SYNTOMIC COHOMOLOGY AND p-ADIC REGULATORS

Jan Nekovář

Version 17/7/98

The aim of this work is to study the image of p-adic regulators for smooth and projective varieties over number fields and local fields. The main tool in this investigation is the syntomic cohomology of Fontaine-Messing. The present work treats only varieties with potentially good reduction at primes dividing p (as does the work of Nizioł[Ni]). An updated version of this article, which is under preparation, will treat also the case of bad reduction, generalizing the results of Langer [La 2].

Apart from the fundamental paper of Fontaine-Messing [FM], the foundational aspects of syntomic topology and cohomology are not very well documented in the literature (with the exception of [Ba], [Br]). An Appendix to the present work, still in preparation, is intended to give a summary of the key properties of syntomic cohomology. We hope that occasional references to the Appendix will not destroy the readability of the current version.

In early stages of work on this project the author was supported by grants from the Czech Grant Agency (201/93/2176) and the Charles University in Prague (GAUK 369). Further progress was made during visits to Université Louis Pasteur in Strasbourg, Université Paris-Sud and University of Minneapolis in Minnesota. The support of all these institutions is gratefully acknowledged. The author would like to thank P. Berthelot, J.-M. Fontaine, H. Gillet, A. Langer and especially W. Messing for helpful discussions on the subject.

Introduction

(0.0) Let X be a smooth projective scheme over a number field F. For an integer i ≥ 0 one defines the L-function L(H^{i}(X), s), associated to the cohomology of degree i of X, by the standard Euler product ^{1}, absolutely convergent^{1} (in particular, without zeroes or poles) in the region Re(s) > i/2 + 1.

This L-function is expected to have a meromorphic continuation to C, with the only possible pole at i/2 + 1 (if i is even), and a functional equation of the form

(L_{∞} · L)(H^{i}(X), s) = a · b^{s} · (L_{∞} · L)(H^{i}(X), i + 1 − s),  (0.0.0)

where L_{∞}(H^{i}(X), s) is a suitable product of Γ-functions.

The value L(H^{i}(X), n) at an integer n > (i + 1)/2 (which is finite and non-zero^{1}, at least if we stay away from the possible pole at n = i/2 + 1) conjecturally depends only on the “motive” M = H^{i}(X)(n) (a “pure motive” of weight i − 2n < −1 over F).

(0.1) According to Beilinson [Be 2], the structure morphism X ⟶ Spec(F) should underlie a morphism of “motivic sites” X_{mot}⟶^{π}Spec(F)_{mot}; mixed motives over F should be abelian sheaves on Spec(F)_{mot}. The fundamental object responsible for the L-value L(H^{i}(X), n) is believed to be not the motive M = R^{i}π_{∗}Q(n) itself, but rather the motivic cohomology group H^{i+1}(X_{mot}, Q(n)). The Leray spectral sequence for π_{∗}

E_{2}^{a,b} = H^{a}(Spec(F)_{mot}, R^{b}π_{∗}Q(n)) ⟹ H^{a+b}(X_{mot}, Q(n))  (0.1.0)

should degenerate at E_{2} to yield isomorphisms (for n ≠ (i + 1)/2)

H^{i+1}(X_{mot}, Q(n)) ⟶^{∼} H^{1}(Spec(F)_{mot}, R^{i}π_{∗}Q(n)) = H^{1}(Spec(F)_{mot}, M)  (0.1.1)

(0.2) While this picture is purely conjectural and possibly too naive, it makes sense in various realizations of motives. Consider first the Hodge realization. Let A be a noetherian subring of R such that A ⊗ Q is a

1 Strictly speaking, Euler factors at primes of F in which X has bad reduction pose a problem. We ignore these difficulties. field. Denote by Spec(F ⊗_{Q} R)_{A−Hodge} the category of mixed A-Hodge structures over F ⊗_{Q} R (cf. [Be 2], Sect. 7).

The Hodge realization of (0.1.0) is the spectral sequence

E_{2}^{a,b} = H^{a}(Spec(F ⊗_{Q} R)_{A−Hodge}, H^{b}((X ⊗_{Q} R)(C), A(n))) ⟹ H_{�D}^{i+1}(X ⊗_{Q} R, A(n)),  (0.2.0)

which degenerates at E_{2} and converges to the Beilinson-Deligne cohomology of X ⊗_{Q} R (called “absolute Hodge cohomology” in [Be 2]). For n ≠ (i+1)/2, the spectral sequence (0.2.0) boils down to an isomorphism

H_{�D}^{i+1}(X ⊗_{Q} R, A(n)) ⟶^{∼} H^{1}(Spec(F ⊗_{Q} R)_{A−Hodge}, M_{A−Hodge}),  (0.2.1)

where

M_{A−Hodge} = H^{i}((X ⊗_{Q} R)(C), A(n)),

which is a pure A-Hodge structure of weight i − 2n over F ⊗_{Q} R.

(0.3) Fix a prime number p and consider p-adic étale realizations. The Leray spectral sequence for X_{et}⟶^{π}Spec(F)_{et} can be identified with the Hochschild-Serre spectral sequence

E_{2}^{a,b} = H^{a}(F, H^{b}(\overline{X}_{et}, Q_{p}(n))) ⟹ H^{a+b}(X_{et}, Q_{p}(n)),  (0.3.0)

where \overline{X} = X ⊗_{F} \overline{F} and we use continuous Galois and étale cohomology (in the sense of [Ja 1]). Denote by F^{.}H^{∗}(X_{et}, Q_{p}(n)) the induced filtration on H^{∗}(X_{et}, Q_{p}(n)). The spectral sequence (0.3.0) degenerates at E_{2} ([De 1]) and Weil’s conjectures - proved by Deligne [De 3] - imply that E_{2}^{0,b} = 0 for b ≠ 2n (cf. [Ja 2], Lemma 3). For n ≠ (i + 1)/2 this gives edge homomorphisms

H^{i+1}(X_{et}, Q_{p}(n)) = F^{1}H^{i+1}(X_{et}, Q_{p}(n)) ⟶ H^{1}(F, M_{p}),  (0.3.1)

where

M_{p} = H^{i}(\overline{X}_{et}, Q_{p}(n)).

This is a continuous p-adic representation of the Galois group G(\overline{F} /F), pure of weight i − 2n at primes of good reduction of X. In fact, all of this is true for proper (not necessarily projective) smooth schemes over F (by [De 4] and [de J]).

(0.4) In this motivic setting, regulator maps should arise as realizations (for n ≠ (i + 1)/2). The Hodge realization r_{∞}:

H^{i+1}(X_{mot}, Q(n)) ⟶^{r_{∞}}  H_{�D}^{i+1}(X ⊗_{Q} R, R(n))
↓^{≀}  ↓^{≀}
H^{1}(Spec(F)_{mot}, M) ⟶^{r_{∞}} H^{1}(Spec(F ⊗_{Q} R)_{R−Hodge}, M_{R−Hodge})

(0.4.0)

The p-adic étale realization r_{p}:

\begin{matrix}H^{i+1}(X_{mot}, Q(n)) & ⟶^{r_{p}} & H^{i+1}(X_{et}, Q_{p}(n)) \\ ↓^{≀} & & ↓ \\ H^{1}(Spec(F)_{mot}, M) & ⟶^{r_{p}} & H^{1}(F, M_{p}) \end{matrix}  (0.4.1)

There is a certain assymetry between r_{∞} and r_{p}; the Beilinson-Deligne cohomology is a local object at archimedean primes of F, while the Galois cohomology group H^{1}(F, M_{p}) is a global invariant. The map r_{p} can be further localized at p, yielding a diagram

H^{i+1}(X_{mot}, Q(n)) ⟶^{r_{p}} H^{i+1}(X_{et}, Q_{p}(n)) ⟶ H^{i+1}((X ⊗_{Q} Q_{p})_{et}, Q_{p}(n))
↓^{≀}  ↓  ↓^{(?)}
H^{1}(Spec(F)_{mot}, M) ⟶^{r_{p}}  H^{1}(F, M_{p})  ⟶^{res_{p}}  H^{1}(F ⊗_{Q} Q_{p}, M_{p})

(0.4.2)

The third vertical arrow is defined only if a local version of E_{2}^{0,i+1} (over Q_{p}) vanishes. This is known to be true (for n ≠ (i + 1)/2) only in a special case when X ⊗_{Q} Q_{p} has a smooth projective model over O_{F} ⊗ Z_{p} (see III.3.5 below).

According to a conjecture of Jannsen ([Ja 2], Conj. 1; see also Prop. II.1.7(1c) below), the restriction map res_{p} should be injective for n < 0, n > i + 1. More generally, the composite map

H^{1}(Spec(F)_{mot}, M)⟶^{r_{p}}H^{1}(F, M_{p})⟶^{res_{p}}H^{1}(F ⊗_{Q} Q_{p}, M_{p})  (0.4.3)

is expected to be injective for all n ≠ (i + 1)/2. This explains why the localized map (0.4.3), not r_{p} itself, is usually called a p-adic regulator.

(0.5) Suppose that X admits a regular model 𝔛, proper and flat over the ring of integers O_{F} ⊂ F. The motivic sites are expected to extend to

X_{mot} ⟶ 𝔛_{mot}, j : Spec(F)_{mot} ⟶ Spec(O_{F})_{mot},

giving a commutative diagram

H^{i+1}(𝔛_{mot}, Q(n))  ⟶ H^{i+1}(X_{mot}, Q(n))
↓  ↓^{≀}
H^{1}(Spec(O_{F})_{mot}, j_{∗}M) ↪ H^{1}(Spec(F)_{mot}, M)

(0.5.0)

Recall the conjectural description of L(H^{i}(X), n), up to a sign:

Conjecture. Suppose that n > (i + 1)/2 (and, if n = i/2 + 1, that L(H^{i}(X), s) does not have a pole at s = n). Then

(1) (Beilinson [Be 1, 2], Deligne [De 5]) The regulator r_{∞} induces an isomorphism

r_{∞} ⊗ 1 : H^{1}(Spec(O_{F})_{mot}, j_{∗}M) ⊗_{Q} R ⟶^{∼} H^{1}(Spec(F ⊗_{Q} R)_{R−Hodge}, M_{R−Hodge})

and L(H^{i}(X), n) is equal to “det(r_{∞} ⊗ 1)”, up to a factor in Q^{∗}.

(2) (Bloch-Kato [BK]) For each prime number p, the p-adic regulator r_{p} induces an isomorphism

r_{p} ⊗ 1 : H^{1}(Spec(O_{F})_{mot}, j_{∗}M) ⊗_{Q} Q_{p} ⟶^{∼} H_{f}^{1}(F, M_{p}) ⊆ H^{1}(F, M_{p}),

where the subspace H_{f}^{1}(F, M_{p}) is the generalized Selmer group defined in [BK]. The p-adic valuation of the undetermined rational factor from (1) can be recovered from the isomorphism r_{p} ⊗ 1.

(0.6) Both parts of this conjecture were originally formulated in terms of a K-theoretic version of motivic cohomology. In this context, the map

H^{i+1}(𝔛_{mot}, Q(n)) ⟶ H^{i+1}(X_{mot}, Q(n))

is replaced by

K_{2n−i−1}(𝔛)_{Q}^{(n)} ⟶ K_{2n−i−1}(X)_{Q}^{(n)},

where the superscript ^{(n)} refers to the subspace of K_{.}(−) ⊗ Q on which all Adams operations ψ^{k} (k ≥ 1) act by k^{n} (cf. [So 2], [Tamme]).

The role of H^{1}(Spec(O_{F})_{mot}, j_{∗}M) is played by
Im [K_{2n−i−1}(𝔛)_{Q}^{(n)} ⟶ K_{2n−i−1}(X)_{Q}^{(n)}]

This group does not depend on the choice of 𝔛, provided at least one 𝔛 exists ([Be 1], 2.4.2). The maps r_{∞} (resp. r_{p}) are given by the Chern character defined on higher K-theory, with values in the Beilinson-Deligne (resp. p-adic étale) cohomology

\begin{aligned}r_{∞} & : K_{2n−i−1}(X)_{Q}^{(n)} ⟶ H_{�D}^{i+1}(X ⊗_{Q} R, R(n)) \\ r_{p} & : K_{2n−i−1}(X)_{Q}^{(n)} ⟶ H^{i+1}(X_{et}, Q_{p}(n)) \end{aligned}  (0.6.0)

See [Gi], [Sc 1] for a general discussion of characteristic classes.

The conjecture of Beilinson (resp. of Bloch-Kato) then predicts that r_{∞} (resp. r_{p}) induces an isomorphism

r_{∞} : (Im [K_{2n−i−1}(𝔛)_{Q}^{(n)} ⟶ K_{2n−i−1}(X)_{Q}^{(n)}]) ⊗_{Q} R ⟶^{∼} H_{�D}^{i+1}(X ⊗_{Q} R, R(n))  (0.6.1)

resp.

r_{p} : (Im [K_{2n−i−1}(𝔛)_{Q}^{(n)} ⟶ K_{2n−i−1}(X)_{Q}^{(n)}]) ⊗_{Q} Q_{p} ⟶^{∼} H_{f}^{1}(F, M_{p})  (0.6.2)

(with a slight modification if n = i/2 + 1 and L(H^{i}(X), s) has a pole at n).

(0.7) The situation at the central point n = (i + 1)/2 of the conjectural functional equation (0.0.0) is slightly different. Let CH^{n}(X) be the Chow group of algebraic cycles of codimension n on X, modulo rational equivalence. Let

CH^{n}(X)_{0} = Ker [CH^{n}(X) ⟶ H^{2n}(\overline{X}_{et}, Q_{p}(n))] =
= Ker [CH^{n}(X) ⟶ H^{2n}(X(C), Q(n))]

be its subgroup represented by cycles which are homologically trivial (modulo torsion).

The p-adic regulator r_{p} is replaced by the p-adic Abel-Jacobi map (see [Ja 3, (9.4.1)])

r_{p} : CH^{n}(X)_{0} ⊗ Q ⟶ F^{1}H^{2n}(\overline{X}_{et}, Q_{p}(n)) ⟶ H^{1}(F, M_{p})

The conjecture of Bloch-Kato predicts that r_{p} induces an isomorphism

r_{p} ⊗ 1 : CH^{n}(X)_{0} ⊗ Q_{p} ⟶^{∼} H_{f}^{1}(F, M_{p})  (0.7.0)

For n = i = 1, this is equivalent to the finiteness of the p-primary part of the Tate-Šafarevič group of the abelian variety A = Pic^{0}(X/F) (over F). Indeed, in this case, M_{p} = V_{p}(A), r_{p} is the usual descent map

A(F) ⊗ Q ↪ H^{1}(F, V_{p}(A))

and H_{f}^{1}(F, V_{p}(A)) is a Q_{p}-version of the standard Selmer group.

(0.8) The conjectures (0.6.2) and (0.7.0) consist of three statements, arranged in an increasing order of difficulty:

Im (r_{p}) ⊆H_{f}^{1}(F, M_{p})  (0.8.0)
Coker (r_{p} ⊗ 1) = 0  (0.8.1)
Ker(r_{p} ⊗ 1) = 0  (0.8.2)

Indeed, (0.8.0) is merely a consistency of the conjectures. (0.8.1) is equivalent to the finiteness of the p-primary part of a suitably generalized Tate-Šafarevič group (see [BK]); this has been verified in certain cases. By contrast, (0.8.2) seems to be beyond reach at present. The only exception is the case n = i = 1 in (0.7), when we are dealing with rational points on the abelian variety A = Pic^{0}(X/F).

(0.9) Let V be an arbitrary p-adic representation of G(\overline{F} /F) (of finite dimension). The Selmer group H_{f}^{1}(F, V) ⊆ H^{1}(F, V) is defined by local conditions, namely by a cartesian diagram

\begin{matrix}H_{f}^{1}(F, V) & ↪ & H^{1}(F, V) \\ ↓ & & ↓ \\ ∏_{v} H_{f}^{1}(F_{v}, V) & ↪ & ∏_{v} H^{1}(F_{v}, V) \end{matrix} ,  (0.9.0)

in which

H_{f}^{1}(F_{v}, V) = \{ \begin{matrix}H_{ur}^{1}(F_{v}, V), & if v |̸ p \\ Ker [H^{1}(F_{v}, V) ⟶ H^{1}(F_{v}, V ⊗_{Q_{p}} B_{cris})] , & if v|p \end{matrix}

(H^{1}(F_{v}, V) vanishes for an archimedean prime v).

(0.10) One of the aims of the present article is to prove a version of (0.8.0), under some mild restrictions on p. In view of (0.9.0), one is naturally led to corresponding local statements over various completions of F.

Changing perspective, we fix a (non-archimedean) prime v of F and consider a proper smooth scheme X_{v} over Spec(F_{v}). Let 𝔛_{v} ⟶ Spec(O_{v}) be a proper model of X_{v} over the ring of integers of F_{v}.

(0.11) Assume first that v |̸ p. Étale Chern classes then define a commutative diagram of p-adic regulators

\begin{matrix}K_{2n−i−1}(𝔛_{v})_{Q} & ⟶ & K_{2n−i−1}(X_{v})_{Q}^{(n)} \\ ↓ & & ↓ \\ H^{i+1}((𝔛_{v})_{et}, Q_{p}(n)) & ⟶ & H^{i+1}((X_{v})_{et}, Q_{p}(n)) \end{matrix}  (0.11.0)

resp. cycle classes (assuming 𝔛_{v} regular)

\begin{matrix}CH^{n}(𝔛_{v}) ⊗ Q & ⟶ & CH^{n}(X_{v}) ⊗ Q \\ ↓ & & ↓ \\ H^{2n}((𝔛_{v})_{et}, Q_{p}(n)) & ⟶ & H^{2n}((X_{v})_{et}, Q_{p}(n)) \end{matrix}  (0.11.1)

The following statement is well known. It is a straightforward consequence of Deligne’s fundamental result in Weil II ([De 4], Thm. 1) and the commutative diagrams (0.11.0-1).

Theorem A. If 𝔛_{v} is proper over Spec(O_{v}) and v |̸ p, then

(1) For n > (i + 1)/2, K_{2n−i−1}(𝔛_{v}) ⊗ Q maps to F^{1}H^{i+1}((X_{v})_{et}, Q_{p}(n)) and

Im [r_{p} : K_{2n−i−1}(𝔛_{v}) ⊗ Q ⟶ H^{1}(F_{v}, H^{i}((\overline{X}_{v})_{et}, Q_{p}(n)))] = 0.

(2) For n = (i + 1)/2 and 𝔛_{v} regular,

Im [r_{p} : CH^{n}(𝔛_{v})_{0} ⊗ Q ⟶ H^{1}(F_{v}, H^{2n−1}((\overline{X}_{v})_{et}, Q_{p}(n)))] = 0.

Here CH^{n}(𝔛_{v})_{0} = Ker [CH^{n}(𝔛_{v}) ⟶ H^{2n}((𝔛_{v} ⊗_{O_{v}} \overline{k(v)})_{et}, Q_{p}(n))].

Remark. If n = (i + 1)/2 and H^{2n−1}(\overline{X}_{et}, Q_{p}) satisfies the purity conjecture for the monodromy filtration at v (e.g. if X has a potentially good reduction at v), then H^{1}(F_{v}, H^{2n−1}((\overline{X}_{v})_{et}, Q_{p}(n))) = 0.

(0.12) For v|p, diagrams (0.11.0-1) no longer exist. Assume that 𝔛_{v} is proper and smooth over Spec(O_{v}). Instead of p-adic étale cohomology of 𝔛_{v}, consider the syntomic cohomology of Fontaine-Messing [FM]. For p > 2 and n ≥ 0, Fontaine and Messing defined maps ^{1}

H^{∗}((𝔛_{v})_{syn}, s_{Q_{p}}(n)) ⟶ H^{∗}((X_{v})_{et}, Q_{p}(n))

Chern classes with values in syntomic cohomology give rise to commutative diagrams of p-adic regulators

\begin{matrix}K_{2n−i−1}(𝔛_{v})_{Q}^{(n)} & ⟶ & K_{2n−i−1}(X_{v})_{Q}^{(n)} \\ ↓ & & ↓ \\ H^{i+1}((𝔛_{v})_{syn}, s_{Q_{p}}(n)) & ⟶ & H^{i+1}((X_{v})_{et}, Q_{p}(n)) \end{matrix}  (0.12.0)

resp. cycle classes

\begin{matrix}CH^{n}(𝔛_{v}) & ⟶ & CH^{n}(X_{v}) \\ ↓ & & ↓ \\ H^{2n}((𝔛_{v})_{syn}, s_{Q_{p}}(n)) & ⟶ & H^{2n}((X_{v})_{et}, Q_{p}(n)) \end{matrix}  (0.12.1)

Our main local result ^{2} (proved in a more general context, when O_{v} is replaced by an arbitrary complete discrete valuation ring of mixed characteristic (0, p), with a perfect residue field) states that there is a canonical isomorphism - compatible with the Hochschild-Serre spectral sequence (0.3.0) for X_{v} -

Ker [H^{i+1}((𝔛_{v})_{syn}, s_{Q_{p}}(n)) ⟶ H^{i+1}((\overline{X}_{v})_{et}, Q_{p}(n))] ⟶^{∼} H_{f}^{1}(F_{v}, H^{i}((\overline{X}_{v})_{et}, Q_{p}(n)))  (0.12.2)

and that

Ker [H^{i+1}((𝔛_{v})_{syn}, s_{Q_{p}}(n)) ⟶ H^{i+1}((X_{v})_{et}, Q_{p}(n))] = 0.  (0.12.3)

This implies that syntomic cohomology of a proper and smooth scheme over Spec(O_{v}) displays all signs of behaviour of a “p-adic Beilinson-Deligne cohomology”, which should be the abutment of a spectral sequence

E_{2}^{a,b} = H_{f}^{a}(F_{v}, H^{b}((\overline{X}_{v})_{et}, Q_{p}(n))) ⟹ H_{�D}^{a+b}(X_{v}, Q_{p}(n))  (0.12.4)

As a corollary, we obtain

Theorem B. If 𝔛_{v} is proper and smooth over Spec(O_{v}) and v|p, then

(1) For n > (i + 1)/2, Im [r_{p} : (K_{2n−i−1}(𝔛_{v}) ⊗ Q)_{0} ⟶ H^{1}(F_{v}, H^{i}((\overline{X}_{v})_{et}, Q_{p}(n)))] ⊆ H_{f}^{1}(F_{v}, H^{i}((\overline{X}_{v})_{et}, Q_{p}(n))).

(2) For n = (i + 1)/2, Im [r_{p} : CH^{n}(X_{v})_{0} ⊗ Q ⟶ H^{1}(F_{v}, H^{2n−1}((\overline{X}_{v})_{et}, Q_{p}(n)))] ⊆ H_{f}^{1}(F_{v}, H^{2n−1}((\overline{X}_{v})_{et}, Q_{p}(n))).

Here (K_{2n−i−1}(𝔛_{v}) ⊗ Q)_{0} = Ker [K_{2n−i−1}(𝔛_{v}) ⊗ Q ⟶ H^{i+1}((\overline{X}_{v})_{et}, Q_{p}(n))].

In fact, Theorem B holds under a weaker assumption, namely that X_{v} has a potentially good reduction. Moreover, it follows from crystalline Weil conjectures for H_{cris}^{i+1}(𝔛_{v}) ([KM], [CLeS]) that (K_{2n−i−1}(𝔛_{v}) ⊗ Q)_{0} = K_{2n−i−1}(𝔛_{v}) ⊗ Q. The statement (2) of Theorem B is crucial for the construction of p-adic height pairings – see [Ne 2], [Ne 3].

(0.13) For 𝔛_{v} smooth and projective over Spec(O_{v}), P. Schneider ([Sc 2], Sect. 7) defined (for n ≥ 0) a cohomology theory H^{∗}((𝔛_{v})_{et}, S_{Q_{p}}(n)) mapping to H^{∗}(X_{et}, Q_{p}(n)). He conjectured that, for i ≠ 2n, 2n − 1, the Hochschild-Serre spectral sequence induces an isomorphism

H^{i+1}((𝔛_{v})_{et}, S_{Q_{p}}(n)) ⟶^{∼} H_{f}^{1}(F_{v}, H^{i}((\overline{X}_{v})_{et}, Q_{p}(n)))  (0.13.0)

It is explained in [Sc 2] that (0.12.2) combined with a result of Kurihara [Ku] implies the following result.

1 The case of p = 2 is treated the work of Tsuji [Ts].

2 For technical reasons, we use the syntomic-étale site of the p-adic completion of 𝔛_{v}. The cohomology of s_{Q_{p}}(n) is the same for both sites.

Theorem C. For i ≠ 2n − 1 and 0 ≤ n < p − 1 Schneider’s conjecture (0.13.0) holds.

(0.14) Our methods also shed some light on the relation between the image of the cycle class map

cl_{X_{v}} : CH^{n}(X_{v}) ⊗ Q ⟶ H^{2n}((X_{v})_{et}, Q_{p}(n))

and the filtration F^{.}H^{2n}((X_{v})_{et}, Q_{p}(n)) on the target (again the case of v |̸ p is well known).

Theorem D. Let X_{v} be proper and smooth over Spec(F_{v}).

(1) If v |̸ p and either r = 1, or r = dim(X), or X_{v} has a potentially good reduction, then Im(cl_{X_{v}}) ∩ F^{1}H^{2n}((X_{v})_{et}, Q_{p}(n)) = 0.

(2) Suppose that v |̸ p and that X_{v} satisfies the condition (∗_{n}) There is a finite extension L/F_{v} such that X_{L} := X_{v} ⊗_{F_{v}} L admits a regular model 𝔛_{L}, proper over Spec(O_{L}), such that the restriction map (CH^{n}(𝔛_{L}) ⊗ Q)_{0} ⟶ (CH^{n}(X_{L}) ⊗ Q)_{0} is surjective. Then Im(cl_{X_{v}}) ∩ F^{1}H^{2n}((X_{v})_{et}, Q_{p}(n)) = 0.

(3) If v|p and if X_{v} has a potentially good reduction, then Im(cl_{X_{v}}) ∩ F^{2}H^{2n}((X_{v})_{et}, Q_{p}(n)) = 0.

The condition (∗_{n}) is known to be satisfied - provided 𝔛_{L} exists - if X_{v} is equidimensional and n = dim(X_{v}).

(0.15) Returning to the global setting, consider the cycle class map

cl_{X} : CH^{n}(X) ⊗ Q ⟶ H^{2n}(X_{et}, Q_{p}(n))

for a smooth and projective scheme X over a number field F. The conjectural injectivity of the Abel-Jacobi map r_{p} from (0.7) predicts that

Im(cl_{X}) ∩ F^{2}H^{2n}(X_{et}, Q_{p}(n)) = 0

Our local results imply

Theorem E. If X is proper and smooth over a number field F and has a potentially good reduction at all primes dividing p, then Im(cl_{X}) ∩ F^{2}H^{2n}(X_{et}, Q_{p}(n)) is a subquotient (as a Q_{p}-vector space) of

Ker [α_{S,Σ} : H^{2}(G_{F,S}, H^{2n−2}(\overline{X}_{et}, Q_{p}(n))) ⟶ ⨁_{v∈Σ} H^{2}(F_{v}, H^{2n−2}(\overline{X}_{et}, Q_{p}(n)))] .

Here G_{F,S} is the Galois group of the maximal extension of F, unramified outside of S = {v|p}∪{primes of bad reduction of X}, and Σ = {v|p} ∪ {v ∈ S | v |̸ p, (∗_{n}) holds for X ⊗_{F} F_{v}}.

Remarks. (i) It is believed that Σ = S. (ii) Conjecturally, Ker(α_{S,S}) = 0. A function field analogue was proved in ([Ja 2], Thm. 4) and ([Ra], Thm. 4.1).

(0.16) There are only a handful of cases in which the vanishing of Ker(α_{S,S}) has been established. Combining Theorem E with results of [Fl], [La 1], [Wi], we obtain

Theorem F. Let E be a modular elliptic curve over Q, without complex multiplication (over \overline{Q}), p > 2 a prime number at which E has good reduction. Then, for every d ≥ 1, the cycle class map

cl_{E^{d}} : CH^{d}(E^{d}) ⟶ H^{2d}((E^{d})_{et}, Q_{p}(d))

satisfies Im(cl_{E^{d}}) ∩ F^{2}H^{2d}((E^{d})_{et}, Q_{p}(d)) = 0.

