On p-adic height pairings

Jan Nekovář

The aim of the present work is to construct p-adic height pairings in a sufficiently general setting, namely for Selmer groups of reasonably behaved p-adic Galois representations over number fields. These pairings should, modulo general conjectures on étale cohomology, induce height pairings between (homologically trivial) algebraic cycles on every proper smooth variety defined over a number field. For smooth projective varieties with good reduction at all primes dividing p, our construction requires only one cohomological assumption: we have to assume that the purity conjecture for certain étale cohomology group holds at all places of bad reduction of the given variety.

More specifically, let X be a proper smooth variety, of dimension d, defined over a number field K. Write CH^{i}(X)_{0} for the group of homologically trivial algebraic cycles of codimension i on X, defined over K, modulo rational equivalence. If i + j = d + 1, then there are “étale Abel-Jacobi maps”

\begin{matrix}Φ_{i} : CH^{i}(X)_{0} & ⟶ H^{1}(K, V) \\ Φ_{j} : CH^{j}(X)_{0} & ⟶ H^{1}(K, V^{∗}(1)), \end{matrix}

where V = H_{et}^{2i−1}(X ⊗ \overline{K}, Q_{p}(i)). For each p-adic representation V of G(\overline{K}/K), there is a locally defined subgroup H_{f}^{1}(K, V) ⊆ H^{1}(K, V), generalizing the classical Selmer group. We shall construct pairings

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

for ‘good’ p-adic Galois representations V and show that, for ‘good’ X, H_{et}^{2i−1}(X ⊗ \overline{K}, Q_{p}(i)) is good and both maps Φ factor through H_{f}^{1}.

In fact, we construct a family of height pairings, depending on the choice of a global p-adic logarithm (which amounts, essentially, to the choice of a Z_{p}-extension of K) and splittings of certain Hodge filtrations associated to V at primes of K dividing p. For V satisfying “Pančiškin’s condition” of [Pe-Ri 3], there is a canonical choice of these splittings, giving rise to a canonical height pairing. Previous constructions of p-adic height pairings worked mainly for abelian varieties (cf. [Co–Gr], [Ma–Ta], [Pe-Ri 1], [Sch 2,3], [Wi], [Za] – the list is by no means exhaustive), only the recent work of B. Perrin-Riou [Pe-Ri 3] treats the subject in a generality similar to ours.

The contents of the paper is the following: in sec. 1 we collect relevant facts about p-adic Galois representations and filtered modules which will be used in the sequel. Although there might be some overlap with the seminar notes [Bu], we preferred to keep the paper reasonably self-contained. In sec. 2 we define abstract height pairings, using the method of Bloch [Bl] and the ideas of Zarhin [Za]. In sec. 3 we describe the linear algebra underlying splittings of local extensions and their relationship to splittings of Hodge filtrations. In sec. 4 we define local height pairings (following closely the constructions in [Sc 2]) and prove that the height pairing constructed in sec. 2 is equal to the sum of the local height pairings. In sec. 5 we discuss the geometric situation. In sec. 6 we study another way of defining height pairings, using splittings coming from universal norms. We show that, in the good reduction case, the canonical height pairing for representations satisfying Pančiškin’s condition can indeed be obtained from universal norms splittings. In sec. 7 we treat the semistable (i.e. the bad reduction) case. In this case, for representations satisfying Pančiškin’s condition, universal norms define a pairing different from the canonical one, and the difference between them can be neatly described (following [Ma–Ta–Te]) in terms of an “extended height pairing”. In sec. 8 we clarify the relationship of our construction to previous definitions for abelian varieties, as well to the work of B.Perrin-Riou [Pe-Ri 3].

This work grew out of numerous conversations with Uwe Jannsen. I thank to him heartily for his generosity with which he shared with me his ideas and insights. My thanks are also due to B.Perrin-Riou, P.Schneider and A.J.Scholl for helpful discussions. Throughout the work on this paper, I was supported by the following institutions: the Max-Planck-Institut für Mathematik in Bonn, Mathematical Institute of the Czechoslovak Academy of Science in Prague and a fellowship from the French Ministry of Research and Technology. I am grateful to all of them.

1. p-adic Galois representations

1.1 In this section we recall basic facts about p-adic Galois representations and associated filtered modules and discuss some complements to [Bl–Ka,ch.3]. The reader should consult [Fo–Il],[Il] for further details. Throughout the paper, all Galois cohomology groups will be continuous cohomology groups, see [Ta],[Ja 2] for further details. 1.2 Let K be a finite extension of Q_{p} with ring of integers O_{K}. Denote by K_{0} its maximal subfield unramified over Q_{p}, by ϕ the absolute Frobenius acting on K_{0} (ϕ induces the p-th power map on the residue field of K_{0}) and by C_{p} the completion of \overline{K}. Write G for the Galois group G(\overline{K}/K), I = G(\overline{K}/K_{ur}) for the inertia group and

χ_{K} : G ⟶ Z_{p}^{∗}

for the cyclotomic character giving the action of G on the Tate module Z_{p}(1) of the multiplicative group. 1.3 Denote by

B_{cris} ⊂ B_{st} ⊂ B_{dR}

the topological rings constructed by Fontaine (see [Fo 2],[Fo–Il],[Fo–Me]). They are equipped with the following structures:

(1) A continuous action of G on all of them

(2) G-equivariant embeddings

K_{0}^{ur} = Q_{p}^{ur} ↪ B_{cris}, \overline{K} ↪ B_{dR}

such that the induced maps

K ⊗_{K_{0}} B_{st} ⟶ B_{dR}, \overline{K} ⊗_{K_{0}^{ur}} B_{st} ⟶ B_{dR}

are injective and B_{cris}^{G} = B_{st}^{G} = K_{0}, B_{dR}^{G} = K.

(3) A decreasing filtration of B_{dR} by \overline{K}-vector spaces (F^{i}B_{dR})_{i∈Z} satisfying

⋃ F^{i}B_{dR} = B_{dR}, ⋂ F^{i}B_{dR} = 0.

(4) A G-equivariant embedding

Q_{p}(1) ↪ F^{1}B_{dR}.

If t is a non-zero element in Q_{p}(1), then

F^{i}B_{dR} = t^{i}F^{0}B_{dR}, F^{i}B_{dR}/F^{i+1}B_{dR} ⟶^{∼} C_{p}(i).

(5) A ϕ-semilinear continuous automorphism f : B_{st} ⟶ B_{st}, preserving B_{cris}.

(6) A B_{cris}-linear derivation N : B_{st} ⟶ B_{st} satisfying Nf = pfN such that

0 ⟶ B_{cris} ⟶ B_{st} ⟶^{N} B_{st} ⟶ 0

is exact. All these structures are canonical, except for N, which depends on the choice of a valuation v : \overline{K}^{∗} ⟶ Q, and the embedding B_{st} ↪ B_{dR}, which depends on the choice of a p-adic logarithm, i.e. of a G-equivariant homomorphism Log : \overline{K}^{∗} ⟶ \overline{K}, which coincides with the usual power series on the group of principal units. For convenience, we also use abbreviation B_{ur} for the completion of the maximal unramified extension of Q_{p}. It is a subring of B_{cris}.

1.4 Let V be a p-adic representation of G, that is a vector space over Q_{p} of finite dimension with a continuous linear action of G. Define for ∗ ∈ {ur, cris, st, dR}

D_{∗}(V) = (V ⊗_{Q_{p}} B_{∗})^{G}.

The map

α_{∗} : D_{∗}(V) ⊗_{K_{∗}} B_{∗} ⟶ V ⊗_{Q_{p}} B_{∗}

(where K_{∗} = B_{∗}^{G}, equal to either K or K_{0}) is always injective, which implies that

dim_{K_{∗}}D_{∗}(V) ≤ dim_{Q_{p}}(V)

with equality iff α_{∗} is bijective. The representation V is called crystalline (resp. semistable, resp. de Rham) if α_{cris} (resp. α_{st}, resp. α_{dR}) is an isomorphism. Such representations form a Q_{p}-linear tensor category Rep_{cris} (resp. Rep_{st}, resp. Rep_{dR}), stable under subquotients, duals and Tate twists. For ∗ = ur, we get the usual category of unramified representations of G(\overline{K}/K), i.e. those on which I acts trivially. 1.5 The groups D_{∗}(V) inherit from B_{∗} the following structures:

• D_{cris}(V) is a vector space over K_{0} equipped with a ϕ-semilinear bijective map f : D_{cris}(V) ⟶ D_{cris}(V)

• D_{st}(V) is a vector space over K_{0} equipped with a ϕ-semilinear bijective map f : D_{st}(V) ⟶ D_{st}(V) and a K_{0}-linear map N : D_{st}(V) ⟶ D_{st}(V) satisfying Nf = pfN.

