟶ H_{ét}^{1}(𝔾_{m,ℚ_{p}}, ℚ_{p}(1))

and the images of the canonical coordinate t under the horizontal maps are the extensions Log_{ét}^{(1)} and Log_{crys}^{(1)}. Locally on 𝔾_{m,ℤ_{p}}^{log}, the upper horizontal map is defined by f ↦ (df, p^{−1} log(f^{p}φ(f)^{−1}))

If we define the lifting of Frobenius on the log scheme 𝔾_{m,ℤ_{p}}^{log} by t ↦ t^{p}, then

using the construction of Log_{crys}^{(1)} in the proof of Proposition 4.8, we obtain the following explicit description of Log_{crys}^{(n)} ([Ban00a] Definition 5.1):

Log_{crys}^{(n)} = ⨁_{0≤r≤n} O · e_{r},
(4.9)  F il^{i}Log_{crys}^{(n)} = ⨁_{i≤−r,0≤r≤n} O · e_{r},

∇(e_{r}) = \begin{cases}e_{r+1} · \frac{dt}{t} & (0 ≤ r ≤ n − 1), \\ 0 & (r = n), \end{cases}
Φ(1 ⊗ e_{r}) = p^{−r}e_{r}.

5. Pol

In this section, we review the realizations of the motivic polylogarithmic prosheaf Pol = (Pol^{(n)})_{n≥1} on ℙ_{ℚ}^{1}\{0, 1, ∞} in the category of smooth ℚ_{p}-sheaves on (ℙ_{ℚ_{p}}\{0, 1, ∞})_{ét} and the category of filtered convergent F -isocrystals on ℙ_{ℤ_{p}}^{1} with the log structure defined by {0, 1, ∞}. The realization in the category of filtered overconvergent F -isocrystals on ℙ_{ℤ_{p}}^{1}\{0, 1, ∞} is constructed by K. Bannai in [Ban00a] and as we will see below, his construction works also for filtered convergent F -isocrystals and syntomic cohomology with log poles. As in [Ban00a], we follow the construction by A. Huber and J. Wildeshaus in [HW98]§2.

Let U denote the scheme ℙ_{ℚ_{p}}^{1}\{0, 1, ∞} and let U^{log} be the scheme ℙ_{ℤ_{p}}^{1} endowed with the log structure defined by the divisor {0, 1, ∞}. The pro-sheaf Pol_{ét} (resp. Pol_{crys}) is a compatible system of an extension of ℚ_{p} (resp. O) by the pullback of Log_{ét}^{(n)} (resp. Log_{crys}^{(n)}) on U (resp. U^{log}) or equivalently a compatible system of a class in H_{ét}^{1}(U, Log_{ét}^{(n)}|_{U}) (resp. H_{syn}^{1}(U^{log}, Log_{crys}^{(n)}|_{U^{log}})). See Proposition 2.2 for the second case. To simplify the notation, we will denote the pull-back of Log_{ét}^{(n)} and Log_{crys}^{(n)} on U and U^{log} by the same symbols.

Let us consider the Gysin exact sequences:

(5.1) 0 → H_{ét}^{1}(𝔾_{m,ℚ_{p}}, Log_{ét}^{(n)}) → H_{ét}^{1}(U, Log_{ét}^{(n)})

→ H_{ét}^{0}(ℚ_{p}, i_{1}^{∗}(Log_{ét}^{(n)}(−1))) → H_{ét}^{2}(𝔾_{m,ℚ_{p}}, Log_{crys}^{(n)})

(5.2) 0 → H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) → H_{syn}^{1}(U^{log}, Log_{crys}^{(n)})

→ H_{syn}^{0}(ℤ_{p}, i_{1}^{∗}(Log_{crys}^{(n)}(−1))) → H_{syn}^{2}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)})

for the unit sections i_{1} : Spec(ℚ_{p}) ↪ 𝔾_{m,ℚ_{p}} and i_{1} : Spec(ℤ_{p}) ↪ 𝔾_{m,ℤ_{p}}^{log}. Since

i_{1}^{∗}(Log_{ét}^{(n)}) and i_{1}^{∗}(Log_{crys}^{(n)}) are canonically isomorphic to ⨁_{0≤r≤n} ℚ_{p}(r) and ⨁_{0≤r≤n} K(r) respectively by definition, the third terms of the above exact sequences are canonically isomorphic to ℚ_{p}. On the other hand, the first and the fourth terms have the following properties.