Under additional hypotheses on p, an analogue of Theorem D can be proved for cohomology with Z_{p}- coefficients. Combined with vanishing results for a Z_{p}-version of α_{S,S} ([Fl]), this can be applied to the torsion in the Chow groups:

Theorem G. Let E be a modular elliptic curve over Q, without complex multiplication (over \overline{Q}). Then there is an explicit integer c_{E} ≥ 1 such that, for every d ≥ 1 and every prime p not dividing (2d)! · c_{E}, we have

(Im [CH^{d}(E^{d}) ⟶ H^{2d}((E^{d})_{et}, Z_{p}(d))])_{tors} = 0.

(0.17) Theorem B was independently proved by Nizioł[Ni] for 𝔛_{v} smooth and projective over Spec(O_{v}). A special case of (0.12.2) is treated by Langer-Saito ([LaSa], Thm. 6.5). Our approach is modelled on a paper of M. Gros [Gros]. However, many arguments in [Gros] are rather sketchy. The Appendix to the present paper will treat the compatibility of syntomic and étale Chern classes in (0.12.0-1) (this is one of the points where the treatment in [Gros] does not seem to be completely satisfactory).

I. Topological background

This chapter sums up technical material that will be needed later.

1. Abstract nonsense

(1.1) Notation. For a category (resp. a site) C, denote by C (resp. C ) the category of presheaves (resp. sheaves) on C (we suppress the dependence on a universe). For a topos E, write Ab(E) for the category of abelian groups in E. If A is a ring in E, Mod(E, A) denotes the category of A-modules in E. Both Ab(E) and Mod(E, A) are abelian categories with enough injectives ([SGA 4], Exp. II, 6.9). If E = (Sets), then we often write (A − Mod) instead of Mod((Sets), A). Write Γ_{E} = Γ(E, −) : E ⟶ (Sets) for the functor of global sections on E. Cohomology groups in E are denoted by H^{q}(E, −) (resp. H^{q}(E; X, −) for an object X of E). Occasionally, we shall abuse the notation and write H^{q}(C, −) instead of H^{q}(C , −) (and similarly for H^{q}(C; X, −)). See Appendix and ([SGA 4], Exp. I–VI) for further topological terminology.

(1.2) Localization. Let A be an abelian category. Denote by Q(A) = A ⊗ Q its localization with respect to all morphisms n · 1_{A} : A ⟶ A (for A ∈ Ob(A), n ≥ 1). It has the same class of objects as A and the identity induces a functor Q : A ⟶ Q(A). We give a list of standard properties of Q(A).

(1) Q(A) is an abelian category.

(2) Q : A ⟶ Q(A) is an additive functor, exact and essentially surjective.

(3) For every A, B ∈ Ob(A),

Hom_{Q(A)}(Q(A), Q(B)) = Hom_{A}(A, B) ⊗ Q

(4) If B is an additive category and u : A ⟶ B an additive functor such that u(n·1_{A}) = n·1_{u(A)} : u(A) ⟶ u(A) is invertible for all A ∈ Ob(A), n ≥ 1, then u factors uniquely as

u : A⟶^{Q}Q(A)⟶^{v}B,

where v is an additive functor. In particular, for every commutative ring R the functor −⊗Q : (R−Mod) ⟶ ((R ⊗ Q) − Mod) factors as

(R − Mod)⟶^{Q}Q(R − Mod)⟶^{ξ}((R ⊗ Q) − Mod),

where ξ is an exact functor.

(5) For A ∈ Ob(A), Q(A) = 0 iff n · Hom_{A}(A, A) = 0 for some n ≥ 1.

(6) An additive functor u : A ⟶ B between abelian categories A, B induces an additive functor Q(u) : Q(A) ⟶ Q(B) such that the diagram

A ⟶^{u} B
↓^{Q_{A}}  ↓^{Q_{B}}
Q(A) ⟶^{Q(u)} Q(B)

is (strictly) commutative.

(7) In the situation of (6), u is exact (resp. left exact, resp. right exact) ⟹ Q(u) is exact (resp. left exact, resp. right exact).

(8) If A has enough injectives (e.g. if A = Mod(E, A)), then Q(A) has enough injectives and

{X ∈ Ob(Q(A)) | X injective} = {Q_{A}(I) | I ∈ Ob(A) injective}

(9) If A has enough injectives and if u : A ⟶ B (as in (6)) preserves injectives, then Q(u) : Q(A) ⟶ Q(B) preserves injectives.

(10) If A has enough injectives and u : A ⟶ B (as in (6)) is left exact, then Q(u) : Q(A) ⟶ Q(B) is left exact, R^{q}u, R^{q}Q(u) are defined for all q ≥ 0 and the diagram

A ⟶^{R^{q}u} B
↓^{Q_{A}}  ↓^{Q_{B}}
Q(A) ⟶^{R^{q}(Q(u))} Q(B)

is commutative (up to a canonical isomorphism).

(11) Denote by C(A) (resp. K(A), resp. D(A)) the category of complexes (resp. the category of complexes up to homotopy, resp. the derived category) of A. The functor Q_{A} : A ⟶ Q(A) extends to triangulated functors Q : K(A) ⟶ K(Q(A)) and Q_{D} : D(A) ⟶ D(Q(A)). If X (resp. Y) is a complex bounded below (resp. bounded above), then the canonical maps

Hom_{C(A)}(X, Y) ⊗ Q ⟶ Hom_{C(Q(A))}(Q(X), Q(Y))
Hom_{K(A)}(X, Y) ⊗ Q ⟶ Hom_{K(Q(A))}(Q(X), Q(Y))

are both isomorphisms.

(12) If A has enough injectives, then the canonical map

Hom_{D(A)}(X, Y) ⊗ Q ⟶ Hom_{D(Q(A))}(Q_{D}(X), Q_{D}(Y))

is an isomorphism for every X ∈ Ob(D^{−}(A)), Y ∈ Ob(D^{+}(A)).

(13) If A has enough injectives and u : A ⟶ B is a left exact functor to an abelian category B, then there is a canonical isomorphism

Q_{D}(Ru(X)) ⟶^{∼} R(Q(u))(Q_{D}(X))

for every X ∈ Ob(D^{+}(A)).

(1.3) Projective systems. Let C be a category. Denote by C^{N} the category of projective systems

[A_{n}]_{n∈N} = [A_{0} ⟵ A_{1} ⟵ A_{2} ⟵ ⋯]

(indexed by N) in C.

If E is a topos, then E^{N} is a topos as well. Indeed, viewing the ordered set N as a category in the usual way, consider the functor Φ : N ⟶ E sending each element of N to the final object e_{E} of E. The pull-back of the fibred topos of arrows F l(E) ⟶ E ([SGA 4], Exp. VI, 7.3.1) by Φ is a fibred topos Φ^{∗}(F l(E)) ⟶ N over N. The associated total topos Top(Φ^{∗}(F l(E)) (cf. [Il 1], VI.5.2) is canonically equivalent to E^{N}. The functor of global sections of E^{N} factors as

Γ_{(E^{N})} : E^{N}⟶^{Γ_{E}^{N}}(Sets)^{N}⟶^{Γ_{N}}(Sets),

where Γ_{E}^{N} ([A_{n}]) = [Γ_{E}(A_{n})] and Γ_{N} = \underleftarrow{lim}_{N} (cf. [SGA 4], Exp. VI, 7.4.15; [Il 1], VI.5.8.1(ii)).

(1.4) Continuous cohomology. For a projective system A = [A_{n}] ∈ Ob(Ab(E^{N})) we shall write, abusively, H^{q}(E, [A_{n}]) instead of H^{q}(E^{N}, [A_{n}]). The spectral sequence of ([SGA 4], Exp. VI, 7.4.15; Appendix 2.3.2) for the composition of functors

Γ_{(E^{N})} = Γ_{N} ∘ Γ_{E}^{N} : Ab(E^{N}) ⟶ (Ab)^{N} ⟶ (Ab)

becomes

0 ⟶ (R^{1} \underleftarrow{lim}_{N}) H^{q−1}(E, A_{n}) ⟶ H^{q}(E, [A_{n}]) ⟶ \underleftarrow{lim}_{N} H^{q}(E, A_{n}) ⟶ 0,  (1.4.1)

as R^{q}Γ_{N} = R^{q} \underleftarrow{lim}_{N} = 0 for q > 1 (in (Ab)^{N}). We denote by H_{naive}^{q}(E, [A_{n}]) the third group in the short

exact sequence (1.4.1).

(1.5) A morphism of topoi u = (u^{∗}, u_{∗}) : E ⟶ E′ induces a morphism of topoi u^{N} : E^{N} ⟶ E^{′N}, given termwise by u. On abelian group objects, the derived functors

R^{q}u_{∗}^{N} : Ab(E^{N}) = Ab(E)^{N} ⟶ Ab(E^{′N}) = Ab(E′)^{N}

can be computed termwise as well:

(R^{q}u_{∗}^{N}) ([A_{n}]) = [R^{q}u_{∗}(A_{n})]  (1.5.1)

The Leray spectral sequence for u^{N} ([SGA 4], Exp. V, 5.3) is given by

E_{2}^{p,q} = H^{p}(E′, [R^{q}u_{∗}(A_{n})]_{n∈N}) ⟹ H^{p+q}(E, [A_{n}]_{n∈N})  (1.5.2)

Denote by F^{i}H^{q}(E, [A_{n}]) the induced filtration on its abutment. The kernel of the edge homomorphism is equal to

Ker [H^{q}(E, [A_{n}]) ⟶ E_{2}^{0,q}] = F^{1}H^{q}(E, [A_{n}])

and (1.5.2) induces a homomorphism

δ : F^{1}H^{q}(E, [A_{n}]) ⟶ E_{∞}^{1,q−1} ↪ E_{2}^{1,q−1}  (1.5.3)

(1.6) Mittag-Leffler conditions. Recall that a projective system [A_{n}] ∈ Ob(Ab(E)^{N}) is a ML-system if, for each n ∈ N, the sequence of subobjects [Im(A_{m+n} ⟶ A_{n})]_{m∈N} of A_{n} becomes stationary for m > m(n). The projective system is ML-zero, if this stationary value is equal to 0 (= the final object of Ab(E)) for all n ∈ N. The projective system is an AR-system (= Artin-Rees), if the value of m(n) can be taken independent of n. If [A_{n}] is ML-zero, then ([Ja 1], Lemma 1.11)

H^{q}(E, [A_{n}]) = 0  (1.6.1)

A ML-isomorphism is a homomorphism f : [A_{n}] ⟶ [B_{n}] such that both Ker(f) and Coker(f) are ML-zero. By (1.6.1), such an f induces an isomorphism

H^{q}(E, [A_{n}]) ⟶^{∼} H^{q}(E, [B_{n}])  (1.6.2)

for all q ≥ 0. A projective system [A_{n}] of abelian groups is called ML-p-adic if it is ML-isomorphic to the system [A ⊗ Z/p^{n}Z] for a suitable abelian group A.

(1.7) Group cohomology. Let G be a pro-finite group and let B_{G} be its classifying topos ([SGA 4], Exp. IV, 2.7). Objects of B_{G} are sets with a discrete left action of G (the stabilizer of each element is an open subgroup of G). Abelian groups in B_{G} are discrete G-modules. For a projective system of discrete G-modules [A_{n}], we write H^{q}(G, [A_{n}]) for H^{q}(B_{G}, [A_{n}]).

The group G acts continuously on the projective limit A = \underleftarrow{lim}_{N} A_{n} (equipped with the topology of a

projective limit of discrete sets) and one can consider continuous cohomology groups H_{cont}^{q}(G, A) in the sense of Tate [Ta], computed using continuous cochains. The basic comparison result, due to Jannsen ([Ja 1], Thm. 2.2), states that for a Mittag-Leffler system [A_{n}] there is a canonical (and functorial) isomorphism

H_{cont}^{q}(G, \underleftarrow{lim}_{N} A_{n}) ⟶^{∼} H^{q}(G, [A_{n}])  (1.7.1)

(1.8) Lemma. Let G be a pro-finite group, p a prime number such that d := cd_{p}(G) (cf. [Se 1], I.3.1) is finite. If [A_{n}] is a projective system of p-power torsion discrete G-modules, then

(1) H^{q}(G, [A_{n}]) = 0, if q > d + 1.

(2) H^{d+1}(G, [A_{n}]) = 0, if [A_{n}] is a ML-system.

Proof. (1) Follows from (1.4.1). For (2), we have to show that (R^{1} \underleftarrow{lim}_{N}) H^{d}(G, [A_{n}]) = 0 (again by (1.4.1)).

In view of (1.6.1), we can assume that [A_{n}] is a surjective system. In this case [H^{d}(G, A_{n})]_{n} is a surjective system as well, hence its (R^{1} \underleftarrow{lim}_{N}) vanishes.

(1.9) Proposition. Let G be a pro-finite group, T a topological G-module. Define a topology on T ⊗ Q as follows: T ⊗ Q = \underrightarrow{lim}_{m} Im(α_{m}), where α_{m} : T ⟶ T ⊗ Q (for m ≥ 1) is given by α_{m}(x) = x ⊗ (1/m). Put

the quotient topology on Im(α_{m}) and the inductive limit topology on T ⊗ Q. If T_{tors} has a finite exponent, then the canonical map

H_{cont}^{q}(G, T) ⊗ Q ⟶ H_{cont}^{q}(G, T ⊗ Q)

is an isomorphism (for each q ≥ 0).

Proof. As in ([Ja 1], Thm. 5.15.c).

(1.10) Remark. If p is a prime number and [A_{n}] is a ML-p-adic projective system of discrete abelian groups, then the projective limit topology on A = \underleftarrow{lim}_{N} A_{n} is the p-adic one (see [Ja 1], Lemma 4.5).

2. Continuous étale cohomology

This is a brief summary of some aspects of Jannsen’s theory [Ja 1].

(2.1) For a scheme X, let X_{et} be its small étale site ([SGA 4], Exp. VII, 1.2). As in 1.4, for an object [A_{n}] of Ab(X_{et}^{∼})^{N}, there are continuous cohomology groups H^{q}(X_{et}, [A_{n}]) sitting in the exact sequence

0 ⟶ (R^{1} \underleftarrow{lim}_{N}) H^{q−1}(X_{et}, A_{n}) ⟶ H^{q}(X_{et}, [A_{n}]) ⟶ \underleftarrow{lim}_{N} H^{q}(X_{et}, A_{n}) ⟶ 0  (2.1.1)

As before, we denote the third group by H_{naive}^{q}(X_{et}, [A_{n}]).

(2.2) Let K be a (commutative) field. Fix a separable closure K_{s} of K and put G_{K} = G(K_{s}/K). Evaluation at \overline{η} : Spec(K_{s}) ⟶ Spec(K) (more precisely, at Spec(L) ⟶ Spec(K) for all finite subextensions K ⊂ L ⊂ K_{s}) defines an equivalence of categories

Spec(K)_{et}^{∼}⟶^{≈}B_{G_{K}} , A ↦ A\overline{η},

inducing canonical isomorphisms

H^{q}(Spec(K)_{et}, [A_{n}]) ⟶^{∼} H^{q}(G_{K}, [(A_{n})\overline{η}]_{n∈N})  (2.2.1)

Let π : X ⟶ Spec(K) be a coherent (= quasi-compact and quasi-separated) morphism (for example, a morphism of finite type). For every abelian sheaf A on X_{et} and q ≥ 0, there is a canonical isomorphism

(R^{q}π_{∗}A)\overline{η} ⟶^{∼} H^{q}(\overline{X}_{et}, A),  (2.2.2)

where \overline{X} = X ⊗_{K} K_{s} ([SGA 4], Exp. VIII, 5.2) (by abuse of notation, we write A instead of f^{∗}A, for f : \overline{X} ⟶ X). If π is not coherent, then the R.H.S. of (2.2.2) has to be replaced by

\underrightarrow{lim}_{L} H^{q}((X ⊗_{K} L)_{et}, A),

with L running through subextensions K ⊂ L ⊂ K_{s}, finite over K. The Leray spectral sequence (1.5.2) becomes – for coherent π – the Hochschild-Serre spectral sequence

E_{2}^{i,j} = H^{i}(G_{K}, [H^{j}(\overline{X}_{et}, A_{n})]_{n∈N}) ⟹ H^{i+j}(X_{et}, [A_{n}]_{n∈N})  (2.2.3)

(2.3) Under additional hypotheses, the spectral sequence (2.2.3) further simplifies ([Ja 1], Rmk. 3.5): (2.3.1) If [H^{j}(\overline{X}_{et}, A_{n})]_{n∈N} is a ML-system for all j ≥ 0 (e.g. if the groups H^{j}(\overline{X}_{et}, A_{n}) are finite for all j, n ≥ 0), then

H^{q}(X_{et}, [A_{n}]) ⟶^{∼} H_{naive}^{q}(X_{et}, [A_{n}])

and (1.7.1) gives

E_{2}^{i,j} = H_{cont}^{i}(G_{K}, H_{naive}^{j}(\overline{X}_{et}, [A_{n}])) ⟹ H^{i+j}(X_{et}, [A_{n}]_{n∈N})  (2.3.1.1)

(2.3.2) If, furthermore, H^{i}(G_{K}, M) is finite for any finite G_{K}-module M and i ≥ 0, then the spectral sequence (2.3.1.1) and its abutment are equal to \underleftarrow{lim}_{N} of the usual spectral sequences

E_{2}^{i,j} = H^{i}(G_{K}, H^{j}(\overline{X}_{et}, A_{n})) ⟹ H^{i+j}(X_{et}, A_{n}),  (2.3.2.1)

as all groups in (2.3.2.1) are finite and \underleftarrow{lim}_{N} is exact on projective systems of finite groups. This happens, for example, if K is a finite field or a finite extension of Q_{ℓ}. (2.3.3) For a prime p different from the characteristic of K we put, as usual,

Z/p^{n}Z(r) = \{ \begin{matrix}μ_{p^{n}}^{⊗r} & if r ≥ 0 \\ Hom(μ_{p^{n}}, Z/p^{n}Z)^{⊗−r} & if r ≤ 0 \end{matrix}

If π : X ⟶ Spec(K) is of finite type, then, according to ([SGA 5], Exp. V, 5.3.1),

[H^{i}(\overline{X}_{et}, Z/p^{n}Z(r))]_{n∈N} is a ML−p − adic system of finite groups  (2.3.3.1)

(for every i ≥ 0, r ∈ Z). This implies that

H^{j}(\overline{X}_{et}, Z_{p}(r)) ⟶^{∼} H_{naive}^{j}(\overline{X}_{et}, Z_{p}(r)).

Following Jannsen [Ja 1], we use the following notation:

H^{q}(X_{et}, Z_{p}(r)) := H^{q}(X_{et}, [Z/p^{n}Z(r)])
H^{q}(X_{et}, Q_{p}(r)) := H^{q}(X_{et}, [Z/p^{n}Z(r)]) ⊗_{Z_{p}} Q_{p}

(2.3.1.1) then gives a spectral sequence

E_{2}^{i,j} = H_{cont}^{i}(G_{K}, H^{i}(\overline{X}_{et}, Z_{p}(r))) ⟹ H^{i+j}(X_{et}, Z_{p}(r)),  (2.3.3.2)

which, on tensoring by Q, becomes – in view of Proposition 1.9 –

E_{2}^{i,j} = H_{cont}^{i}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) ⟹ H^{i+j}(X_{et}, Q_{p}(r))  (2.3.3.3)

Note that the topology on H^{i}(\overline{X}_{et}, Z_{p}(r)) (resp. on H^{i}(\overline{X}_{et}, Q_{p}(r))) is the p-adic one, according to (1.10) and (2.3.3.1),

If follows from (2.3.3.1) and Lemma 1.8 that

E_{2}^{i,j} = 0  for i > cd_{p}(K)  (2.3.3.4)

(for both spectral sequences (2.3.3.2–3)).

If X is smooth and projective over K, then the spectral sequence (2.3.3.3) degenerates at E_{2}, by ([De 1], Prop. 2.4) and the existence of a good formalism for the derived category of Q_{p}-sheaves (Ekedhal [Ek], Jannsen (unpublished)). It follows from de Jong’s theorem on alterations [de J] that E_{2} = E_{∞} for X proper (not necessarily projective) and smooth over K. (2.3.4) If, furthermore, K is as in (2.3.2), then (2.3.3.2) is the projective limit of spectral sequences

E_{2}^{i,j} = H^{i}(G_{K}, H^{i}(\overline{X}_{et}, Z/p^{n}Z(r))) ⟹ H^{i+j}(X_{et}, Z/p^{n}Z(r)),  (2.3.4.1)

with all groups in (2.3.4.1) being finite.

(2.4) Proper base change. Let S be the spectrum of a henselian local ring, s ↪ S the inclusion of the closed point. Let π : X ⟶ S be a proper morphism with a special fibre X_{s} = X ⊗_{S} s. Denote by i : X_{s} ↪ X the immersion of X_{s} into X. If A = [A_{n}] is a projective system of torsion abelian sheaves on X_{et}, then i induces an isomorphism

i^{∗} : H^{q}(X_{et}, [A_{n}]) ⟶^{∼} H^{q}((X_{s})_{et}, [i^{∗}A_{n}])  (∀q ≥ 0)  (2.4.1)

This follows from (2.1.1) and the usual proper base change theorem. The latter also implies that

(R^{q}π_{∗}^{N}A)\overline{s} := [(R^{q}π_{∗}A_{n})\overline{s}]_{n∈N} ⟶^{∼} [H^{q}((X\overline{s})_{et}, \overline{i}^{∗}A_{n})]_{n∈N}  (2.4.2)

(2.5) Let p be a prime number. Write “Z_{p}” for the projective system [Z/p^{n}Z]_{n∈N}; it is a ring in (Sets)^{N}. For any topos E, the morphism of topoi Γ_{E}^{N} : E^{N} ⟶ (Sets)^{N} defines a ring (Γ_{E}^{N})^{∗}(“Z_{p}”) in E^{N}, which will be again denoted by “Z_{p}”. By abuse of language, we put

Mod(E, “Z_{p}”) := Mod(E^{N}, “Z_{p}”), Mod(E, “Q_{p}”) := Q(Mod(E, “Z_{p}”))

These are abelian categories with enough injectives (cf. (1.1), (1.2.8)).

If K is a field of characteristic char(K) ≠ p and G = G_{K}, denote by “Z_{p}(r)” (for any r ∈ Z) the projective system of discrete G-modules [Z/p^{n}Z(r)]_{n∈N}. It is a flat object of Mod(B_{G}, “Z_{p}”). Write “Q_{p}(r)” for Q(“Z_{p}(r)”), which is a flat object of Mod(B_{G}, “Q_{p}”).

Tensor products X ↦ X ⊗_{“Z_{p}”} “Z_{p}(r)” (resp. X ↦ X ⊗_{“Q_{p}”} “Q_{p}(r)”) define exact endofunctors X ↦ X(r) (“Tate twists”) of Mod(B_{G}, “Z_{p}”) (resp. Mod(B_{G}, “Q_{p}”)). For every X and r, q ∈ Z, there are canonical isomorphisms

(X(r))(q) ⟶^{∼} X(r + q),  (X(r))(−r) ⟶^{∼} X  (2.5.1)

3. Some homological algebra

This section should be skipped on the first reading. We shall need Proposition 3.5 for our main comparison results in Chapters III and IV.

(3.1) Suppose we are given the following data:

(0) An integer i ∈ Z.

(1) Abelian categories with enough injectives C, \overline{C}, D.

(2) Left exact functors u : \overline{C} ⟶ C, Ψ : C ⟶ D, Φ = Ψ ∘ u : \overline{C} ⟶ D.

(3) Distinguished triangles Δ : A ⟶ B ⟶ C ⟶ A[1] (resp. \overline{Δ} : \overline{A} ⟶ \overline{B} ⟶ \overline{C} ⟶ \overline{A}[1]) in D^{+}(C) (resp. D^{+}(\overline{C})).

(4) A morphism of triangles Δ ⟶ (Ru)(\overline{Δ}) in D^{+}(C).

(5) Objects E ∈ Ob(D^{+}(C)), \overline{E} ∈ Ob(D^{+}(\overline{C})) and an isomorphism ρ : E ⟶^{∼} (Ru)(\overline{E}).

(6) Morphisms μ : A ⟶ E (resp. \overline{μ} : \overline{A} ⟶ \overline{E}) in D^{+}(C) (resp. D^{+}(\overline{C})) such that the diagram

A  ⟶^{μ}  E
↓  ↓^{≀ρ}
(Ru)(\overline{A}) ⟶^{Ru(\overline{μ})} (Ru)(\overline{E})

is commutative. Denote the composition

(R^{i+1}Ψ)(A)⟶^{μ}(R^{i+1}Ψ)(E) ⟶^{∼} (R^{i+1}Φ)(\overline{E})

by ν and write λ^{j} for the map (R^{j}Ψ)(B) ⟶ (R^{j}Ψ)(C).

We assume the following axioms hold: (A1) Ker [H^{j}(\overline{A}) ⟶ H^{j}(\overline{B})] = 0 for j = i, i + 1.

(A2) \overline{μ} induces an isomorphism τ_{≤i}\overline{A} ⟶^{∼} τ_{≤i}\overline{E} and a monomorphism Φ(H^{i+1}(\overline{A})) ↪ Φ(H^{i+1}(\overline{E})). (A3) The canonical maps (R^{j}Ψ)(B) ⟶ Φ(H^{j}(\overline{B})), (R^{j}Ψ)(C) ⟶ Φ(H^{j}(\overline{C})) (for j = i, i + 1) induce

isomorphisms

Coker(λ^{i}) ⟶^{∼} Coker [Φ(H^{i}(\overline{B})) ⟶ Φ(H^{i}(\overline{C}))]
Ker(λ^{i+1}) ⟶^{∼} Ker [Φ(H^{i+1}(\overline{B})) ⟶ Φ(H^{i+1}(\overline{C}))]

Note that (A2) resp. (A3) follow from stronger axioms (A2’) \overline{μ} induces an isomorphism τ_{≤i+1}\overline{A} ⟶^{∼} τ_{≤i+1}\overline{E}. (A3’) The canonical maps (R^{j}Ψ)(B) ⟶ Φ(H^{j}(\overline{B})), (R^{j}Ψ)(C) ⟶ Φ(H^{j}(\overline{C})) are isomorphisms for j =

i, i + 1.

(3.2) The spectral sequence E_{2}^{p,q} = (R^{p}Φ)(H^{q}(\overline{A})) ⟹ (R^{p+q}Φ)(\overline{A}) defines a decreasing filtration F^{i}(R^{j}Φ)(\overline{A}) and homomorphisms

\begin{aligned}edge & : (R^{j}Φ)(\overline{A}) ⟶ E_{∞}^{0,j} ↪ E_{2}^{0,j} = Φ(H^{j}(\overline{A})) \\ δ & : F^{1}(R^{j}Φ)(\overline{A}) = Ker(edge) ⟶ E_{∞}^{1,j−1} ↪ E_{2}^{1,j−1} = (R^{1}Φ)(H^{j−1}(\overline{A})) \end{aligned}

(and similarly for \overline{B}, \overline{C}). The maps in (A3) are given by

(R^{j}Ψ)(B) = H^{j}(RΨ(B)) ⟶^{(3.1.4)}H^{j}(RΨRu(\overline{B})) ⟶^{∼} H^{j}(RΦ(\overline{B})) = (R^{j}Φ)(\overline{B})⟶^{edge}Φ(H^{j}(\overline{B}))

Proposition. For every j ∈ Z, the following diagram is commutative:

Ker [(R^{j}Φ)(\overline{C})⟶^{edge}Φ(H^{j}(\overline{C})) ⟶ Φ(H^{j+1}(\overline{A}))]  ⟶ F^{1}(R^{j+1}Φ)(\overline{A})
↓^{edge}  ↓^{δ}
Ker [Φ(H^{j}(\overline{C})) ⟶ Φ(H^{j+1}(\overline{A}))] ⟶^{∂}(R^{1}Φ)(Im[H^{j}(\overline{A}) ⟶ H^{j}(\overline{B})]) ⟵ (R^{1}Φ)(H^{j}(\overline{A}))

Here ∂ is the coboundary map in the cohomology exact sequence of

0 ⟶ Im[H^{j}(\overline{A}) ⟶ H^{j}(\overline{B})] ⟶ H^{j}(\overline{B}) ⟶ Ker[H^{j}(\overline{C}) ⟶ H^{j+1}(\overline{A})] ⟶ 0

Proof. This is a derived category version of [Ja 3, Lemma 9.5].

