p-adic Abel-Jacobi maps and p-adic heights

Jan Nekovář

This is a slightly expanded version of the author’s lecture at the 1998 Banff Conference “The Arithmetic and Geometry of Algebraic Cycles”. Its aim is to give a survey of foundational results on p-adic Abel-Jacobi maps and p-adic height pairings for algebraic cycles on varieties over number fields. Some of the results are new; their proofs will be given elsewhere.

1. p-adic cycle class maps

(1.0) Let X be a proper and smooth variety over a field K. Fix a separable closure K^{sep} of K and put \overline{X} = X ×_{K} K^{sep}, G_{K} = Gal(K^{sep}/K). We also fix a prime number p ≠ char(K). Throughout this article, all cohomology groups with ℚ_{p}-coefficients refer to continuous cohomology in the sense of [Ja 1].

(1.1) The fundamental object of our interest is the p-adic cycle class map

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

with values in arithmetic étale cohomology of X. The link to geometric étale cohomology V^{j} = H^{j}(\overline{X}_{et}, ℚ_{p}) is provided by the Hochschild-Serre spectral sequence

etE_{2}^{i,j} = H^{i}(G_{K}, V^{j}(n)) ⟹ H^{i+j}(X_{et}, ℚ_{p}(n)),  (1.1.1)

which can also be viewed as a Leray spectral sequence for the structure morphism X ⟶ Spec(K).

(1.2) It is a non-trivial fact that (1.1.1) degenerates at E_{2}, i.e. that ^{et}E_{2}^{i,j} = ^{et}E_{∞}^{i,j}. For X smooth and projective over K, this follows from Deligne’s Thesis [De 1] and his proof of the Hard Lefschetz Theorem [De 2] (together with the existence of an appropriate formalism of derived categories for continuous cohomology [Ek]). The general case reduces to the previous one by using de Jong’s theorem on alterations [deJ].

As a consequence of this degeneration, the filtration F^{∗}H^{∗}(X_{et}, ℚ_{p}(n)) defined by (1.1.1) satisfies

gr_{F}^{j} H^{m}(X_{et}, ℚ_{p}(n)) ⟶^{∼} H^{j}(G_{K}, V^{m−j}(n)).

One obtains, therefore, a sequence of induced maps

cl_{0,X} : CH^{n}(X) ⟶ H^{0}(G_{K}, V^{2n}(n))
cl_{1,X} : Ker(cl_{0,X}) ⟶ H^{1}(G_{K}, V^{2n−1}(n))
cl_{j,X} : Ker(cl_{j−1,X}) ⟶ H^{j}(G_{K}, V^{2n−j}(n))
cl_{j,X} ⊗ ℚ_{p} : Ker(cl_{j−1,X}) ⊗ ℚ_{p} ⟶ H^{j}(G_{K}, V^{2n−j}(n))

(1.3) The subgroup

CH^{n}(X)_{hom∼0} := Ker(cl_{0,X}) ⊆ CH^{n}(X)

is the subgroup of homologically trivial cycles (modulo torsion) and cl_{1,X} is the p-adic Abel-Jacobi map.

If char(K) = 0, then one can assume without loss of generality that there is an embedding σ : K^{sep} ↪ ℂ. Putting X_{ℂ} = \overline{X} ×_{K^{sep},σ} ℂ and using standard comparison isomorphisms between étale and singular cohomology over ℂ

CH^{n}(\overline{X}) ⟶ H^{2n}(\overline{X}_{et}, ℚ_{p}(n)) ⟶_{σ_{∗}}^{∼} H^{2n}((X_{ℂ})_{et}, ℚ_{p}(n))
↓^{σ_{∗}}  ↓^{≃}
CH^{n}(X_{ℂ})  ⟶  H^{2n}(X_{ℂ}^{an}, ℚ(n)) ⊗_{ℚ} ℚ_{p}

we see that Ker(cl_{0,X}) is independent on the choice of p.

If char(K) > 0, then the independence of Ker(cl_{0,X}) on p is unknown in general.

2. Varieties over number fields

(2.0) From now on, K will be a finite extension of ℚ (hence H^{i}(G_{K}, V^{j}(n)) = 0 for i > 2). In this case the cycle class maps are expected to behave reasonably well.

(2.1) Conjecture. (1) Im (cl_{0,X} ⊗ ℚ_{p}) =^{?} H^{0}(G_{K}, V^{2n}(n)) (a special case of Tate’s conjectures [Ta]).

(2) Ker(cl_{1,X} ⊗ ℚ_{p}) =^{?} 0 and Im (cl_{1,X} ⊗ ℚ_{p}) =^{?} H_{f}^{1}(G_{K}, V^{2n−1}(n))

(a special case of conjectures of Bloch-Kato [BK]).