Proposition 5.3. The homomorphisms induced by the projections:

H_{ét}^{1}(𝔾_{m,ℚ_{p}}, Log_{ét}^{(n+1)}) → H_{ét}^{1}(𝔾_{m,ℚ_{p}}, Log_{ét}^{(n)}),
H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n+1)}) → H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)})

are 0 for all n ≥ 1 and we have

H_{ét}^{2}(𝔾_{m,ℚ_{p}}, Log_{ét}^{(n)}) = 0 (n ≥ 2),
H_{syn}^{2}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) = 0 (n ≥ 1).

To prove this proposition, we first calculate the geometric étale cohomology of Log_{ét}^{(n)} and the crystalline cohomology of Log_{crys}^{(n)}.

Proposition 5.4. (1) The natural homomorphisms

ℚ_{p}(n) = H_{ét}^{0}(𝔾_{m,\overline{ℚ_{p}}}, ℚ_{p}(n)) → H_{ét}^{0}(𝔾_{m,\overline{ℚ_{p}}}, Log_{ét}^{(n)})
H_{ét}^{1}(𝔾_{m,\overline{ℚ_{p}}}, Log_{ét}^{(n)}) → H_{ét}^{1}(𝔾_{m,\overline{ℚ_{p}}}, ℚ_{p}) ≅ ℚ_{p}(−1)

are isomorphisms and

H_{ét}^{i}(𝔾_{m,\overline{ℚ_{p}}}, Log_{ét}^{(n)}) = 0 (i ≥ 2)

for all n ≥ 1.

(2) The Hodge spectral sequence for H_{crys}^{∗}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) degenerates at E_{1} and these cohomology groups can be naturally regard as filtered φ-modules over ℚ_{p} in the sense of Fontaine. Furthermore, the natural homomorphisms of filtered φ-modules:

K(n) = H_{crys}^{0}(𝔾_{m,ℤ_{p}}^{log}, K(n)) → H_{crys}^{0}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)})
H_{crys}^{1}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) → H_{crys}^{1}(𝔾_{m,ℤ_{p}}^{log}, K) ≅ K(−1)

are isomorphisms and

H_{syn}^{i}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) = 0 (i ≥ 2)

for all n ≥ 1.

Proof. We will give a proof of (2). The proof of (1) is similar. It is well-known that the Hodge spectral sequence for the de Rham cohomology of ℙ_{ℚ_{p}}^{1} with log poles along {0, 1, ∞} degenerates at E_{1}. The crystalline cohomology H_{crys}^{i}(𝔾_{m,ℤ_{p}}^{log}, O) is isomorphic to K if i = 0, K(−1) if i = 1, and 0 if i ≥ 2 as filtered φ-modules. By definition, the class of Log_{crys}^{(1)} in H_{crys}^{1}(𝔾_{m,ℤ_{p}}^{log}, O) is non-trivial. Hence, by looking at the long exact sequence of crystalline cohomology associated to the extension, we obtain the claim for n = 1. One can prove the claim for the general n by induction using the commutative diagram

0 ⟶ O(n) ⟶  Log_{crys}^{(n)}  ⟶ Log_{crys}^{(n−1)} ⟶ 0
∥  ↑^{∪}  ↑^{∪}
0 ⟶ O(n) ⟶ Log_{crys}^{(1)}(n − 1) ⟶ K(n − 1) ⟶ 0

and the non-triviality of the lower extension.