(3.3) The axiom (A1) implies that

Im [(R^{i}Φ)(\overline{C}) ⟶ (R^{i+1}Φ)(\overline{A})] ⊆ F^{1}(R^{i+1}Φ)(\overline{A}),  (3.3.1)

the sequence

0 ⟶ H^{i}(\overline{A}) ⟶ H^{i}(\overline{B}) ⟶ H^{i}(\overline{C}) ⟶ 0

is exact and the maps

Φ(H^{j}(\overline{A})) ⟶ Ker [Φ(H^{j}(\overline{B})) ⟶ Φ(H^{j}(\overline{C}))]  (3.3.2)

are isomorphisms for j = i, i + 1. According to the axiom (A2), \overline{μ} induces isomorphisms

\begin{matrix}(R^{j}Φ)(\overline{A}) ⟶^{∼} (R^{j}Φ)(\overline{E}), H^{j}(\overline{A}) ⟶^{∼} H^{j}(\overline{E})  ( & ∀j ≤ i) \\ F^{1}(R^{j}Φ)(\overline{A}) ⟶^{∼} F^{1}(R^{j}Φ)(\overline{E})  ( & ∀j ≤ i + 1) \end{matrix}  (3.3.3)

and a monomorphism (which is an isomorphism, if (A2’) holds)

F^{1}(R^{i+1}Φ)(\overline{A}) ↪ F^{1}(R^{i+1}Φ)(\overline{E}).  (3.3.4)

By transport of structure, we obtain an exact sequence

0 ⟶ H^{i}(\overline{E}) ⟶ H^{i}(\overline{B}) ⟶ H^{i}(\overline{C}) ⟶ 0

Denote by

∂ : Φ(H^{i}(\overline{C})) ⟶ (R^{1}Φ)(H^{i}(\overline{E}))

the corresponding boundary map.

(3.4) Corollary. Assuming (A1), (A2), the following diagram is commutative

Coker [(R^{i}Φ)(\overline{B})) ⟶ (R^{i}Φ)(\overline{C}))] ⟶^{(3.3.1)} F^{1}(R^{i+1}Φ)(\overline{A}) ⟶^{\overline{μ}} F^{1}(R^{i+1}Φ)(\overline{E})
↓^{edge}  ↓^{δ}  ↓^{δ}
Coker [Φ(H^{i}(\overline{B})) ⟶ Φ(H^{i}(\overline{C}))] ↪^{∂} (R^{1}Φ)(H^{i}(\overline{A})) ⟶^{\overline{μ}} (R^{1}Φ)(H^{i}(\overline{E}))

Both maps \overline{μ} are isomorphisms, by (3.3.3).

(3.5) Proposition. Suppose the axioms (A1)–(A3) hold. Then

(1) There is a commutative diagram with exact rows, with vertical maps induced by ν:

0 ⟶  Coker(λ^{i})  ⟶ (R^{i+1}Ψ)(A) ⟶ Ker(λ^{i+1}) ⟶ 0
↓^{β}  ↓^{ν}  ↓^{α}
0 ⟶ F^{1}(R^{i+1}Φ)(\overline{E}) ⟶ (R^{i+1}Φ)(\overline{E}) ⟶^{edge} Φ(H^{i+1}(\overline{E}))
↓^{δ}
(R^{1}Φ)(H^{i}(\overline{E}))

(2) The composition δ ∘ β is equal to the map ∂ (composed with the isomorphism of (A3)). In particular, Ker(δ ∘ β) = 0.

(3) Ker(α) = 0. If (A2’) holds, then α is an isomorphism and Coker(edge) = 0.

(4) Ker(ν) = 0.

Proof. (1),(2) The first row is the cohomology sequence of Δ. The commutativity of the right square is clear. The commutativity of the left square follows from Corollary 3.4 and the commutative diagram

Coker(λ^{i})  ⟶  (R^{i+1}Ψ)(A)
↓  ↓
Coker [(R^{i}Φ)(\overline{B})) ⟶ (R^{i}Φ)(\overline{C}))] ↪ F^{1}(R^{i+1}Φ)(\overline{A}) ↪ (R^{i+1}Φ)(\overline{A})

(3) The map α is the composition of the monomorphism Φ(H^{i+1}(\overline{A})) ↪ Φ(H^{i+1}(\overline{E})) (which is an isomorphism, if (A2’) is satisfied), and the isomorphisms (3.3.2) and (A3).

(4) Follows from Ker(α) = Ker(β) = 0.

(3.6) Define a filtration on (R^{j}Ψ)(A) by

F^{0}(R^{j}Ψ)(A) = (R^{j}Ψ)(A)
F^{1}(R^{j}Ψ)(A) = Ker [(R^{j}Ψ)(A) ⟶ (R^{j}Ψ)(B)] ⟵^{∼} Coker(λ^{j−1})
F^{m}(R^{j}Ψ)(A) = 0  (m > 1)

Proposition 3.5 can then be reformulated as follows:

Proposition. (1) For every j ≥ 0, we have ν(F^{j}(R^{i+1}Ψ)(A)) ⊆ F^{j}(R^{i+1}Φ)(\overline{E})) and the induced map gr_{F}^{j} (ν) : gr_{F}^{j} (R^{i+1}Ψ)(A) ⟶ gr_{F}^{j} (R^{i+1}Φ)(\overline{E}) = E_{∞}^{j,i+1−j} is a monomorphism.

(2) Im [gr_{F}^{1} (R^{i+1}Ψ)(A) ↪ E_{∞}^{1,i} ↪ E_{2}^{1,i}] = Im(∂).

(3) If (A2’) is satisfied, then the map

gr_{F}^{0} (R^{i+1}Ψ)(A) ↪ E_{∞}^{0,i+1} ↪ E_{2}^{0,i+1} = Φ(H^{i+1}[\overline{E}))

is an isomorphism and E_{∞}^{0,i+1} = E_{2}^{0,i+1}.

II. Local situation at ℓ ≠ p

There is nothing new in this chapter. We simply summarize well-known results on étale cohomology and p-adic regulators for varieties over ℓ-adic fields.

1. Étale cohomology

(1.1) Geometric situation. Let ℓ ≠ p be prime numbers, K/Q_{ℓ} a finite extension, O_{K} ⊂ K the ring of integers, k the residue field of O_{K}, I_{K} ⊂ G_{K} the inertia subgroup. For a morphism π : 𝔛 ⟶ Spec(O_{K}), denote by π_{η} : X ⟶ Spec(K) = η (resp. π_{s} : Y ⟶ Spec(k) = s) its generic (resp. special) fibre. Denote the corresponding open (resp. closed) immersion by j : X ↪ 𝔛 (resp. i : Y ↪ 𝔛).

(1.2) Cohomology. For a fixed r ∈ Z, there is a projective system of abelian sheaves [Z/p^{n}Z(r)]_{n∈N} on 𝔛_{et}. The Leray spectral sequences (1.5.2) for π, π_{η}, π_{s} (tensored by Q) are

E_{2}^{a,b} = H^{a}(Spec(O_{K})_{et}, [R^{b}π_{∗}(Z/p^{n}Z(r))]_{n∈N}) ⊗ Q ⟹ H^{a+b}(𝔛_{et}, Q_{p}(r))
sE_{2}^{a,b} = H^{a}(Spec(k)_{et}, [R^{b}π_{s∗}(Z/p^{n}Z(r))]_{n∈N}) ⊗ Q ⟹ H^{a+b}(Y_{et}, Q_{p}(r))
ηE_{2}^{a,b} = H^{a}(Spec(K)_{et}, [R^{b}π_{η∗}(Z/p^{n}Z(r))]_{n∈N}) ⊗ Q ⟹ H^{a+b}(X_{et}, Q_{p}(r))

(1.2.1)

(see (2.3.3) for the notation). If π is a morphism of finite type, then

\begin{aligned}^{s}E_{2}^{a,b} & = H_{cont}^{a}(G_{k}, H^{b}(\overline{Y}_{et}, Z_{p}(r))) ⊗ Q ⟶^{∼} H_{cont}^{a}(G_{k}, H^{b}(\overline{Y}_{et}, Q_{p}(r))) \\ ^{η}E_{2}^{a,b} & = H_{cont}^{a}(G_{K}, H^{b}(\overline{X}_{et}, Z_{p}(r))) ⊗ Q ⟶^{∼} H_{cont}^{a}(G_{K}, H^{b}(\overline{X}_{et}, Q_{p}(r))) \end{aligned}  (1.2.2)

by (I.1.9) and (I.2.3.3.2). According to (I.2.3.3–4), we have

H^{q}(Z_{et}, Z_{p}(r)) ⟶^{∼} H_{naive}^{q}(Z_{et}, Z_{p}(r))  (1.2.3)

for all Z = X, \overline{X}, Y, \overline{Y}.

(1.3) Base change. The immersions i, j induce homomorphisms

\begin{matrix}^{s}E_{c}^{a,b}⟵^{i^{∗}} & E_{c}^{a,b}⟶^{j^{∗}}^{η}E_{c}^{a,b} \\ H^{a+b}(Y_{et}, Q_{p}(r))⟵^{i^{∗}} & H^{a+b}(𝔛_{et}, Q_{p}(r))⟶^{j^{∗}}H^{a+b}(X_{et}, Q_{p}(r)) \end{matrix}  (1.3.1)

As cd_{p}(k) = 1 and cd_{p}(K) = 2, (I.2.3.3.4) implies that ^{s}E_{2}^{a,b} = 0 for a > 1 and ^{η}E_{2}^{a,b} = 0 for a > 2. Thus sE_{2}^{a,b} = ^{s}E_{∞}^{a,b}.

The filtrations F^{.} induced by the spectral sequences (1.2.1) on their abutments are compatible with the maps i^{∗}, j^{∗} in (1.3.1). From now on, assume that π is proper. Then i^{∗} induces an isomorphism of spectral sequences E_{c}^{a,b} and ^{s}E_{c}^{a,b}, by (I.2.4.1–2). This means that F^{2}H^{i}(𝔛_{et}, Q_{p}(r)) = 0 and that the map δ of (I.1.5.3) (for π_{s}) is an isomorphism, sitting in a commutative diagram

\begin{matrix}F^{1}H^{i+1}(Y_{et}, Q_{p}(r)) ⟵^{∼} & F^{1}H^{i+1}(𝔛_{et}, Q_{p}(r)) & ⟶ F^{1}H^{i+1}(X_{et}, Q_{p}(r)) \\ ↓^{≀} & & ↓^{δ} \\ H_{cont}^{1}(G_{k}, H^{i}(\overline{Y}_{et}, Q_{p}(r))) & ⟶ & H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) \end{matrix}  (1.3.2)

All of its entries (as well as H^{i}(\overline{X}_{et}, Q_{p}(r)) and H^{i}(\overline{Y}_{et}, Q_{p}(r))) are Q_{p}-vector spaces of finite dimension, by (I.2.3.4). Write V_{s}^{i} (resp. V_{η}^{i}) for H^{i}(\overline{Y}_{et}, Q_{p}) (resp. H^{i}(\overline{X}_{et}, Q_{p})). The graded pieces for the filtrations F^{.} satisfy

\begin{aligned}gr_{F}^{0} H^{i+1}(𝔛_{et}, Q_{p}(r)) & = E_{∞}^{0,i+1} ⟶^{∼} ^{s}E_{∞}^{0,i+1} = ^{s}E_{2}^{0,i+1} = (V_{s}^{i+1}(r))^{G_{k}} \\ gr_{F}^{1} H^{i+1}(𝔛_{et}, Q_{p}(r)) & = E_{∞}^{1,i} ⟶^{∼} ^{s}E_{∞}^{1,i} = ^{s}E_{2}^{1,i} = H_{cont}^{1}(G_{k}, V_{s}^{i}(r)) \\ gr_{F}^{j} H^{i+1}(𝔛_{et}, Q_{p}(r)) & = 0  (∀j > 1) \end{aligned}  (1.3.3.1)

and

\begin{aligned}gr_{F}^{0} H^{i+1}(X_{et}, Q_{p}(r)) & = ^{η}E_{∞}^{0,i+1} ↪ ^{η}E_{2}^{0,i+1} = (V_{η}^{i+1}(r))^{G_{K}} \\ gr_{F}^{1} H^{i+1}(X_{et}, Q_{p}(r)) & = ^{η}E_{∞}^{1,i} = ^{η}E_{2}^{1,i} = H_{cont}^{1}(G_{K}, V_{η}^{i}(r)) \end{aligned}  (1.3.3.2)

The homomorphisms

\begin{aligned}gr_{F}^{0} (j^{∗}) & : (V_{s}^{i+1}(r))^{G_{k}} ⟶ (V_{η}^{i+1}(r))^{G_{K}} \\ gr_{F}^{1} (j^{∗}) & : H_{cont}^{1}(G_{k}, V_{s}^{i}(r)) ⟶ H_{cont}^{1}(G_{K}, V_{η}^{i}(r)) \end{aligned}  (1.3.4)

are induced by the specialization maps

sp^{j} : V_{s}^{j} ⟶ (V_{η}^{j})^{I_{K}}  (1.3.5)

for j = i + 1 (resp. j = i). In particular, the bottom arrow in (1.3.2) is equal to gr_{F}^{1} (j^{∗}), hence factors through the unramified Galois cohomology:

H_{cont}^{1}(G_{k}, V_{s}^{i}(r)) ⟶ H_{cont}^{1}(G_{k}, (V_{η}^{i}(r))^{I_{K}}) = H_{ur}^{1}(G_{K}, V_{η}^{i}(r)) ↪ H_{cont}^{1}(G_{K}, V_{η}^{i}(r))  (1.3.6)

According to Deligne ([De 4], Thm. 1), V_{s}^{i}(r) is mixed of weights ≤ i − 2r. Consequently,

\begin{matrix}gr_{F}^{0} H^{i+1}(𝔛_{et}, Q_{p}(r)) = 0,  if r > (i + 1)/2 \\ gr_{F}^{1} H^{i+1}(𝔛_{et}, Q_{p}(r)) = 0,  if r > i/2 \end{matrix}  (1.3.7)

If π is proper and smooth, then I_{K} acts trivially on V_{η}^{i} and sp^{i} is an isomorphism (for all i ≥ 0), by the proper and smooth base change theorems.

(1.4) Proposition. Let π : 𝔛 ⟶ Spec(O_{K}) be proper.

(1) If r > (i + 1)/2, then H^{i+1}(𝔛_{et}, Q_{p}(r)) = 0.

(2) If r = (i + 1)/2, then the edge homomorphism H^{2r}(𝔛_{et}, Q_{p}(r)) ⟶ V_{s}^{2r}(r)^{G_{k}} is an isomorphism.

Proof. Everything follows from (1.3.7) and (1.3.3.1).

(1.5) Purity conjecture for the monodromy filtration. Suppose that π : 𝔛 ⟶ Spec(O_{K}) is proper and its generic fibre π_{η} : X ⟶ Spec(K) is smooth. According to Grothendieck ([ST], Appendix), there is a finite extension K′/K such that I_{K′} acts on H := H^{i}(\overline{X}_{et}, Q_{p}) unipotently, through the p-part of its tame quotient. This gives rise ([De 4], Prop. 1.6.1) to a monodromy operator N : H ⟶ H(−1) and the corresponding monodromy filtration M_{i}H (increasing, exhaustive and separating). It is characterized by

N(M_{j}H) ⊆ M_{j−2}(−1) (j ∈ Z); N^{j} : Gr_{j}^{M}(H) ⟶^{∼} Gr_{−j}^{M}(H)(−j)

The purity conjecture for the monodromy filtration ([De 2], 8.1) predicts that

H = M_{i}H (⟺ M_{−i−1}H = 0)  (1.5.1)
Over K′, Gr_{j}^{M}(H) is unramified and pure of weight i + j (∀j ∈ Z)  (1.5.2)

The characteristic polynomial det(1 − T · F r_{k′} | Gr_{j}^{M}(H)) has coefficients in Q

and is independent of p  (1.5.3)

If true, then the weights of

H^{i}(\overline{X}_{et}, Q_{p})^{I_{K}} ⊆ H^{i}(\overline{X}_{et}, Q_{p})^{I′_{K}} = Ker(N : H ⟶ H(−1))  (1.5.4)

are contained in the set {0, 1, …, i} and all weights of

H^{i}(\overline{X}_{et}, Q_{p})_{I_{K}} ⟵ H^{i}(\overline{X}_{et}, Q_{p})_{I′_{K}} = Coker(N(1) : H(1) ⟶ H)  (1.5.5)

are contained in {i, i + 1, …, 2i}.

(1.6) The conjecture (1.5.1–3) has been proved in the following cases: (1.6.0) i = 0 (trivial). (1.6.1) X has good reduction ([De 3]). In this case H is pure of weight i: H = M_{0}H and M_{−1}H = 0. (1.6.2) i = 1 ([SGA 7], Exp. IX, Thm. 3.6). (1.6.3) dim(X) = 2 ([RZ], Thm. 2.13, assuming that X has semistable reduction; it follows from de Jong’s theorem on alterations [de J] that this assumption is unnecessary). (1.6.4) X is an abelian variety (by (1.6.2) and the fact that the cohomology ring of \overline{X} is the exterior algebra over H^{1}). (1.6.5) If X is equidimensional of dim(X) = d, then (1.5.1–3) is valid for i iff it is valid for 2d − i, by the Poincaré duality. (1.6.6) If X has a potentially semistable reduction, then (1.5.1) is proved in [RZ] (cf. [Il 2, Cor. 3.3]). As in 1.6.3, the general case follows from de Jong’s theorem.

(1.7) The following statement is well-known ([Ja 2], [So 1]). It shows that, in many cases, the cohomology group H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) vanishes.

Proposition. (1) If X is proper and smooth over K and the purity conjecture for the monodromy filtration holds for H^{i}(\overline{X}_{et}, Q_{p}), then

(a) H^{0}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = H_{ur}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0  for  \{ \begin{matrix}r < 0 \\ r > i/2 \end{matrix}

In particular, the ”local Euler factor” of H^{i}(X) has no pole for these values of s = r.

(b) H_{cont}^{2}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0  for  \{ \begin{matrix}r < i/2 + 1 \\ r > i + 1 \end{matrix}

(c) H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0  for  \{ \begin{matrix}r < 0 \\ r = (i + 1)/2 \\ r > i + 1 \end{matrix}

(2) If, furthermore, X has good reduction, then

(a) H^{0}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = H_{ur}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0 for r ≠ i/2.

(b) H_{cont}^{2}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0 for r ≠ i/2 + 1.

(c) H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r))) = 0 for r ≠ i/2, i/2 + 1.

(d) H^{i+1}(X_{et}, Q_{p}(r)) = 0 for r ≠ i/2, (i + 1)/2, i/2 + 1.

(e) H^{i+1}(𝔛_{et}, Q_{p}(r)) = H^{i+1}(Y_{et}, Q_{p}(r)) = 0 for r ≠ i/2, (i + 1)/2. (here 𝔛 is a model of X, proper and smooth over Spec(O_{K}), and Y its special fibre). In fact, the statements (a)–(d) are true if X has only a potentially good reduction.

Proof. (1) This is a standard weight argument. Putting V := H^{i}(\overline{X}_{et}, Q_{p}(r)), we see that V^{I_{K}} has no invariants (resp. coinvariants) by the Frobenius for r < 0 or r > i/2, by (1.5.4), proving (a). By local duality ([Se 1], II.5.2.2), H_{cont}^{2}(G_{K}, V) is dual to H^{0}(G_{K}, V^{∗}(1)). The statement (b) follows from (1.5.5), as V^{∗}(1)^{I_{K′}} = (V_{I_{K′}})^{∗}(1) has no invariants under Frobenius for the indicated values of r. For (c), we use the formula for the local Euler characteristic ([Se 1], II.5.4.17): dim H^{0}(G_{K}, V) − dim H^{1}(G_{K}, V) + dim H^{2}(G_{K}, V) = 0.

(2) In (a)–(d), all the groups involved satisfy Galois descent for finite Galois extensions L/K. This means that all the statements for X with a potentially good reduction are reduced, by a suitable base change, to the case of a good reduction. The argument for (a)–(c) is then the same as in (1), simplified by the fact that V is unramified and pure of weight i − 2r. (d) follows from the Hochschild-Serre spectral sequence and (a)–(c). For (e), observe that the two cohomology groups are isomorphic by the proper base change (2.4.1). As in (1.3), the Hochschild-Serre spectral sequence gives an exact sequence

0 ⟶ H_{cont}^{1}(G_{k}, H^{i}(\overline{Y}_{et}, Q_{p}(r))) ⟶ H^{i+1}(Y_{et}, Q_{p}(r)) ⟶ H^{i+1}(\overline{Y}_{et}, Q_{p}(r))^{G_{k}} ⟶ 0

and we apply the vanishing statements (a) (resp. (c)) for the first (resp. the third) term.

(1.8) If π is semistable, then the specialization maps sp^{i} can be analysed using the spectral sequence of vanishing cycles and an explicit construction of the monodromy operator N [RZ], [Il 2]. For example, if the purity conjecture (1.5.2) for the monodromy filtration on H^{i}(\overline{X}_{et}, Q_{p}) holds, then sp^{i} is surjective.

(1.9) The discussion in (1.3.1-6) remains valid if one replaces everywhere Q_{p} by Z_{p} (and V_{s}^{i} (resp. V_{η}^{i}) by T_{s}^{i} = H^{i}(\overline{Y}_{et}, Z_{p}) (resp. T_{η}^{i} = H^{i}(\overline{X}_{et}, Z_{p}))). In particular,

Im [δ ∘ j^{∗} : F^{1}H^{i+1}(𝔛_{et}, Z_{p}(r)) ⟶ H_{cont}^{1}(G_{K}, T_{η}^{i}(r))] ⊆ H_{ur}^{1}(G_{K}, T_{η}^{i}(r))

If π is proper and smooth, then T_{η}^{i} is unramified and sp^{i} : T_{s}^{i} ⟶^{∼} T_{η}^{i} is an isomorphism (for all i ≥ 0). Consequently,

gr_{F}^{0} (j^{∗}) : (T_{s}^{i+1}(r))^{G_{k}} ⟶ (T_{η}^{i+1}(r))^{G_{K}}

is an isomorphism and

gr_{F}^{1} (j^{∗}) : H_{cont}^{1}(G_{k}, T_{s}^{i}(r)) ⟶ H_{cont}^{1}(G_{K}, T_{η}^{i}(r))

is injective, with image equal to H_{ur}^{1}(G_{K}, T_{η}^{i}(r)). This implies that the restriction map

j^{∗} : H^{i+1}(𝔛_{et}, Z_{p}(r)) ⟶ H^{i+1}(X_{et}, Z_{p}(r))  (1.9.1)

is injective and that

F^{k}H^{i+1}(𝔛_{et}, Z_{p}(r)) = (j^{∗})^{−1} (F^{k}H^{i+1}(X_{et}, Z_{p}(r))) ,

for every k ≥ 0. In particular,

Im(j^{∗}) ∩ F^{2}H^{i+1}(X_{et}, Z_{p}(r)) = 0.  (1.9.2)

2. Regulators

(2.1) K-theory. In the situation of (1.1), suppose that π : 𝔛 ⟶ Spec(O_{K}) is of finite type. Then, for r > (i + 1)/2, there are étale Chern classes ([Gi])

K_{2r−i−1}(𝔛)  ⟶^{j^{∗}}  K_{2r−i−1}(X)
↓^{c_{i+1,r}}  ↓^{c_{i+1,r}}
H^{i+1}(𝔛_{et}, Z/p^{n}Z(r)) ⟶^{j^{∗}} H^{i+1}(X_{et}, Z/p^{n}Z(r))

which factor through K_{2r−i−1}(𝔛, Z/p^{n}Z) (resp. K_{2r−i−1}(X, Z/p^{n}Z)). Taking projective limit and using (1.2.3), we obtain a commutative diagram

\begin{matrix}K_{2r−i−1}(𝔛) ⊗ Q & ⟶^{j^{∗}} & K_{2r−i−1}(X) ⊗ Q \\ ↓^{ch_{i+1,r}} & & ↓^{ch_{i+1,r}} \\ H^{i+1}(𝔛_{et}, Q_{p}(r)) & ⟶^{j^{∗}} & H^{i+1}(X_{et}, Q_{p}(r)) \end{matrix}  (2.1.1)

with ch_{i+1,r} = (−1)^{r−1}/(r − 1)! · c_{i+1,r} (the scalar factor ensures that the Chern character ch is compatible with products). This map was denoted by r_{p} in the Introduction.

(2.2) Theorem. If π : 𝔛 ⟶ Spec(O_{K}) is proper and r > (i + 1)/2, then

Im [K_{2r−i−1}(𝔛) ⊗ Q⟶^{ch∘j^{∗}}H^{i+1}(X_{et}, Q_{p}(r))] = 0

Proof. In the commutative diagram (2.1.1), the group H^{i+1}(𝔛_{et}, Q_{p}(r)) vanishes, by Proposition 1.4.1.

(2.3) Algebraic cycles. Let us now consider the “central point” r = (i + 1)/2. Assume that π : 𝔛 ⟶ Spec(O_{K}) is a separated morphism of finite type with 𝔛 regular. Cycle classes define a commutative diagram

\begin{matrix}CH^{r}(𝔛) ⊗ Q & ⟶^{j^{∗}} & CH^{r}(X) ⊗ Q \\ ↓^{cl_{𝔛}} & & ↓^{cl_{X}} \\ H^{2r}(𝔛_{et}, Q_{p}(r)) & ⟶^{j^{∗}} & H^{2r}(X_{et}, Q_{p}(r)) \end{matrix}  (2.3.1)

The vertical maps can be defined in several ways (see [Grot], [Sa] in the smooth case); one possibility is to use K-theory:

\begin{aligned}cl_{𝔛} : CH^{r}(𝔛) ⊗ Q & ⟶^{∼} K_{0}(𝔛)_{Q}^{(r)} ⟶ H^{2r}(𝔛_{et}, Q_{p}(r)) \\ cl_{X} : CH^{r}(X) ⊗ Q & ⟶^{∼} K_{0}(X)_{Q}^{(r)} ⟶ H^{2r}(X_{et}, Q_{p}(r)) \end{aligned}

These isomorphisms are due to Grothendieck ([SGA 6], Exp. XIV, 4.1) and Soulé ([So 2], Thm. 4(iv); [GiSo], Thm. 8.2); the maps from K-theory to étale cohomology are given by (−1)^{r−1}/(r − 1)! · c_{r}, where c_{r} is the r-th Chern class of a vector bundle ([SGA 5], Exp. VII, Prop. 3.4; [Ja 1], Thm. 6.12). Denote the subgroups of homologically trivial cycles by

(CH^{r}(X) ⊗ Q)_{0} :=Ker [CH^{r}(X) ⊗ Q ⟶ H^{2r}(\overline{X}_{et}, Q_{p}(r))]
(CH^{r}(𝔛) ⊗ Q)_{0} :=Ker [CH^{r}(𝔛) ⊗ Q ⟶ H^{2r}((𝔛 ⊗_{O_{K}} O_{K}^{ur})_{et}, Q_{p}(r))]

Suppose, furthermore, that π is proper. Then

H^{2r}((𝔛 ⊗_{O_{K}} O_{K}^{ur})_{et}, Q_{p}(r)) ⟶^{∼} H^{2r}(\overline{Y}_{et}, Q_{p}(r)) = V_{s}^{2r}(r)  (2.3.2)

by the proper base change (I.2.4.1).