(2.2) Let us recall the definition of Bloch-Kato style Selmer groups [BK] appearing in Conjecture 2.1.2. Let V be a ℚ_{p}-representation of G_{K}, i.e. a finite dimensional vector space over ℚ_{p} with a continuous ℚ_{p}-linear action of G_{K} (e.g. V = V^{j}(n)). The subspace H_{f}^{1}(G_{K}, V) ⊂ H^{1}(G_{K}, V) is defined by the cartesian square

\begin{matrix}H_{f}^{1}(G_{K}, V) & ↪ & H^{1}(G_{K}, V) \\ ↓ & ■ & ↓ \\ ∏_{v} H_{f}^{1}(G_{v}, V) & ↪ & ∏_{v} H^{1}(G_{v}, V), \end{matrix}

in which v runs through all non-archimedean primes of K, G_{v} = Gal(\overline{K}_{v}/K_{v}) and

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

Replacing B_{cris} by B_{st} (resp. by B_{dR}), one obtains – for v|p – groups H_{st}^{1}(G_{v}, V) (resp. H_{g}^{1}(G_{v}, V)) satisfying

H_{f}^{1}(G_{v}, V) ⊆ H_{st}^{1}(G_{v}, V) ⊆ H_{g}^{1}(G_{v}, V) ⊆ H^{1}(G_{v}, V).

We refer to [Fo 1] and [Fo 2] for more details on Fontaine’s rings B_{cris} ⊂ B_{st} ⊂ B_{dR}.

(2.3) Examples. (1) n = 1. In this case V^{1}(1) = V_{p}(A) = T_{p}(A) ⊗_{ℤ_{p}} ℚ_{p}, where A = Pic_{X/K}^{0} is the connected Picard variety of X. Descent theory for the abelian variety A gives maps into classical Selmer groups

A(K) ⊗ ℤ/p^{n}ℤ ↪ Sel(A/K, p^{n}) ⊂ H^{1}(G_{K}, A_{p^{n}}).

Passing to the projective limit and tensoring with ℚ_{p} one obtains the cycle class map for homologically trivial divisors

cl_{1,X} ⊗ ℚ_{p} : A(K) ⊗ ℚ_{p} ↪ H_{f}^{1}(G_{K}, V_{p}(A)) ⊂ H^{1}(G_{K}, V_{p}(A))

(as A(K) is finitely generated by the Mordell-Weil theorem, and

(\underleftarrow{lim}_{n} Sel(A/K, p^{n})) ⊗_{ℤ_{p}} ℚ_{p} = H_{f}^{1}(G_{K}, V_{p}(A))

by [BK, 3.11]). The conjectural equality 2.1.2

Im (cl_{1,X} ⊗ ℚ_{p}) =^{?} H_{f}^{1}(G_{K}, V_{p}(A))

is equivalent to the finiteness of � (A/K)_{p^{∞}}, the p-primary part of the Tate-Šafarevič group of A over K.

(2) n = d = dim(X) (assuming that X is equidimensional). The Albanese variety Alb_{X/K} can be identified with the dual Â of A = Pic_{X/K}^{0} and V^{2d−1}(d) with V_{p}(Â). The cycle class map cl_{d,X} ⊗ℚ_{p} is the composition of the (surjective) Albanese map

CH_{0}(X) ⊗ ℚ_{p} −↠ Â(K) ⊗ ℚ_{p}

and the (injective) descent map Â(K) ⊗ ℚ_{p} ↪ H_{f}^{1}(G_{K}, V_{p}(Â)). The injectivity of the Albanese map is a very difficult open problem. The surjectivity of the descent map is, as before, equivalent to the finiteness of � (A/K)_{p^{∞}}.

(2.4) As a basic check of consistency of the Conjecture 2.1.2 one would like to show that

Im (cl_{1,X} ⊗ ℚ_{p}) ⊆^{?} H_{f}^{1}(G_{K}, V^{2n−1}(n))  (2.4.1)

For n = 1 or n = dim(X) this follows from the discussion in Examples 2.3. In general, (2.4.1) boils down to a local question at each non-archimedean prime v of K. Changing perspective, we fix v and consider a proper and smooth variety X_{v} over K_{v}. The question is whether

Im (cl_{1,X_{v}} ⊗ ℚ_{p}) ⊆^{?} H_{f}^{1}(G_{v}, V^{2n−1}(n))  (2.4.2)

(now with V^{j} = H^{j}((\overline{X}_{v})_{et}, ℚ_{p})). At primes not dividing p the situation is more or less under control.

(2.5) Proposition. Let v |̸ p. If X_{v} has potentially good reduction or, more generally, if the purity conjecture for the monodromy filtration ([Ja 2], [RZ]) on V^{2n−1} holds, then H^{1}(G_{v}, V^{2n−1}(n)) = 0.

Proof. Tate’s local Euler characteristic formula [Mi, Thm. I.2.8], [Se, II.5.7 Thm. 5] gives (for V = V^{2n−1}(n))

dim_{ℚ_{p}} H^{1}(G_{v}, V) = dim_{ℚ_{p}} H^{0}(G_{v}, V) + dim_{ℚ_{p}} H^{0}(G_{v}, V^{∗}(1)).  (2.5.1)

Our assumptions, together with Weil conjectures proved by Deligne [De 2], imply that the geometric Frobenius F r(v) ∈ G_{v}/I_{v} acts on both V^{I_{v}} and (V^{∗}(1))^{I_{v}} with weights < 0. It follows that the R.H.S. of (2.5.1) is equal to zero.