Proof of Proposition 5.3. By Propositions 5.4 (2) and (2.3), we have a short exact sequence

0 → H_{syn}^{1}(ℤ_{p}, K(n)) → H_{syn}^{1}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) → H_{syn}^{0}(ℤ_{p}, K(−1)) → 0

and the last term vanishes. Furthermore the projection Log_{crys}^{(n+1)} → Log_{crys}^{(n)} sends K(n+1) to 0. This implies the claim for the first syntomic cohomology. By Propositions 5.4 (2) and (2.3) again, we see H_{syn}^{2}(𝔾_{m,ℤ_{p}}^{log}, Log_{crys}^{(n)}) ≅ H_{syn}^{1}(ℤ_{p}, K(−1)) = 0. One can verify the claim for étale cohomology similarly using Proposition 5.4 (1) and the Leray spectral sequence. The difference from the syntomic case arises from the non-vanishing of H_{ét}^{2}(ℚ_{p}, ℚ_{p}(1)). Note H_{ét}^{2}(ℚ_{p}, ℚ_{p}(r)) = 0 for r ≥ 2.

By taking the projective limit of the Gysin exact sequences (5.1), (5.2) with respect to n and using Proposition 5.3, we obtain isomorphisms:

(5.5)  \underleftarrow{lim}_{n}H_{ét}^{1}(U, Log_{ét}^{(n)}) →^{∼} ℚ_{p},
(5.6)  \underleftarrow{lim}_{n}H_{syn}^{1}(U^{log}, Log_{crys}^{(n)}) →^{∼} ℚ_{p}.

Definition 5.7. We define Pol_{ét} = (Pol _{ét}^{(n)})_{n≥1} and Pol_{crys} = (Pol _{crys}^{(n)})_{n≥1} to be the projective systems of extensions corresponding to 1 by the above isomorphisms (5.5) and (5.6) respectively.

Let U_{1}^{log} be the p-adic formal completion of the scheme ℙ_{ℤ_{p}}^{1}\{1} endowed with the log structure defined by the divisor {0, ∞} and let F_{U_{1}^{log}} be the lifting of Frobenius defined by t ↦ t^{p}, where t denotes the canonical coordinate. Note that one can also define a lifting of Frobenius on ℙ_{ℤ_{p}}^{1} in the same way, but it does not give a lifting of Frobenius on U^{log} because 1 − t^{p} ≠ (1 − t)^{p}·unit. K. Bannai gave the following explicit description of Pol _{crys}^{(n)}. Strictly speaking, he proved this theorem for the realization in the category of filtered overconvergent F -isocrystals but his proof still works in our settings with a slight modification.

Theorem 5.8 (K. Bannai [Ban00a] Theorem 2). The restriction of Pol _{crys}^{(n)} on U_{1}^{log} with F_{U_{1}^{log}} is explicitly described as follows. (See (4.9) for the explicit description

of Log_{crys}^{(n)}.)

Pol _{crys}^{(n)} = O · e ⊕ Log_{crys}^{(n)},
F il^{i}(Pol _{crys}^{(n)}) = \begin{cases}O · e ⊕ F il^{i}(Log_{crys}^{(n)}) & (i ≤ 0), \\ 0 & (i > 0), \end{cases}

∇(e) = \frac{dt}{t − 1}e_{1},
Φ(1 ⊗ e) = e + ∑_{r=1}^{n}(−1)^{r−1}l_{r}^{(p)}(t)e_{r},

where l_{r}^{(p)}(t) denote the p-adic polylogarithmic functions defined inductively by the following differential equations:

l_{1}^{(p)}(0) = 0, dl_{1}^{(p)}(t) = (1 − \frac{F_{U_{1}^{log}}^{∗}}{p} )(\frac{dt}{1 − t}),

l_{r+1}^{(p)}(0) = 0, dl_{r+1}^{(p)}(0) = l_{r}^{(p)}(t) ⊗ \frac{dt}{t} (r ≥ 1).