(2.4) Theorem. Assume π : 𝔛 ⟶ Spec(O_{K}) is proper and 𝔛 is regular. Then

Im [(CH^{r}(𝔛) ⊗ Q)_{0}⟶^{j^{∗}}(CH^{r}(X) ⊗ Q)_{0}⟶^{cl_{X}}H^{2r}(X_{et}, Q_{p}(r))] = 0.

Proof. By (2.3.2) and Proposition 1.4.2, we have (CH^{r}(𝔛) ⊗ Q)_{0} = Ker(cl_{𝔛}). We conclude by (2.3.1).

(2.5) As in 1.5, assume that π is proper with smooth generic fibre π_{η}. A folklore conjecture states that

cl_{X}((CH^{r}(X) ⊗ Q)_{0}) = 0.  (2.5.1)

According to (1.3.3.2),

(CH^{r}(X) ⊗ Q)_{0} = cl_{X}^{−1}(F^{1}H^{2r}(X_{et}, Q_{p}(r))),

which means that (2.5.1) is equivalent to

Im(cl_{X}) ∩ F^{1}H^{2r}(X_{et}, Q_{p}(r)) = 0.  (2.5.2)

Note that the purity conjecture for the monodromy filtration on H^{2r−1}(\overline{X}_{et}, Q_{p}) would imply that

F^{1}H^{2r}(X_{et}, Q_{p}(r)) = F^{2}H^{2r}(X_{et}, Q_{p}(r)),  (2.5.3)

by (1.3.3.2) and Proposition 1.7.1(c).

(2.6) Proposition. Let π be proper and π_{η} smooth. The conjecture (2.5.1) holds in the following cases.

(i) X is equidimensional of dimension dim(X) = r (zero cycles). (ii) r = 1 (divisors). (iii) 𝔛 is regular and the map j^{∗} : (CH^{r}(𝔛) ⊗ Q)_{0} ⟶ (CH^{r}(X) ⊗ Q)_{0} is surjective. (iv) π is proper and smooth (good reduction).

Proof. (i) [Be 3, 2.2], [Ra, Prop. 3.2]. (ii) There is a canonical isomorphism α : A(K) ⊗ Q ⟶^{∼} (CH^{1}(X) ⊗ Q)_{0}, where A = Pic^{0}(X/F). The composition cl_{X} ∘ α is the descent map

A(K) ⊗ Q ⟶ A(K)ˆ⊗Q_{p} ↪ H_{cont}^{1}(G_{K}, V_{p}(A)).

The group A(K) has a subgroup of finite index, isomorphic to the direct sum of several copies of Z_{ℓ}. As ℓ ≠ p, this implies that the completed tensor product A(K)ˆ⊗Q_{p} vanishes. (iii) This follows from Theorem 2.4. (iv) In this case the map j^{∗} is surjective, since sp^{2r} is an isomorphism. We conclude by (iii).

(2.7) Consider the base change map u : X ⊗_{K} L ⟶ X for a finite field extension L/K of degree d. There are contravariant (resp. covariant) maps u^{∗} (resp. u_{∗}) between Chow groups, étale cohomology and the terms of the Hochschild-Serre spectral sequences of X resp. X ⊗_{K} L. They satisfy the usual identitites:

u_{∗} ∘ u^{∗} = d · 1; u^{∗} ∘ u_{∗} = ∑_{g∈G(L/K)} g (if L/K is Galois)

This implies that the restriction maps

u^{∗} : CH^{r}(X) ⊗ Z[1/d] ⟶ CH^{r}(X ⊗_{K} L) ⊗ Z[1/d]  (2.7.1)

and

u^{∗} : F^{j}H^{m}(X_{et}, Z_{p}(r)) ⊗ Z[1/d] ⟶ F^{j}H^{m}((X ⊗_{K} L)_{et}, Z_{p}(r)) ⊗ Z[1/d]  (2.7.2)

are both injective. Moreover, for L/K Galois, the image of u^{∗} coincides with G(L/K)-invariants of the R.H.S. It follows from the injectivity of (2.7.2) that the conditions (iii) and (iv) of Proposition 2.6 can be replaced by (iii’) There is a finite extension L/K and a proper regular model 𝔛_{L} of X ⊗_{K} L over Spec(O_{L}) such that the map

j^{∗} : (CH^{r}(𝔛_{L}) ⊗ Q)_{0} ⟶ (CH^{r}(X ⊗_{K} L) ⊗ Q)_{0}

is surjective. (iv’) X has potentially good reduction.

(2.8) Proposition. If L/K is a finite extension of degree prime to p such that X ⊗_{K} L has good reduction, then

Im(cl_{X}) ∩ F^{2}H^{2n}(X_{et}, Z_{p}(n)) = 0.

Proof. For X ⊗_{K} L this follows from (1.9.2); we can descend to X thank to the injectivity of (2.7.2).

(2.9) It follows from a recent work of K. Künnemann [Kn] that (2.5.1) holds for abelian varieties with potentially totally toric reduction. Another (partial) result in this direction is proved in ([GrSc], Prop. 7.2), where the authors show that the class of a modified diagonal on the triple product of a semistable curve lies in the image of j^{∗}.

(2.10) Proof of Theorems A and D(1-2). Theorem A is a combination of Theorems 2.2 and 2.4. Theorem D(1-2) follows from Proposition 2.6 and remarks in 2.7.

III. Local situation at p

In this chapter we prove a comparison result (Theorem 3.2) that relates syntomic cohomology (in the case of good reduction) to the Bloch-Kato exponential map. Similar result (Theorem 5.2) is proved for cohomology with Z_{p}-coefficients, under more restrictive hypotheses. As a consequence we obtain upper bounds on the images of p-adic regulators.

1. p-Adic Galois representations

In this section we recall Fontaine’s machinery relating crystalline and étale cohomology and also the exponential map of Bloch-Kato.

(1.1) Notation. Let k be a perfect field of characteristic char(k) = p > 0; W = W (k) the ring of Witt vectors of k; K_{0} = W [p^{−1}] the fraction field of W; σ the lifting of the absolute Frobenius a ↦ a^{p} to W and K_{0}; K a finite totally ramified extension of K_{0}. We fix an algebraic closure \overline{K} of K and write G for the Galois group G(\overline{K}/K). For any scheme S over Spec(W) put S_{n} = S ⊗_{W} W_{n}.

(1.2) Fontaine [Fo 2, 4] constructed topological K_{0}-algebras B_{cris}, B_{dR} equipped with the following structure:

(i) A continuous K_{0}-linear action of G on B_{cris} and B_{dR}. (ii) A commutative diagram of (G-equivariant and injective) continuous K_{0}-algebra homomorphisms

K_{0}^{ur} ↪ B_{cris}^{+} ↪ B_{cris}
∩ ↓  ∩ ↓  ∩ ↓
\overline{K} ↪ B_{dR}^{+} ↪ B_{dR}

(iii) A σ-linear continuous bijective Q_{p}-algebra homomorphism f : B_{cris} ⟶ B_{cris}, commuting with G and

preserving B_{cris}^{+}. (iv) An exhaustive and separated decreasing filtration F^{i}B_{dR} (by \overline{K}[G]-submodules of B_{dR}) with B_{dR}^{+} =

F^{0}B_{dR}. Warning: B_{cris}^{+} ≠ B_{dR}^{+} ∩ B_{cris}.

(v) A Q_{p}[G]-equivariant injective map

Q_{p}(1) = (\underleftarrow{lim}_{n} μ_{p^{n}}(\overline{K})) ⊗_{Z_{p}} Q_{p} ↪ B_{cris}^{+}

such that, for any 0 ≠ t ∈ Q_{p}(1) ⊂ B_{cris}^{+} ⊂ B_{dR}^{+}, we have f(t) = pt, F^{n}B_{dR} = t^{n}B_{dR}^{+} (n ∈ Z), B_{cris} = B_{cris}^{+}[t^{−1}] and B_{dR} = B_{dR}^{+}[t^{−1}].

(vi) (B_{cris})^{G} = (B_{cris}^{+})^{G} = K_{0}, (B_{dR})^{G} = (B_{dR}^{+})^{G} = K.

(1.3) These rings have a crystalline interpretation (see Appendix for the notation). For a subextension L/K of \overline{K}/K and n, r ≥ 0, put

B_{n,L} :=H^{0}((Spec(O_{L})/W_{n})_{cris}, O_{Spec(O_{L})/W_{n}}) = H^{0}(Spf(O_{L})_{syn−et}, O_{n}^{cris})
J_{n,L}^{[r]} :=H^{0}((Spec(O_{L})/W_{n})_{cris}, J_{Spec(O_{L})/W_{n}}^{[r]}) = H^{0}(Spf(O_{L})_{syn−et}, J_{n}^{[r]})

(here O_{L} is the ring of integers of L and Spf(O_{L}) denotes the formal p-adic completion of Spec(O_{L})). B_{n,L} is a W_{n}-algebra and J_{n,L}^{[r]} an ideal of B_{n,L} with a divided power structure. Put

\begin{aligned}B_{n} & := \underrightarrow{lim}_{L} B_{n,L}, \\ J_{B_{n}}^{[r]} & := \underrightarrow{lim}_{L} J_{n,L}^{[r]}, \end{aligned}  \begin{aligned}A_{cris} & := \underleftarrow{lim}_{n} B_{n} \\ J_{A_{cris}}^{[r]} & := \underleftarrow{lim}_{n} J_{B_{n}}^{[r]}, \end{aligned}

where L runs through all finite extesions of K inside \overline{K}. By [FM, II.2.2] we have

B_{n} = B_{n,\overline{K}}, J_{B_{n}}^{[r]} = J_{n,\overline{K}}^{[r]}

It is proved in ([Fo 3], Thm. 1(ii)) that

B_{n} = A_{cris} ⊗ Z/p^{n}Z (n ≥ 0)

and the same argument shows that

J_{B_{n}}^{[r]} = J_{A_{cris}}^{[r]} ⊗ Z/p^{n}Z (n ≥ 0)

The rings B_{cris}^{+}, B_{dR}^{+} are defined as

B_{cris}^{+} := A_{cris} ⊗ Q = A_{cris} ⊗_{W} K_{0}
B_{dR}^{+} := \underleftarrow{lim}_{r} (Q ⊗ \underleftarrow{lim}_{n} (B_{n}/J_{B_{n}}^{[r]})) = \underleftarrow{lim}_{r} (Q ⊗ (A_{cris}/J_{A_{cris}}^{[r]}))

The embedding B_{cris}^{+} ↪ B_{dR}^{+} is induced by the canonical maps

A_{cris} = \underleftarrow{lim}_{n} B_{n} ⟶ \underleftarrow{lim}_{n} (B_{n}/J_{n}^{[r]})

The ring B_{dR} has a canonical topology, defined in [Fo 4]. Its subrings A_{cris}, B_{cris}, B_{dR}^{+} are equipped with the induced topology. The topology induced on \overline{K}, A_{cris} is coarser than the p-adic topology. Recall from [FM, III.3.1] that there are homomorphisms

μ_{p^{n}} ⟶ s_{n}(1) ⟶ J_{n}^{[1]}

of sheaves on Spec(W_{n})_{syn}. They define maps μ_{p^{n}}(L) ⟶ J_{n,L}^{[1]} and, passing to the limit, homomorphisms

Z_{p}(1) ⟶ J_{A_{cris}}^{[1]} ↪ A_{cris},  Q_{p}(1) ⟶ B_{cris}^{+}  (1.3.1)

If t ≠ 0 is any element in the image of Q_{p}(1), then B_{cris} = B_{cris}^{+}[t^{−1}] and B_{dR} = B_{dR}^{+}[t^{−1}]. The action of the crystalline Frobenius f on B_{n,L} defines, in the limit, the operator f : B_{cris}^{+} ⟶ B_{cris}^{+}. By definition of s_{n}(1), f = p on Q_{p}(1) ⊂ B_{cris}^{+}; it extends to B_{cris} by putting f(t^{−1}) = p^{−1}t^{−1}.

(1.4) Denote by Rep(G) the abelian category of p-adic representations of G (= vector spaces of finite dimension over Q_{p} with a continuous linear action of G) and by MF_{K} the additive category of filtered Dieudonné modules over K ^{1}. Recall that an object of MF_{K} is a vector space D of finite dimension over K_{0}, equipped with a σ-linear bijective endomorphism f : D ⟶ D and with a separated exhaustive decreasing filtration F^{i}D_{K} of D_{K} = D ⊗_{K_{0}} K by K-subspaces. Morphisms in MF_{K} are K_{0}-linear maps compatible with f and the filtrations F^{i}. Fontaine [Fo 1, 4] defines a functor V : MF_{K} ⟶ Rep(G) by

V (D) = (D ⊗_{K_{0}} B_{cris})^{f=1} ∩ F^{0} (D_{K} ⊗_{K} B_{dR})  (1.4.1)

Here f = f ⊗ f on D ⊗_{K_{0}} B_{cris} and

F^{i} (D_{K} ⊗_{K} B_{dR}) = ∑_{a+b=i} Im (F^{a}D_{K} ⊗_{K} F^{b}B_{dR} ↪ D_{K} ⊗_{K} B_{dR}) .  (1.4.2)

For V ∈ Ob(Rep(G)) he defines

D(V) = (V ⊗_{Q_{p}} B_{cris})^{G} ,  D_{dR}(V) = (V ⊗_{Q_{p}} B_{dR})^{G}  (1.4.3)

1 A terminology suggested in [Og] is “Fontaine modules”.

D(V) is a vector space over K_{0} = (B_{cris})^{G} with a σ-linear action of f = 1 ⊗ f. D_{dR}(V) is a vector space over K = (B_{dR})^{G} with a decreasing filtration by K-subspaces

F^{i}D_{dR}(V) = (V ⊗_{Q_{p}} F^{i}B_{dR})^{G}  (1.4.4)

It is shown in [Fo 1] that the canonical map D(V) ⊗_{K_{0}} K ⟶ D_{dR}(V) is injective and that

dim_{K_{0}}(D(V)) ≤ dim_{K}(D_{dR}(V)) ≤ dim_{Q_{p}}(V)  (1.4.5)

The full subcategory of crystalline representations Rep_{cris}(G) ⊂ Rep(G) consists of those V which satisfy

dim_{K_{0}}(D(V)) = dim_{Q_{p}}(V) ( ⟹^{(1.4.5)}= dim_{K}(D_{dR}(V)))

Then D becomes a functor D : Rep_{cris}(G) ⟶ MF_{K} with

D(V)_{K} = D_{dR}(V), F^{i}D(V)_{K} = F^{i}D_{dR}(V).

(1.5) The full subcategory of admissible filtered Dieudonné modules MF_{K}^{ad} ⊂ MF_{K} is, by definition, the essential image of D. According to a fundamental result of Fontaine ([Fo 1], Thm. 3.6.5), the functors V and D are quasi-inverse to each other and define an equivalence of categories Rep_{cris}(G) ≈ MF_{K}^{ad}.

More precisely, for V ∈ Ob(Rep_{cris}(G)) and D = D(V) (resp. D ∈ Ob(MF_{K}^{ad}) and V = V (D)), there is a canonical isomorphism

D ⊗_{K_{0}} B_{cris} ⟶^{∼} V ⊗_{Q_{p}} B_{cris},  (1.5.1)

compatible with the action of G and f. Here g ∈ G acts by 1 ⊗ g (resp. g ⊗ g) on the L.H.S. (resp. R.H.S.) and f acts as f ⊗ f (resp. 1 ⊗ f) on the L.H.S. (resp. R.H.S.). The induced isomorphism

D_{K} ⊗_{K} B_{dR} = D ⊗_{K_{0}} B_{dR} ⟶^{∼} V ⊗_{Q_{p}} B_{dR}  (1.5.2)

is compatible with the filtrations: F^{i}(D_{K} ⊗_{K} B_{dR}), defined in (1.4.2), corresponds to V ⊗_{Q_{p}} F^{i}B_{dR}. The functors D ↦ F^{i}D_{K}, D ↦ D_{K}/F^{i}D_{K} are exact on MF_{K}^{ad} and (1.5.2) induces isomorphisms

F^{i}D_{K} ⟶^{∼} (V ⊗_{Q_{p}} F^{i}B_{dR})^{G} ,  D_{K}/F^{i}D_{K} ⟶^{∼} (V ⊗_{Q_{p}} (B_{dR}/F^{i}B_{dR}))^{G}  (1.5.3)

(1.6) Both Rep_{cris}(G) and MF_{K}^{ad} are Q_{p}-linear rigid tensor categories and the functors D, V preserve this structure. For example, we have

\begin{aligned}D(V_{1} ⊗_{Q_{p}} V_{2}) & = D(V_{1}) ⊗_{K_{0}} D(V_{2})  (with f = f ⊗ f) \\ D_{dR}(V_{1} ⊗_{Q_{p}} V_{2}) & = D_{dR}(V_{1}) ⊗_{K} D_{dR}(V_{2}) \end{aligned}  (1.6.1)

with the convolution filtration (defined as in (1.4.2)) and

D(V^{∗}) = Hom_{K_{0}}(D(V), K_{0})  (with f(d^{∗}) = σ ∘ d^{∗} ∘ f^{−1})
D_{dR}(V^{∗}) = Hom_{K}(D_{dR}(V), K) (1.6.2)

with F^{i}(Hom_{K}(D_{dR}(V), K)) = (F^{1−i}D_{dR}(V))^{⊥}.

The unit object of Rep(G) (resp. MF_{K}^{ad}) is Q_{p} with trivial action of G (resp. D(Q_{p}) = K_{0} with f = σ, D_{dR}(Q_{p}) = K = F^{0} ⊃ F^{1} = 0). The representation on p-power roots of unity

Q_{p}(1) = Z_{p}(1) ⊗_{Z_{p}} Q_{p} = (\underleftarrow{lim}_{n} μ_{p^{n}}(\overline{K})) ⊗_{Z_{p}} Q_{p}

is crystalline, hence all its tensor powers Q_{p}(n) = Q_{p}(1)^{⊗n} (n ∈ Z) are crystalline as well. The corresponding filtered modules are

\begin{aligned}D(Q_{p}(n)) & = K_{0} e_{−n} with f = p^{−n}σ \\ D_{dR}(Q_{p}(n)) & = K e_{−n} = F^{−n} ⊃ F^{−n+1} = 0 \end{aligned}  (1.6.3)

If V is a crystalline representation, then its Tate twists V (n) = V ⊗_{Q_{p}} Q_{p}(n) (n ∈ Z) are crystalline as well and (1.6.1-3) give

D(V (n)) = D(V) ⊗_{K_{0}} K_{0} e_{−n} ⟶^{∼} D(V)  (with f replaced by p^{−n}f)
D_{dR}(V (n)) = D_{dR}(V) ⊗_{K} K e_{−n} ⟶^{∼} D_{dR}(V)

(1.6.4)

with F^{i}D_{dR}(V (n)) = F^{i+n}D_{dR}(V) ⊗_{K} K e_{−n} ⟶^{∼} F^{i+n}D_{dR}(V). In fact, the formulas (1.6.4) can be used to define Tate twists D{n} for objects of MF_{K}.

(1.7) Bloch-Kato theory. Let V be a crystalline representation of G. For q ≥ 0, define

H_{f}^{q}(G, V) := Ext_{Rep_{cris}(G)}^{q}(Q_{p}, V) (⟶^{∼} Ext_{MF_{K}^{ad}}^{q}(D(Q_{p}), D(V)))  (1.7.1)

These Yoneda Ext-groups can be interpreted in terms of derived functors as follows ([Hu], Thm. 2.6): the Ind-category A = Ind(Rep_{cris}(G)) (see [SGA 4], Exp. I, 8.2.1) is an abelian category with enough injectives, the functor F : A ⟶ (Ab) given by

F ((A_{i})_{i∈I}) = \underrightarrow{lim}_{I} Hom_{Rep_{cris}(G)}(Q_{p}, V)

(for any small filtered category I) is left exact and, for every q ≥ 0,

H_{f}^{q}(K, −) = R^{q}F |_{Rep_{cris}(G)}  (1.7.2)

It is shown in [BK] (see also [FoPR], Prop. 3.3.7) that

\begin{aligned}H_{f}^{0}(K, V) & = H^{0}(G, V) = V^{G} \\ H_{f}^{1}(K, V) & = Ker [H_{cont}^{1}(G, V) ⟶ H_{cont}^{1}(G, V ⊗_{Q_{p}} B_{cris})] \\ H_{f}^{q}(K, V) & = 0  (q ≥ 2) \end{aligned}  (1.7.3)

More precisely, H_{f}^{q}(K, V) are cohomology groups of the complex

C_{f}^{.}(V) = [D(V)⟶^{(1−f,can)}D(V) ⊕ D_{dR}(V)/F^{0}]  (1.7.4)

The isomorphism

V^{G} ⟶^{∼} H^{0}(C_{f}^{.}(V)) = D(V)^{f=1} ∩ F^{0}D_{dR}(V)

follows from (1.2.(vi)), (1.4.1); the isomorphism H_{f}^{1}(K, V) ⟶^{∼} H^{1}(C_{f}^{.}(V)) can be described as follows [Ne 2]: given an extension in Rep_{cris}(G)

0 ⟶ V ⟶ E ⟶ Q_{p} ⟶ 0,

chose a K_{0}-linear section s of the surjection D(E) ⟶ D(Q_{p}) = K_{0} and associate to (the extension class of) E the class of [(f − 1)s(1), −s(1) mod F^{0}D_{dR}(E)] ∈ C_{f}^{1}(V) in H^{1}(C_{f}^{.}(V)), which does not depend on the choice of s; observe that D_{dR}(V)/F^{0} ⟶^{∼} D_{dR}(E)/F^{0} (cf. [Ne 2], Prop. 1.21).

The original approach of Bloch and Kato (which makes sense for representation V that are de Rham, i.e. those satisfying dim_{Q_{p}}(V) = dim_{K}(D_{dR}(V))) was based on the exact sequence ([BK], 1.17.2)

0 ⟶ Q_{p} ⟶ B_{cris}⟶^{(1−f,can)}B_{cris} ⊕ B_{dR}/B_{dR}^{+} ⟶ 0,  (1.7.5)

which, on tensoring with V, gives a commutative diagram with exact rows (cf. 1.5)

0 ⟶ V ⟶ V ⊗_{Q_{p}} B_{cris}  ⟶^{(1−1⊗f,can)} (V ⊗_{Q_{p}} B_{cris}) ⨁ (V ⊗_{Q_{p}} (B_{dR}/B_{dR}^{+})) ⟶ 0
∥  ↓^{≀}  ↓^{≀}
0 ⟶ V ⟶ D(V) ⊗_{K_{0}} B_{cris} ⟶^{(1−f⊗f,can)} (D(V) ⊗_{K_{0}} B_{cris}) ⨁ \frac{D_{dR}(V)⊗_{K}B_{dR}}{F^{0}(D_{dR}(V)⊗_{K}B_{dR})} ⟶ 0

(1.7.6)

The map (1 − f, can) in (1.7.5) admits a continuous section ([BK], 1.18), thus (1.7.6) gives rise to an exact cohomology sequence ([BK], 3.8.4)

\begin{matrix}0 ⟶ & H^{0}(G, V) ⟶ D(V)⟶^{(1−f,can)}D(V) ⊕ D_{dR}(V)/F^{0}⟶^{∂} \\ ⟶ Ker [H_{cont}^{1}(G, V) ⟶ H_{cont}^{1}(G, V ⊗_{Q_{p}} B_{cris})] ⟶ 0 \end{matrix}  (1.7.7)

The coboundary map ∂ is the exponential map of Bloch-Kato

exp_{V} : D(V) ⊕ D_{dR}(V)/F^{0} ⟶ H_{f}^{1}(K, V)  (1.7.8)

Replacing V by V (r) (for r ∈ Z), we get exact sequences (cf. (1.6.4))

0 ⟶ V (r) ⟶D(V) ⊗_{K_{0}} B_{cris} ⟶^{(1−p^{−r}f⊗f,can)}(D(V) ⊗_{K_{0}} B_{cris}) ⨁ \frac{D_{dR}(V) ⊗_{K} B_{dR}}{F^{r}(D_{dR}(V) ⊗_{K} B_{dR})} ⟶ 0 (1.7.9)
0 ⟶H^{0}(G, V (r)) ⟶ D(V)⟶^{(1−p^{−r}f,can)}D(V) ⊕ D_{dR}(V)/F^{r} ⟶ H_{f}^{1}(K, V (r)) ⟶ 0 (1.7.10)

If T ⊂ V is a Z_{p}-lattice stable by G_{K}, then H_{f}^{1}(K, T) is defined as the preimage of H_{f}^{1}(K, V) in H^{1}(K, T).

2. Geometric situation

In this section we recall Fontaine’s crystalline conjecture and its ”syntomic” proof. We then make preparations for the proof of comparison results of Section 3.

(2.1) In the notation of (1.1), let 𝔛 ⟶ Spec(O_{K}) be a proper smooth morphism with a generic (resp. special) fibre X = 𝔛 ⊗_{O_{K}} K (resp. Y = 𝔛 ⊗_{O_{K}} k). Write 𝔛̂ = \underrightarrow{lim}_{n}(𝔛 ⊗ Z/p^{n}Z) (resp. ˆ \overline{𝔛}) for the p-adic

completion of 𝔛 (resp. of \overline{𝔛} = 𝔛 ⊗_{O_{K}} O\overline{K}) and put \overline{X} = X ⊗_{K} \overline{K}. We recall from Appendix various syntomic-étale ([FM], III.4) and étale topoi and morphisms between them:

𝔛̂_{syn−et}^{∼} ⟶ 𝔛_{syn−et}^{∼} ⟵ X_{et}^{∼},  (2.1.1)

projective limits of (2.1.1) for 𝔛 ⊗_{O_{K}} O_{L} (for finite extensions K ⊂ L ⊂ \overline{K})

ˆ\overline{𝔛}_{syn−et}^{∼} ⟶ \overline{𝔛}_{syn−et}^{∼} ⟵ \overline{X}_{et}^{∼}  (2.1.2)

and a G-equivariant version of (2.1.2)

(ˆ\overline{𝔛}_{syn−et}^{∼}, G) ⟶ (\overline{𝔛}_{syn−et}^{∼}, G) ⟵ (\overline{X}_{et}^{∼}, G)  (2.1.3)

There is a commutative diagram of morphisms of topoi

\begin{matrix}ˆ\overline{𝔛}_{syn−et}^{∼} & ⟶^{v} & (ˆ\overline{𝔛}_{syn−et}^{∼}, G) & ⟶^{π} & 𝔛̂_{syn−et}^{∼} \\ ↓^{Γ\overline{𝔛}} & & ↓^{Γ\overline{𝔛}} & & ↓^{Γ_{𝔛}} \\ (Sets) & ⟶^{v} & B_{G} & ⟶^{Γ_{G}} & (Sets) \end{matrix}  (2.1.4)

and similar diagrams for 𝔛_{syn−et}^{∼} and X_{et}^{∼}. In (2.1.4), B_{G} is the category of discrete G-sets, v^{∗} = “forget the action of G”, (Γ_{G})_{∗} = (−)^{G} , (Γ_{G})^{∗} = “let G act trivially”. The étale version of π defines an equivalence of categories π^{∗} : X_{et}^{∼} ≈ (\overline{X}_{et}^{∼}, G), which underlies the Hochschild-Serre spectral sequence. See Appendix for more details.