• DR(V) := D_{dR}(V) is a vector space over K with a decreasing filtration by subspaces (F^{i}DR(V))_{i∈Z} satisfying ⋃ F^{i}DR(V) = DR(V), ⋂ F^{i}DR(V) = 0. They are related as follows:

• D_{cris}(V) = D_{st}(V)^{N=0} = Ker(N)

• The canonical map D_{st}(V) ⊗_{K_{0}} K ⟶ DR(V) is injective

• dim_{K_{0}}D_{cris}(V) ≤ dim_{K_{0}}D_{st}(V) ≤ dim_{K}DR(V) ≤ dim_{Q_{p}}(V)

• V is crystalline ⟹ V is semistable ⟹ V is de Rham

• The Tate twist D(n) is given by tensoring with

D_{cris}(Q_{p}(n)) = D_{st}(Q_{p}(n)) = (K_{0}, f = p^{−n}ϕ, N = 0)

resp. DR(Q_{p}(n)) = (K, F^{−n}DR = K, F^{−n+1}DR = 0). It will be more convenient to consider the monodromy operator N as a f-equivariant map

N : D_{st}(V) ⟶ D_{st}(V)(−1) = D_{st}(V (−1)).

1.6 According to the fundamental results of Fontaine [Fo 1],[Fo-Il], a crystalline resp. semistable representation V can be recovered from D_{cris}(V) resp. D_{st}(V) as follows:

Let MF_{K}(f, N) be the category of finite-dimensional vector spaces D over K_{0} equipped with

• A ϕ-semilinear bijective map f : D ⟶ D

• A K_{0}-linear map N : D ⟶ D satisfying Nf = pfN

• A decreasing filtration F^{i}D_{K} by subspaces of D_{K} = D ⊗_{K_{0}} K satisfying ⋃ F^{i}D_{K} = D_{K}, ⋂ F^{i}D_{K} = 0 (morphisms being Q_{p}-linear maps compatible with f, N, F^{.}). The functor

D_{st} : Rep_{st}(G) ⟶ MF_{K}(f, N)

is fully faithful and induces a tensor equivalence between Rep_{st}(G) and its essential image MF_{K}^{ad}(f, N) (“admissible filtered (f, N)-modules”). A quasi-inverse functor V_{st} of D_{st} is given by

V_{st}(D) = (D ⊗_{K_{0}} B_{st})^{f=1,N=0} ∩ F^{0}(D_{K} ⊗_{K} B_{dR}).

Similarly, the filtered f-modules with N = 0 form a category MF_{K}^{f} and

D_{cris} : Rep_{cris}(G) ⟶ MF_{K}^{f}

induces a tensor equivalence between Rep_{cris}(G) and MF_{K}^{ad} := MF_{K}^{ad}(f, N) ∩ MF_{K}^{f}, a quasi-inverse of D_{cris} being

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

1.7 The abstract correspondence between p-adic Galois representations and filtered modules is a realization of Grothendieck’s “mysterious functor” between étale and de Rham cohomology.

Let X_{/K} be a proper smooth K-scheme with good reduction, let X be a proper smooth O_{K}-scheme with generic fibre X ⊗ K isomorphic to X. Write X_{s} for the special fibre of X. Fix an integer m ≥ 0. Then • V := H_{et}^{m}(X ⊗ \overline{K}, Q_{p}) is a p-adic representation of G

• The crystalline cohomology of the special fiber D := H_{cris}^{m}(X_{s}/O_{K_{0}}) ⊗_{O_{K_{0}}} K_{0} admits a ϕ-semilinear bijective endomorphism f

• [Be–Og] There is a canonical isomorphism

D ⊗_{K_{0}} K ⟶^{∼} H_{dR}^{m}(X_{/K})

and the Hodge filtration on H_{dR}^{m} makes D a filtered f-module.

1.8 Theorem. [Fa] In the situation of 1.7, V is a crystalline representation, D is an admissible f-module and there are canonical isomorphisms

D ⟶^{∼} D_{cris}(V), V ⟶^{∼} V_{cris}(D),

inducing an isomorphism

(DR(V), F^{.}) ⟶^{∼} (H_{dR}^{m}(X_{/K}), Hodge filtration).

1.9 Suppose that X is a proper flat O_{K}-scheme with semistable reduction (i.e. the special fibre X_{s} is a reduced divisor with normal crossing) and generic fibre X. Choose a prime element π ∈ O_{K}, which in turn determines a p-adic logarithm Log_{π} characterized by Log_{π}(π) = 0, hence an embedding B_{st} ↪ B_{dR}. Fix an integer m ≥ 0. Then

• V := H_{et}^{m}(X ⊗ \overline{K}, Q_{p}) is a p-adic representation of G

• The cohomology of the de Rham-Witt complex with logarithmic singularities along the special fiber

D := H^{m}(X_{s}, W ω_{X_{s}}^{.}) ⊗_{O_{K_{0}}} K_{0}

admits a ϕ-semilinear bijective endomorphism f and a nilpotent automorphism N satisfying Nf = pfN.

• [Hy–Ka] There is a canonical isomorphism (depending on the choice of π)

D ⊗_{K_{0}} K ⟶^{∼} H_{dR}^{m}(X_{/K})

and the Hodge filtration on the second group makes D a filtered (N, f)-module.

1.10 Theorem. [Ka] In the situation of 1.9, suppose that m < (p − 1)/2. Then V is a semistable representation, D is an admissible (N, f)-module and there are canonical isomorphisms

D ⟶^{∼} D_{st}(V), V ⟶^{∼} V_{st}(D)

inducing an isomorphism

(DR(V), F^{.}) ⟶^{∼} (H_{dR}^{m}(X_{/K}), Hodge filtration).

It is conjectured [Fo–Il] that the statement of the theorem is valid for all m and X, after passing to a suitable finite extension L/K.

1.11 Theorem. [Fa] Let X be a smooth K-scheme, m ≥ 0 an integer. Then V := H_{et}^{m}(X ⊗ \overline{K}, Q_{p}) is a de Rham representation and there is a canonical isomorphism

(DR(V), F^{.}) ⟶^{∼} (H_{dR}^{m}(X/K), Hodge filtration).

It is conjectured [Fo 2] that the same should be true without the smoothness assumption.

1.12 We now recall the main points of [Bl–Ka, ch.3] and add some remarks concerning the semistable case. Let V be a p-adic representation of G. Define for ∗ ∈ {ur, cris, st, dR}

H_{∗}^{1}(K, V) = Ker(H^{1}(K, V) ⟶ H^{1}(K, V ⊗ B_{∗})).

If V is a ∗-representation, then H_{∗}^{1}(K, V) classifies those extensions

0 ⟶ V ⟶ E ⟶ Q_{p}(0) ⟶ 0

in which E is again a ∗-representation. We use abbreviations of [Bl–Ka] H_{f}^{1} := H_{cris}^{1}, H_{g}^{1} := H_{dR}^{1} and define also

H_{e}^{1}(K, V) = Ker(H^{1}(K, V) ⟶ H^{1}(K, V ⊗ B_{cris}^{f=1})).

We have obvious inclusions

H_{ur}^{1} ⊆ H_{f}^{1} ⊆ H_{st}^{1} ⊆ H_{g}^{1}

and H_{ur}^{1}(K, V) = H^{1}(G/I, V^{I}) is the usual group of unramified cohomology classes (cf. [Fl]). If T ⊂ V is a G(\overline{K}/K)-equivariant Z_{p}-lattice in V, define H_{∗}^{1}(K, T) to be the inverse image of H_{∗}^{1}(K, V) in H^{1}(K, T). For example, for V = Q_{p}(1) we have H^{1}(K, Q_{p}(1)) = K^{∗} ˆ⊗Q_{p}, where the symbol Aˆ⊗Q_{p} stands for (\underleftarrow{lim}_{n}(A/p^{n}A)) ⊗_{Z_{p}} Q_{p}, and

H_{e}^{1}(K, Q_{p}(1)) = H_{f}^{1}(K, Q_{p}(1)) = O_{K}^{∗} ˆ⊗Q_{p} ⊆ H^{1}(K, Q_{p}(1)) = K^{∗} ˆ⊗Q_{p}.

1.13 Recall that according to [Se, II.5.2 Th.2] the cup product

H^{i}(K, V) × H^{2−i}(K, V^{∗}(1)) ⟶^{∪} H^{2}(K, Q_{p}(1)) ⟶^{∼} Q_{p}

is a perfect duality for any i = 0, 1, 2 and that H^{i}(K, V) = 0 for i ≥ 3.

1.14 Theorem. [Bl–Ka, 3.8] If V is a de Rham representation, then in the perfect duality

H^{1}(K, V) × H^{1}(K, V^{∗}(1)) ⟶^{∪} H^{2}(K, Q_{p}(1)) ⟶^{∼} Q_{p}

the annihilators of various subgroups of H^{1}(K, V) are

H_{e}^{1}(V)^{⊥} = H_{g}^{1}(V^{∗}(1)), H_{f}^{1}(V)^{⊥} = H_{f}^{1}(V^{∗}(1)), H_{g}^{1}(V)^{⊥} = H_{e}^{1}(V^{∗}(1))

(V^{∗}(1) = Hom(V, Q_{p}(1)) is the Tate twist of the representation contragredient to V).