6. p-adic periods and comparison

Since Log_{crys}^{(n)} (resp. Pol _{crys}^{(n)}) is a successive extension of O(r) (0 ≤ r ≤ n), it comes from a crystalline sheaf on (𝔾_{m,ℚ_{p}})_{ét} (resp. (ℙ_{ℚ_{p}}^{1}\{0, 1, ∞})_{ét}) by Theorem

3.3. In this section, we will calculate the p-adic periods of Log_{crys}^{(n)} and Pol _{crys}^{(n)} and show that the crystalline sheaf associated to Pol _{crys}^{(n)} coincides with Pol _{ét}^{(n)}.

Let U_{1,∞}^{log} be the p-adic formal completion of ℙ_{ℤ_{p}}^{1}\{1, ∞} endowed with the log structure defined by 0 and let A be its ring of coordinates. We define the lifting of Frobenius F on U_{1,∞}^{log} by t ↦ t^{p}. Choose an algebraic closure of Frac(A) and define \overline{A} as in §3. We set G = Gal(Frac(\overline{A})/Frac(A)) as in §3 and simply write B_{crys} for B_{crys}(\overline{A}) in the following. By the proof of Theorem 3.3, the p-adic crystalline representation of G corresponding to Log_{crys}^{(n)} is

(6.1) F il^{0}(B_{crys} ⊗_{ℚ_{p}⊗A} Log_{crys}^{(n)})^{∇=0,φ=1}.

If we use the explicit description (4.9) of Log_{crys}^{(n)}, we have

F il^{0}(B_{crys} ⊗ Log_{crys}^{(n)}) = ⨁_{0≤r≤n} F il^{r}B_{crys} · e_{r}

and for its element f = ∑_{r=0}^{n} f_{r} · e_{r} (f_{r} ∈ F il^{r}B_{crys}), we have:

(6.2)  ∇(f) = 0 ⟺ ∇(f_{0}) = 0, ∇(f_{r}) = −f_{r−1} · \frac{dt}{t} (1 ≤ r ≤ n).
(6.3)  φ(f) = f ⟺ φ(f_{r}) = p^{r}f_{r}. (0 ≤ r ≤ n)

Choose a compatible system \{t_{n}\}_{n≥1}, t_{n+1}^{p} = t_{n}, t_{0} = t of p^{n}-th roots of t in \overline{A} and define the element \underline{t} of R\overline{A} to be (t_{n} mod p)_{n≥0}. Then the element [\underline{t}] of W (R\overline{A}) ⊂

A_{crys} = A_{crys}^{∇=0} satisfies φ([\underline{t}]) = [\underline{t}]^{p} and its image in A_{crys}/F il^{1}A_{crys} ⊂ ℚ_{p} ⊗ ˆ \overline{A} is t. Set u := [\underline{t}]t^{−1}. Then, thanks to the log structure at the point 0, u is contained in 1 + F il^{1}A_{crys} and log(u) converges in A_{crys} p-adically. We have log(u) ∈ F il^{1}A_{crys} and

∇(log(u)) = −\frac{dt}{t}, φ(log(u)) = p log(u).

Using the fact that the dimension of (6.1) is n + 1, we obtain the following proposition:

Proposition 6.4. Choose a non-zero element ε of ℚ_{p}(1). Then

\{ε^{⊗r}(e_{r} + log(u)e_{r+1} + ⋯ + \frac{1}{(n − r)!} log(u)^{n−r}e_{n})|0 ≤ r ≤ n\}

is a basis of the ℚ_{p}-vector space (6.1).

Next we will calculate a lifting of 1 ∈ ℚ_{p} = F il^{0}(B_{crys})^{∇=0,φ=1} in the p-adic crystalline representation of G associated to Pol _{crys}^{(n)}:

(6.5) F il^{0}(B_{crys} ⊗_{ℚ_{p}⊗A} Pol _{crys}^{(n)})^{∇=0,φ=1}.