(2.2) The crystalline conjecture. For i ≥ 0, denote by

D^{i} := H^{i}((Y/K_{0})_{cris}, O_{Y/K_{0}}) = H^{i}(RΓ_{N}RΓ((Y/W_{n})_{cris}, O_{Y/W_{n}})) ⊗_{W} K_{0}
D̃^{i} := H^{i}((𝔛_{1}/K_{0})_{cris}, O_{𝔛_{1}/K_{0}}) = H^{i}(RΓ_{N}RΓ((𝔛_{1}/W_{n})_{cris}, O_{𝔛_{1}/W_{n}})) ⊗_{W} K_{0}

the crystalline cohomology of Y (resp. 𝔛_{1}) with coefficients in K_{0}. The canonical isomorphism

D_{K}^{i} = D^{i} ⊗_{K_{0}} K ⟶^{∼} H_{dR}^{i}(X/K) = H^{i}(X_{Zar}, Ω_{X/K}^{.})

proved in ([BO 2], Cor. 2.5) makes D^{i} an object of the category MF_{K}; the filtration F^{r}D_{K}^{i} is given by the Hodge filtration

F^{r}H_{dR}^{i}(X/K) = H^{i}(X_{Zar}, σ_{≥r}Ω_{X/K}^{.})

Put

V^{i} := H^{i}(\overline{X}_{et}, Q_{p}) ⟶^{∼} H_{naive}^{i}(\overline{X}_{et}, Z_{p}) ⊗_{Z_{p}} Q_{p}

(cf. I.2.3.3). The crystalline conjecture of Fontaine [Fo 2] asserts that (2.2.1) D^{i} is admissible. (2.2.2) V^{i} is a crystalline representation of G. (2.2.3) There are canonical isomorphisms D(V^{i}) ⟶^{∼} D^{i}, V (D^{i}) ⟶^{∼} V^{i}.

This conjecture was proved under some restrictions (K = K_{0} and p > dim(X)) in [FM] and in full generality in [Fa]. Combining the methods of [FM] with results on p-adic vanishing cycles, the conjecture was proved in [KaM] (resp. [Ts]) for p > 2 dim(X) + 1 (resp. all p).

(2.3) Various syntomic sheaves of crystalline origin form “Z_{p}”-modules (cf. I.2.5, Appendix):

O_{“Z_{p}”}^{cris} = [O_{n}^{cris}]_{n∈N} , J_{“Z_{p}”}^{[r]} = [J_{n}^{[r]}]_{n∈N} , J_{“Z_{p}”}^{′[r]} = [J_{n}^{′[r]}]_{n∈N} , s_{“Z_{p}”}(r) = [s_{n}(r)]_{n∈N}

are objects of Mod(𝔛̂_{syn−et}^{∼}, “Z_{p}”). We denote by O_{“Q_{p}”}^{cris} = Q(O_{“Z_{p}”}^{cris}), J_{“Q_{p}”}^{[r]} = Q(J_{“Z_{p}”}^{[r]}) etc. their images

in Mod(𝔛̂_{syn−et}^{∼}, “Q_{p}”). Recall from [FM, III.4.1] and Appendix that all of these sheaves have the same cohomology on the syntomic-etale site of 𝔛̂ as on the syntomic site of 𝔛 (and similarly for \overline{𝔛}).

Proposition. (1) For 0 ≤ r < p there is an exact sequence in Mod(𝔛̂_{syn−et}^{∼}, “Z_{p}”)

0 ⟶ s_{“Z_{p}”}(r) ⟶ J_{“Z_{p}”}^{[r]}⟶^{1−f_{r}}O_{“Z_{p}”}^{cris} ⟶ 0

(2) For every r ≥ 0 there is a commutative diagram with exact rows in Mod(𝔛̂_{syn−et}^{∼}, “Q_{p}”)

0 ⟶ s_{“Q_{p}”}(r) ⟶ J_{“Q_{p}”}^{[r]}  ⟶^{1−p^{−r}f}  O_{“Q_{p}”}^{cris}  ⟶ 0
∥  ∩ ↓  ↓^{(id,0)}
0 ⟶ s_{“Q_{p}”}(r) ⟶ O_{“Q_{p}”}^{cris} ⟶^{(1−p^{−r}f,can)} O_{“Q_{p}”}^{cris} ⊕ O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]} ⟶ 0

Proof. (1) This follows from [FM, III.1.1] and the fact that (i_{n+r})_{∗} : (𝔛_{n+r}^{∼})_{syn} ⟶ 𝔛̂_{syn−et}^{∼} is an exact functor ([FM, III.4.1], Appendix). (2) The exactness of the first row is obtained as in (1), observing that J_{“Z_{p}”}^{[r]}/J_{“Z_{p}”}^{′[r]} is killed by p^{r}; an easy diagram chase proves the exactness of the second row.

(2.4) Fix r ≥ 0. According to Proposition 2.3, there is a distinguished triangle

s_{“Q_{p}”}(r) ⟶ O_{“Q_{p}”}^{cris} ⟶^{(1−p^{−r}f,can)}O_{“Q_{p}”}^{cris} ⊕ O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]} ⟶ s_{“Q_{p}”}(r)[1]  (2.4.1)

in D^{+}(Mod(𝔛̂_{syn−et}^{∼}, “Q_{p}”)). Applying R(Q(Γ^{N}(𝔛̂_{syn−et}))), we obtain a distinguished triangle

R(Q(Γ^{N}(𝔛̂_{syn−et})))(s_{“Q_{p}”}(r)) ⟶ R(Q(Γ^{N}(𝔛̂_{syn−et})))(O_{“Q_{p}”}^{cris})⟶^{(1−p^{−r}f,can)}
⟶R(Q(Γ^{N}(𝔛̂_{syn−et})))(O_{“Q_{p}”}^{cris}) ⊕ R(Q(Γ^{N}(𝔛̂_{syn−et})))(O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]})⟶ R(Q(Γ^{N}(𝔛̂_{syn−et})))(s_{“Q_{p}”}(r))[1]
(2.4.2)

in D^{+}(Mod(Sets, “Q_{p}”)), which will be denoted by

Δ_{r} : A_{r} ⟶ B_{r} ⟶ C_{r} ⟶ A_{r}[1]  (2.4.3)

In fact, all A_{r}, B_{r}, C_{r} are objects of D^{b}(Mod(Sets, “Q_{p}”)), (see Appendix).

Apply the functor π^{∗} from (2.1.4) to the triangle (2.4.1) and take R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et}))). We obtain a distinguished triangle

R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(s_{“Q_{p}”}(r)) ⟶ R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(O_{“Q_{p}”}^{cris})⟶^{(1−p^{−r}f,can)}
⟶R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(O_{“Q_{p}”}^{cris})⊕R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]})⟶ R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(s_{“Q_{p}”}(r))[1]
(2.4.4)

in D^{+}(Mod(B_{G}, “Q_{p}”)), which will be denoted by

\overline{Δ}_{r} : \overline{A}_{r} ⟶ \overline{B}_{r} ⟶ \overline{C}_{r} ⟶ \overline{A}_{r}[1]  (2.4.5)

Note that C_{r} = B_{r} ⊕ C′_{r}, \overline{C}_{r} = \overline{B}_{r} ⊕ \overline{C}′_{r} for

C′_{r} = R(Q(Γ^{N}(𝔛̂_{syn−et})))(O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]}), \overline{C}′_{r} = R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]})

(2.5) Proposition. For every q, r ≥ 0, there are canonical maps between distinguished triangles

(Q(Γ_{G}^{N})^{∗}Δ_{r}) (q) ⟶ \overline{Δ}_{r+q}, Q(Γ_{G}^{N})^{∗}Δ_{r} ⟶ \overline{Δ}_{r+q}(−q)

in Mod(B_{G}, “Q_{p}”). Proof. For q = 0 there is nothing to prove: for every abelian sheaf A on 𝔛̂_{syn−et} there is a canonical map

Γ_{G}^{∗}RΓ(𝔛̂_{syn−et}, A)⟶^{can}RΓ(ˆ\overline{𝔛}_{syn−et}, π^{∗}A)

induced by π. Suppose that q ≥ 1. The canonical map

“Z_{p}(1)” = [μ_{p^{n}}(\overline{K})]_{n∈N} ⟶ [H^{0}(ˆ\overline{𝔛}_{syn−et}, s_{n}(1))]_{n∈N}

in Mod(B_{G}, “Z_{p}”) is compatible with products, defining a map

“Z_{p}(q)” ⟶ [H^{0}(ˆ\overline{𝔛}_{syn−et}, s_{n}(q))]_{n∈N} ⟶ RΓ(ˆ\overline{𝔛}_{syn−et}, s_{“Z_{p}”}(q))

in D^{+}(Mod(B_{G}, “Z_{p}”)). Together with the cup product

\overline{A}_{r}⊗^{L}_{“Q_{p}”}\overline{A}_{q} = R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(s_{“Q_{p}”}(r))⊗^{L}R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(s_{“Q_{p}”}(q))⟶^{∪}

⟶^{∪}R(Q(Γ^{N}(ˆ\overline{𝔛}_{syn−et})))(s_{“Q_{p}”}(r + q)) = \overline{A}_{r+q},

this induces a map

α_{r,q} : (Q(Γ_{G}^{N})^{∗}A_{r}) (q)⟶^{can}\overline{A}_{r}(q) = \overline{A}_{r}⊗^{L}_{“Q_{p}”}“Q_{p}(q)” ⟶ \overline{A}_{r}⊗^{L}_{“Q_{p}”}\overline{A}_{q}⟶^{∪}\overline{A}_{r+q}

(see Appendix for the discussion of cup products and comments on the simultaneous appearance of RΓ and

⊗^{L}).

In the same vein, the morphisms s_{“Z_{p}”}(q) ⟶ J_{“Z_{p}”}^{[q]} ⟶ O_{“Z_{p}”}^{cris} and the cup product associated to O_{“Z_{p}”}^{cris} × O_{“Z_{p}”}^{cris} ⟶ O_{“Z_{p}”}^{cris} give rise to a map

β_{r,q} : (Q(Γ_{G}^{N})^{∗}B_{r}) (q)⟶^{can}\overline{B}_{r}(q) = \overline{B}_{r}⊗^{L}_{“Q_{p}”}“Q_{p}(q)” ⟶ \overline{B}_{r}⊗^{L}_{“Q_{p}”}\overline{A}_{q} ⟶ \overline{B}_{r}⊗^{L}_{“Q_{p}”}\overline{B}_{q}⟶^{∪}\overline{B}_{r+q},

compatible with α_{r,q}. As f acts as p^{q} on the image of “Q_{p}(q)” in O_{“Q_{p}”}^{cris}, we have

1 − p^{r+q}f ∘ β_{r,q} = 1 − p^{−r}β_{r,q} ∘ f  (2.5.1)

Finally, the cup product associated to

O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r]} × J_{“Q_{p}”}^{[q]} ⟶ J_{“Q_{p}”}^{[q]}/J_{“Q_{p}”}^{[r+q]} ⟶ O_{“Q_{p}”}^{cris}/J_{“Q_{p}”}^{[r+q]}

and s_{“Q_{p}”}(q) ⟶ J_{“Q_{p}”}^{[q]} define a map

γ′_{r,q} : (Q(Γ_{G}^{N})^{∗}C′_{r}) (q)⟶^{can}\overline{C}′_{r}(q) = \overline{C}′_{r}⊗^{L}_{“Q_{p}”}“Q_{p}(q)” ⟶ \overline{C}′_{r}⊗^{L}_{“Q_{p}”}RΓ(ˆ\overline{𝔛}_{syn−et}, J_{“Q_{p}”}^{[q]})⟶^{∪}\overline{C}′_{r+q},

Thank to (2.5.1), the map

γ_{r,q} = (β_{r,q}, γ′_{r,q}) : (Q(Γ_{G}^{N})^{∗}C_{r}) (q) ⟶ \overline{C}_{r+q}

together with α_{r,q} and β_{r,q} define a map of triangles

(Q(Γ_{G}^{N})^{∗}Δ_{r}) (q) ⟶ \overline{Δ}_{r+q}

Using (I.2.5.1), we get the second map.

(2.6) Let u : (E′, A′) ⟶ (E, A) be a morphism of ringed topoi such that u^{∗} : Mod(E, A) ⟶ Mod(E′, A′) is exact ( ⟺ A′ is a flat u^{−1}(A)-module). The “trivial duality” ([Ber], V.3.3.1; [BO1], Prop. 7.7) gives a canonical (and functorial) isomorphism

Hom_{D(Mod(E′,A′))}(u^{∗}(X), Y′) ⟶^{∼} Hom_{D(Mod(E,A))}(X, Ru_{∗}(Y′))  (2.6.1)

for all X ∈ Ob(D^{−}(Mod(E, A))), Y′ ∈ Ob(D^{+}(Mod(E′, A′))). From (I.1.2.12–13) we get a canonical isomorphism

Hom_{D(Q(Mod(E′,A′)))}(Q(u)^{∗}(Q_{D}(X)), Q_{D}(Y′)) ⟶^{∼} Hom_{D(Q(Mod(E,A)))}(Q_{D}(X), RQ(u)_{∗}(Q_{D}(Y′))) (2.6.2)

For (E′, A′) = ((B_{G})^{N}, “Z_{p}”), (E, A) = ((Sets)^{N}, “Z_{p}”) and u = Γ_{G}^{N}, we know that all constituents A_{r}, B_{r}, C_{r} of Δ_{r} (resp. \overline{A}_{r+q}, \overline{B}_{r+q}, \overline{C}_{r+q} of \overline{Δ}_{r+q}) are of the form Q_{D}(X) for some X ∈ Ob(D^{b}(Mod(E, A))) (resp. Q_{D}(Y′) for some Y′ ∈ Ob(D^{+}(Mod(E′, A′)))) (2.4). It follows from Proposition 2.5 and (2.6.2) that, for all r, q ≥ 0, there is a canonical map of distinguished triangles

Δ_{r} ⟶ RQ(Γ_{G}^{N})_{∗} (\overline{Δ}_{r+q}(−q))  (2.6.3)

in D^{+}(Q(Mod((Sets)^{N}, “Z_{p}”))) = D^{+}(Mod(Sets, “Q_{p}”)).

(2.7) There is a diagram of functors

Mod(B_{G}, “Q_{p}”) ⟶^{Q(Γ_{N})} Q(Z_{p}[G] − Mod) ⟶^{ξ_{G}} (Q_{p}[G] − Mod)
↓^{Q((Γ_{G}^{N})_{∗})}  ↓^{Q((Γ_{G})_{∗})}  ↓^{(Γ_{G})_{∗}}
Mod(Sets, “Q_{p}”) ⟶^{Q(Γ_{N})} Q(Z_{p} − Mod) ⟶^{ξ} (Q_{p} − Mod),

(2.7.1)

in which Γ_{G} = (−)^{G}, Γ_{N} = \underleftarrow{lim}_{N} and ξ, ξ_{G} are the (exact) functors from (I.1.2.4). The first square of (2.7.1) is commutative; in the second square, there is a canonical morphism of functors

ξ ∘ Q((Γ_{G})_{∗}) ⟶ (Γ_{G})_{∗} ∘ ξ_{G}  (2.7.2)

such that

ξ ∘ Q((Γ_{G})_{∗})(X) ⟶ (Γ_{G})_{∗} ∘ ξ_{G}(X)

is a monomorphism for every X ∈ Ob(Q(Z_{p}[G] − Mod)) (this is just the canonical map Y^{G} ⊗_{Z_{p}} Q_{p} ⟶ (Y ⊗_{Z_{p}} Q_{p})^{G}). Put

Ψ = ξ ∘ Q(Γ_{N}) : Mod(Sets, “Q_{p}”) ⟶ (Q_{p} − Mod)
Φ = Ψ ∘ Q((Γ_{G}^{N})_{∗}) : Mod(B_{G}, “Q_{p}”) ⟶ (Q_{p} − Mod)
Θ = ξ_{G} ∘ Q(Γ_{N}) : Mod(B_{G}, “Q_{p}”) ⟶ (Q_{p}[G] − Mod)

(2.7.3)

(all of these functors are left exact) and write v^{∗} : (Q_{p}[G] − Mod) ⟶ (Q_{p} − Mod) for the functor “forget the action of G”. There are canonical morphisms of functors (induced by (2.7.2))

Φ ⟶ (Γ_{G})_{∗} ∘ Θ ⟶ v^{∗} ∘ Θ  (2.7.4)

such that both arrows in

Φ(X) ⟶ (Γ_{G})_{∗} ∘ Θ(X) ⟶ v^{∗} ∘ Θ(X)  (2.7.5)

are monomorphisms (for every X ∈ Ob(Mod(B_{G}, “Q_{p}”))).

(2.8) We show that syntomic(-étale) cohomology fits into the abstract framework of I.3.1.

(0) Fix an integer i ∈ Z. (1),(2) We have categories C = Mod(Sets, “Q_{p}”), \overline{C} = Mod(B_{G}, “Q_{p}”), D = (Q_{p} − Mod) and functors u = Q((Γ_{G}^{N})_{∗}), Ψ, Φ from (2.7.3). (3),(4) Fix r ≥ 0 and choose q ≥ 0 such that r + q > i (as cd_{p}(\overline{X}_{et}) = 2 dim(X), we can treat all relevant i’s simultaneously by taking q such that r + q > 2 dim(X)). By (2.4) and (2.6.3), there are distinguished triangles Δ = Δ_{r}, \overline{Δ} = (\overline{Δ})_{r+q}(−q) and a morphism of triangles Δ ⟶ (Ru)(\overline{Δ}). As in (2.4), we have C = B ⊕ C′, \overline{C} = \overline{B} ⊕ \overline{C}′.

(5) Étale cohomology defines objects E = R(Q(Γ^{N}(X_{et})))(“Z_{p}(r)”) ∈ Ob(D^{+}(C)), \overline{E} = R(Q(Γ^{N}(\overline{X}_{et}))) (“Z_{p}(r)”) ∈ Ob(D^{+}(\overline{C})). The isomorphism ρ : E ⟶^{∼} (Ru)(\overline{E}) comes from the equivalence of categories π^{∗} : X_{et}^{∼} ≈ (\overline{X}_{et}^{∼}, G).

(6) The morphisms μ : A ⟶ E (resp. \overline{μ} : \overline{A} ⟶ \overline{E}) are furnished by the Fontaine-Messing map ([FM], III.5.1; Appendix). The diagram in I.3.1.6 is commutative, as the Fontaine-Messing map is compatible with cup products, the Galois action and with the map μ_{p^{n}} ⟶ s_{n}(1).

(2.9) In Sections 2.9–14 we show that objects defined in 2.8 verify the axioms (A1)–(A3) of I.3.1. Almost by definition, we have

R^{j}Ψ(A) = H^{j}(𝔛̂_{syn−et}, s_{Q_{p}}(r))  (∀j ≥ 0)  (2.9.1)

According to ([KaM], Lemma 4.5(1)),

R^{j}Ψ(C′) = D_{K}^{j}/F^{r}  (∀j ≥ 0)  (2.9.2)

It is shown in ([KaM], Prop. 1.2, Prop. 1.3(1)) that (for j ≥ 0)

\begin{aligned}Θ(H^{j}(\overline{B}_{r+q})) & = D^{j} ⊗_{K_{0}} B_{cris}^{+} \\ Θ(H^{j}(\overline{C}′_{r+q})) & = (D_{K}^{j} ⊗_{K} B_{dR}^{+})/F^{r+q}(D_{K}^{j} ⊗_{K} B_{dR}^{+}), \end{aligned}  (2.9.3)

which implies that, for j ≤ r + q,

\begin{aligned}Θ(H^{j}(\overline{B})) & = D^{j} ⊗_{K_{0}} B_{cris}^{+}(−q) ⟶^{∼} D^{j} ⊗_{K_{0}} t^{−q}B_{cris}^{+} ↪ D^{j} ⊗_{K_{0}} B_{cris} \\ Θ(H^{j}(\overline{C}′)) & = \frac{D_{K}^{j} ⊗_{K} B_{dR}^{+}}{F^{r+q}(D_{K}^{j} ⊗_{K} B_{dR}^{+})}(−q) ⟶^{∼} \frac{D_{K}^{j} ⊗_{K} t^{−q}B_{dR}^{+}}{F^{r}(D_{K}^{j} ⊗_{K} t^{−q}B_{dR}^{+})} ↪ \frac{D_{K}^{j} ⊗_{K} B_{dR}}{F^{r}(D_{K}^{j} ⊗_{K} B_{dR})} \end{aligned}  (2.9.4)

Here t is a generator of Q_{p}(1) ⊂ B_{cris}^{+}. Note that the last map in (2.9.4) is indeed injective: for i ≤ r + q we have

F^{r}(D_{K}^{j} ⊗_{K} F^{−q}B_{dR}) = ∑_{k≤j} F^{k}D_{K}^{j} ⊗_{K} F^{r−k}(F^{−q}B_{dR}) = ∑_{k≤j} F^{k}D_{K}^{j} ⊗_{K} F^{r−k}B_{dR} = F^{r}(D_{K}^{j} ⊗_{K} B_{dR})

(2.10) Vanishing cycles and the crystalline conjecture. Consider the map induced by μ on cohomology:

H^{j}(\overline{A}) = Q([H^{j}(ˆ\overline{𝔛}_{syn−et}, s_{n}(r + q))(−q)]_{n∈N}) ⟶ Q([H^{i}(\overline{X}_{et}, Z/p^{n}Z(r))]_{n∈N}) = H^{j}(\overline{E})  (2.10.1)

A fundamental theorem on vanishing cycles, proved in an increasing degree of generality by Kato [Ka], Kurihara [Ku] and Tsuji [Ts], asserts that (2.10.1) is an isomorphism for all j ≤ r + q, proving thus (A2). According to (I.2.3.3.1), the projective system [H^{j}(\overline{X}_{et}, Z/p^{n}Z(r))]_{n∈N} is ML-p-adic, hence the last group in (2.10.1) is isomorphic to Q([T^{j}(r) ⊗ Z/p^{n}Z]_{n∈N}), where

T^{j} = H^{j}(\overline{X}_{et}, Z_{p}) ⟶^{∼} \underleftarrow{lim}_{n} H^{j}(\overline{X}_{et}, Z/p^{n}Z)

This implies that (2.10.1) induces an isomorphism

Θ(H^{j}(\overline{A})) ⟶^{∼} V^{j}(r),  (∀j ≤ i + 1)  (2.10.2)

which, together with (2.9.4), shows that the complex

Θ(H^{j}(\overline{A})) ⟶ Θ(H^{j}(\overline{B})) ⟶ Θ(H^{j}(\overline{C}))

is isomorphic to the first row of the following commutative diagram

V^{j}(r) ⟶^{α} D^{j} ⊗_{K_{0}} t^{−q}B_{cris}^{+} ⟶^{(1−p^{−r},can)} (D^{j} ⊗_{K_{0}} t^{−q}B_{cris}^{+}) ⊕ \frac{D_{K}^{j}⊗_{K}t^{−q}B_{dR}^{+}}{F^{r}(D_{K}^{j}⊗_{K}t^{−q}B_{dR}^{+})}
∩ ↓  ∩ ↓
0 ⟶ V (D^{j}(r)) ⟶ D^{j} ⊗_{K_{0}} B_{cris}  ⟶^{(1−p^{−r},can)} (D^{j} ⊗_{K_{0}} B_{cris}) ⊕ \frac{D_{K}^{j}⊗_{K}B_{dR}}{F^{r}(D_{K}^{j}⊗_{K}B_{dR})}
(2.10.3)

(here V is Fontaine’s functor defined in (1.4.1)). Let us rapidly recall how one deduces from (2.10.3) the crystalline conjecture (2.2) (see [KaM, 3.2], [Ts] for more details): an argument based on Poincaré duality ([FM], 6.3) shows that the map α in (2.10.3) is injective; a dimension count then implies that α induces isomorphisms

V^{j}(r) ⟶^{∼} V (D^{j}(r)),  V^{j} ⊗_{Q_{p}} B_{cris} ⟶^{∼} D^{j} ⊗_{K_{0}} B_{cris},  (∀j ≤ i + 1)

which proves the crystalline conjecture (taking, e.g., i = 2 dim(X)).

(2.11) Put K^{j} := Ker[Φ(H^{j}(\overline{A})) ⟶ Φ(H^{j}(\overline{B}))]. As (2.8.4) and α in (2.10.3) are injective, it follows that

Φ(K^{j}) = Ker [Φ(H^{j}(\overline{A})) ⟶ Φ(H^{j}(\overline{B}))] = 0  (∀j ≤ i + 1)  (2.11.1)

However, K^{j} ⊆ H^{j}(\overline{A}) ⟶^{∼} Q([T^{j}(r) ⊗ Z/p^{n}Z]_{n∈N}), with T^{j} a Z_{p}-module of finite type. For such K^{j}, (2.11.1) implies that K^{j} = 0 for all j ≤ i + 1. This proves the axiom (A1) of I.3.1.

(2.12) As crystalline cohomology depends only on reduction (mod p), we have

R^{j}Ψ(B) = H^{j}((𝔛/W)_{cris}, O_{𝔛/W}) ⊗_{W} K_{0} = H^{j}((𝔛_{1}/W)_{cris}, O_{𝔛_{1}/W}) ⊗_{W} K_{0} = D̃^{j}

Our aim is to compare D̃^{j} and D^{j}, using the “Frobenius trick” of [BO 2]. For a scheme T over Spec(F_{p}) denote by F_{T} : T ⟶ T its absolute Frobenius morphism. Consider the commutative diagram

Y ↪^{g} 𝔛_{1}
↓^{F_{Y}}  ↓^{F_{𝔛_{1}}}
Y ↪^{g} 𝔛_{1}

and the induced maps on crystalline cohomology

D^{j} ⟵^{g^{∗}}  D̃^{j}
↑^{f}  ↑^{f′}
D^{j} ⟵^{g^{∗}}  D̃^{j}

For sufficiently large integer n >> 0 there is a morphism ρ : 𝔛_{1} ⟶ Y (depending on n, which will be fixed) such that

ρ ∘ g = F_{Y}^{n}, g ∘ ρ = F_{𝔛_{1}}^{n}.

The induced map on cohomology ρ^{∗} : D^{j} ⟶ D̃^{j} satisfies f′ ∘ ρ^{∗} = ρ^{∗} ∘ f, g^{∗} ∘ ρ = f^{n}, ρ^{∗} ∘ g^{∗} = f^{′n}. The crystalline Frobenius f : D^{j} ⟶ D^{j} is bijective, which implies that g^{∗} : D̃^{j} ⟶ D^{j} is surjective. According to [BO 2], the canonical map

D̃^{j} = R^{j}Ψ(B) ⟶ R^{j}Ψ(C′) =^{(2.9.2)} D_{K}^{j}/F^{r}

factors as D̃^{j}⟶^{g^{∗}}D^{j}⟶^{can}D_{K}^{j}/F^{r}.

(2.13) Lemma. For every 0 ≠ α ∈ K_{0}, the map g^{∗} induces isomorphisms

(D̃^{j})^{αf′=1} ⟶^{∼} (D^{j})^{αf=1},  D̃^{j}/(αf′ − 1)D̃^{j} ⟶^{∼} D^{j}/(αf − 1)D^{j}

and a quasi-isomorphism

[D̃^{j} ⟶^{(αf′−1,can∘g^{∗})}D̃^{j} ⊕ D_{K}^{j}/F^{r}] ⟶^{Qis} [D^{j}⟶^{(αf−1,can)}D^{j} ⊕ D_{K}^{j}/F^{r}] .

Proof. (Following a remark of P. Berthelot) We apply the Snake lemma to the diagram

0 ⟶ Ker(g^{∗}) ⟶ D̃^{j} ⟶^{g^{∗}} D^{j} ⟶ 0
↓^{αf′−1}  ↓^{αf′−1}  ↓^{αf−1}
0 ⟶ Ker(g^{∗}) ⟶ D̃^{j} ⟶^{g^{∗}} D^{j} ⟶ 0

We first observe that g^{∗} indeed induces an isomorphism between the kernels (resp. cokernels) of the second and third vertical maps (it is invertible, its inverse being α^{n}ρ^{∗}). This implies that the first vertical arrow must be an isomorphism. Another application of the Snake lemma, this time to

0 ⟶ Ker(g^{∗}) ⟶  D̃^{j}  ⟶^{g^{∗}}  D^{j}  ⟶ 0
↓^{αf′−1}  ↓^{(αf′−1,can∘g^{∗})}  ↓^{(αf−1,can)}
0 ⟶ Ker(g^{∗}) ⟶ D̃^{j} ⊕ D_{K}^{j}/F^{r} ⟶^{(g^{∗},id)} D^{j} ⊕ D_{K}^{j}/F^{r} ⟶ 0,

concludes the proof.