3. Varieties over p-adic fields

(3.0) In this section, v|p (i.e. K_{v} is a finite extension of ℚ_{p}) and X_{v} is proper and smooth over K_{v}. Our goal is to verify (2.4.2).

(3.1) Theorem. (i) If X_{v} has potentially good reduction, then (2.4.2) holds. (ii) In general,

Im (cl_{1,X_{v}} ⊗ ℚ_{p}) ⊆ H_{st}^{1}(G_{v}, V^{2n−1}(n)) = H_{g}^{1}(G_{v}, V^{2n−1}(n)).

If the p-adic version of the purity conjecture for the monodromy filtration for V^{2n−1} holds [Mo, Conj. 3.27], then H_{st}^{1}(G_{v}, V^{2n−1}(n)) = H_{f}^{1}(G_{v}, V^{2n−1}(n)) and (2.4.2) again holds.

(3.2) The purity conjecture for the monodromy filtration is known to hold (for all V^{j} and all v) for abelian varieties and for compactified Kuga-Sato varieties over modular curves [Sa]. The validity of (2.4.1) in the latter case implies that the main result of [Ne 1] holds for all p.

(3.3) We now explain an approach to the proof of Theorem 3.1 based on [Ne 3]. The part (i) was proved independently by Nizioł[Ni] (she assumed X_{v} to be smooth and projective, but the general case follows by the same argument as in 3.9 below). A special case of (ii) is due to A. Langer [La]. Let us distinguish two cases.

(3.4) Good reduction case. Assume that X_{v} has a model 𝔛_{v}, proper and smooth over O_{v} (the ring of integers of K_{v}). According to Fontaine’s conjecture C_{cris}, which is now a theorem ([FM], [Fa], [KaM], [Ts]), all V^{j} are crystalline representations of G_{v} ([Fo 1], [Fo 2]). One defines

H_{f}^{i}(G_{v}, −) = Ext_{Rep_{cris}(G_{v})}^{i}(ℚ_{p}, −),

where Rep_{cris}(G_{v}) – a full subcategory of Rep(G_{v}) – is the category of crystalline representations of G_{v} (for i = 1 this is compatible with the definition given in 2.2). For an object V of Rep_{cris}(G_{v}) one has

H_{f}^{0}(G_{v}, V) = H^{0}(G_{v}, V), H_{f}^{1}(G_{v}, V) ⊆ H^{1}(G_{v}, V), H_{f}^{i}(G_{v}, V) = 0 (i > 1).

The key ingredient of the proof of Theorem 3.1.in the good reduction case is the syntomic cohomology H^{i}((𝔛_{v})_{syn}, s_{ℚ_{p}}(n)) of Fontaine-Messing [FM], which is a p-adic analogue of Deligne cohomology (more precisely, of Beilinson’s “absolute Hodge cohomology” [Be]). Recall that there are two kinds of cohomology theories, geometric and arithmetic (= absolute), that come in pairs [Ja 3, 10.5]. For example, we have:

Geometric theory  Arithmetic theory
étale cohomology of \overline{X}  étale cohomology of X
Betti cohomology (over ℂ)  absolute Hodge cohomology
crystalline cohomology of 𝔛_{v}  syntomic cohomology

Given the limited scope of this article, we have to treat syntomic cohomology as a black box. One of its important features is the existence of a commutative diagram of cycle class maps

cl_{𝔛_{v}}^{syn} : CH^{n}(𝔛_{v}) ⟶ H^{2n}((𝔛_{v})_{syn}, s_{ℚ_{p}}(n))
↓^{res}  ↓^{α_{F M}}
cl_{X_{v}}^{et} : CH^{n}(X_{v}) ⟶ H^{2n}((X_{v})_{et}, ℚ_{p}(n)),

(3.4.1)

where α_{F M} is the (highly non-trivial) Fontaine-Messing map (for p > 2 it was constructed in [FM]; the case of p = 2 is treated in [Ts]).

One possible definition of the syntomic cycle class cl_{𝔛_{v}}^{syn} uses the fact that CH^{n}(𝔛_{v}) ⊗ ℚ is canonically isomorphic to the weight n eigenspace of K_{0}(𝔛_{v}) ⊗ ℚ with respect to Adams operations ψ^{k}; the usual formula cl_{𝔛_{v}}^{syn} = (−1)^{n−1}c_{n}^{syn}/(n − 1)! then makes sense. Similarly, one has cl_{X_{v}}^{et} = (−1)^{n−1}c_{n}^{et}/(n − 1)!. The compatibility between the syntomic c_{n}^{syn} and étale c_{n}^{et} Chern classes of vector bundles with respect to the Fontaine-Messing map reduces, by the splitting principle, to the compatibility of the two c_{1}’s of a line bundle, which can be verified explicitly. See [Ts] for more details, which are also valid in the logarithmic case discussed in 3.5 below.