By the explicit description of K. Bannai (Theorem 5.8), we have

F il^{0}(B_{crys} ⊗_{ℚ_{p}⊗A} Pol _{crys}^{(n)}) = F il^{0}B_{crys} · e ⊕ ⨁_{0≤r≤n} F il^{r}B_{crys} · e_{r}

and for its element e + ∑_{0≤r≤n} f_{r}e_{r}, we have:

(6.6) ∇(f) = 0 ⟺ ∇(f_{0}) = 0, ∇(f_{1}) = −\frac{dt}{t − 1} − f_{0} \frac{dt}{t},

∇(f_{r}) = −f_{r−1} \frac{dt}{t} (2 ≤ r ≤ n),

(6.7) φ(f) = f ⟺ φ(f_{0}) = f_{0}, (1 − p^{−r}φ)(f_{r}) = (−1)^{r−1}l_{r}^{(p)} (1 ≤ r ≤ n).

The equations for f_{0} imply f_{0} ∈ ℚ_{p}. Hence by subtracting an element of (6.1), we may assume f_{0} = 0.

Proposition 6.8. There exist l_{r} ∈ F il^{r}B_{crys} (r ≥ 1) satisfying the following equations:

∇(l_{1}) = \frac{dt}{1 − t}, ∇(l_{r}) = l_{r−1} \frac{dt}{t} (r ≥ 2), (1 − p^{−r}φ)(l_{r}) = l_{r}^{(p)} (r ≥ 1),

where l_{r}^{(p)} are as in Theorem 5.8. Choose one solution {l_{r} ∈ F il^{r}B_{crys}|r ≥ 1}. Then

e + ∑_{r=1}^{n}(−1)^{r−1}l_{r} · e_{r}

is an element of (6.5) lifting 1 in ℚ_{p} = F il^{0}B_{crys}^{∇=0,φ=1}.

Proof. Choose a non-zero element ε of ℚ_{p}(1) ⊂ F il^{1}B_{crys}. Then ε is invertible in B_{crys}, ∇(ε) = 0, φ(ε) = pε and the multiplication by ε^{r} induces an isomorphism F il^{0}B_{crys} →^{∼} F il^{r}B_{crys}. Hence by Proposition 3.1 (2), we have the following exact sequence:

0 ⟶ ℚ_{p}(r) ⟶ F il^{r}B_{crys}

⟶^{α} (F il^{r−1}B_{crys} ⊗_{A} Ω_{A}^{1}(log)) ⊕ B_{crys} ⟶^{β} B_{crys} ⊗_{A} Ω_{A}^{1}(log) ⟶ 0,

where α(x) = (∇(x), (1 − p^{−r}φ)(x)) and β((y, z)) = (1 − p^{−r}φ)(y) − ∇(z). Since (1 − p^{−1}φ)(\frac{dt}{1−t}) = ∇(l_{1}^{(p)}), there exists a solution l_{1} unique up to addition by an element of ℚ_{p}(1). For r ≥ 2, if we are given a solution l_{s} for 1 ≤ s ≤ r − 1, then (1 − p^{−r}φ)(l_{r−1} \frac{dt}{t}) = (1 − p^{−(r−1)}φ)(l_{r−1})\frac{dt}{t} = l_{r−1}^{(p)} \frac{dt}{t} = ∇(l_{r}^{(p)}). Hence there exists a solution l_{r} unique up to addition by an element of ℚ_{p}(r). The second claim follows from (6.6) and (6.7).

As the proof shows, the solution \{l_{r}\} is not unique. If we choose a point \overline{0}: Spec(\overline{ℤ_{p}}) → Spec(\overline{A}) over the point 0: Spec(ℤ_{p}) → Spec(A), then there exists a unique solution such that each l_{r} is “regular at \overline{0}” and its “value at \overline{0}” is 0.

If we choose a compatible system of p^{n}-th roots of 1 − t in \overline{A} and define the element [\underline{1 − t}] in B_{crys}(\overline{A}) similarly as [\underline{t}], then log([\underline{1 − t}](1 − t)^{−1}) ∈ F il^{1}B_{crys} is well defined and satisfies the equations for l_{1}.