(2.14) Observe that, for j ≤ i + 1, the composite map

\begin{matrix}D̃^{j} = & R^{j}Ψ(B) ⟶ Φ(H^{j}(\overline{B})) ↪^{incl} (Γ_{G})_{∗} ∘ Θ(H^{j}(\overline{B}))⟶^{(2.9.4)∼} \\ ⟶^{∼} (D^{j} ⊗_{K_{0}} t^{−q}B_{cris}^{+})^{G} ↪^{incl} (D^{j} ⊗_{K_{0}} B_{cris})^{G} = D^{j} \end{matrix}

resp.

\begin{matrix}D_{K}^{j}/F^{r} = & R^{j}Ψ(C′) ⟶ Φ(H^{j}(\overline{C}′)) ↪^{incl} (Γ_{G})_{∗} ∘ Θ(H^{j}(\overline{C}′))⟶^{(2.9.4)∼} \\ ⟶^{∼} (\frac{D_{K}^{j} ⊗_{K} F^{−q}B_{dR}}{F^{r}(D_{K}^{j} ⊗_{K} F^{−q}B_{dR})})^{G} ↪^{incl} (\frac{D_{K}^{j} ⊗_{K} B_{dR}}{F^{r}(D_{K}^{j} ⊗_{K} B_{dR})})^{G} =^{(1.5.3)} D_{K}^{j}/F^{r} \end{matrix}

is equal to g^{∗} resp. to the identity map. This implies that all four inclusion maps ↪^{incl} above are equalities. The axiom (A3) then follows from Lemma 2.13 (for α = p^{−r}).

3. The Main Comparison Theorem

In this section we prove Theorems B, C and D(3).

(3.1) The assumptions of (2.1) (including p > 2) are in force. For i, r ≥ 0, denote the map

D^{i}⟶^{(1−p^{−r},can)}D^{i} ⊕ D_{K}^{i}/F^{r}

by λ_{i,r}. From (2.9.1–2) and the cohomology sequence of Δ_{r} we obtain an exact sequence

0 ⟶ Coker(λ_{i,r}) ⟶ H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) ⟶ Ker(λ_{i+1,r}) ⟶ 0  (3.1.1)

The crystalline conjecture (2.2) and (1.7.3–4) yield isomorphisms

Ker(λ_{i+1,r}) ⟶^{∼} H^{0}(G_{K}, V^{i+1}(r))  (3.1.2.1)
exp_{V^{i}(r)} : Coker(λ_{i,r}) ⟶^{∼} H_{f}^{1}(K, V^{i}(r))  (3.1.2.2)

The Hochschild-Serre spectral sequence gives rise to an exact sequence

\begin{aligned}0 & ⟶ & F^{1}H^{i+1}(X_{et}, Q_{p}(r)) & ⟶ & H^{i+1}(X_{et}, Q_{p}(r)) & ⟶^{edge} & H^{0}(G_{K}, V^{i+1}(r)) \\ ↓ & δ \\ & & H_{cont}^{1}(G_{K}, V^{i}(r)) \end{aligned}  (3.1.3)

We are now ready to state our main p-adic comparison result, relating (3.1.1) and (3.1.3).

(3.2) Theorem. Let i, r ≥ 0. The Fontaine-Messing map ν : H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) ⟶ H^{i+1}(X_{et}, Q_{p}(r)) is injective and gives rise to a commutative diagram with exact rows

0 ⟶  Coker(λ_{i,r})  ⟶ H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) ⟶  Ker(λ_{i+1,r})  ⟶ 0
↓^{β}  ↓^{ν}  ↓^{α}
0 ⟶ F^{1}H^{i+1}(X_{et}, Q_{p}(r)) ⟶  H^{i+1}(X_{et}, Q_{p}(r))  ⟶ H^{0}(G_{K}, V^{i+1}(r)) ⟶ 0
↓^{δ}
H_{cont}^{1}(G_{K}, V^{i}(r))

in which α is an isomorphism, δ ∘ β = exp_{V^{i}(r)} is injective and Im(δ ∘ β) = H_{f}^{1}(K, V^{i}(r)). Proof. Everything follows from Proposition I.3.5, applied to the data (2.8). The axioms of I.3.1 were verified in (2.9)–(2.12).

(3.3) Denote by F_{et}^{j} the filtration induced on H^{i+1}(X_{et}, Q_{p}(r)) by the Hochschild-Serre spectral sequence. As cd_{p}(k) = 1 ([Se 1], II.2.2.3), we have cd_{p}(K) ≤ 2 ([Se 1], II.4.3.12), hence F_{et}^{j} = 0 for j > 2 (I.2.3.3.4) and E_{2}^{1,b} = E_{∞}^{1,b} for all b ≥ 0 (recall that, for X smooth and projective over K, the Hochschild-Serre spectral sequence degenerates at E_{2} (I.2.3.3)).

As in I.3.6, we define a filtration F_{syn}^{j} on H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) by

\begin{aligned}F_{syn}^{0} & = H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) \\ F_{syn}^{1} & = Ker [H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r))⟶^{h}H^{i+1}(𝔛̂_{syn−et}, O_{Q_{p}}^{cris})] \\ F_{syn}^{2} & = 0 \end{aligned}

(the map h is induced by s_{n}(r) ↪ J_{n}^{[r]} ↪ O_{n}^{cris}). From I.3.6 we get

Proposition. (1) For all j ≥ 0, ν(F_{syn}^{j}) ⊆ F_{et}^{j}.

(2) gr_{F}^{0} (ν) : F_{syn}^{0}/F_{syn}^{1} ⟶^{∼} F_{et}^{0}/F_{et}^{1} = E_{∞}^{0,i+1} = E_{2}^{0,i+1} = H^{0}(G_{K}, V^{i+1}(r)) is an isomorphism.

(3) gr_{F}^{1} (ν) : F_{syn}^{1} ↪ F_{et}^{1}/F_{et}^{2} = E_{∞}^{1,i} = E_{2}^{1,i} = H_{cont}^{1}(G_{K}, V^{i}(r)) is injective, with image equal to H_{f}^{1}(K, V^{i}(r)).

(3.4) p-adic Beilinson-Deligne cohomology. In an ideal world, one would hope that there is a canonical object RΓ_{cris}(\overline{X}_{et}, Q_{p}(r)) in D^{b}(Rep_{cris}(G)) with cohomology groups H^{i}(\overline{X}_{et}, Q_{p}(r)); the p-adic Beilinson-Deligne cohomology of X would then be defined as

H_{�D}^{i} (X, Q_{p}(r)) = Hom_{D^{b}(Rep_{cris}(G))}(Q_{p}, RΓ_{cris}(\overline{X}_{et}, Q_{p}(r))[i])

The associated spectral sequence

E_{2}^{a,b} = H_{f}^{a}(K, V^{b}(r)) ⟹ H_{�D}^{a+b}(X, Q_{p}(r))

would degenerate into exact sequences

0 ⟶ H_{f}^{1}(K, V^{i}(r)) ⟶ H_{�D}^{i+1}(X, Q_{p}(r)) ⟶ H^{0}(G_{K}, V^{i+1}(r)) ⟶ 0.

Theorem 3.2 shows that H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) behaves exactly like this conjectural “p-adic Beilinson-Deligne cohomology”, as it sits in an exact sequence

0 ⟶ H_{f}^{1}(K, V^{i}(r)) ⟶ H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) ⟶ H^{0}(G_{K}, V^{i+1}(r)) ⟶ 0.

It would be interesting to carry out Beilinson’s construction [Be 2] of the Beilinson-Deligne cohomology (called ”absolute Hodge cohomology” in [Be 2]) in the p-adic context.

(3.5) If K is a finite extension of Q_{p} ( ⟺ k is a finite field), then the crystalline Weil conjectures predict that the action of f on D^{i} should be pure of weight i, i.e. all eigenvalues of the semi-simplification of the K_{0}-linear map f^{[K_{0}:Q_{p}]} : D^{i} ⟶ D^{i} should be algebraic integers with all archimedean absolute values equal to p^{[K_{0}:Q_{p}]i/2}. This was proved by Katz and Messing [KM] for 𝔛 smooth and projective over Spec(O_{K}), and by Chiarellotto and Le Stum [CLeS] in general. As a consequence, the crystalline conjecture, (1.7.4) and local duality imply that

\begin{aligned}H^{0}(G_{K}, V^{i}(r)) & = 0  for r ≠ i/2 \\ H_{cont}^{2}(G_{K}, V^{i}(r)) & = 0  for r ≠ i/2 + 1 \end{aligned}  (3.5.1)

It follows from (3.5.1) that the Hochschild-Serre spectral sequence induces isomorphisms

δ : H^{i+1}(X_{et}, Q_{p}(r)) ⟶^{∼} H_{cont}^{1}(G_{K}, V^{i}(r))  (r ≠ (i + 1)/2)  (3.5.2)

and Theorem 3.2 reduces to

(3.6) Corollary. Suppose that K is a finite extension of Q_{p} and that 𝔛 is proper and smooth over Spec(O_{K}). Then, for every r ≠ (i+1)/2, the isomorphism (3.5.2) and the Fontaine-Messing map ν induce an isomorphism

δ ∘ ν : H^{i+1}(𝔛̂_{syn−et}, s_{Q_{p}}(r)) ⟶^{∼} H_{f}^{1}(K, V^{i}(r))

(3.7) It is explained in Appendix how to construct Chern classes

K_{j}(𝔛) ⟶ H^{2i−j}(𝔛̂_{syn−et}, s_{Q_{p}}(r)),

which are homomorphisms for j > 0 and are compatible with p-adic étale Chern classes on X. The corresponding Chern character maps ([Sc 1], p. 28) define syntomic regulators (resp. cycle classes), which are multiplicative and compatible with p-adic étale regulators (resp. cycle classes) on X. For r > (i + 1)/2, put

K_{2r−i−1}(𝔛)_{0} := Ker [K_{2r−i−1}(𝔛) ⟶ H^{0}(G_{K}, H^{i+1}(\overline{X}_{et}, Q_{p}(r)))]

and write

δ ∘ r_{p} ∘ j^{∗} : K_{2r−i−1}(𝔛)_{0} ⟶ H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r)))

for the map induced by the p-adic étale regulator and the Hochschild-Serre spectral sequence.

(3.8) Theorem. Under the assumptions of (2.1),

(1) Im [CH^{r}(X)_{0}⟶^{r_{p}}H_{cont}^{1}(G_{K}, H^{2r−1}(\overline{X}_{et}, Q_{p}(r)))] ⊆ H_{f}^{1}(K, H^{2r−1}(\overline{X}_{et}, Q_{p}(r))).

(2) The cycle class map cl_{X} : CH^{r}(X) ⟶ H^{2r}(X_{et}, Q_{p}(r)) satisfies Im(cl_{X}) ∩ F^{2}H^{2r}(X_{et}, Q_{p}(r)) = 0.

(3) For r > (i + 1)/2, Im [δ ∘ r_{p} ∘ j^{∗} : K_{2r−i−1}(𝔛)_{0} ⟶ H_{cont}^{1}(G_{K}, H^{i}(\overline{X}_{et}, Q_{p}(r)))] ⊆ H_{f}^{1}(K, H^{i}(\overline{X}_{et}, Q_{p}(r))).

(4) If K is a finite extension of Q_{p}, then K_{2r−i−1}(𝔛)_{0} = K_{2r−i−1}(𝔛) for all r > (i + 1)/2.

Proof. (1),(2) This follows from Theorem (3.2), the compatibility of étale and syntomic cycle classes (Appendix) and the fact that j^{∗} : CH^{r}(𝔛) ⟶ CH^{r}(X) is surjective.

(3) Follows from Theorem 3.2 and the compatibility of étale and syntomic regulators (Appendix).

(4) In this case, H^{0}(G_{K}, H^{i+1}(\overline{X}_{et}, Q_{p}(r))) = 0 vanishes, by crystalline Weil conjectures (3.5.1).

(3.9) If L/K is a finite Galois extension, then the canonical maps

\begin{matrix}H^{0}(K, V) & ⟶ H^{0}(L, V)^{G(L/K)} \\ H_{f}^{1}(K, V) & ⟶ H_{f}^{1}(L, V)^{G(L/K)} \end{matrix}

are isomorphisms (for every Q_{p}-representation V of G_{K}). This implies that the statements (1), (2) of Theorem 3.8 hold true under a weaker assumption that X has a potentially good reduction.

(3.10) Proof of Theorems B, C, D(3). The statements of Theorem B and Theorem D(3) follow from Theorem 3.8 and (3.9). Theorem C follows from Corollary 3.6 and [Ku], as explained in ([Sc 2], Sect. 7).

4. Integral theory

What follows is an integral version of results of Sect. 2. Throughout Sect. 4, K = K_{0}. From 4.13 on, p > 2.

(4.1) We recall the main results of the theory of Fontaine-Laffaille [FL]. Denote by MF_{W,tf} the category of triples (M, F^{i}M, φ_{i}), where

(0) M is a W -module of finite type.

(1) (F^{i}M)_{i∈Z} is a decreasing filtration of M by W -submodules, which are direct summands and satisfy F^{i}M = M (resp. F^{i}M = 0) for i << 0 (resp. i >> 0).

(2) φ_{i} : F^{i}M ⟶ M are σ-linear maps with φ_{i}|_{F^{i+1}M} = pφ_{i+1}.

(3) ∑_{i∈Z} φ_{i}(F^{i}M) = M. A morphism (M, F^{i}M, φ_{i,M}) ⟶ (N, F^{i}N, φ_{i,N}) is a W -linear map α : M ⟶ N satisfying α(F^{i}M) ⊆ F^{i}N, α ∘ φ_{i,M} = φ_{i,N} ∘ α (for all i ∈ Z). Denote by MF_{W,lf} the full subcategory of MF_{W,tf} consisting of objects with length_{W} (M) < ∞ (⟺ M is W -torsion). Both of these categories are abelian and Z_{p}-linear. All morphisms are strictly compatible with filtrations and all functors M ↦ F^{i}M are exact.

(4.2) For a pair of integers a ≤ b, denote by MF_{W}^{[a,b]} (resp. MF_{W,tors}^{[a,b]}) the full subcategory of MF_{W,tf}

(resp. of MF_{W,lf}) consisting of objects with F^{a}M = M, F^{b+1} = 0. Denote by MF_{W}^{[a,b[} (resp. MF_{W}^{]a,b]}) the full subcategory of MF_{W}^{[a,b]} consisting of M with no non-zero quotients N satisfying N = F^{b}N (resp. no non-zero subobjects satisfying F^{a+1}N = 0). (And similarly with MF_{W,tors}^{[a,b[}, resp. MF_{W,tors}^{]a,b]}). For every n ∈ Z there is a functor M ↦ M{n} (“Tate twist”), given by

F^{i}(M{n}) = F^{i+n}M, φ_{i,M{n}} = φ_{i+n,M}

This defines an equivalence of categories between MF_{W}^{[a,b]} and MF_{W}^{[a−n,b−n]}. The category MF_{W,tf} admits internal Hom and ⊗. The unit object for ⊗ is

11 = M, F^{i}M = \{ \begin{matrix}W, & if i ≤ 0 \\ 0, & if i > 0 \end{matrix} , φ_{i} = \{ \begin{matrix}p^{−i}σ, & if i ≤ 0 \\ 0, & if i > 0 \end{matrix}

(4.3) The formulas D := M ⊗_{W} K, F^{i}D := F^{i}M ⊗_{W} K, f|_{F^{i}D} := φ_{i} ⊗ p^{i}σ define a functor

− ⊗_{W} K : MF_{W,tf} ⟶ MF_{K},  (4.3.1)

compatible with internal Hom, ⊗ and Tate twists. According to Laffaille ([L], Thm. 3.2), the essential image of (4.3.1) is the category of weakly admissible filtered Dieudonné modules MF_{K}^{f}. This category was defined by Fontaine, who also proved ([Fo 1], Prop. 4.4.5) that MF_{K}^{f} contains MF_{K}^{ad} and conjectured that these two categories coincide. If we define MF_{K}^{[a,b]} and its (weakly) admissible versions as in 4.2, then [FL, Thm. 8.4] says that

MF_{K}^{f,[a,b]} = MF_{K}^{ad,[a,b]},  if 0 ≤ b − a < p  (4.3.2)

(4.4) We recall the definition and basic properties of functors ψ_{r} and Λ_{r} studied by Kato [Ka].

(4.4.1) Fix n ≥ 0. Recall (1.3) that B_{n} = A_{cris} ⊗ Z/p^{n}Z has a decreasing filtration by J_{B_{n}}^{[r]} (r ≥ 0) and that, for 0 ≤ r < p, the crystalline Frobenius f defines σ-linear maps f_{r} : J_{B_{n}}^{[r]} ⟶ B_{n} such that p^{r}f_{r} = f.

Let M be an object of MF_{W,tors}^{[0,p−1]}, killed by p^{n}. Put on M ⊗_{W} A_{cris} = M ⊗_{W_{n}} B_{n} the convolution filtration

F^{r}(M ⊗_{W_{n}} B_{n}) = ∑_{0≤i≤r} ((F^{i}M) ⊗_{W_{n}} J_{B_{n}}^{[r−i]})

and define, for 0 ≤ r < p, σ-linear maps f_{r} : F^{r}(M ⊗_{W} A_{cris}) ⟶ M ⊗_{W} A_{cris} by

f_{r}|(F^{i}M) ⊗_{W_{n}} J_{B_{n}}^{[r−i]} = φ_{i,M} ⊗ f_{r−i} 0 ≤ i ≤ r)

This is well-defined and independent on n, as all quotients J_{B_{n}}^{[i]}/J_{B_{n}}^{[i+1]} are free W_{n}-modules.

(4.4.2) Definition. For M ∈ Ob(MF_{W,tors}^{[0,p−1]}) and 0 ≤ r < p, define a Z_{p}[G_{K}]-module ψ_{r}(M) (resp. Λ_{r}(M)) to be the kernel (resp. the cokernel) of the map

1 − f_{r} : F^{r}(M ⊗_{W} A_{cris}) ⟶ M ⊗_{W} A_{cris}

For M ∈ Ob(MF_{W}^{[0,p−1]}) and 0 ≤ r < p, put ψ_{r}(M) = \underleftarrow{lim}_{n}(ψ_{r}(M/p^{n}M)).

(4.4.3) By ([Ka], II.3.2), a short exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

in MF_{W,tors}^{[0,p−1]} induces an exact sequence of Z_{p}[G_{K}]-modules

0 ⟶ ψ_{r}(M′) ⟶ ψ_{r}(M) ⟶ ψ_{r}(M′′) ⟶ Λ_{r}(M′) ⟶ Λ_{r}(M) ⟶ Λ_{r}(M′′) ⟶ 0

(for every 0 ≤ r < p).

(4.5) Lemma. Let M ∈ Ob(MF_{W}^{[0,p−1]}), 0 ≤ r < p. Then

(1) If r > 0, then the canonical map ψ_{r−1}(M)(1) ⟶ ψ_{r}(M) (induced by μ_{p^{n}}(O\overline{K}) ↪ J_{B_{n}}^{[1]}) is injective. If F^{r}M = 0, then it is an isomorphism.

(2) Assume that r < p − 1, length_{W} (M) < ∞ and either: (a) F^{r+1}M = 0, or (b) F^{r}M = M. Then Λ_{r}(M) = 0.

Proof. First of all, we can assume that k = \overline{k}, as replacing M by M ⊗_{W} W (\overline{k}) ∈ Ob(MF_{W (\overline{k})}^{[0,p−1]}) does not change ψ_{r}(M) nor Λ_{r}(M). In this case, all statements are proved in ([Ka], II.3.4.1, 3.7.1, 3.8.1) for simple objects of MF_{W (\overline{k})}^{[0,p−1]} (note that the statements of ([Ka], II.3.4, 3.7) are true even for r = p − 1, by ([FL], Thm. 5.3)). The general case follows by dévissage and (4.4.3).

(4.6) We are now ready to formulate a covariant version of the main result of Fontaine and Laffaille ([FL], Thm. 5.3, Thm. 6.1) (cf . [Fo 3], [Ka]). Denote by Rep_{Z_{p}}(G_{K}) the category of Z_{p}-modules of finite type with a continuous linear action of G_{K} (recall that K = K_{0}).

Proposition. (1) The formula T (M) := ψ_{p−1}(M)(1 − p) defines an exact and faithful functor T : MF_{W}^{[0,p−1]} ⟶ Rep_{Z_{p}}(G_{K}).

(2) If M ∈ Ob(MF_{W,tors}^{[0,p−1]}), then T (M) is finite and length_{W} (M) = length_{Z_{p}}(T (M)).

(3) If M ∈ Ob(MF_{W}^{[0,p−1]}) is free over W, then T (M) is free over Z_{p} and rk_{W} (M) = rk_{Z_{p}}(T (M)).

(4) The restriction of T to MF_{W}^{[0,p−1[} (resp. MF_{W}^{]0,p−1]}) is fully faithful, inducing an equivalence of categories between MF_{W}^{[0,p−1[} (resp. MF_{W}^{]0,p−1]}) and its essential image.

(5) For every M ∈ Ob(MF_{W}^{[0,p−1]}), the filtered Dieudonné module D := W ⊗_{K} M is admissible and there is a canonical isomorphism V (D) ⟶^{∼} T (M) ⊗_{Z_{p}} Q_{p}.

For example, for 0 ≤ r < p, 11{−r} is an object of MF_{W}^{[0,p−1]} and T (11{−r}) = Z_{p}(−r).

The isomorphism in (5) is given by

ψ_{p−1}(M)(1 − p) ⊗_{Z_{p}} Q_{p} ⟶^{∼} (F^{p−1}(D ⊗_{K_{0}} B_{cris}^{+})(1 − p))^{f_{p−1}=1} ⟶^{∼} F^{0}(D ⊗_{K_{0}} t^{1−p}B_{cris}^{+})^{f=1} ⟶^{∼}

⟶^{∼} F^{0}(D ⊗_{K_{0}} B_{cris})^{f=1}

(4.7) For M ∈ Ob(MF_{W,tf}), the complex

C^{.}(M) = [F^{0}M⟶^{1−φ_{0}}M]

(in degrees 0, 1) computes the Ext-groups Ext(11, M) in the category MF_{W,tf}. The isomorphisms between H^{i}(C^{.}(M)) and Ext_{MF_{W,tf}}^{i} (11, M) can be described as follows.

For i = 0, a morphism α : 11 ⟶ M is uniquely determined by its value α(1) ∈ (F^{0}M)^{φ_{0}=1} = H^{0}(C^{.}(M)).

For i = 1, an extension 0 ⟶ M ⟶ E ⟶ 11 ⟶ 0 gives an exact sequence of W -modules

0 ⟶ F^{0}M ⟶ F^{0}E ⟶ W ⟶ 0

Choosing a W -linear splitting s : W ⟶ F^{0}E, we obtain an element m = (φ_{0,E} − 1)(s(1)) ∈ M, the class of which in M/(1 − φ_{0,M})F^{0}M = H^{1}(C^{.}(M)) depends only on the isomorphism class of E. As H^{1}(C^{.}(M)) is right exact, it follows that Ext^{i}(11, M) vanish for i > 1.

Replacing M by M{r}, we see that the complex

[F^{r}M⟶^{1−φ_{r}}M]

computes Ext(11, M{r}) = Ext^{i}(11{−r}, M) (for every r ∈ Z).

Put D := M ⊗_{W} K, which is an object of MF_{K}^{f}. The maps incl : F^{0}D ↪ D, (id, 0) : D ⟶ D ⊕ D/F^{0}D define a quasi-isomorphism between C^{.}(M) ⊗_{W} K = [F^{0}D⟶^{1−f}D] and the complex

[D⟶^{(1−f,can)}D ⊕ D/F^{0}D]

of 1.7.4.

(4.8) For M ∈ Ob(MF_{W}^{[0,p−1]}), put T := T (M). If 0 ≤ r < p, then the functor T induces homomorphisms (by the discussion in 4.7)

α_{r,M} : (F^{r}M)^{φ_{r}=1} ⟶^{∼} Hom_{MF_{W,tf}} (11{−r}, M) = Hom_{MF_{W}^{[0,p−1]}}(11{−r}, M) ↪
↪ Hom_{Rep_{Zp} (G_{K})}(Z_{p}(−r), T) = T (r)^{G_{K}}

(injective) and

β_{r,M} : Coker [F^{r}M⟶^{1−φ_{r}}M] ⟶^{∼} Ext_{MF_{W,tf}}^{1} (11{−r}, M) = Ext_{MF_{W}^{[0,p−1]}}^{1}(11{−r}, M) ⟶
⟶ Ext_{Rep_{Zp} (G_{K})}^{1}(Z_{p}(−r), T) = H_{cont}^{1}(G_{K}, T (r))

The filtered Dieudonné module D := M ⊗_{W} K is admissible (4.3.2) and the canonical map V := T ⊗_{Z_{p}} Q_{p} ⟶ V (D) is an isomorphism (4.6.5). Comparing (4.7) with (1.7.4), we see that the following diagrams are commutative (the vertical maps are given by − ⊗_{Z_{p}} Q_{p}):

(F^{r}M)^{φ_{r}=1} ⟶^{α_{r,M}} T (r)^{G_{K}}
↓  ↓
(F^{r}D)^{p^{−r}f=1} ⟶^{1.7} V (r)^{G_{K}}
Coker [F^{r}M⟶^{1−φ_{r}}M]  ⟶^{β_{r,M}}  H_{cont}^{1}(G_{K}, T (r))
↓  ↓
Coker [F^{r}D ⟶^{1−p^{−r}f}D] ⟶^{∼} Coker [D ⟶^{(1−p^{−r}f,can)}D ⊕ D/F^{r}D] ⟶^{exp_{V (r)}} H_{cont}^{1}(G_{K}, V (r))

As Im(exp_{V (r)}) = H_{f}^{1}(K, V (r)), this shows that Im(β_{r,M}) ⊆ H_{f}^{1}(K, T (r)).

If M ∈ Ob(MF_{W}^{[0,p−1[}) and 0 ≤ r < p − 1 (i.e. 11{−r} ∈ Ob(MF_{W}^{[0,p−1[})), then α_{r,M} is an isomorphism and β_{r,M} is injective, by Proposition 4.6.4.

(4.9) Lemma. ([BK], Lemma 4.5) Let M ∈ Ob(MF_{W}^{[0,p−1[}), 0 ≤ r < p − 1. Put T := T (M). If M is torsion-free (⟺ T is torsion-free), then β_{r,M} induces an isomorphism

Coker [F^{r}M⟶^{1−φ_{r}}M] ⟶^{∼} H_{f}^{1}(K, T (r)) ⊆ H_{cont}^{1}(G_{K}, T (r))

Proof. [BK] ^{1} One must show that the cokernel of the map

Coker [F^{r}M⟶^{1−φ_{r}}M] ⟶ H_{cont}^{1}(K, T (r))

is torsion-free. This follows from the commutative diagram with exact rows

M/(1 − φ_{r})F^{r}M ⟶^{p} M/(1 − φ_{r})F^{r}M ⟶ (M/pM)/(1 − φ_{r})(F^{r}(M/pM)) ⟶ 0
↓^{β_{r,M}}  ↓^{β_{r,M}}  ↓^{β_{r,M/pM}}
H_{cont}^{1}(G_{K}, T (r)) ⟶^{p} H_{cont}^{1}(G_{K}, T (r)) ⟶  H^{1}(G_{K}, T/pT (r))

and the injectivity of all vertical arrows β_{r,−} (by (4.8)).

(4.10) Geometric case. Let 𝔛 be a proper and smooth scheme over Spec(W), X = 𝔛⊗_{W} K, \overline{X} = X⊗_{K} \overline{K}, 𝔛_{n} = 𝔛 ⊗ Z/p^{n}Z, Y = 𝔛_{1}, \overline{𝔛} = 𝔛 ⊗_{W} O\overline{K}, \overline{Y} = \overline{𝔛} ⊗_{O\overline{K}} \overline{k}. Put d = dim(Y).