(3.5) Semistable reduction case. Assume that X_{v} has a proper semistable model 𝔛_{v} over O_{v} (i.e., étale locally, 𝔛_{v} is given by Spec(O_{v}[T_{1}, …, T_{n}]/(T_{1} … T_{m} − π_{v})), where m ≤ n and π_{v} is a uniformizer of O_{v}). In this case 𝔛_{v} has a canonical log-structure 𝔛_{v}^{×} induced from X_{v} (see [HK] for a crash course on log-structures). According to Fontaine-Jannsen’s conjecture C_{st}, proved in ([Ka], [Ts]), all V^{j} are semistable representations of G_{v} ([Fo 1], [Fo 2]). As in 3.4, one defines

H_{st}^{i}(G_{v}, −) = Ext_{Rep_{st}(G_{v})}^{i}(ℚ_{p}, −),

where Rep_{st}(G_{v}) – a full subcategory of Rep(G_{v}) – is the category of semistable representations of G_{v} (again, for i = 1 this is compatible with the definition given in 2.2). For an object V of Rep_{st}(G_{v}) one has [FPR, Chap. I, Prop. 3.3.8]

H_{st}^{0}(G_{v}, V) = H^{0}(G_{v}, V), H_{st}^{1}(G_{v}, V) ⊆ H^{1}(G_{v}, V),
H_{st}^{2}(G_{v}, V) −↠ H^{2}(G_{v}, V), H_{st}^{i}(G_{v}, V) = 0 (i > 2).

More precisely, [FPR] gives an explicit complex computing cohomology H_{st}^{i}(G_{v}, V). The map

H_{st}^{2}(G_{v}, V) −↠ H^{2}(G_{v}, V)

is surjective, being dual to the inclusion

H^{0}(G_{v}, V^{∗}(1)) = D_{st}(V^{∗}(1))^{N=0,φ=1} ∩ F^{0}D_{dR}(V^{∗}(1)) ↪ D_{st}(V^{∗}(1))^{N=0,φ=1}.

Here D_{st}(V^{∗}(1)) = H^{0}(G_{v}, V^{∗}(1) ⊗_{ℚ_{p}} B_{st}) denotes the filtered (φ, N)-module associated to V^{∗}(1) and F^{i}D_{dR}(V^{∗}(1)) = H^{0}(G_{v}, V^{∗}(1) ⊗_{ℚ_{p}} F^{i}B_{dR}).

There is a logarithmic version of syntomic cohomology and of the diagram (3.4.1):

CH^{n}(𝔛_{v}) ⟶ H^{2n}((𝔛_{v}^{×})_{syn}, s_{ℚ_{p}}^{log}(n))
↓^{res}  ↓^{α_{F M}}
CH^{n}(X_{v}) ⟶ H^{2n}((X_{v})_{et}, ℚ_{p}(n)),

(3.5.1)

Again, the log-syntomic cycle class map can be defined using Chern classes with values in log-syntomic cohomology [Ts].

(3.6) Theorem. If X_{v} has good (resp. semistable) reduction and if 𝔛_{v} is a proper and smooth (resp. proper and semistable) model of X_{v} over O_{v}, then there is a canonical spectral sequence

synE_{∞}^{i,j} = ^{syn}E_{2}^{i,j} = H_{f}^{i}(G_{v}, V^{j}(n)) ⟹ H^{i+j}((𝔛_{v})_{syn}, s_{ℚ_{p}}(n))

(resp.

log−synE_{2}^{i,j} = H_{st}^{i}(G_{v}, V^{j}(n)) ⟹ H^{i+j}((𝔛_{v}^{×})_{syn}, s_{ℚ_{p}}^{log}(n)))

compatible (via α_{F M}) with the Hochschild-Serre spectral sequence

etE_{2}^{i,j} = H^{i}(G_{v}, V^{j}(n)) ⟹ H^{i+j}((X_{v})_{et}, ℚ_{p}(n)).

(3.7) See [Ne 3] for the proof in the good reduction case. The semistable reduction case will be treated in an updated version of [Ne 3]; a special case was proved by A. Langer [La]. In the good reduction case there is also a version with ℤ_{p}-coefficients, valid for K_{v}/ℚ_{p} unramified and p − 1 > max(i + j, n). These results have interesting applications to the study of the image of the cycle class map:

(3.8) Corollary. If X_{v} has potentially good reduction, then

Im (cl_{X_{v}} : CH^{n}(X_{v}) ⟶ H^{2n}((X_{v})_{et}, ℚ_{p}(n))) ∩ F^{2}H^{2n}((X_{v})_{et}, ℚ_{p}(n)) = 0.

(3.9) Proof of Theorem 3.1. (assuming Theorem 3.6). For every finite Galois extension K′_{v}/K_{v} and ∗ ∈ {∅, f, g, st} one has

H_{∗}^{1}(\overline{K}_{v}/K_{v}, V) = H_{∗}^{1}(\overline{K}_{v}/K′_{v}, V)^{Gal(K′_{v}/K_{v})}.

One is free, therefore, to replace K_{v} by K′_{v} and X_{v} by its base change to K′_{v}. This reduces (i) to the case of good reduction, in which case everything follows from the commutative diagram (3.4.1) and Theorem 3.6 (as the restriction map CH^{n}(𝔛_{v}) ⟶ CH^{n}(X_{v}) is surjective).