Here it is worthwhile to mention that the equations (6.2), (6.3), (6.6) and (6.7) depend on the special choice F of liftings of Frobenius, but their solutions do not. Indeed, if we choose another lifting of Frobenius, then the Frobenius endomorphism on B_{crys} ⊗ E is twisted by the canonical isomorphism:

A_{φ′} ⊗_{A} (B_{crys} ⊗ E) ≅ A_{φ} ⊗_{A} (B_{crys} ⊗ E)

induced by the connection. However, for any horizontal element x ∈ (B_{crys}⊗E)^{∇=0}, the image of 1 ⊗ x is again 1 ⊗ x (cf. (1.1)).

In order to calculate the residue at 1 of the crystalline sheaf associated to Pol _{crys}^{(n)}, we need to work on the p-adic formal completion of an open log subscheme of U^{log} containing 1. Let U_{0,∞}^{log} be the p-adic completion of ℙ_{ℤ_{p}}^{1}\{0, ∞} endowed with the log structure defined by {1}. Then (t − 1) ↦ (t − 1)^{p} defines a lifting of Frobenius on U_{0,∞}^{log}, which we denote by F′. By the construction of Log_{crys}^{(1)} in the proof of Proposition 4.8 (or by calculating the twist of the Frobenius endomorphism directly), we see that the Frobenius of Log_{crys}^{(1)} with respect to F′ is given by

φ(e_{0}) = e_{0} + p^{−1} log(F^{′∗}(t)t^{−p})e_{1}.

By taking its symmetric powers, we see that the Frobenius of Log_{crys}^{(n)} is given by

φ(e_{r}) = p^{−r} ∑_{r≤s≤n} \frac{1}{(s − r)!}(p^{−1} log(F^{′∗}(t)t^{−p}))^{s−r} · e_{s} (0 ≤ r ≤ n).

The solution of (6.1) for U_{0,∞}^{log} is again given by the same formula as Proposition 6.4 using log([\underline{t}]t^{−1}) in B_{crys} associated to U_{0,∞}^{log}.

The Frobenius endomorphism of Pol _{crys}^{(2)} with respect to F′ is simply given by φ(e) = e and we see that e + log([\underline{1 − t}](1 − t)^{−1})e_{1} is an element of (6.5) for U_{0,∞}^{log} and n = 2. Note that log([\underline{1 − t}](1−t)^{−1}) is well defined thanks to the log structure at 1.

Theorem 6.9. The sheaf Pol _{ét}^{(n)} is crystalline and we have a canonical isomorphism D_{crys}(Pol _{ét}^{(n)}) ≅ Pol _{crys}^{(n)}.

Proof. Let F^{(n)} be the crystalline sheaf on ℙ_{ℚ_{p}}^{1}\{0, 1, ∞} associated to Pol _{crys}^{(n)}.

Then by Proposition 4.8, F^{(n)} is an extension of ℚ_{p} by Log_{ét}^{(n)} and is compatible with n. Hence by the definition of Pol _{ét}^{(n)}, it suffices to prove that the residue of F^{(2)} at 1 is 1. The pull-back of F^{(2)} on Spec(A_{0,∞}[(p(t − 1))^{−1}]) is given by the p-adic representation (6.5) for U_{0,∞}^{log} and n = 2. Here A_{0,∞} denotes the ring of coordinates of U_{0,∞}^{log}. Since the natural injection ℤ_{p}[t, t^{−1}, (t−1)^{−1}] → ℚ_{p}[[t−1]][(t−1)^{−1}] factors through A_{0,∞}[(p(t − 1))^{−1}], the inertia group acts through this representation. For the element g of the inertia at 1 corresponding to an element ε of ℤ̂(1), we have g(log([\underline{1 − t}](1 − t)^{−1})) = ε + log([\underline{1 − t}](1 − t)^{−1}). Hence, by the remark before Theorem 6.9 the residue at 1 of F^{(2)} is 1.

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Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

   E-mail address: t-tsuji@ms.u-tokyo.ac.jp