For q, r, n ≥ 0, denote

M_{n}^{q} := H^{q}((𝔛_{n})_{syn}, O_{n}^{cris}) = H^{q}((Y/W_{n})_{cris}, O_{Y/W_{n}}) = H_{dR}^{q}(𝔛_{n}/W_{n})
F^{r}M_{n}^{q} := H^{q}((𝔛_{n})_{syn}, J_{n}^{[r]}) = H^{q}((𝔛_{n})_{Zar}, σ_{≥r}Ω_{𝔛_{n}/W_{n}}^{.})
T_{n}^{q} := H^{q}(\overline{X}_{et}, Z/p^{n}Z);  T^{q} := H^{q}(\overline{X}_{et}, Z_{p})
M^{q} := H^{q}(𝔛_{syn}, O_{Z_{p}}^{cris}) = H^{q}((Y/W)_{cris}, O_{Y/W})
F^{r}M^{q} := H^{q}(𝔛_{syn}, J_{Z_{p}}^{[r]})

By ([BO 1], 7.24.3), we have

M^{q} ⟶^{∼} \underleftarrow{lim}_{n} (M_{n}^{q}) , F^{r}M^{q} ⟶^{∼} \underleftarrow{lim}_{n} (F^{r}M_{n}^{q}) , T^{q} ⟶^{∼} \underleftarrow{lim}_{n} (T_{n}^{q})

and the projective system [M_{n}^{q}]_{n∈N} (resp. [T_{n}^{q}]_{n∈N}) is AR-isomorphic to [M^{q} ⊗Z/p^{n}Z] (resp. [T^{q} ⊗Z/p^{n}Z]). All F^{r}M^{q} are W -modules of finite type, T^{q} are objects of Rep_{Z_{p}}(G_{K}).

1 Note that there is a misprint in ([BK], Lemma 4.5); it should read H_{f}^{1}, not H_{e}^{1}.

(4.11) Proposition. If min(q, d) < p, then

(1) The canonical maps F^{r+1}M_{n}^{q} ⟶ F^{r}M_{n}^{q}, F^{r+1}M^{q} ⟶ F^{r}M^{q} are injective ∀n, r ≥ 0).

(2) (M_{n}^{q}, F^{r}M_{n}^{q}, φ_{r} = f_{r}) (resp. (M^{q}, F^{r}M^{q}, φ_{r} = f_{r})) is an object of MF_{W,tors}^{[0,min(q,d)]} (resp. MF_{W}^{[0,min(q,d)]}).

Proof. ([FM], II.2.7), ([Ka], II.2.5, I.1.8), ([Fo 3], Prop. 1.3).

(4.12) Proposition. Let q, n ≥ 0, 0 ≤ r < p. Then

(1) H^{q}(\overline{𝔛}_{syn}, O_{n}^{cris}) ⟶^{∼} H_{dR}^{q}(𝔛_{n}/W_{n}) ⊗_{W_{n}} B_{n} = M_{n}^{q} ⊗_{W_{n}} B_{n}.

(2) If min(q + 1, d) < p, then

(i) H^{q}(\overline{𝔛}_{syn}, J_{n}^{[r]}) ⟶^{∼} F^{r}(H_{dR}^{q}(𝔛_{n}/W_{n}) ⊗_{W_{n}} B_{n}) = F^{r}(M_{n}^{q} ⊗_{W_{n}} B_{n}).

(ii) There is an exact sequence

0 ⟶ Λ_{r}(M_{n}^{q−1}) ⟶ H^{q}(\overline{𝔛}_{syn}, s_{n}(r)) ⟶ ψ_{r}(M_{n}^{q}) ⟶ 0

(iii) If p − 1 > r ≥ min(q − 1, d), then H^{q}(\overline{𝔛}_{syn}, s_{n}(r)) ⟶^{∼} ψ_{r}(M_{n}^{q}).

Proof. (1) ([FM], III.1.3), ([Ka], I.4.6, I.1.8). (2)(i) ([FM], III.1.5.i), ([Ka], II.4.1, I.1.8). We recall the argument: there are spectral sequences

\begin{aligned}E_{1}^{a,b} & = H^{b}(𝔛_{n}, Ω_{𝔛_{n}/W_{n}}^{a}) ⟹ M_{n}^{a+b} \\ ′E_{1}^{a,b} & = E_{1}^{a,b} ⊗_{W_{n}} J_{B_{n}}^{[r−a]} ⟹ H^{a+b}((\overline{𝔛}_{n})_{syn}, J_{n}^{[r]}) \end{aligned}

According to Proposition 4.11.1, we have E_{1}^{a,b} = E_{∞}^{a,b} for a + b < p − 1 (or even for all a, b ≥ 0, provided d < p). As J_{B_{n}}^{[r−a]} is flat over W_{n}, ′E_{c}^{a,b} = E_{c}^{a,b} ⊗_{W_{n}} J_{B_{n}}^{[r−a]} for all a, b ≥ 0, c ≥ 1. This implies the claim. (ii) This follows form (1), (i) and the exact cohomology sequence of 0 ⟶ s_{n}(r) ⟶ J_{n}^{[r]}⟶^{1−f_{r}}O_{n}^{cris} ⟶ 0. (iii) By Lemma 4.5.2, Λ_{r}(M_{n}^{q−1}) = 0.

(4.13) Proposition. ([Ka]) For 0 ≤ q ≤ r < p − 1 and n ≥ 0, the Fontaine-Messing map induces isomorphisms

\begin{aligned}H^{q}(\overline{𝔛}_{syn}, s_{n}(r)) & ⟶^{∼} H^{q}(\overline{X}_{et}, Z/p^{n}Z(r)) = T_{n}^{q}(r) \\ H^{q}(\overline{𝔛}_{syn}, s_{Z_{p}}(r)) & ⟶^{∼} H^{q}(\overline{X}_{et}, Z_{p}(r)) = T^{q}(r) \end{aligned}

Proof. The first statement is proved by Kato ([Ka], I.4.3). The second statement follows by passing to the projective limit, as R^{1} \underleftarrow{lim}_{n} T_{n}^{q}(r) = 0 by I.2.3.3.1.

(4.14) Corollary. For 0 ≤ q ≤ r < p − 1 and n ≥ 0, the Fontaine-Messing map and (4.12) induce isomorphisms

\begin{aligned}T_{n}^{q} & ⟶^{∼} ψ_{r}(M_{n}^{q})(−r) ⟶^{∼} T (M_{n}^{q}) \\ T^{q} & ⟶^{∼} ψ_{r}(M^{q})(−r) ⟶^{∼} T (M^{q}) \end{aligned}

(The second isomorphism comes from Lemma 4.5.1.)

Proof. Combine Proposition 4.13, Proposition 4.12.2.(iii) and Lemma 4.5.1.

(4.15) As in (2.8), we invoke the axiomatic setting of I.3.1.

(0) Fix an integer i ∈ Z (to be specified later). (1),(2) We have categories C = Mod(Sets, “Z_{p}”), \overline{C} = Mod(B_{G}, “Z_{p}”), D = (Z_{p} − Mod) and functors u = (Γ_{G}^{N})_{∗}, Ψ = Γ_{N} = \underleftarrow{lim}_{N}, Φ = Ψ ∘ u = \underleftarrow{lim}_{N}(−)^{G}.

(3),(4) For an integer 0 ≤ r < p, there are distinguished triangles

Δ_{r} :RΓ^{N}(𝔛̂_{syn−et}, s_{“Z_{p}”}(r)) ⟶ RΓ^{N}(𝔛̂_{syn−et}, J_{“Z_{p}”}^{[r]})⟶^{1−f_{r}}RΓ^{N}(𝔛̂_{syn−et}, O_{“Z_{p}”}^{cris}) ⟶
⟶ RΓ^{N}(𝔛̂_{syn−et}, s_{“Z_{p}”}(r))[1]

in D^{+}(C) and

\overline{Δ}_{r} :RΓ^{N}(ˆ\overline{𝔛}_{syn−et}, s_{“Z_{p}”}(r)) ⟶ RΓ^{N}(ˆ\overline{𝔛}_{syn−et}, J_{“Z_{p}”}^{[r]})⟶^{1−f_{r}}RΓ^{N}(ˆ\overline{𝔛}_{syn−et}, O_{“Z_{p}”}^{cris}) ⟶
⟶ RΓ^{N}(ˆ\overline{𝔛}_{syn−et}, s_{“Z_{p}”}(r))[1]

in in D^{+}(\overline{C}). By Appendix, Δ_{r} lies in D^{b}(C). If q ≥ 0 is an integer satisfying r + q < p, then the same argument as in (2.5) gives morphisms of triangles

(Γ_{G}^{N})^{∗} Δ_{r} ⟶ \overline{Δ}_{r+q}(−q)

in in D^{+}(\overline{C}) and

Δ_{r} ⟶ R (Γ_{G}^{N})_{∗} (\overline{Δ}_{r+q}(−q))

in D^{+}(C). This defines a morphism Δ ⟶ (Ru)(\overline{Δ}) for Δ := Δ_{r}, \overline{Δ} := \overline{Δ}_{r+q}(−q).

(5) As before, étale cohomology defines objects E = RΓ^{N}(X_{et}, “Z_{p}(r)”) (resp. \overline{E} = RΓ^{N}(\overline{X}_{et}, “Z_{p}(r)”)) of D^{+}(C) (resp. D^{+}(\overline{C})) and an isomorphism ρ : E ⟶^{∼} (Ru)(\overline{E}).

(6) The morphisms μ : A ⟶ E (resp. \overline{μ} : \overline{A} ⟶ \overline{E}) are given by the Fontaine-Messing map ([FM], III.5.1; Appendix).

(4.16) We shall verify the axioms (A1), (A2’), (A3’) of I.3.1, under the assumptions

0 ≤ i + 1, r < p − 1

We choose q ≥ 0 such that i + 1 ≤ r + q < p − 1.

By Proposition 4.11 and 4.12,

\begin{aligned}H^{j}(\overline{B}) & = [F^{r+q}(M_{n}^{j} ⊗_{W_{n}} B_{n})(−q)]_{n∈N}  (∀j ≤ p − 2) \\ H^{j}(\overline{C}) & = [M_{n}^{j} ⊗_{W_{n}} B_{n}(−q)]_{n∈N}  (∀j ≥ 0) \end{aligned}  (4.16.1)

Lemma 4.5.2 and Proposition 4.11 then imply

Ker [H^{j}(\overline{A}) ⟶ H^{j}(\overline{B})] = Coker [H^{j−1}(\overline{B}) ⟶ H^{j−1}(\overline{C})] = 0

for all j ≤ r + q + 1, proving (A1). According to Proposition 4.13, μ induces isomorphisms

H^{j}(\overline{A}) ⟶^{∼} H^{j}(\overline{E}) (∀j ≤ r + q),

which gives (A2’). Now to (A3’): by definition,

(R^{j}Ψ)(B) = F^{r}M^{j},  (R^{j}Ψ)(C) = M^{j}  (∀j ≥ 0)  (4.16.2)

It follows from (4.16.1) that

Φ(H^{j}(\overline{B})) = \underleftarrow{lim}_{n} (F^{r+q}(M_{n}^{j} ⊗_{W} A_{cris})(−q))^{G}  (∀j ≤ p − 2)
Φ(H^{j}(\overline{C})) = \underleftarrow{lim}_{n} (M_{n}^{j} ⊗_{W} A_{cris}(−q))^{G}  (∀j ≥ 0)

(4.16.3)

Recall that, for n ≥ 0 and 0 ≤ q < p − 1,

(A_{cris} ⊗ Z/p^{n}Z(−q))^{G} = Z/p^{n}Z · t^{q},  (4.16.4)

where t is a Z_{p}-basis of Z_{p}(1) ↪ A_{cris}. Combining (4.16.2–4), we get (A3’).

(4.17) In fact, the assumptions of (4.16) can be slightly altered, to cover the value i + 1 = p − 1 as well. Assume that

0 ≤ r < p − 1,  dim(𝔛) < p  (4.17.1)

Then all M^{j} are objects of MF_{W}^{[0,p−2]}, by Proposition 4. 11(2). Consider the objects defined in (4.15), for q = p − 2 − r = i − r. The formulas (4.16.1) hold for all j ≥ 0 and the same arguments as in the rest of (4.16) give (A1), (A3’) and show that \overline{μ} induces an isomorphism τ_{≤i}\overline{A} ⟶^{∼} τ_{≤i}\overline{E}. Our aim is to show that Φ(H^{i+1}(\overline{A})) ⟶ Φ(H^{i+1}(\overline{E})) is injective. Observing that both maps

α_{r,M^{p−1}} : Ker(λ′_{p−1,r}) = (F^{r}M^{p−1})^{φ_{r}=1} ⟶ Φ(H^{i+1}(\overline{A})) = (T (M^{p−1})(r))^{G_{K}}
Ker(λ_{p−1,r}) = Ker(λ′_{p−1,r}) ⊗_{Z_{p}} Q_{p} ⟶ Φ(H^{i+1}(\overline{E})) ⊗_{Z_{p}} Q_{p} = H^{0}(G_{K}, V^{i+1}(r))

are isomorphisms, by (4.8) and (3.1.2.1) respectively, we see that (A2) is satisfied, provided

F^{r}M^{p−1} is torsion − free  (4.17.2)

5. The Integral Comparison Theorem

(5.1) The assumptions of (4.1) (including p > 2) are in force. For 0 ≤ r < p and j ≥ 0, denote the map 1 − φ_{r} : F^{r}M^{j} ⟶ M^{j} by λ′_{j,r}. As in (3.1), the cohomology sequence of Δ = Δ_{r} becomes

0 ⟶ Coker(λ′_{i,r}) ⟶ H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r)) ⟶ Ker(λ′_{i+1,r}) ⟶ 0

According to (4.8) and Proposition 4.11, for j satisfying min(j, d) < p there are injective homomorphisms

α_{r,M^{j}} : Ker(λ′_{j,r}) ↪ H^{0}(K, T (M^{j})(r))
β_{r,M^{j}} : Coker(λ′_{j,r}) ↪ H_{f}^{1}(K, T (M^{j})(r))

Moreover, Corollary 4.14 says that T (M^{j}) is canonically isomorphic to T^{j}, provided 0 ≤ j < p − 1.

(5.2) Theorem. Assume that 0 ≤ i + 1, r < p − 1. Then:

(1) The Fontaine-Messing map ν : H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r)) ⟶ H^{i+1}(X_{et}, Z_{p}(r)) is injective and gives rise to a commutative diagram with exact rows

0 ⟶  Coker(λ′_{i,r})  ⟶ H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r)) ⟶ Ker(λ′_{i+1,r}) ⟶ 0
↓^{β}  ↓^{ν}  ↓^{α}
0 ⟶ F^{1}H^{i+1}(X_{et}, Z_{p}(r)) ⟶  H^{i+1}(X_{et}, Z_{p}(r))  ⟶ H^{0}(G, T^{i+1}(r)) ⟶ 0
↓^{δ}
H_{cont}^{1}(G, T^{i}(r))

(2) α = α_{r,M^{i+1}} is an isomorphism, δ ∘ β = β_{r,M^{i}} is injective and Im(δ ∘ β) ⊆ H_{f}^{1}(K, T^{i}(r)). If M^{i} is torsion-free (⟺ T^{i} is torsion-free), then Im(δ ∘ β) = H_{f}^{1}(K, T^{i}(r)).

Proof. This follows from Proposition I.3.5 (applied to the data (4.15)), (4.8) and Lemma 4.9. The axioms of I.3.1 were verified in (4.16).

(5.3) As in (3.3), Theorem 5.2 can be reformulated in terms of the filtrations F_{et}^{j} = F^{j}H^{i+1}(X_{et}, Q_{p}(r)) (with F_{et}^{j} = 0 for j > 2) and F_{syn}^{j} defined by

F_{syn}^{0} = H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r))
F_{syn}^{1} = Ker [H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r)) ⟶ H^{i+1}(𝔛̂_{syn−et}, J_{Z_{p}}^{[r]})] =^{(4.11.1)}
= Ker [H^{i+1}(𝔛̂_{syn−et}, s_{Z_{p}}(r)) ⟶ H^{i+1}(𝔛̂_{syn−et}, O_{Z_{p}}^{cris})]
F_{syn}^{2} = 0

Proposition. Let 0 ≤ i + 1, r < p − 1. Then the Fontaine-Messing map ν satisfies

(1) ν(F_{syn}^{j}) ⊆ F_{et}^{j} (∀j ≥ 0).

(2) gr_{F}^{0} (ν) : F_{syn}^{0}/F_{syn}^{1} ⟶^{∼} F_{et}^{0}/F_{et}^{1} = E_{∞}^{0,i+1} = E_{2}^{0,i+1} = H^{0}(G, T^{i+1}(r)) is an isomorphism.

(3) gr_{F}^{1} (ν) : F_{syn}^{1} ↪ F_{et}^{1}/F_{et}^{2} = E_{∞}^{1,i} = E_{2}^{1,i} = H_{cont}^{1}(G_{K}, T^{i}(r)) is injective, with image contained in H_{f}^{1}(K, T^{i}(r)).

(4) If T^{i} is torsion-free (⟺ M^{i} is torsion-free), then the image of gr_{F}^{1} (ν) is equal to H_{f}^{1}(K, T^{i}(r)).

(5.4) Suppose that p > dim(𝔛) − 1, 0 ≤ r < p. The map

(−1)^{r−1}c_{r}^{syn}/(r − 1)! : (K_{0}(𝔛) ⊗ Z [\frac{1}{(dim(𝔛) − 1)!}])^{(r)} ⟶ H^{2r}(𝔛̂_{syn−et}, s_{Z_{p}}(r)),

composed with the isomorphism ([So 2], Thm. 4(iv); [GiSo], Thm. 8.2)

CH^{r}(𝔛) ⊗ Z [\frac{1}{(dim(𝔛) − 1)!}] ⟶^{∼} (K_{0}(𝔛) ⊗ Z [\frac{1}{(dim(𝔛) − 1)!}])^{(r)}

defines a cycle class map cl_{𝔛} which makes the following diagram commutative:

CH^{r}(𝔛) ⊗ Z [\frac{1}{(dim(𝔛)−1)!}] ⟶^{j^{∗}} CH^{r}(X) ⊗ Z [\frac{1}{(dim(𝔛)−1)!}]
↓^{cl_{𝔛}}  ↓^{cl_{X}}
H^{2r}(𝔛̂_{syn−et}, s_{Z_{p}}(r))  ⟶^{ν}  H^{2r}(X_{et}, Z_{p}(r))

(5.4.1)

(5.5) Proposition. Suppose that p > max(2r + 1, dim(𝔛) − 1). Then

Im(cl_{X}) ∩ F^{2}H^{2r}(X_{et}, Z_{p}(r)) = 0.

Proof. This follows from Proposition 5.3, the diagram (5.4.1) and the surjectivity of the map j^{∗} in (5.4.1).

(5.6) Assume that 0 ≤ r < p − 1, i + 1 = p − 1, dim(𝔛) < p and that F^{r}H_{dR}^{p−1}(𝔛/W) is torsion-free. Then the conclusions of (5.2)–(5.5) are still valid with the following modifications: in Theorem 5.2, the map α is injective and there is no zero at the end of the second row in (1). In Proposition 5.3(2), the map gr_{F}^{0} (ν) is injective. In Proposition 5.5, we take r = (i + 1)/2. All this follows again from Proposition I.3.5 and 4.17.

IV. Cycle classes and filtrations on Chow groups

In this chapter we treat varieties over number fields, by putting together local results of Chapters II and III,

1. Beilinson’s philosophy

(1.1) let F/Q be a finitely generated extension of transcendence degree d ≥ 0. If π : X ⟶ Spec(F) is a smooth and projective scheme, then the conjectural spectral sequence (0.1.0)

E_{2}^{a,b} = H^{a}(Spec(F)_{mot}, R^{b}π_{∗}Q(n)) ⟹ H^{a+b}(X_{mot}, Q(n))

is expected to degenerate at E_{2}. It is also expected that E_{2}^{a,b} = 0 for a > d+1 ([Ja 4], 4.12(c)). Consequently, the induced filtration F^{.}H^{∗}(X_{mot}, Q(n)) should satisfy

\begin{aligned}gr_{F}^{j} H^{m}(X_{mot}, Q(n)) & ⟶^{∼} H^{j}(Spec(F)_{mot}, R^{m−j}π_{∗}Q(n)) \\ F^{d+2}H^{m}(X_{mot}, Q(n)) & = 0 \end{aligned}  (1.1.1)

The corresponding p-adic étale realization, given by the Hochschild-Serre spectral sequence (0.3.0), degenerates at E_{2} and defines a filtration on H^{∗}(X_{et}, Q_{p}(n)) satisfying

gr_{F}^{j} H^{m}(X_{et}, Q_{p}(n)) ⟶^{∼} H_{cont}^{j}(G_{F} , H^{m−j}(\overline{X}_{et}, Q_{p}(n)))

The p-adic realization map

H^{m}(X_{mot}, Q(n)) ⟶ H^{m}(X_{et}, Q_{p}(n))  (1.1.2)

is expected to be strictly compatible with the filtrations, thus inducing injective maps

gr_{F}^{j} H^{m}(X_{mot}, Q(n)) ↪ gr_{F}^{j} H^{m}(X_{et}, Q_{p}(n))  (1.1.3)

(1.2) In order to make (1.1.3) a testable statement, one replaces motivic cohomology by its K-theoretic version

H^{m}(X_{mot}, Q(n)) =^{?} K_{2n−m}(X)_{Q}^{(n)}

We shall be particularly interested in the case m = 2n, when

K_{0}(X)_{Q}^{(n)} ⟶^{∼} CH^{n}(X) ⊗ Q

and the map (1.1.2) is replaced by the cycle class map

cl_{X} : CH^{n}(X) ⊗ Q ⟶ H^{2n}(X_{et}, Q_{p}(n))

The filtration F^{i}H^{2n}(X_{mot}, Q(n)) should then correspond to the mysterious filtration on CH^{n}(X) ⊗ Q, studied extensively by Beilinson, Bloch, Jannsen, Murre, Raskind, S. Saito and many other people (see [Ja 4] and [Ra] for references). The conjecture (1.1.1) predicts that

F^{d+2}(CH^{n}(X) ⊗ Q) = 0  (1.2.1)

and 1.1.3 can be reformulated as

F^{j}(CH^{n}(X) ⊗ Q) = cl_{X}^{−1} (F^{j}H^{2n}(X_{et}, Q_{p}(n)))  (1.2.2)

This implies, among other things, that

Ker(cl_{X}) = 0  (1.2.3)
Im(cl_{X}) ∩ F^{d+2}H^{2n}(X_{et}, Q_{p}(n)) = 0  (1.2.4)

The injectivity of cl_{X} is a very difficult problem. We shall investigate (1.2.4) in the special case d = 0 (i.e. when F is a number field).

2. Cycle class maps and étale cohomology

(2.1) Fix a prime number p. Let F/Q be a finite extension, π_{F} : X ⟶ Spec(F) a proper and smooth map. Let S be a finite set of primes of F such that {v|p} ⊆ S and that there is a proper and smooth model π : 𝔛 ⟶ Spec(O_{F,S}) = Spec(O_{F}) − S of X. Let G_{F,S} be the Galois group of the maximal extension of F unramified outside of S ∪ {v|∞}. Denote the map Spec(F) ⟶ Spec(O_{F,S}) by j.

For every i, n ≥ 0, r ∈ Z, the sheaf A = R^{i}π_{∗}(Z/p^{n}Z(r)) on Spec(O_{F,S})_{et} is locally constant and constructible, thus A = j_{∗}j^{∗}A, j^{∗}A = R^{i}π_{F ∗}(Z/p^{n}Z(r)) = H^{i}(\overline{X}_{et}, Z/p^{n}Z(r)) (which is a finite G_{F,S}- module) and

H^{∗}(Spec(O_{F,S})_{et}, A) = H^{∗}(G_{F,S}, H^{i}(\overline{X}_{et}, Z/p^{n}Z(r)))

([Mi, II.2.9]). Using I.2.3, we get a similar statement for continuous cohomology

H^{∗}(Spec(O_{F,S})_{et}, [R^{i}π_{∗}(Z/p^{n}Z(r))]_{n∈N}) = H_{cont}^{∗}(G_{F,S}, T^{i}(r))  (2.1.1)

Here T^{i}(r) = H^{i}(\overline{X}_{et}, Z_{p}). Of course, as G_{F,S} satisfies the condition I.2.3.2, the continuous cohomology group H_{cont}^{j}(G_{F,S}, T^{i}(r)) coincides with the naive one and is a Z_{p}-module of finite type.

(2.2) It follows from (2.1.1) that the Leray spectral sequence for π_{∗} is just

E_{2}^{a,b} = H_{cont}^{a}(G_{F,S}, T^{b}(r)) ⟹ H^{a+b}(𝔛_{et}, Z_{p}(r))

(cf. [Ja 2, Lemma 6]). It defines a filtration F^{.}H^{n,r} on H^{n,r} = H^{n}(𝔛_{et}, Z_{p}(r)). Similarly, the Leray spectral sequence for π_{F ∗}

′E_{2}^{a,b} = H_{cont}^{a}(G_{F} , T^{b}(r)) ⟹ H^{a+b}(X_{et}, Z_{p}(r))

defines a filtration F^{.}H^{n,r} on H^{n,r} = H^{n}(X_{et}, Z_{p}(r)). The restriction map j^{∗} : H^{n,r} ⟶ H^{n,r} is compatible with the filtrations. As cd_{p}(G_{F}) = cd_{p}(G_{F,S}) = 2, we have

\begin{aligned}gr_{F}^{0} (H^{n,r}) & = E_{∞}^{0,n} = E_{3}^{0,n} ↪ E_{2}^{0,n} \\ gr_{F}^{1} (H^{n,r}) & = E_{∞}^{1,n−1} = E_{2}^{1,n−1} \\ gr_{F}^{2} (H^{n,r}) & = F^{2}H^{n,r} = E_{∞}^{2,n−2} = E_{3}^{2,n−2} \\ F^{3}H^{n,r} & = 0 \end{aligned}  (2.2.1)

(and similarly for H^{n,r}). It follows from Weil’s conjectures [De 3] that E_{2}^{0,n} = ′E_{2}^{0,n} are finite groups for n ≠ 2r, thus

E_{∞}^{2,n−2} differs from E_{2}^{2,n−2} by a finite group  (n ≠ 2r − 1)  (2.2.2)

(and similarly for ′E_{2}). It follows from remarks at the end of (2.1) that all groups E_{2}^{a,b}, H^{n,r} are Z_{p}-modules of finite type. As

\begin{matrix}E_{2}^{0,n} = T^{n}(r)^{G_{F,S}} & = T^{n}(r)^{G_{F}} = ′E_{2}^{0,n} \\ E_{2}^{1,n−1} = H_{cont}^{1}(G_{F,S}, T^{n−1}(r)) & ↪^{inf} H_{cont}^{1}(G_{F} , T^{n−1}(r)) = ′E_{2}^{1,n−1}, \end{matrix}

we see that both maps

\begin{aligned}gr_{F}^{0} (j^{∗}) & : gr_{F}^{0} (H^{n,r}) ⟶ gr_{F}^{0} (H^{n,r}) \\ gr_{F}^{1} (j^{∗}) & : gr_{F}^{1} (H^{n,r}) ⟶ gr_{F}^{1} (H^{n,r}) \end{aligned}

are injective. As a result, we have

F^{k}H^{n,r} = (j^{∗})^{−1}(F^{k}H^{n,r})  (k = 0, 1, 2)  (2.2.3)

(2.3) For every r ≥ 0, there is a commutative diagram of cycle class maps (cf. [Sa], Sect. 5)

CH^{r}(𝔛)  ⟶^{j^{∗}}  CH^{r}(X)
↓^{cl_{𝔛}}  ↓^{cl_{X}}
H^{2r,r} = H^{2r}(𝔛_{et}, Z_{p}(r)) ⟶^{j^{∗}} H^{2r}(X_{et}, Z_{p}(r)) = H^{2r,r},

in which the top horizontal arrow is surjective. Combined with (2.2.3), this implies that

Im(cl_{X}) ∩ F^{k}H^{2r,r} = j^{∗}(Im(cl_{𝔛}) ∩ F^{k}H^{2r,r})  (k = 0, 1, 2)  (2.3.1)

(2.4) We are also interested in torsion phenomena. Recall a well-known

Lemma. Let F be a field finitely generated over Q, X a separated scheme of finite type over F.