1.15 Theorem. [Bl–Ka, 3.8.4] Let V be a de Rham representation. Then there is a commutative diagram with exact rows

0 ⟶ H^{0}(K, V) ⟶ D ⟶^{(1−f,i)} D ⊕ DR(V)/F^{0} ⟶ H_{f}^{1}(K, V) ⟶ 0
∥  ↑ ∪  ↑ ∪  ↑ ∪
0 ⟶ H^{0}(K, V) ⟶ D^{f=1} ⟶^{i}  DR(V)/F^{0}  ⟶^{exp} H_{e}^{1}(K, V) ⟶ 0

in which D = D_{cris}(V) and i is induced by the inclusion D_{cris}(V) ↪ DR(V).

It comes from exact sequences

0 ⟶ Q_{p} ⟶ B_{cris} ⟶^{(1−f,i)} B_{cris} ⊕ B_{dR}/F^{0} ⟶ 0
∥  ↑ ∪  ↑ ∪
0 ⟶ Q_{p} ⟶ B_{cris}^{f=1}  ⟶^{i}  B_{dR}/F^{0}  ⟶ 0

on tensoring with V and taking cohomology.

1.16 Corollary. There is a canonical exact sequence

0 ⟶ H_{e}^{1}(K, V) ⟶ H_{f}^{1}(K, V) ⟶ D_{cris}(V)/(f − 1)D_{cris}(V) ⟶ 0

1.17 Theorem. If V is a de Rham representation, denote D := D_{st}(V) and define X by the cartesian square

X  ⟶^{α}  D
↓^{β}  ↓^{−N}
D(−1) ⟶^{1−f} D(−1)

Then there is a commutative diagram with exact rows and columns

0  0  0
↓  ↓  ↓
0 ⟶ H^{0}(K, V) ⟶ D_{cris}(V) ⟶^{(1−f,i)} D ⊕ DR(V)/F^{0} ⟶ H_{f}^{1}(K, V) ⟶ 0
∥  ↓  ↓  ↓
0 ⟶ H^{0}(K, V) ⟶  D  ⟶^{(j,i)} X ⊕ DR(V)/F^{0} ⟶ H_{st}^{1}(K, V) ⟶ 0
↓^{−N}  ↓^{β}
D(−1) =  D(−1)

in which i is induced by the inclusion D_{cris}(V) ↪ DR(V) and j : D ⟶ X is characterized by α ∘ j = 1 − f, β ∘ j = −N.

Proof. Define Y by the cartesian square

Y ⟶^{α} B_{st}
↓^{β}  ↓^{−N}
B_{st} ⟶^{1−pf} B_{st}

and let j : B_{st} ⟶ Y be the unique map satisfying α ∘ j = 1 − f, β ∘ j = −N. Then the exact sequence [Bl–Ka, 1.17.2] can be completed to a commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ Q_{p} ⟶ B_{cris} ⟶^{(1−f,i)} B_{cris} ⊕ B_{dR}/F^{0} ⟶ 0
∥  ↓  ↓
0 ⟶ Q_{p} ⟶ B_{st}  ⟶^{(j,i)} Y ⊕ B_{dR}/F^{0} ⟶ 0
↓^{−N}  ↓^{β}
B_{st} =  B_{st}
↓  ↓
0  0

On tensoring with V and taking cohomology we obtain the desired diagram (of course, we must check that each epimorphism in the diagram admits a continuous section; for N this follows from the explicit description of B_{st} as the polynomial ring in one variable over B_{cris} (see [Fo–Il]), for (1 − f, i) this is proved in [Bl–Ka, 1.18] and the claim for the remaining two maps β and (j, i) immediately follows).

1.18 Corollary. If V is a semistable representation, then there is a canonical exact sequence

0 ⟶ H_{f}^{1}(K, V) ⟶ H_{st}^{1}(K, V) ⟶ (D_{cris}(V^{∗}(1))^{∗})^{f=1} ⟶ 0

Proof. Snake lemma applied to 1.17 gives an exact sequence

0 ⟶ H_{f}^{1}(K, V) ⟶ H_{st}^{1}(K, V) ⟶ Im(β)/Im(N) ⟶ 0,

but the third group is equal to (D_{st}(V (−1))/ND_{st}(V))^{f=1}. Since D_{st} commutes with Tate twists and duality on Rep_{st}, the group in the brackets is dual to D_{st}(V^{∗}(1))^{N=0} = D_{cris}(V^{∗}(1)).

1.19 Let V be a de Rham representation. Define two complexes (of K_{0}-vector spaces)

C_{cris}^{.}(V) := [D_{cris}(V)⟶^{(f−1,−i)}D_{cris}(V) ⊕ DR(V)/F^{0}]
C_{st}^{.}(V) := [D_{st}(V) ⟶^{(f−1,N,−i)}D_{st}(V) ⊕ D_{st}(V (−1)) ⊕ DR(V)/F^{0} ⟶^{(N,1−f,0)}D_{st}(V (−1))]

in degrees 0, 1 and 0, 1, 2 respectively (note the change of signs with respect to 1.15 and 1.17). We get from 1.15 and 1.17 isomorphisms

α_{V} : H^{0}(K, V) ⟶^{∼} H^{0}(C_{cris}^{.}(V)) = H^{0}(C_{st}^{.}(V))
β_{V,cris} : H^{1}(C_{cris}^{.}(V)) ⟶^{∼} H_{f}^{1}(K, V)
β_{V,st} : H^{1}(C_{st}^{.}(V)) ⟶^{∼} H_{st}^{1}(K, V)

(with β’s induced by boundary maps in cohomology exact sequences). These isomorphisms are functorial (i.e. α_{W} ∘ϕ = ϕ∘α_{V}, β_{W} ∘ϕ = ϕ∘β_{V} for each homomorphism ϕ : V ⟶ W between de Rham representations) and the map induced on H^{1} by inclusion C_{cris}^{.}(V) ↪ C_{st}^{.}(V) is compatible (via β’s) with the natural map H_{f}^{1}(K, V) ⟶ H_{st}^{1}(K, V).

1.20 Let V be a semistable representation. Suppose we are given an extension

0 ⟶ D_{st}(V) ⟶ D ⟶ K_{0} ⟶ 0

in the category of filtered (f, N)-modules (where K_{0} stands for D_{st}(Q_{p}(0))). Writing D_{K} for D ⊗_{K_{0}} K, we have a commutative diagram with exact rows

0 ⟶ DR(V) ⟶ D_{K} ⟶ K ⟶ 0
↓  ↓^{π}  ↓
0 ⟶ DR(V)/F^{0} ⟶^{∼} D_{K}/F^{0} ⟶ 0 ⟶ 0.

Choose a K_{0}-linear section s : K_{0} ⟶ D and define a triple

[x, y, z] := [(f − 1)s(1), Ns(1), −π ∘ i(s(1))] ∈ D_{st}(V) ⊕ D_{st}(V (−1)) ⊕ DR(V)/F^{0},

which is in fact a cycle (in degree 1) in the complex C_{st}^{.}(V). Moreover, its class in H^{1}(C_{st}^{.}(V)), to be denoted ρ(D), does not depend on the choice of s. It is easily seen that ρ in fact defines an injective Q_{p}-linear map

ρ_{V,st} : Ext_{MF_{K}(f,N)}^{1}(K_{0}, D_{st}(V)) ↪ H^{1}(C_{st}^{.}(V)).

Since admissible (f, N)-modules form a full subcategory, the canonical map

τ_{V,st} : H_{st}^{1}(K, V) = Ext_{MF_{K}^{ad}(f,N)}^{1}(K_{0}, D_{st}(V)) ↪ Ext_{MF_{K}(f,N)}^{1}(K_{0}, D_{st}(V))

is injective as well.

Restricting ourselves to filtered modules with N = 0, we obtain, for crystalline V, an injective homomorphism

ρ_{V,cris} : Ext_{MF_{K}}^{1} (K_{0}, D_{cris}(V)) ↪ H^{1}(C_{cris}^{.}(V)).

1.21 Proposition. If V is crystalline (resp. semistable), then the composite

ρ_{V,cris} ∘ τ_{V,cris} ∘ β_{V,cris} : H^{1}(C_{cris}^{.}(V)) ⟶ H^{1}(C_{cris}^{.}(V))

resp.

ρ_{V,st} ∘ τ_{V,st} ∘ β_{V,st} : H^{1}(C_{st}^{.}(V)) ⟶ H^{1}(C_{st}^{.}(V))

is the identity map.