(ii) According to de Jong’s theorem on alterations [deJ] there is – possibly after a finite extension of scalars K′_{v}/K_{v} – an alteration π : X′_{v} ⟶ X_{v} such that X′_{v} has semistable reduction. The existence of trace maps in étale cohomology implies that the induced maps π^{∗} : V^{j} ⟶ V^{′j} are split injective. It follows that the statement

Im (cl_{1,X_{v}} ⊗ ℚ_{p}) ⊆^{?} H_{st}^{1}(G_{v}, V^{2n−1}(n))

reduces to a similar statement for X′_{v}, in which case we conclude as before by the commutative diagram (3.5.1) and Theorem 3.6. The quotient H_{st}^{1}(G_{v}, V)/H_{f}^{1}(G_{v}, V) (for V = V^{2n−1}(n)) is dual to D_{st}(V^{∗}(1))^{N=0,φ=1} ([Ne 2, Cor. 1.18]). The p-adic version of the purity conjecture for the monodromy filtration for V^{2n−1} ([Mo, Conj. 3.27]; cf. [Il, Quest. 3.13.2], [Fo 3, Conj. 2.4.3-5]) implies that φ acts on D_{st}(V^{∗}(1))^{N=0} with weights < 0; thus D_{st}(V^{∗}(1))^{N=0,φ=1} vanishes. The equality H_{st}^{1} = H_{g}^{1} for potentially semistable representations is proved in [Hy] (see also [Ne 2, Prop. 1.24.3]). This concludes the proof of Theorem 3.1.

(3.10) Perhaps it is worth pointing out that in the good reduction case the weights of the action of φ on D_{cris}(V^{j}) = H^{0}(G_{v}, V^{j} ⊗_{ℚ_{p}} B_{cris}) are equal to j (“crystalline Weil conjectures”). For 𝔛_{v} smooth and projective over Spec(O_{v}) this was proved by Katz and Messing [KM], the general case by Chiarellotto and Le Stum [CLeS].

(3.11) If Z ⊂ X_{v} is a homologically trivial cycle of codimension n, the statement

cl_{1,X_{v}}(Z) ∈^{?} H_{st}^{1}(G_{v}, V^{2n−1}(n))

would follow if one could prove directly that H^{2n−1}((\overline{X_{v} − |Z|})_{et}, ℚ_{p}) is a semistable representation of Gal(\overline{K}_{v}/K′_{v}) for some finite extension K′_{v}/K_{v} (i.e. it is potentially semistable). This is because of an alternative description of cl_{1,X_{v}}(Z) in terms of the extension class of Galois representations arising from the exact cohomology sequence

0 ⟶ H^{2n−1}(\overline{X}_{v}, ℚ_{p})(n) ⟶ H^{2n−1}((\overline{X_{v} − |Z|})_{et}, ℚ_{p})(n) ⟶ H_{|\overline{Z}|}^{2n}(\overline{X}_{v}, ℚ_{p})(n) ⟶ ⋯

(see [Ja 3, Lemma 9.4]).

4. p-adic height pairings

(4.0) As in Sect. 2, let X be a proper and smooth variety over a number field K, equidimensional of dimension dim(X) = d. Fix integers i, j ≥ 1 satisfying i+j = d+1. In the notation of 1.1, put V = V^{2i−1}(i). Poincaré duality

V^{2i−1}(i) × V^{2j−1}(j) ⟶ V^{2d}(d + 1)⟶^{T r} ℚ_{p}(1)

then identifies V^{2j−1}(j) with Hom_{ℚ_{p}}(V, ℚ_{p}(1)) = V^{∗}(1).

(4.1) We assume that the following assumption is satisfied: (A) At each non-archimedean prime v of K where X has bad reduction, the purity conjecture for the

monodromy filtration for V^{2i−1} (or, equivalently, for V^{2j−1}) holds. It follows from Theorem 3.1 that under the assumption (A) there are p-adic Abel-Jacobi maps

\begin{matrix}CH^{i}(X)_{hom∼0} & ⟶ H_{f}^{1}(G_{K}, V) \\ CH^{j}(X)_{hom∼0} & ⟶ H_{f}^{1}(G_{K}, V^{∗}(1)) \end{matrix}

(4.2) Theorem. ([Ne 2]) Under the assumption (A) there are height pairings

⟨ , ⟩ : H_{f}^{1}(G_{K}, V) × H_{f}^{1}(G_{K}, V^{∗}(1)) ⟶ ℚ_{p}

depending on the following additional data:

(a) A continuous homomorphism ℓ : 𝔸_{K}^{∗}/K^{∗} ⟶ ℚ_{p}.

(b) ℚ_{p}-linear splittings of Hodge filtrations

F^{i}H_{dR}^{2i−1}(X_{v}/K_{v}) ⟶^{⟵} H_{dR}^{2i−1}(X_{v}/K_{v})

for all primes v|p (where X_{v} = X ×_{K} K_{v}).

(4.3) Corollary. There are p-adic height pairings

CH^{i}(X)_{hom∼0} × CH^{j}(X)_{hom∼0} ⟶ ℚ_{p}

depending on the data (a), (b) and factoring through p-adic Abel-Jacobi maps.

(4.4) Remarks. (1) The assumptions of [Ne 2, 2.1.2, 7.2], under which the height pairings are constructed, follow from (A), possibly after a finite extension of the field K (cf. the discussion in 3.9-10 above).