(1) For every a ≥ 0 there is an integer C′ ≥ 1 such that H^{a}(\overline{X}_{et}, Z_{p})_{tors} is finite for every prime number p and vanishes for p |̸ C′(X, a).

(2) If X is proper and smooth over F, then, for every a ≥ 0, b ∈ Z, a ≠ 2b, there is an integer C(X, a, b) ≥ 1 such that (H^{a}(\overline{X}_{et}, Z_{p}(b)) ⊗ Q_{p}/Z_{p})^{G_{F}} is finite for every prime number p and vanishes for p |̸ C(X, a, b).

Proof. (1) Choose an embedding σ : \overline{F} ↪ C and put Y = \overline{X} ⊗_{\overline{F} ,σ} C. The smooth base change and comparison theorem with classical cohomology tell us that

H^{a}(\overline{X}_{et}, Z_{p}) = H^{a}(Y (C), Z) ⊗_{Z} Z_{p}

However, H^{a}(Y (C), Z) is a finitely generated abelian group, so its torsion is finite.

(2) There exists a subring R ↪ F, finitely generated as a Z-algebra, and a proper and smooth model 𝔛 ⟶ Spec(R) of X. Let v be a closed point of Spec(R), with residue field k(v). Let p be a prime, different from the characteristic ℓ_{v} of k(v). Then H^{a}(\overline{X}_{et}, Z_{p}(b)) is isomorphic to H^{a}((\overline{X} ⊗_{R} \overline{k}(v))_{et}, Z_{p}(b)), by the smooth and proper base change theorems. By Weil’s conjectures [De 4], the polynomial P_{v}(T) = det(1 − F r(v)T | H^{a}(\overline{X}_{et}, Q_{p}(b)) has coefficients in Z[1/ℓ_{v}], is independent of p and P_{v}(1) ≠ 1 (as H^{a}(\overline{X}_{et}, Q_{p}(b)) is pure of weight a − 2b ≠ 0 at v). The group (H^{a}(\overline{X}_{et}, Z_{p}(b)) ⊗ Q_{p}/Z_{p})^{G_{F}} is annihilated by P_{v}(1) by the Cayley-Hamilton theorem, so we can take C(X, a, b) = ℓ_{v} · (numerator of P_{v}(1)).

(2.5) Lemma. In the situation of 2.1, we have

\begin{aligned}(gr_{F}^{0} H^{n}(X_{et}, Z_{p}(r)))_{tors} & = (gr_{F}^{0} H^{n}(𝔛_{et}, Z_{p}(r)))_{tors} = 0  (p |̸ C′(X, n)) \\ (gr_{F}^{1} H^{n}(X_{et}, Z_{p}(r)))_{tors} & = (gr_{F}^{1} H^{n}(𝔛_{et}, Z_{p}(r)))_{tors} = 0  (p |̸ C(X, n − 1, r)C′(X, n − 1), n ≠ 2r + 1) \end{aligned}

Proof. As (E_{2}^{0,n})_{tors} = (′E_{2}^{0,n}))_{tors} = H^{n}(X_{et}, Z_{p}(r))_{tors}^{G_{F}}, the first statement follows from Lemma 2.4.(1) and (2.2.1). For the second statement, observe that

(E_{2}^{1,n−1})_{tors} ↪ (′E_{2}^{1,n−1})_{tors} = H^{1}(G_{F} , H^{n−1}(\overline{X}_{et}, Z_{p}(r))_{tors}) ⊕ H^{1}(G_{F} , H^{n−1}(\overline{X}_{et}, Z_{p}(r))/tors)_{tors}

The first term vanishes for p |̸ C′(X, n − 1), by Lemma 2.4.(1). The second term is isomorphic to ([Ta], Prop. 2.3)

(H^{n−1}(\overline{X}_{et}, Z_{p}(r)) ⊗ Q_{p}/Z_{p})^{G_{F}} /Div,

which vanishes for p |̸ C(X, n − 1, r), n ≠ 2r + 1, by Lemma 2.4.(2). We conclude again by (2.2.1).

(2.6) Corollary. If X is equidimensional of dimension d and A = Alb(X) is the Albanese variety of X, then

(gr_{F}^{0} H^{2d}(X_{et}, Z_{p}(d)))_{tors} = 0  (for all p)
(gr_{F}^{1} H^{2d}(X_{et}, Z_{p}(d)))_{tors} = 0  (for p |̸ ♯(A(F)_{tors}))

Proof. The trace map induces an isomorphism H^{2d}(\overline{X}_{et}, Z_{p}(d)) ⟶^{∼} Z_{p}^{c}, thus C′(X, 2d) = 1. As

H^{2d−1}(\overline{X}_{et}, Z_{p}(d)) ⟶^{∼} T_{p}(A),

we have

(H^{2d−1}(\overline{X}_{et}, Z_{p}(d)) ⊗ Q_{p}/Z_{p})^{G_{F}} = A(F)_{p^{∞}}.

Both claims now follow from Lemma 2.5.

3. Proof of Theorems E, F, G

(3.1) We begin with Theorem E, following its notation and assumptions. Fix a proper and smooth model 𝔛 ⟶ Spec(O_{F,S}) of X as in (2.1). Writing V^{i} for H^{i}(\overline{X}_{et}, Q_{p}), then we have (by I.2.3.3 and (2.2.2))

\begin{aligned}F^{2}H^{2n}(X_{et}, Q_{p}(n)) & = H_{cont}^{2}(G_{F} , V^{2n−2}(n)) \\ F^{2}H^{2n}(𝔛_{et}, Q_{p}(n)) & = H_{cont}^{2}(G_{F,S}, V^{2n−2}(n)) \end{aligned}

For every prime v of F, the restriction map

res_{v} : H^{2n}(X_{et}, Q_{p}(n)) ⟶ H^{2n}((X_{v})_{et}, Q_{p}(n))

(where X_{v} = X ⊗_{F} F_{v}) is compatible with the filtrations F^{j} on both sides. Again by I.2.3.3, we have

F^{2}H^{2n}((X_{v})_{et}, Q_{p}(n)) = H_{cont}^{2}(G_{F_{v}}, V^{2n−2}(n))

Applying II.2.5.3 and III.3.8-9, we see that

Im(cl_{𝔛}) ∩ F^{2}H^{2n}(𝔛_{et}, Q_{p}(n)) ⊆ Ker(α_{S,Σ})

Theorem E now follows from a Q_{p}-version of 2.3.1.

(3.2) For Theorem F, we apply Theorem E, with F = Q and X = E^{d}. According to Proposition II.2.6(i), we have Σ = S. Poitou-Tate duality and Poincaré duality for \overline{X} show that Ker(α_{S,S}) is the Q_{p}-dual of

Ker [β_{V} : H_{cont}^{1}(G_{Q,S}, V) ⟶ ⨁_{ℓ∈S} H_{cont}^{1}(G_{Q_{ℓ}}, V)] ,

where V = H^{2}(\overline{X}_{et}, Q_{p}(1)). Künneth formula gives

V = Q_{p}^{\binom{d+1}{2}} ⊕ W^{\binom{d}{2}},

where

W = Sym^{2} (H^{1}(\overline{E}_{et}, Q_{p})) (1).

Class field theory shows that Ker(β_{Q_{p}}) = 0. The vanishing of Ker(β_{W}) follows from the results of [Fl], [La 1, Lemma 2.5], [Wi].

(3.3) We now turn to the proof of Theorem G, with the following value of c_{E}:

c_{E} = 2N_{E}(deg(Φ))(♯(E(Q)_{tors})) ∏_{ℓ∈T_{1}}(ℓ − 1)(−ord_{ℓ}(j(E))) ∏_{ℓ∈T_{2}} ℓ  (3.3.1)

Here N_{E} is the conductor of E, Φ : X_{0}(N_{E}) ⟶ E some modular parametrization of E and

T_{1} = {ℓ | E has a potentially multiplicative reduction at ℓ} T_{2} = {ℓ | G_{Q} ⟶ Aut(E_{ℓ}) is not surjective}

Note that, according to ([Se 2], Thm. 2), the set T_{2} is finite. If E is semistable (i.e. N_{E} is square-free), then T_{2} ⊆ {2, 3, 5, 7} ([Se 2], Prop. 21). If d = 1, then CH^{1}(E) = Pic(E), CH^{1}(E)_{0} = Pic^{0}(E) = E(Q) and the kernel of cl_{E} : CH^{1}(E) ⟶ H^{2}(E_{et}, Z_{p}(1)) is equal to the prime-to-p torsion in E(Q); thus Im(cl_{E})_{tors} = E(Q)_{p^{∞}}. Suppose now that d > 1 and p |̸ (2d)!·c_{E} (in particular, p > 3). Put X = E^{d} (= Alb(X)). As p |̸ ♯((E^{d})_{tors}), Cor. 2.6 implies that

H^{2d}(𝔛_{et}, Z_{p}(d))_{tors} ⊆ (F^{2}H^{2d}(𝔛_{et}, Z_{p}(d)))_{tors}

Here we take 𝔛 as in (2.1), with S = \{ℓ|pN_{E}\}. Write, as before, T^{i} = H^{i}(\overline{X}_{et}, Z_{p}). Each T^{i} is torsion-free by Künneth formula, thus the spectral sequence in (2.2) satisfies E_{2}^{0,i} = 0 for i ≠ 2d. This shows that

F^{2}H^{2d}(𝔛_{et}, Z_{p}(d)) = E_{2}^{2,2d−2} = H_{cont}^{2}(G_{Q,S}, T^{2d−2}(d))

In fact the same argument applies locally: for any n ≥ 1 and any prime ℓ, we have

(V^{2n−1}(n))^{G_{Q_{ℓ}}} = 0,

by II.1.6.4 and II.1.7(1a) (resp. III.3.5.1) for ℓ ≠ p (resp. ℓ = p). As T^{2n−1} is torsion-free, we obtain

F^{2}H^{2n}((X_{ℓ})_{et}, Z_{p}(n)) = H_{cont}^{2}(G_{Q_{ℓ}}, T^{2n−2}(n))  (3.3.2)

(here X_{ℓ} = X ⊗_{Q} Q_{ℓ}). Fix now ℓ ∈ S and consider the restriction map

res_{ℓ} : H_{cont}^{2}(G_{Q,S}, T^{2d−2}(d)) ⟶ H_{cont}^{2}(G_{Q_{ℓ}}, T^{2d−2}(d))

By Künneth formula,

T^{2d−2}(d) = Z_{p}(1)^{d} ⊕ (T_{p}(E))^{⊗2})^{\binom{d}{2}} .

It follows that

H_{cont}^{2}(G_{Q_{ℓ}}, T^{2d−2}(d))_{tors} = H_{cont}^{2}(G_{Q_{ℓ}}, (T_{p}(E))^{⊗2})\binom{d}{2}_{tors}.  (3.3.3)

Write H for Im(cl_{𝔛})_{tors}. We claim that

res_{ℓ}(H) = 0  (∀ℓ ∈ S)  (3.3.4)

Case ℓ = p: In this case (3.3.4) follows from (3.3.2) and Proposition III.5.5. Case ℓ ≠ p: By ([Si], IV.10.3 and V.5.3), the greatest common divisor of degrees [K : Q_{ℓ}] of extensions over which E acquires semistable reduction divides 24. As p > 3, we can choose such an extension K/Q_{ℓ} of degree d prime to p. Subcase ℓ ∈ T_{1}: By Lemma 3.4 below, the group (3.3.3) vanishes. Subcase ℓ ∉ T_{1} ∪ {p}: E has a potentially good reduction at ℓ. As above, we can choose an extension K/Q_{ℓ} of degree prime to p over which E acquires good reduction. In this case (3.3.4) follows from Proposition II.2.8 and (3.3.2). So we see that H is contained in (the torsion subgroup of)

Ker [α : H_{cont}^{2}(G_{Q,S}, T) ⟶ ⨁_{ℓ∈S} H_{cont}^{2}(G_{Q_{ℓ}}, T)]

for T = T^{2d−2}(d). By Poitou-Tate duality, Ker(α) is the Pontryagin dual of

Ker [β : H_{cont}^{1}(G_{Q,S}, T^{∗}(1) ⊗ Q_{p}/Z_{p}) ⟶ ⨁_{ℓ∈S} H_{cont}^{1}(G_{Q_{ℓ}}, T^{∗}(1) ⊗ Q_{p}/Z_{p})]

We have

T^{∗}(1) = Z_{p}^{\binom{d+1}{2}} ⊕ (Sym^{2}(T_{p}(E))(−1))^{\binom{d}{2}}

The first factor does not contribute to Ker(β) by class field theory; as for the second factor,

Ker [H_{cont}^{1}(G_{Q,S}, Sym^{2}(T_{p}(E))(−1) ⊗ Q_{p}/Z_{p}) ⟶ ⨁_{ℓ∈S} H_{cont}^{1}(G_{Q_{ℓ}}, Sym^{2}(T_{p}(E))(−1) ⊗ Q_{p}/Z_{p})]

vanishes by ([Fl], Thm. 1). Theorem G is proved.

(3.4) Lemma. Let K be a finite extension of Q_{ℓ}, k the residue field of K, E an elliptic curve over K with a potentially multiplicative reduction. For a prime p ≠ ℓ, put T = T_{p}(E), V = T ⊗ Q. If p |̸ (♯(k^{∗}))(−ord_{K}(j(E))) (here ord_{K} : K^{∗} ⟶ Z is a surjective valuation and j(E) the j-invariant of E), then

(1) H^{0}(G_{K}, V^{⊗2}/T^{⊗2}(−1)) ⟶^{∼} Q_{p}/Z_{p}.

(2) H_{cont}^{2}(G_{K}, T^{⊗2}) ⟶^{∼} Z_{p}.

Proof. It is sufficient to prove (1); the statement (2) then follows by local duality, as T^{∗}(1) ⟶^{∼} T. Our assumptions imply that E is a quadratic twist of a Tate curve. Such a twist does not change the Galois representation T^{⊗2}, so we can assume that E itself is a Tate curve over K, with Tate’s parameter q_{E} ∈ K^{∗}. The boundary map

δ : Q_{p}/Z_{p} ⟶ H^{1}(G_{K}, Q_{p}/Z_{p}(1)) = K^{∗} ⊗ Q_{p}/Z_{p}

associated to the standard exact sequence of G_{K}-modules

0 ⟶ Q_{p}/Z_{p}(1)⟶^{α}V/T ⟶^{β}Q_{p}/Z_{p} ⟶ 0

is given by δ(a) = q_{E} ⊗ a. As p |̸ (♯(k^{∗})), the valuation ord_{K} defines an isomorphism ord_{K} ⊗ id : K^{∗} ⊗ Q_{p}/Z_{p} ⟶^{∼} Q_{p}/Z_{p}. This implies that δ is an isomorphism as well, since p |̸ ord_{K}(q_{E}) = −ord_{K}(j(E)). Observing that

H^{0}(G_{K}, Q_{p}/Z_{p}(1)) = H^{0}(G_{K}, Q_{p}/Z_{p}(−1)) = 0

(again by p |̸ (♯(k^{∗}))), we see that various cohomology sequences associated to the diagram

0  0  0
↓  ↓  ↓
0 ⟶ Q_{p}/Z_{p}(1) ⟶^{id⊗α(−1)}  V/T  ⟶^{id⊗β(−1)}  Q_{p}/Z_{p}  ⟶ 0
↓^{α(−1)⊗id}  ↓^{α(−1)⊗id}  ↓^{α(−1)⊗id}
0 ⟶  V/T  ⟶^{id⊗α(−1)} V^{⊗2}/T^{⊗2}(−1) ⟶^{id⊗β(−1)} V/T (−1) ⟶ 0
↓^{β(−1)⊗id}  ↓^{β(−1)⊗id}  ↓^{β(−1)⊗id}
0 ⟶ Q_{p}/Z_{p}  ⟶^{id⊗α(−1)}  V/T (−1)  ⟶^{id⊗β(−1)} Q_{p}/Z_{p}(−1) ⟶ 0
↓  ↓  ↓
0  0  0

give H^{0}(G_{K}, V/T) = Ker(δ) = 0 and isomorphisms

\begin{aligned}& & Q_{p}/Z_{p} \\ ↓ & ≀ \\ H^{0}(G_{K}, V^{⊗2}/T^{⊗2}(−1)) & ⟶^{∼} & H^{0}(G_{K}, V/T (−1)), \end{aligned}  (3.3.5)

which proves the statement (1) of the Lemma (to see that the horizontal map in (3.3.5) is an isomorphism, note that the boundary map

H^{0}(G_{K}, V/T (−1)) ⟶ H^{1}(G_{K}, V/T)

composed with the isomorphism H^{0}(G_{K}, Q_{p}/Z_{p}) ⟶^{∼} H^{0}(G_{K}, V/T (−1)) vanishes for trivial reasons).

References

[Ba] W. Bauer, On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic p > 0, Invent. Math. 108 (1992), 263–287.

[Be 1] A. A. Beilinson, Vysšije reguljatory i značenija L-funkcij, in: Itogi Nauki i Techniki, Sovremennyje Problemy Matematiki 24 , VINITI, Moskva, 1984, pp. 181–238 (English translation: Higher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036–2070).

[Be 2] A. A. Beilinson, Notes on absolute Hodge cohomology, in: Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, Proceedings of a Summer Research Conference held June 12-18, 1983, in Boulder, Colorado, Contemporary Mathematics 55 , American Mathematical Society, Providence, Rhode Island, pp. 35–68.

[Be 3] A. A. Beilinson, Height pairing between algebraic cycles, in: K-theory, Arithmetic and Geometry; Seminar, Moscow 1984-86 (Yu.I. Manin, ed.), Lect. Notes in Math. 1289 , Springer, Berlin, Heidelberg, New York, 1987, pp. 1–26.

[Ber] P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0, Lect. Notes in Math. 407 , Springer, Berlin, Heidelberg, New York, 1974.

[BO 1] P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press, Princeton, NJ, 1978. [BO 2] P. Berthelot, A. Ogus, F -Isocrystals and De Rham Cohomology. I, Invent. Math. 72 (1983), 159–199.

[BK] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift I, Progress in Mathematics 86 , Birkhäuser, Boston, Basel, Berlin, 1990, pp. 333–400.

[Br] C. Breuil, Cohomologie étale de p-torsion et cohomologie cristalline en réduction semi-stable, preprint

Orsay 1997.

[CLeS] B. Chiarellotto, B. Le Stum, Sur la pureté de la cohomologie cristalline, C. R. Acad. Sci Paris, Série I, t. 326 (1998), 961–963.

[De 1] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. de

l’I.H.E.S. 35 (1968), 107–126. [De 2] P. Deligne, Théorie de Hodge I, Actes CIM Nice 1970, t. I, Gauthier-Villars, 1971, pp. 425–430. [De 3] P. Deligne, La conjecture de Weil I, Publ. Math. de l’I.H.E.S. 43 (1974), 273–308. [De 4] P. Deligne, La conjecture de Weil II, Publ. Math. de l’I.H.E.S. 52 (1980), 137–252. [De 5] P. Deligne, Lettre à C. Soulé (1/20/1985).

[Ek] T. Ekedahl, On The Adic Formalism, in: The Grothendieck Festschrift, Vol. II, Progress in Math. 87 , Birkhäuser, Boston, Basel, 1990, pp. 197–218.

[Fa] G. Faltings, Crystalline cohomology and p-adic Galois representations, in: Algebraic Analysis, Geometry, and Number Theory (Igusa, J.-I., ed.), John Hopkins University Press, Baltimore, 1990, pp. 25–79.

[Fl] M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992),

307–327. [Fo 1] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in: Astérisque 65 , Soc.

Math. de France, Paris, 1979, pp. 3–80. [Fo 2] J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local;

construction d’un anneau de Barsotti-Tate, Ann. Math. 115 (1982), 529–577. [Fo 3] J.-M. Fontaine, Cohomologie de de Rham, cohomologie cristalline et représentations p-adiques, in: Al-

gebraic Geometry, Lect. Notes in Math. 1016 , Springer, Berlin, Heidelberg, New York, 1983, pp. 86–108.

[Fo 4] J.-M. Fontaine, Le corps des périodes p-adiques, in: [Bures], Exposé II, pp. 59–101. [Fo 5] J.-M. Fontaine, Représentations p-adiques semistables, in: [Bures], Exposé III, pp. 113–184.

[FL] J.-M. Fontaine, G. Laffaille, Construction de représentations p-adiques, Ann. Sci. E.N.S. (4) 15 (1982), 547–608.

[FM] J.-M. Fontaine, W. Messing, p-adic periods and p-adic étale cohomology, in: Current Trends in Arithmetic Algebraic Geometry, Contemporary Mathematics 67 , American Mathematical Society, Providence, Rhode Island, 1987, pp. 179–207.

[FoPR] J.-M. Fontaine and B. Perrin-Riou, Autour des conjectures de Bloch and Kato: cohomologie galoisienne et valeurs de fonctions L, in: Motives, Proceedings of AMS Summer Research Conference in Seattle (July 1991), Proc. Symposia in Pure Math. 55/I , American Mathematical Society, Providence, Rhode Island, 1994, pp. 599–706.

[Gi] H. Gillet, Riemann-Roch theorems for higher algebraic K-theory, Adv.in Math. 40 (1981), 203–289.

[GiSo] H. Gillet, C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), 243–278.

[Gros] M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques I, Invent. Math. 99 (1990), 293–320.

[GrSc] B.H. Gross, C. Schoen, The modified diagonal cycle on the triple product of a pointed curve, Ann. Inst. Fourier, Grenoble 45, 3 (1995), 649-679.

[Grot] A. Grothendieck, La classe de cohomologie associée à un cycle, [SGA 4\frac{1}{2}], pp. 129–153.

[Hu] A. Huber, Calculation of Derived Functors via Ind-Categories, J. Pure Appl. Algebra 90 (1993), 39–48. [Il 1] L. Illusie, Complexe Cotangent et Déformations II, Lect. Notes in Math. 283 , Springer, Berlin,

Heidelberg, New York, 1972. [Il 2] L. Illusie, Autour du théorème de monodromie local, in: [Bures], Exposé I, pp. 9–57. [Ja 1] U. Jannsen, Continuous étale cohomology, Mathematische Annalen 280 (1988), 207–245. [Ja 2] U. Jannsen, On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology, in:

Galois Groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Mathematical Sciences Research Institute Publications 16, Proceedings of a Workshop held March 23-27, 1987, Springer, Berlin, Heidelberg, New York, 1989, pp. 315–360.

[Ja 3] U. Jannsen, Mixed Motives and Algebraic K-Theory, Lect. Notes in Math. 1400 , Springer, Berlin,

Heidelberg, New York, 1990. [Ja 4] U. Jannsen, Motivic sheaves and filtrations on Chow groups, in: Motives, Proceedings of AMS Sum-

mer Research Conference in Seattle (July 1991), Proc. Symposia in Pure Math. 55/I , American Mathematical Society, Providence, Rhode Island, 1994, pp. 245–302.

[de J] A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. de l’I.H.E.S. 83 (1996), 51–93.

[Ka] K. Kato, On p-Adic Vanishing Cycles, in: Algebraic Geometry, Sendai 1985, Adv. Studies in Pure

Math. 10 , 1987, pp. 207–251.

[KaM] K. Kato, W. Messing, Syntomic cohomology and p-adic étale cohomology, Tôhoku Math. J. 44 (1992), 1–9.

[KM] N. Katz, W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.

[Kn] K. Künnemann, Projective regular models for abelian varieties, semi-stable reduction, and the height pairing, preprint 1996.

[Ku] M. Kurihara, A Note on p-adic Etale Cohomology, Proc. Japan Acad., Ser. A, 63 (1987), 275–278.

[L] G. Laffaille, Groupes p-divisibles et modules filtrées: le cas peu ramifié, Bull. Soc. Math. France 108

(1980), 187–206. [La 1] A. Langer, Selmer groups and torsion zero cycles on the self-product of a semistable elliptic curve, Doc.

Math. 2 (1997), 47–59. [La 2] A. Langer, Local points of motives in semi-stable reduction, Comp. Math., to appear.

[LaSa] A. Langer, S. Saito, Torsion zero-cycles on the self-product of a modular elliptic curve, Duke Math. J.

85 (1996), 315–357.

[Mi] J. S. Milne, Arithmetic Duality Theorems, Perspectives in Mathematics 1 , Academic Press, Boston,

1986. [Ne 1] J. Nekovář, Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992),

99–125. [Ne 2] J. Nekovář, On p-adic height pairings, in: Séminaire de Théorie des Nombres de Paris 1990/91, Progress

in Math. 108 , (S. David, ed.), Birkhäuser, Boston, 1993, pp. 127–202. [Ne 3] J. Nekovář, On the p-adic height of Heegner cycles, Math. Annalen 302 (1995), 609–686.

[Ni] W. Nizioł, On the image of p-adic regulators, Invent. Math. 127 (1997), 375–400.

[Og] A. Ogus, F -crystals, Griffiths transversality, and the Hodge decomposition, Astérisque 221 , Soc. Math. de France, Paris, 1994.

[RZ] M. Rapoport, T. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.

[Ra] W. Raskind, Higher ℓ-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J. 78 (1995), 33–57.

[Sa] S. Saito, On the cycle class map for torsion algebraic cycles of codimension two, Invent. Math. 106 (1991), 443–460.

[Sc 1] P. Schneider, Introduction to the Beilinson Conjectures, in: Beilinson’s Conjectures on Special Values of L-functions, (M. Rapoport, N. Schappacher, P. Schneider, eds.), Perspectives in Mathematics 4 , Academic Press, Boston, 1988, pp. 1–35.

[Sc 2] P. Schneider, p-adic points of motives, in: Motives, Proceedings of AMS Summer Research Conference in Seattle (July 1991), Proc. Symposia in Pure Math. 55/II , American Mathematical Society, Providence, Rhode Island, 1994, pp. 225–249.

[Se 1] J.-P. Serre, Cohomologie Galoisienne, Lect. Notes in Math. 5 , Springer, Berlin, Göttingen, Heidelberg,

New York, 1964. [Se 2] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15

(1972), 259–331.

[ST] J.-P. Serre, J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517.

[Si] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math. 151 ,

Springer, New York, 1994. [So 1] C. Soulé, The rank of étale cohomology of varieties over p-adic or number fields, Comp. Math. 53

(1984), 113–131. [So 2] C. Soulé, Opérations en K-théorie algébrique, Canadian J. of Math. 37 (1985), 488–550.

[T] G. Tamme, The theorem of Riemann-Roch, in: Beilinson’s Conjectures on Special Values of L-functions, (M. Rapoport, N. Schappacher, P. Schneider, eds.), Perspectives in Mathematics 4 , Academic Press, Boston, 1988, pp. 103–168.

[Ta] J. Tate, Relations between K_{2} and Galois cohomology, Invent. Math. 36 (1976), 257–274.

[Ts] T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, preprint 1996.

[Wi] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443–551.

[Bures] Périodes p-adiques, Astérisque 223 , Soc. Math. de France, Paris, 1994. [SGA 4] Théorie dee Topos et Cohomologie Étale des Schémas, Lect. Notes in Math. 269 , 270 , 305 , Springer,

Berlin, Heidelberg, New York, 1972/3.

[SGA 4\frac{1}{2}] Cohomologie Étale, Lect. Notes in Math. 569 , Springer, Berlin, Heidelberg, New York, 1977. [SGA 5] Cohomologie ℓ-adique et Fonctions L, Lect. Notes in Math. 589 , Springer, Berlin, Heidelberg, New

York, 1977. [SGA 6] Théorie des Intersections et Théorème de Riemann-Roch, Lect. Notes in Math. 225 , Springer, Berlin,

Heidelberg, New York, 1971.

[SGA 7/I] Groupes de Monodromie en Géométrie Algébrique, Lect. Notes in Math. 288 , Springer, Berlin, Heidelberg, New York, 1972.

Department of Pure Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 1SB United Kingdom

e-mail: nekovar@pmms.cam.ac.uk