Proof. We shall treat only the semistable case. Let [x, y, z] ∈ D_{st}(V)⊕D_{st}(V (−1))⊕DR(V)/F^{0} represent an element in H^{1}(C_{st}^{.}(V)). Then, in the notation of 1.17, [x, y] ∈ X = H^{0}(K, V ⊗ Y), z ∈ H^{0}(K, V ⊗ B_{dR}/F^{0}). Define

γ = (γ_{1}, γ_{2}) : Q_{p} ⟶ (V ⊗ Y) ⊕ (V ⊗ B_{dR}/F^{0})

by γ_{1}(1) = [x, y], γ_{2}(1) = z. The boundary β_{V,st}([x, y, z]) is then represented by the extension

0 ⟶ V ⊗ Q_{p} ⟶ E ⟶ Q_{p}(0) ⟶ 0,

which is the pullback of

0 ⟶ V ⊗ Q_{p} ⟶ V ⊗ B_{st} ⟶^{id⊗(j,i)}(V ⊗ Y) ⊕ (V ⊗ B_{dR}/F^{0}) ⟶ 0

via γ. Choose u ∈ V ⊗ B_{st} which is mapped to γ(1) = [x, y, z]. Then

E = (V ⊗ Q_{p}) + Q_{p} · u ↪ V ⊗ B_{st}.

This embedding is G-equivariant and the action of g ∈ G is given by

g(v ⊗ 1) = gv ⊗ 1, gu − u = v(g) ⊗ 1,

where v(g) ∈ V and g ↦ v(g) is a 1-cocycle representing β_{V,st}([x, y, z]). We have

D_{st}(E) = (E ⊗ B_{st})^{G} = (V ⊗ Q_{p} ⊗ B_{st} + u ⊗ B_{st})^{G} ↪ V ⊗ B_{st} ⊗ B_{st}.

The action of f resp. N on D_{st}(E) is given by id ⊗ id ⊗ f resp. id ⊗ id ⊗ N. Denote by s_{23} : V ⊗ B_{st} ⊗ B_{st} ⟶ V ⊗ B_{st} ⊗ B_{st} the operator interchanging the two factors B_{st}. The canonical embedding D_{st}(V) ↪ D_{st}(E) can be written as d ↦ s_{23}(d ⊗ 1), if we consider D_{st}(V) as a subgroup of V ⊗ B_{st}. As for each g ∈ G

(g − 1)s_{23}(u ⊗ 1) = v(g) ⊗ 1 ⊗ 1 = (g − 1)(u ⊗ 1)

and s_{23}(u ⊗ 1) ∈ V ⊗ Q_{p} ⊗ B_{st}, the difference u ⊗ 1 − s_{23}(u ⊗ 1) lies in D_{st}(E). Its image under the map D_{st}(E) ⟶ D_{st}(Q_{p}) is equal to 1, hence we can choose a section s as in 1.20 satisfying s(1) = u⊗1−s_{23}(u⊗1). Since f, N act on D_{st}(E) ⊂ V ⊗ B_{st} ⊗ B_{st} via the third factor, we have

\begin{aligned}(f − 1)s(1) & = s_{23}((1 − f)u ⊗ 1) = s_{23}(x ⊗ 1) \\ Ns(1) & = −Ns_{23}(u ⊗ 1) = −s_{23}(Nu ⊗ 1) = s_{23}(y ⊗ 1) \end{aligned}

and analogously, using embedding DR(E) ↪ V ⊗ B_{st} ⊗ B_{dR},

−s(1)(mod F^{0}) = s_{23}(u(mod F^{0}) ⊗ 1) = s_{23}(z ⊗ 1).

This proves that the triple [x, y, z] ∈ Y ⊕ DR(V)/F^{0} coincides with [(f − 1)s(1), Ns(1), −i(s(1))] = ρ_{V,st} ∘ τ_{V,st} ∘ β_{V,st}([x, y, z]), which was to be proved.

1.22 Corollary. If V is semistable, then each extension

0 ⟶ D_{st}(V) ⟶ D ⟶ K_{0} ⟶ 0

in the category of filtered (f, N)-modules is admissible, i.e. comes from an extension

0 ⟶ V ⟶ E ⟶ Q_{p}(0) ⟶ 0

of semistable Galois representations.

1.23 Theorem. Suppose that 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 is an exact sequence of crystalline (resp. semistable) Galois representations over K. Then 0 ⟶ C_{∗}^{.}(X) ⟶ C_{∗}^{.}(Y) ⟶ C_{∗}^{.}(Z) ⟶ 0 is an exact sequence of complexes (for ∗ = cris resp. st) and the corresponding boundary map δ_{C} fits into a commutative diagram

H^{0}(K, Z) ⟶^{α_{Z}} H^{0}(C_{∗}^{.}(Z))
↓^{δ}  ↓^{δ_{C}}
H_{∗}^{1}(K, X) ⟵^{β_{X,∗}} H^{1}(C_{∗}^{.}(X))

(where δ is the usual boundary map in the cohomology sequence). Proof. Exactness of the sequence of C^{.}’s is obvious. Let z ∈ H^{0}(K, Z). Define a Q_{p}-linear map γ : Q_{p} ⟶ Z by γ(1) = z (hence γ is a morphism of representations), and take the pullback of Y by γ, obtaining an extension

0 ⟶ X ⟶ E ⟶ Q_{p}(0) ⟶ 0.

By [Bo, X.7.6.5 Cor 1.a], the class [E] ∈ H_{∗}^{1}(K, V) of this extension is equal to δ(z). We know that α is functorial, i.e. γ ∘ α_{Q_{p}(0)} = α_{Z} ∘ γ. We are thus reduced to prove that β_{X,∗} ∘ δ_{C} ∘ α_{Q_{p}(0)}(1) = [E], where δ_{C} is computed for the extension E. By definition of ρ in 1.20, we have δ_{C} ∘ α_{Q_{p}(0)}(1) = ρ_{X,∗} ∘ τ_{X,∗}([E]), and 1.21 tells us that β_{X,∗} ∘ δ_{C} ∘ α_{Q_{p}(0)}(1) = β_{X,∗} ∘ ρ_{X,∗} ∘ τ_{X,∗}([E]) = [E] as claimed.

1.24 Proposition. Define, for a p-adic Galois representation V of G, h_{∗}^{i}(V) := dim_{Q_{p}}H_{∗}^{i}(K, V). Then (1) h^{1}(V) = h^{0}(V) + h^{2}(V) + [K : Q_{p}] dim_{Q_{p}}(V), h^{i}(V) = h^{2−i}(V^{∗}(1)), h_{ur}^{1}(V) = h^{0}(V)

(2) If V is a de Rham representation, then

h_{f}^{1}(V) = h^{0}(V) + [K : Q_{p}]dim_{K}(DR(V)/F^{0})
h_{e}^{1}(V) = h_{f}^{1}(V) − dim_{Q_{p}}D_{cris}(V)^{f=1}
h_{g}^{1}(V) = h_{f}^{1}(V) + dim_{Q_{p}}D_{cris}(V^{∗}(1))^{f=1}

(3) If V is a semistable representation, then

H_{st}^{1}(K, V) = H_{g}^{1}(K, V)

(this is proved by a different method in [Hy]).

Proof. (1) Follows from the local duality [Se, II.5.2 Th.2] and the formula for the local Euler characteristic [Se, II.5.7. Th.5].

(2) Follows from 1.15 and 1.14, taking into account (1) and equalities dim_{K}F^{0}DR(V^{∗}(1)) = dim_{K}DR(V)/F^{0}, dim_{K}(DR(V)) = dim_{Q_{p}}(V).

(3) We know that H_{st}^{1}(K, V) is a subspace of H_{g}^{1}(K, V), but according to 1.18 and (2) their dimensions are equal.

1.25 Proposition. Let ∗ ∈ {ur, cris, st, dR}. If

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0

is an exact sequence of de Rham representations such that D_{∗}(Y) ⟶ D_{∗}(Z) is surjective (in particular if X, Y, Z are ∗-representations), then the cohomology sequence induces an exact sequence

0 ⟶ H^{0}(K, X) ⟶ H^{0}(K, Y) ⟶ H^{0}(K, Z) ⟶ H_{∗}^{1}(K, X) ⟶ H_{∗}^{1}(K, Y) ⟶ H_{∗}^{1}(K, Z).

If ∗ ∈ {ur, cris}, then this exact sequence can be extended by zero on the right.

Proof. The first statement follows from the commutative diagram with exact rows

H^{0}(Z)  ⟶  H^{1}(X)  ⟶  H^{1}(Y)  ⟶  H^{1}(Z)
↓  ↓  ↓  ↓
H^{0}(Z ⊗ B_{∗}) ⟶^{0} H^{1}(X ⊗ B_{∗}) ⟶ H^{1}(Y ⊗ B_{∗}) ⟶ H^{1}(Z ⊗ B_{∗})

For ∗ ∈ {ur, cris}, the alternating sum of dimensions

h^{0}(X) − h^{0}(Y) + h^{0}(Z) − h_{∗}^{1}(X) + h_{∗}^{1}(Y) − h_{∗}^{1}(Z) = 0

vanishes by 1.24, which proves the second claim.