(2) It is still an open problem whether the real-valued height pairing factors through the complex-analytic Abel-Jacobi map (if min(i, j) > 1).

(3) Sometimes the assumption (A) can be verified only for a suitable direct factor W ⊂ V coming from a “submotive” of h^{2i−1}(X)(i). In that case one gets only pairings

H_{f}^{1}(G_{K}, W) × H_{f}^{1}(G_{K}, W^{∗}(1)) ⟶ ℚ_{p}

depending on a similar set of data.

(4) By class field theory, ℓ can be viewed as a ℤ_{p}-linear map Gal(K_{∞}/K) ⟶ ℚ_{p}, where K_{∞} is the compositum of all ℤ_{p}-extensions of K inside \overline{K}. The dependence of ⟨ , ⟩ on ℓ is linear, which means that once the choices of splittings in (b) are fixed one obtains a “universal” height pairing with values in the finite-dimensional ℚ_{p}-vector space Gal(K_{∞}/K) ⊗_{ℤ_{p}} ℚ_{p}.

(5) The dependence on ℓ can be seen already in the following simple example: let X = ℙ_{K}^{1}, x ∈ K^{∗} \ {1}, Z = (0) − (∞), Z′ = (1) − (x). Consider first the real valued height: as CH^{1}(X)_{hom∼0} = 0, the global height ⟨Z, Z′⟩ = 0 ∈ ℝ. However, the local heights ⟨Z, Z′⟩_{v} = log |x|_{v} are non-zero (unles x is a root of unity) and

⟨Z, Z′⟩ = ∑_{v} ⟨Z, Z′⟩_{v} = log (∏_{v} |x|_{v}) = 0

vanishes by the product formula. In the p-adic situation, ℓ defines continuous homomorphisms ℓ_{v} : K_{v}^{∗} ⟶ ℚ_{p} for all primes v, and the local p-adic heights are equal to ⟨Z, Z′⟩_{v} = ℓ_{v}(x). The global height

⟨Z, Z′⟩ = ∑_{v} ⟨Z, Z′⟩_{v} = ∑_{v} ℓ_{v}(x) = 0

vanishes, as ℓ|_{K^{∗}} = 0.

(6) In certain cases there are canonical splittings of the Hodge filtrations in (b), which then define “canonical p-adic height pairings”. This happens whenever there is, for each prime v|p, an exact sequence of representations of G_{v}

0 ⟶ V_{v}^{+} ⟶ V ⟶^{π} V_{v}^{−} ⟶ 0

with the property that π induces an isomorphism F^{0}D_{dR}(V) ⟶^{∼} D_{dR}(V_{v}^{−}). For example, for i = 1 this happens iff A = Pic_{X/K}^{0} has ordinary reduction at all v|p (see [Ne 2] for more details).

5. p-adic height pairings in families

(5.0) The p-adic heights discussed in Sect. 4 appear in the p-adic version of the conjecture of Birch and Swinnerton-Dyer (and its higher-dimensional analogue). In the formulation of [PR 1], [PR 2] both sides of the conjecture depend on the data (a), (b) from Theorem 4.2. Traditionally, p-adic L-functions have characters of Gal(K_{∞}/K) as arguments. The homomorphism ℓ then defines a direction in which derivatives of the p-adic L-function are taken. Perrin-Riou’s p-adic L-function depends on an extra parameter, of which the splittings (b) are a special case.

(5.1) Apart from Perrin-Riou’s new parameter, p-adic L-functions tend to depend on additional variables. For example, Hida’s theory of ordinary families of p-adic modular forms [Hi] produces two-variable p-adic L-functions, with weight as the second variable [GS], [Ki].

On the algebraic side, the theory gives p-adic families of representations of G_{K} of the following type (for K = ℚ):

(1) R is a complete local ring, finite and flat over Λ = O_{F} [[X]], where O_{F} is the ring of integers in a finite extension F of ℚ_{p}.

(2) T is an R-module of finite type, flat over Λ, with an R-linear continuous action of G_{K}. The action is unramified outside a finite set S of primes of K. One can assume that S contains all primes dividing p and all archimedean primes of K.

(3) For every prime v|p in K there is an exact sequence of R[G_{v}]-modules

0 ⟶ T_{v}^{+} ⟶ T ⟶ T_{v}^{−} ⟶ 0,

with T_{v}^{±} flat over Λ.

(5.2) In concrete terms, T constructed by Hida interpolates Galois representations associated to modular forms as follows: for every even integer k ≥ 2 in a fixed residue class modulo 2(p − 1) and a prime ideal 𝔭_{k} ⊂ R above (X + 1 − (1 + p)^{k−2}) ⊂ Λ, V_{k} = (T/𝔭_{k}T) ⊗_{O_{F}} F is a two-dimensional representation of G_{ℚ} over F, associated to - up to a Tate twist - a Hecke eigenform of weight k on Γ_{1}(Np) (for p |̸ 6N). Our normalization of the Tate twist (cf. [Gr]) is such that V_{k} is a direct factor of H^{k−1}(\overline{X}_{et}, ℚ_{p}(k/2)), where X is a compactified Kuga-Sato variety of dimension k − 1, fibered over the modular curve X_{1}(Np). In this case

∧_{F}^{2} V_{k} ⟶^{∼} F (1),  Hom_{F} (V_{k}, F (1)) ⟶^{∼} V_{k}.  (5.2.1)

The ring R is a local component of the ordinary part of the projective limit of Hecke algebras (with ℤ_{p}- coefficients) acting on the Jacobians J_{1}(Np^{r}) of modular curves X_{1}(Np^{r}). Up to a twist, T is the Pontrjagin dual of the R-part of

\underrightarrow{lim}_{r} J_{1}(Np^{r})_{p^{∞}}(\overline{ℚ})^{ord},

where the superscript “ord” refers to the subgroup on which the Hecke operator T (p) is not topologically nilpotent [Hi].