1.26 Proposition. Let ∗ ∈ {ur, cris, st, dR}, let X, Y be ∗-representations. Denote by Ext_{K,∗}^{1}(X, Y) ⊆ Ext_{K}^{1}(X, Y) the Q_{p}-vector space of those extensions

0 ⟶ Y ⟶ E ⟶ X ⟶ 0

for which E is a ∗-representation. Then there is a canonical isomorphism

Ext_{K,∗}^{1}(X, Y) ⟶^{∼} H_{∗}^{1}(K, Hom(X, Y)).

Proof. Since the internal Ext^{1} between p-adic Galois representations vanishes, we have Ext_{K}^{1}(X, Y) ⟶^{∼} H^{1}(K, Hom(X, Y)). This isomorphism can be described explicitly as follows: starting from

0 ⟶ Y ⟶ E ⟶ X ⟶ 0,

we first apply Hom(X, −) to this extension and then take the pullback via Q_{p}.1_{X} ↪ Hom(X, X). As the category of ∗-representations is stable under duals, tensor products and subobjects, Ext_{K,∗}^{1}(X, Y) maps to H_{∗}^{1}(K, Hom(X, Y)). Conversely, suppose that

0 ⟶ Y ⟶ E ⟶ X ⟶ 0

maps to an element in H_{∗}^{1}(K, Hom(X, Y)). We must show that E is a ∗-representation. The class of E vanishes in H^{1}(K, Hom(X, Y) ⊗ B_{∗}), hence also in H^{1}(K, Hom(X ⊗ B_{∗}, Y ⊗ B_{∗})). This implies that the boundary map H^{0}(K, X ⊗ B_{∗}) ⟶ H^{1}(K, Y ⊗ B_{∗}) is trivial, hence

dim_{K_{∗}}D_{∗}(E) = dim_{K_{∗}}D_{∗}(X) + dim_{K_{∗}}D_{∗}(Y) = dim_{Q_{p}}(X) + dim_{Q_{p}}(Y) = dim_{Q_{p}}(E),

which means that E is a ∗-representation and we are done.

1.27 Corollary. If X, Z are semistable representations, then

(1) Every extension 0 ⟶ D_{st}(X) ⟶ D ⟶ D_{st}(Z) ⟶ 0 in the category of filtered (f, N)-modules comes from an extension 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 of semistable Galois representations.

(2) If a de Rham representation Y sits in an exact sequence 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0, then Y is semistable.

Proof. (1) follows from 1.22 and 1.26, while (2) from 1.24.3 and 1.26.

1.28 Proposition. Suppose that in an exact sequence 0 ⟶ X ⟶ V ⟶ Y ⟶ 0 of p-adic Galois representations both X and Y are de Rham and satisfy F^{0}DR(X) = DR(Y)/F^{0} = 0. Then

(1) V is de Rham as well and the embedding X ↪ V induces an isomorphism DR(X) ⟶^{∼} DR(V)/F^{0}.

(2) If X, Y are semistable, so is V.

(3) D_{st}(Y (−1))^{f=1} = D_{st}(X^{∗})^{f=1} = 0, H^{0}(K, Y) = D_{st}(Y)^{f=1}, H^{0}(K, X^{∗}(1)) = D_{st}(X^{∗}(1))^{f=1}.

(4) H_{ur}^{1}(K, Y) = H_{f}^{1}(K, Y) = H_{st}^{1}(K, Y) = D_{cris}(Y)/(f − 1) = D_{st}(Y)/(f − 1).

Proof. (1) Put Z := Hom(Y, X). This is a de Rham representation and Ext_{K}^{1}(Y, X) = H^{1}(K, Z), Ext_{K,dR}^{1}(Y, X) = H_{g}^{1}(K, Z) by 1.26. As

F^{0}DR(Z) = {z ∈ Hom_{K}(DR(Y), DR(X)) | z(F^{i}DR(Y)) ⊆ F^{i}DR(X) ∀i} = 0,

we have also H^{0}(K, Z) = 0, hence, by 1.24,

h^{1}(Z) = h^{0}(Z^{∗}(1)) + [K : Q_{p}]dim_{Q_{p}}(Z)
h_{g}^{1}(Z) = dim_{Q_{p}}(D_{cris}(Z^{∗}(1))^{f=1} + [K : Q_{p}]dim_{K}(DR(Z)/F^{0}).

Since DR(Z^{∗}(1))/F^{0} = 0, we have also H^{0}(K, Z^{∗}(1)) = D_{cris}(Z^{∗}(1))^{f=1}, which implies that H^{1}(K, Z) = H_{g}^{1}(K, Z), hence V is de Rham. The kernel resp. cokernel of DR(X) ⟶ DR(V)/F^{0} is then F^{0}DR(X) = 0 resp. DR(Y)/F^{0} = 0.

(2) Follows from (1) and 1.27.2.

(3) Suppose that 0 ≠ x ∈ D_{st}(Y (−1))^{f=1}. Then N^{i}x ≠ 0, N^{i+1}x = 0 for some i ≥ 0. As DR(Y (−i − 1)) = F^{i+1}DR(Y (−i − 1)), N^{i}x lies in D_{st}(Y (−i − 1))^{f=1,N=0} ∩ F^{0} = H^{0}(K, Y (−i − 1)), defining an injective map Q_{p}(i + 1) ↪ Y, in contradiction to our assumption DR(Y)/F^{0} = 0. This proves that D_{st}(Y (−1))^{f=1} = 0, hence N vanishes on D_{st}(Y)^{f=1}. Consequently, H^{0}(K, Y) = D_{st}(Y)^{f=1,N=0} ∩ F^{0} = D_{st}(Y)^{f=1}. The same arguments applied to 0 ⟶ Y^{∗}(1) ⟶ V^{∗}(1) ⟶ X^{∗}(1) ⟶ 0 prove the statements about X.

(4) We have H_{ur}^{1} ⊆ H_{f}^{1} ⊆ H_{g}^{1}. By 1.24, h_{ur}^{1}(Y) = h_{f}^{1}(Y). The formulas in (3) imply that (D_{st}(Y (−1))/ ND_{st}(Y))^{f=1} = 0, hence H_{f}^{1}(K, Y) = H_{st}^{1}(K, Y) by 1.18 and H_{ur}^{1}(K, Y) = H_{st}^{1}(K, Y) by counting dimensions. By 1.16, H_{f}^{1}(K, Y) = D_{cris}(Y)/(f − 1), and the canonical map D_{cris}(Y)/(f − 1) ⟶ D_{st}(Y)/(f − 1) is an isomorphism: it is injective, as D_{st}(Y (−1))^{f=1} = 0, and both groups have the same dimension equal to h^{0}(Y).

1.29 Definition. A p-adic Galois representation V is called ordinary, if it admits a decreasing filtration by subrepresentations ⋯ F^{i}V ⊇ F^{i+1}V ⋯ such that ⋃ F^{i}V = V, ⋂ F^{i}V = 0 and all graded factors are of the form

F^{i}V/F^{i+1}V = A_{i}(i)

with A_{i} unramified, i.e. A_{i} = A_{i}^{I}.

1.30 Theorem. Every ordinary representation is semistable.

Proof. Suppose that V = F^{i}V and F^{j}V = 0 for some i < j. If j − i = 1, then V = A_{i}(i) is a Tate twist of an unramified representation, hence crystalline. Suppose the claim has been proved for all ordinary representations with j − i ≤ k. If, for our V, j − i = k + 1, write V (−i) as an extension

0 ⟶ (F^{i+1}V)(−i) ⟶ V (−i) ⟶ A_{i}(0) ⟶ 0.

By induction hypothesis, both extreme terms are semistable. Since DR(A_{l}(l)) has pure filtration −l, DR((F^{i+1}V)(−i)), which is a consecutive extension of terms DR(A_{l}(l − i)) with l ≥ i + 1, satisfies F^{0}DR((F^{i+1}V)(−i)) = 0. As DR(A_{i}(0))/F^{0} = 0, V (−i) is semistable by 1.28.2., and so must be V.

1.31 In fact, the proof of 1.30 tells us that for each ordinary representation V, the exact sequence

0 ⟶ F^{1}V ⟶ V ⟶ V/F^{1}V ⟶ 0

satisfies the assumptions of 1.28 and gives an isomorphism DR(F^{1}V) ⟶^{∼} DR(V)/F^{0}.

1.32 Proposition. Suppose that, in the situation of 1.28, V is a semistable representation satisfying D_{cris}(V)^{f=1} = D_{cris}(V^{∗}(1))^{f=1} = 0. Then

(1) H_{st}^{1}(K, X) = H^{1}(K, X)

(2) N induces an isomorphism H^{0}(K, Y) = D_{st}(Y)^{f=1} ⟶^{∼} D_{st}(X(−1))^{f=1}

(3) The cohomology sequence

0 ⟶ H^{0}(K, Y) ⟶ H^{1}(K, X) ⟶ H_{st}^{1}(K, V) ⟶ 0

is exact. Proof. (1) As DR(X^{∗}(1))/F^{0} = 0, H_{e}^{1}(K, X^{∗}(1)) = 0, hence H^{1}(K, X) = H_{g}^{1}(K, X) = H_{st}^{1}(K, X) by 1.14 and 1.24.3.

(2) By 1.28.3, there is a commutative diagram with exact rows

0 ⟶  0  ⟶  D_{st}(V)^{f=1}  ⟶^{∼} D_{st}(Y)^{f=1} ⟶ 0
↓  ↓^{N}  ↓^{N}
0 ⟶ D_{st}(X(−1))^{f=1} ⟶ D_{st}(V (−1))^{f=1} ⟶  0  ⟶ 0.

Our assumption on V implies that the only non-trivial vertical map is an isomorphism, hence we conclude by snake lemma.

(3) The exactness, possibly without the zero on the right, follows from 1.25 and (1). We get from 1.24, 1.28.3 and vanishing of H^{0}(K, V) ⊆ D_{cris}(V)^{f=1} = 0

h_{st}^{1}(V) = h_{g}^{1}(V) = [K : Q_{p}]dim_{K}DR(V)/F^{0} = [K : Q_{p}]dim_{Q_{p}}(X)
h^{1}(X) = h^{0}(X^{∗}(1)) + [K : Q_{p}]dim_{Q_{p}}(X).

According to 1.28.3 and (2), h^{0}(Y) = dim_{Q_{p}}D_{st}(X(−1))^{f=1} = dim_{Q_{p}}D_{st}(X^{∗}(1))^{f=1} = h^{0}(X^{∗}(1)), hence h^{1}(X) = h_{st}^{1}(V) + h^{0}(Y) and we are done by counting dimensions.

1.33 Lemma. If V is a de Rham representation, then (V/V^{I})^{I} = 0.

Proof. Let 0 ≠ x ∈ (V/V^{I})^{I}. Let A ⊆ (V/V^{I})^{I} be the G(\overline{K}/K)-submodule generated by Q_{p} · x and take the pullback

0 ⟶ V^{I} ⟶ V ⟶ V/V^{I} ⟶ 0
∥  ↑ ∪  ↑ ∪
0 ⟶ V^{I} ⟶ E ⟶ A ⟶ 0.

As V is de Rham, so is E. Applying 1.28.4 and 1.26, we see that E is unramified, i.e. E ⊆ V^{I} – a contradiction.

1.34 Recall from [Bl–Ka 3.10-11] the relation between 1.15 and classical exponential maps. If A is an abelian variety over K and T = T_{p}(A), then H_{e}^{1}(K, T) = H_{f}^{1}(K, T) = A(K)ˆ⊗Z_{p}, DR(V)/F^{0} is canonically identified with the tangent space Lie(A) of A at the origin and the map exp : DR(V)/F^{0} ⟶ H_{e}^{1}(K, V) is the composition of the usual exponential map Lie(A) ⟶ A(K)⊗Q with the “Kummer map” A(K)ˆ⊗Z_{p} ⟶ H^{1}(K, T), which is obtained as a projective limit of boundary maps arising from

0 ⟶ A(\overline{K})_{p^{n}} ⟶ A(\overline{K})⟶^{p^{n}}A(\overline{K}) ⟶ 0.

Similarly, for T = Z_{p}(1), DR(Q_{p}(1))/F^{0} = K(1) is the tangent space to the formal multiplicative group at the origin and exp : K(1) ⟶ H_{e}^{1}(K, Q_{p}(1)) = O_{K}^{∗} ˆ⊗Q_{p} is the usual exponential, provided we identify K(1) with K by means of the canonical invariant differential dz/z. We shall keep track of the Tate twist, however. As a result, we shall consider both the standard logarithm ( = inverse of exp) log : O_{K}^{∗} ˆ⊗Q_{p} ⟶^{∼} K(1) and its extension Log : K^{∗} ˆ⊗Q_{p} ⟶ K(1) as K(1)-valued. Later on, we shall encounter algebraic logarithms, i.e. homomorphisms l : K^{∗} ˆ⊗Q_{p} ⟶ Q_{p}. We say that l is ramified, if its restriction to O_{K}^{∗} ˆ⊗Q_{p} does not vanish. If this is the case, then there is a unique analytic logarithm Log and a unique Q_{p}-linear map λ : K(1) ⟶ Q_{p} such that l = λ ∘ Log.

1.35 Let us recall explicit description of D_{st} for Kummer extensions: if

0 ⟶ Q_{p}(1) ⟶ V ⟶ Q_{p}(0) ⟶ 0

is an extension corresponding to [V] = u ∈ K^{∗} ˆ⊗Q_{p} = H^{1}(K, Q_{p}(1)), then there is a unique f-equivariant decomposition

D_{st}(V) = D_{st}(Q_{p}(1)) ⊕ D_{st}(Q_{p}(0)) = K_{0}(1) ⊕ K_{0},

inducing DR(V) = K(1) ⊕ K. The Hodge filtration is determined by F^{0}DR(V) = K · (Log(u), 1) and monodromy operator is given by

N : K(1) ⊕ K ⟶ K⟶^{v(u)}K ↪ K ⊕ K(−1),

where v, Log are the valuation resp. analytic logarithm used for the definition of B_{st} ↪ B_{dR}.

The complex C_{st}^{.}(Q_{p}(1)) is quasi-isomorphic to [K_{0} ⊕ K(1) ⟶^{(f−1,0)}K_{0}][−1] and the isomorphism

β_{Q_{p}(1),st} : H^{1}(C_{st}^{.}(Q_{p}(1))) = Q_{p} ⊕ K(1) ⟶^{∼} H^{1}(K, Q_{p}(1))

is given by

β^{−1}(u) = (v(u), Log(u)).

2. General p-adic pairings for Galois representations

2.1.1 We start with the following data:

• K – a finite extension of Q

• V – a p-adic Galois representation over K, i.e. a finite dimensional Q_{p}-vector space with a continuous linear action of G(\overline{K}/K).

2.1.2 We assume that the following conditions are satisfied: (A) V is unramified outside a finite set S_{0} of primes of K

(B)  L_{v}(V, 0)L_{v}(V^{∗}(1), 0) ≠ 0  (∀v|̸p)
(C)  V_{℘} is a crystalline representation of G_{℘} = G(\overline{K_{℘}}/K_{℘})  (∀℘|p)
(D)  L_{℘}(V, 0)L_{℘}(V^{∗}(1), 0) ≠ 0  (∀℘|p)

Here V_{v} denotes V considered as a representation of G_{v} = G(\overline{K_{v}}/K_{v}) and the local L-factors are

L_{v}(V, s) = det(1 − F r(v)(Nv)^{−s} | V^{I_{v}})  (v|̸p)
L_{℘}(V, s) = det(1 − f^{[K_{℘}^{0}:Q_{p}]}(N℘)^{−s} | D_{cris}(V_{℘})) (℘|p),

where F r(v) is a representative of the geometric Frobenius element at v.

2.1.3 Note the following consequences of (B)–(D):

(B) ⇒  H^{i}(K_{v}, V) = H^{i}(K_{v}, V^{∗}(1)) = 0  (∀v|̸p, i = 0, 1, 2)
(D) ⇒  D_{cris}(V_{℘})^{f=1} = D_{cris}(V^{∗}(1)_{℘})^{f=1} = 0  (∀℘|p),

by local duality and the formula for the local Euler charakteristic.

2.1.4 Denote by O_{K} the ring of integers in K. Let us fix a finite set S ⊇ S_{0} ∪ {℘|p} of primes of K and denote by G_{S} the Galois group of the maximal extension of K unramified outside S. Define for v|̸p

H_{f}^{1}(K_{v}, V) = H_{ur}^{1}(K_{v}, V)

and put

H_{f}^{1}(G_{S}, V) = {x ∈ H^{1}(G_{S}, V) | x_{v} ∈ H_{f}^{1}(K_{v}, V) ∀v ∈ S }

and similarly for V^{∗}(1). It follows from the condition (B) that the groups H^{1}(G_{S}, V), H^{1}(G_{S}, V^{∗}(1)) do not depend on S (cf. [Ja 1, Lemma 4]) and the same is true for H_{f}^{1}. In fact, they are equal to H^{1}(K, −) and

H_{f}^{1}(K, −) = {x ∈ H^{1}(K, −) | x_{v} ∈ H_{f}^{1}(K_{v}, −) ∀v }

respectively. We may now formulate our main result.

2.2 Theorem. Suppose we are given a non-trivial continuous homomorphism l : A_{K}^{∗}/K^{∗} ⟶ Q_{p}. One may then define a family of continuous bilinear pairings

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

depending on the choice of splittings of Hodge filtrations 0 ⟶ F^{0}DR(V_{℘}) ⟶ DR(V_{℘}) ⟶ DR(V_{℘})/F^{0} ⟶ 0 for primes ℘|p.