(5.3) In the abstract setup of 5.1.1-3, put ω_{R} = Hom_{Λ}(R, Λ) and

T^{∗} = Hom_{Λ}(T, Λ) = Hom_{R}(T, ω_{R}), (T^{∗})_{v}^{±} = Hom_{Λ}(T_{v}^{∓}, Λ) = Hom_{R}(T_{v}^{∓}, ω_{R}).

In the situation of 5.2 there is a canonical skew-symmetric isomorphism T^{∗}(1) ⟶^{∼} T, which interpolates the isomorphisms (5.2.1).

Plater [Pl] defined Selmer groups

H_{f}^{1}(G_{K}, T)^{0} ⊂ H_{f}^{1}(G_{K}, T) ⊂ H^{1}(G_{K}, T)

and - under certain assumptions - height pairings

H_{f}^{1}(G_{K}, T)^{0} × H_{f}^{1}(G_{K}, T^{∗}(1))^{0} ⟶ R  (5.3.1)

that interpolate the height pairings

H_{f}^{1}(G_{K}, V_{k}) × H_{f}^{1}(G_{K}, V_{k}^{∗}(1)) ⟶ F

of Theorem 4.2.

(5.4) Selmer complexes It seems that the correct setup for duality theorems and p-adic height pairings involves Selmer complexes [Ne 4] rather than Selmer groups. The idea is that one should consider local conditions for complexes rather than for cohomology groups, as in 2.2. In the situation of 5.1, let K_{S} be the maximal extension of K (inside \overline{K}) unramified outside S and G_{K,S} = Gal(K_{S}/K). Denote by C_{cont}^{•}(G_{K,S}, T) the complex of non-homogeneous continuous cochains (with respect to the pro-finite topologies) on G_{K,S} with values in T and put, for every non-archimedean prime v ∈ S,

U_{v}^{+}(T) = \{ \begin{matrix}C_{cont}^{•}(G_{v}/I_{v}, T^{I_{v}}), & if v |̸ p \\ C_{cont}^{•}(G_{v}, T_{v}^{+}), & if v|p \end{matrix}

The Selmer complex ℝΓ̃_{f}(G_{K,S}, T) is defined as the simple complex associated to

C_{cont}^{•}(G_{K,S}, T) ⟶ ⨁_{v∈S_{f}} C_{cont}^{•}(G_{v}, T)
↑
⨁_{v∈S_{f}} U_{v}^{+}(T)

(where S_{f} ⊂ S is the subset of non-archimedean primes of S). Its cohomology sits in an exact sequence

⋯ ⟶ H̃_{f}^{i}(G_{K,S}, T) ⟶ H^{i}(G_{K,S}, T) ⊕ ⨁_{v∈S_{f}} H^{i}(U_{v}^{+}(T)) ⟶

⟶ ⨁_{v∈S_{f}} H^{i}(G_{v}, T) ⟶ H̃_{f}^{i+1}(G_{K,S}, T) ⟶ ⋯

The appropriate generalization of (5.3.1) is given by the following

(5.5) Theorem. [Ne 4] In the situation of 5.1.1-3, assume that either p > 2 or K is totally imaginary. Then, for every continuous homomorphism ℓ : Gal(K_{∞}/K) ⟶ R, there is a “height pairing”

ℝΓ̃_{f}(G_{K,S}, T) ⊗^{𝕃}_{R} ℝΓ̃_{f}(G_{K,S}, T^{∗}(1)) ⟶ ω_{R}[−2],

which depends R-linearly on ℓ.

(5.6) Selmer complexes ℝΓ̃_{f}(G_{K,S}, T) are better than classical Selmer groups even for elliptic curves (when R and Λ are replaced by ℤ_{p}). For example, if K = ℚ and T = T_{p}(E) is the Tate module of an elliptic curve E over ℚ with split multiplicative reduction at p, then T_{p}^{+} = ℤ_{p}(1) and T_{p}^{−} = ℤ_{p}. The group H^{0}(G_{p}, T_{p}^{−}) = ℤ_{p} then contributes to the kernel of H̃_{f}^{1}(G_{K,S}, T) ⟶ H^{1}(G_{K,S}, T); there is an exact sequence

0 ⟶ ℚ_{p} ⟶ H̃_{f}^{1}(G_{K,S}, T_{p}(E)) ⊗_{ℤ_{p}} ℚ_{p} ⟶ H^{1}(G_{K}, V_{p}(E)) ⟶ 0.