Proof. Assume that a ∈ H_{f}^{1}(K, V) = H_{f}^{1}(G_{S}, V), a′ ∈ H_{f}^{1}(K, V^{∗}(1)) = H_{f}^{1}(G_{S}, V^{∗}(1)). Then a represents an extension

0 ⟶ V ⟶ E ⟶ Q_{p}(0) ⟶ 0

of continuous G_{S}-modules with E_{℘} (= E as a module over G_{℘}) crystalline for all ℘|p. Dualizing, we get an extension

0 ⟶ Q_{p}(1) ⟶ E^{∗}(1) ⟶ V^{∗}(1) ⟶ 0

with E^{∗}(1)_{℘} crystalline as well.

2.3 Lemma. There is an exact sequence

0 ⟶ H_{f}^{1}(G_{S}, Q_{p}(1)) ⟶ H_{f}^{1}(G_{S}, E^{∗}(1)) ⟶ H_{f}^{1}(G_{S}, V^{∗}(1)) ⟶ 0,

in which H_{f}^{1}(G_{S}, Q_{p}(1)) = O_{K}^{∗} ⊗ Q_{p}.

Proof. According to [Sch 1], H^{1}(G_{S}, Q_{p}(1)) = O_{S}^{∗} ⊗Q_{p} and by 1.12 H_{f}^{1}(K_{℘}, Q_{p}(1)) = O_{℘}^{∗} ˆ⊗Q_{p}, which proves that H_{f}^{1}(G_{S}, Q_{p}(1)) = O_{K}^{∗} ⊗ Q_{p}. Consider the commutative diagram with exact rows:

0 ⟶  O_{S}^{∗} ⊗ Q_{p}  ⟶ H^{1}(G_{S}, E^{∗}(1)) ⟶ H^{1}(G_{S}, V^{∗}(1)) ⟶ Q_{p}^{♯S−1}
↓  ↓  ↓  ∩ ↓
0 ⟶ ⨁_{v∈S} H^{1}(K_{v}, Q_{p}(1)) ⟶ ⨁_{v∈S} H^{1}(K_{v}, E^{∗}(1)) ⟶ ⨁_{v∈S} H^{1}(K_{v}, V^{∗}(1)) ⟶^{δ} Q_{p}^{♯S}

We know that H^{1}(K_{v}, V^{∗}(1)) = H^{1}(K_{v}, V) = 0 for all v|̸p as a consequence of the condition (B). It follows that

E^{∗}(1)_{v} = V^{∗}(1)_{v} ⊕ Q_{p}(1)

as a representation of G(\overline{K_{v}}/K_{v}), hence

H_{f}^{1}(K_{v}, E^{∗}(1)) = H_{f}^{1}(K_{v}, Q_{p}(1)) = 0.

Let ℘|p. As

H^{0}(K_{℘}, V^{∗}(1)) ⊆ D_{cris}(V^{∗}(1)_{℘})^{f=1} = 0,

we get from 1.23 an exact sequence

0 ⟶ H_{f}^{1}(K_{℘}, Q_{p}(1)) ⟶ H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0.

An easy diagram chase then shows that the global sequence in question is exact if the zero on the right is removed. To verify the surjectivity it suffices to check that the composition

H_{f}^{1}(G_{S}, V^{∗}(1)) ⟶ H^{1}(K_{℘}, V^{∗}(1))⟶^{δ}Q_{p}

vanishes for all ℘|p. The boundary map δ is given by the cup product with the extension class a_{℘} lying in H_{f}^{1}(K_{℘}, V), hence vanishes on H_{f}^{1}(K_{℘}, V^{∗}(1)) by 1.14. Lemma is proved.

2.4 It follows from the previous discusion that the following commutative diagram has exact rows:

0 ⟶ O_{K}^{∗} ⊗ Q_{p} ⟶ H_{f}^{1}(G_{S}, E^{∗}(1)) ⟶^{π} H_{f}^{1}(G_{S}, V^{∗}(1)) ⟶ 0
↓^{i}  ↓^{j}  ↓
0 ⟶ ⨁_{℘|p} O_{℘}^{∗} ˆ⊗Q_{p} ⟶ ⨁_{℘|p} H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ ⨁_{℘|p} H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0

↓^{l}  . ; . ; . ; ↓_{l′}
Q_{p}  =  Q_{p}

and l ∘ i = 0. Suppose we are given a splitting of this diagram, i.e. a homomorphism l′ making the lower square commutative. The symbol

h(a, a′) := l′ ∘ j ∘ π^{−1}(a′) ∈ Q_{p}

is then well defined and is linear in a′. We are looking for l′ “functorial” in a, so that h(a, a′) becomes linear in a as well. To do so, fix ℘|p and look at the sequence

0 ⟶ H_{f}^{1}(K_{℘}, Q_{p}(1)) ⟶ H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0.

2.5 Lemma. The exponential maps of 1.15 define a canonical isomorphism between two exact sequences

0 ⟶ H_{f}^{1}(K_{℘}, Q_{p}(1)) ⟶ H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0
↑^{≀}  ↑^{≀}  ↑^{≀}
0 ⟶ DR(Q_{p}(1)) ⟶ DR(E^{∗}(1)_{℘})/F^{0} ⟶ DR(V^{∗}(1)_{℘})/F^{0} ⟶ 0

Proof. As D_{cris}(Q_{p}(1))^{f=1} = D_{cris}(V^{∗}(1)_{℘})^{f=1} = 0, D_{cris}(E^{∗}(1)_{℘})^{f=1} must vanish as well. It follows then from 1.15, 1.16 that

exp : DR(−)/F^{0} ⟶^{∼} H_{e}^{1}(K_{℘}, −) = H_{f}^{1}(K_{℘}, −)

for each of the modules Q_{p}(1), E^{∗}(1), V^{∗}(1).

2.6 Define, for a K_{℘}-vector space A, D(A) = Hom_{K_{℘}}(A, K_{℘}(1)). Applying D to the bottom row of the diagram in Lemma 2.5, we get an exact sequence

0 ⟶ F^{0}DR(V_{℘}) ⟶ F^{0}DR(E_{℘}) ⟶ F^{0}DR(Q_{p}(0)) = K_{℘} ⟶ 0,

which will be denoted by σ(e). Recall that H_{f}^{1}(K_{℘}, V_{℘}) is equal to the extension group Ext_{cris}^{1}(Q_{p}(0), V_{℘}) in the category of crystalline representations of G_{℘}. Define the set of rigidified extensions to be

Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) := {(e, w) | e ∈ Ext_{cris}^{1}(Q_{p}(0), V_{℘}) , w a splitting of σ(e)},

where by a splitting we mean a K_{℘}-linear retraction F^{0}DR(V_{℘}) ⟶^{⟵^{w}} F^{0}DR(E_{℘}). The set Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) becomes an abelian group under a slightly modified Baer sum (see [Za]).

2.7 Lemma. The forgetful functor induces an exact sequence

0 ⟶ F^{0}DR(V_{℘}) ⟶ Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ Ext_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ 0,

which is canonically isomorphic to the sequence

0 ⟶ F^{0}DR(V_{℘}) ⟶ DR(V_{℘}) ⟶ DR(V_{℘})/F^{0} ⟶ 0.

Proof. Exactness of the first sequence follows easily from the definitions and the fact that rigidifications on V_{℘} ⊕ Q_{p} correspond to Hom_{K_{℘}}(K_{℘}, F^{0}DR(V_{℘})) = F^{0}DR(V_{℘}). In order to identify the two exact sequences we recall a down-to-earth description of the exponential morphism

DR(−)/F^{0} ⟶ H_{e}^{1}(K_{℘}, −)

(equal to β_{cris} in our case) given in 1.21. We define an isomorphism Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶^{∼} DR(V_{℘}) as follows: as D_{cris}(V)^{f=1} = 0, for each crystalline extension

0 ⟶ V_{℘} ⟶ E ⟶ Q_{p}(0) ⟶ 0

the corresponding extension of filtered modules

0 ⟶ D_{cris}(V) ⟶ D(E) ⟶ K_{℘}^{0} ⟶ 0

admits a unique section s : K_{℘}^{0} ⟶ D_{cris}(E) satisfying (f − 1)s(1) = 0. A splitting

0 ⟶ F^{0}DR(V_{℘}) ⟶^{⟵^{w}} F^{0}DR(E) ⟶ K ⟶ 0

determines in turn a section t : K ⟶ F^{0}DR(E). The difference x(E) := t(1) − i(s(1)) then lies in DR(V_{℘}) and one checks, using 1.21, that the map E ⟼ x(E) defines a homomorphism Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ DR(V_{℘}) making the diagram

0 ⟶ F^{0}DR(V_{℘}) ⟶ Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ Ext_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ 0
∥  ↓  ↑^{≀exp}
0 ⟶ F^{0}DR(V_{℘}) ⟶  DR(V_{℘})  ⟶  DR(V_{℘})/F^{0}  ⟶ 0

commutative.

2.8 We can now conclude the proof of Theorem 2.2: suppose that we are given for each ℘|p a Q_{p}-linear splitting of

0 ⟶ F^{0}DR(V_{℘}) ⟶ DR(V_{℘}) ⟶^{⟵} DR(V_{℘})/F^{0} ⟶ 0.