This is consistent with the phenomenon discovered by Mazur, Tate and Teitelbaum [MTT]: the term H^{0}(G_{p}, T_{p}^{−}) is responsible for a trivial zero of the p-adic L-function and

H̃_{f}^{1}(G_{K,S}, V_{p}(E)) = H̃_{f}^{1}(G_{K,S}, T_{p}(E)) ⊗_{ℤ_{p}} ℚ_{p}

is the corresponding “extended Selmer group”. A variant of Theorem 5.5 then defines a height pairing

H̃_{f}^{1}(G_{K,S}, T_{p}(E)) × H̃_{f}^{1}(G_{K,S}, T_{p}(E)) ⟶ ℤ_{p}.

(5.7) Coleman [Co] constructed p-adic analytic families of non-ordinary modular forms. It would be interesting to extend previous constructions to this case.

References

[Be] A.A. Beilinson, Notes on absolute Hodge cohomology, in: Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, (Boulder, Colorado, 1983), Contemp. Math. 55/I , Amer. Math. Soc., Providence, R.I., 1986, 35–68.

[BK] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift,

Vol. I, Progress in Mathematics 86 , Birkhäuser, Boston, Basel, 1990, pp. 333–400.

[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.

[Co] R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479. [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, La conjecture de Weil II, Publ. Math. de l’I.H.E.S. 52 (1980), 137–252.

[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. [Fo 1] J.-M. Fontaine, Le corps des périodes p-adiques, in: [Bures], Exposé II, pp. 59–101. [Fo 2] J.-M. Fontaine, Représentations p-adiques semistables, in: [Bures], Exposé III, pp. 113–184. [Fo 3] J.-M. Fontaine, Représentations ℓ-adiques potentiellement semistables, in: [Bures], Exposé VIII, pp.

321–347.

[FI] J.-M. Fontaine, L. Illusie, p-adic periods: a survey, in: Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi, 1993, pp. 57–93.

[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.

[FPR] J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et

valeurs de fonctions L, in: [Motives], Part I, pp. 599–706.

[Gr] R. Greenberg, Elliptic curves and p-adic deformations, in: Elliptic curves and related topics, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994, pp. 101–110.

[GS] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447.

[Hi] H. Hida, Elementary theory of L-functions and Eisenstein series, London Math. Soc. Student Texts 26 , Cambridge University Press, Cambridge, 1993.

[Hy] O. Hyodo, H_{g}^{1}(K, V) = H_{st}^{1}(K, V), unpublished manuscript.

[HK] O. Hyodo, K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, in: [Bures],

Exposé V, pp. 221–268.

[Il] L. Illusie, Autour du théorème de monodromie locale, 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.

[deJ] A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. de l’I.H.E.S. 83 (1996), 51–93.

[Ka] K. Kato, Semi-stable reduction and p-adic étale cohomology, in: [Bures], Exposé VI, pp. 269–293.

[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.

[Ki] K. Kitagawa, On standard p-adic L-functions of families of elliptic cusp forms, in: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemp. Math. 165 , Amer. Math. Soc., Providence, RI, 1994, 81–110.

[La] A. Langer, Local points of motives in semi-stable reduction, Comp. Math., to appear.

[MTT] B. Mazur, J. Tate,, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer,

Invent. Math. 84 (1986), 1–48.

[Mi] J. S. Milne, Arithmetic Duality Theorems, Perspectives in Mathematics 1 , Academic Press, Boston, 1986.

[Mo] A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), 301–337.

[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, Syntomic cohomology and p-adic regulators, preprint 1998; available at

http://www.dpmms.cam.ac.uk/~nekovar/papers/

[Ne 4] J. Nekovář, Selmer complexes, in preparation.

[Ni] W. Nizioł, On the image of p-adic regulators, Invent. Math. 127 (1997), 375–400. [PR 1] B. Perrin-Riou, Fonctions L p-adiques d’une courbe elliptique et points rationelles, Ann. Inst. Fourier

43 (1993), 945–995. [PR 2] B. Perrin-Riou, Fonctions L p-adiques des représentations p-adiques, Astérisque 229 , Soc. Math. de

France, Paris, 1995.

[Pl] A. Plater, Height pairings in families of deformations, J. Reine Angew. Math. 486 (1997), 97–127.

[RZ] M. Rapoport, T. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Inv. Math. 68 (1982), 21–101.

[Sa] T. Saito, Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997), 607–620.

[Se] J.-P. Serre, Cohomologie Galoisienne, Lect. Notes in Math. 5 , Springer, Berlin, Göttingen, Heidelberg, New York, 1964.

[Ta] J. Tate, Algebraic cycles and poles of zeta functions, in: Arithmetical Algebraic Geometry, Proc. Conf. Purdue Univ., 1963, (O.F.G. Schilling, ed.), Harper & Row, New York, 1965, pp. 93–110.

[Ts] T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, preprint

1996.

[Bures] Périodes p-adiques, Astérisque 223 , Soc. Math. de France, Paris, 1994.

[Motives] Motives, Proceedings of AMS Summer Research Conference in Seattle (July 1991), Proc. Symposia in Pure Math. 55/I,II , American Mathematical Society, Providence, Rhode Island, 1994.

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