By lemma 2.4, it can be identified with a splitting r_{℘} of

0 ⟶ F^{0}DR(V_{℘}) ⟶ Extrig_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶^{⟵^{r℘}} Ext_{cris}^{1}(Q_{p}(0), V_{℘}) ⟶ 0

and one defines the map

l′ : ⨁_{℘|p} H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ Q_{p}

to be

l′ = l ∘ ⨁_{℘|p} w_{℘},

where w_{℘} is the splitting

0 ⟶ H_{f}^{1}(K_{℘}, Q_{p}(1)) ⟶^{⟵^{w℘}} H_{f}^{1}(K_{℘}, E^{∗}(1)) ⟶ H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0

corresponding to the splitting r_{℘}(a_{℘}) of the dual exact sequence. It follows from the previous discussion that h(a, a′) ∈ Q_{p} is additive in the variable a, which concludes the proof of Theorem 2.2.

2.9 In fact, it suffices to have splittings r_{℘} for those primes ℘|p in which l is ramified (= does not vanish on O_{℘}^{∗}). It may be surprising that the data required for the construction of the pairings are Q_{p}-linear and not K_{℘}-linear splittings of Hodge filtrations.

3. Splittings of local extensions

3.1 This section deals with relations between splittings of Hodge filtrations and splittings of certain cohomology sequences. We shall work with p-adic Galois representations over a field K_{℘} of finite degree over Q_{p}. In 3.2-4 we retain the assumptions (C),(D) of 2.1.2, namely that V is a crystalline p-adic Galois representation over K_{℘} satisfying

D_{cris}(V)^{f=1} = D_{cris}(V^{∗}(1))^{f=1} = 0,

which implies that we have isomorphisms

exp : DR(V)/F^{0} ⟶^{∼} H_{e}^{1}(K_{℘}, V) = H_{f}^{1}(K_{℘}, V)

and similarly for V^{∗}(1).

3.2 If A is a Q_{p}-vector space of finite dimension, write A(n) = A ⊗ Q_{p}(n) for A as a Galois representation, on which G_{℘} acts by the n-th power of the cyclotomic character χ_{K}. Suppose we are given an extension of crystalline representations

0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0.

Its class [E_{1}] lies in

Ext_{K_{℘},f}^{1}(A(0), V) = Hom(A, H_{f}^{1}(K_{℘}, V)) ⟶^{∼} Hom(A, DR(V)/F^{0}).

Denote by α : A ⊗ K_{℘} ⟶ DR(V)/F^{0} its K_{℘}-linear extension and by μ : DR(V) ⟶ DR(V)/F^{0} the canonical projection. As D_{cris}(V)^{f=1} = 0, the exact sequence

0 ⟶ D_{cris}(V) ⟶ D_{cris}(E_{1}) ⟶ D_{cris}(A(0)) = A ⊗ K_{℘}^{0} ⟶ 0

admits a unique f-equivariant splitting

D_{cris}(E_{1}) = D_{cris}(V) ⊕ (A ⊗ K_{℘}^{0}),

inducing

DR(E_{1}) = DR(V) ⊕ (A ⊗ K_{℘}).

According to 1.21,

F^{0}DR(E_{1}) = {(x, a) | x ∈ DR(V), a ∈ A ⊗ K_{℘}, μ(x) = α(a)},

i.e. the exact sequence

0 ⟶ F^{0}DR(V) ⟶ F^{0}DR(E_{1}) ⟶ A ⊗ K_{℘} ⟶ 0

is the pullback of

0 ⟶ F^{0}DR(V) ⟶ DR(V) ⟶ DR(V)/F^{0} ⟶ 0

via α : A ⊗ K_{℘} ⟶ DR(V)/F^{0}.

3.3 Suppose that we are given an extension of crystalline representations

0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0,

where B is another Q_{p}-vector space of finite dimension. Applying 3.2 to E_{2}^{∗}(1), we get a unique f-equivariant splitting

D_{cris}(E_{2}^{∗}(1)) = D_{cris}(V^{∗}(1)) ⊕ (B^{∗} ⊗ K_{℘}^{0}),

inducing

D_{cris}(E_{2}) = D_{cris}(V) ⊕ (B ⊗ K_{℘}^{0}(1))
DR(E_{2}) = DR(V) ⊕ (B ⊗ K_{℘}(1)).

Write μ′ : DR(V^{∗}(1)) ⟶ DR(V^{∗}(1))/F^{0} for the canonical projection and β′ : B^{∗} ⊗ K_{℘} ⟶ DR(V^{∗}(1))/F^{0} for the K_{℘}-linear extension of [E_{2}^{∗}(1)] ∈ Hom(B^{∗}, DR(V^{∗}(1))/F^{0}). Let β : F^{0}DR(V) ⟶ B ⊗ K_{℘}(1) be the map obtained from β by applying the functor D of 2.6. According to 3.2,

F^{0}DR(E_{2}^{∗}(1)) = {(x′, b′) | x′ ∈ DR(V^{∗}(1)), b′ ∈ B^{∗} ⊗ K_{℘}, μ′(x′) = β′(b′)},

which implies that

F^{0}DR(E_{2}) = (F^{0}DR(E_{2}^{∗}(1)))^{⊥} = {(x, −β(x)) | x ∈ F^{0}DR(V)}.

In analogy to 3.2, this means that there is a commutative diagram with exact rows

0 ⟶ F^{0}DR(V) ⟶  DR(V)  ⟶ DR(V)/F^{0} ⟶ 0
↓^{β}  ↓  ∥
0 ⟶ B ⊗ K_{℘}(1) ⟶ DR(E_{2})/F^{0} ⟶ DR(V)/F^{0} ⟶ 0
↓^{≀ exp}  ↓^{≀ exp}  ↓^{≀ exp}
0 ⟶ H_{f}^{1}(K_{℘}, B(1)) ⟶ H_{f}^{1}(K_{℘}, E_{2}) ⟶ H_{f}^{1}(K_{℘}, V) ⟶ 0

It follows that any splitting of the Hodge filtration

0 ⟶ F^{0}DR(V) ⟶ DR(V) ⟶^{⟵} DR(V)/F^{0} ⟶ 0

induces a splitting of

0 ⟶ H_{f}^{1}(K_{℘}, B(1)) ⟶ H_{f}^{1}(K_{℘}, E_{2}) ⟶^{⟵} H_{f}^{1}(K_{℘}, V) ⟶ 0.

In down-to-earth terms, it is simply the composition

DR(V)/F^{0} ⟶ DR(V) ↪ DR(E_{2}) ⟶ DR(E_{2})/F^{0}.

3.4 Suppose that V is merely semistable, but still satisfies D_{cris}(V)^{f=1} = D_{cris}(V^{∗}(1))^{f=1} = 0. If

\begin{matrix}0 & ⟶ & V & ⟶ & E_{1} & ⟶ & A(0) & ⟶ & 0 \\ 0 & ⟶ & B(1) & ⟶ & E_{2} & ⟶ & V & ⟶ & 0 \end{matrix}

are extensions of semistable representations, then our condition on V implies that there are unique (f, N)- equivariant decompositions

D_{st}(E_{1}) = D_{st}(V) ⊕ (A ⊗ K_{℘}^{0})
D_{st}(E_{2}) = D_{st}(V) ⊕ (B ⊗ K_{℘}^{0}(1)).

The extension classes [E_{1}] ∈ Hom(A, H_{st}^{1}(K_{℘}, V)) = Hom(A, DR(V)/F^{0}) and [E_{2}^{∗}(1)] define, as before, K_{℘}-linear maps

α : A ⊗ K_{℘} ⟶ DR(V)/F^{0}, β : F^{0}DR(V) ⟶ B ⊗ K_{℘}(1)

and F^{0}DR(E_{1}), F^{0}DR(E_{2}) are given by the same formulas as in 3.2, 3.3.

3.5 The rest of sec.3 collects some preliminary material for sec.7 and should be skipped in the first reading. Recall from 1.23 that the complex

C_{st}^{.}(W) = [D_{st}(W) ⟶^{(f−1,N,−i)}D_{st}(W) ⊕ D_{st}(W (−1)) ⊕ DR(W)/F^{0} ⟶^{(N,1−f,0)}D_{st}(W (−1))]

computes functorially H^{0}(K_{℘}, W) and H_{st}^{1}(K_{℘}, W) for a semistable representation W. An exact sequence of complexes

0 ⟶ [D_{st}(W (−1)) ⊕ DR(W)/F^{0} ⟶^{(1−f,0)}D_{st}(W (−1))][−1] ⟶ C_{st}^{.}(W) ⟶

⟶ [D_{st}(W)⟶^{f−1}D_{st}(W)] ⟶ 0

induces a long exact sequence

0 ⟶ H^{0}(K_{℘}, W) ⟶ D_{st}(W)^{f=1} ⟶^{(N,−i)}D_{st}(W (−1))^{f=1} ⊕ DR(W)/F^{0} ⟶ H_{st}^{1}(K_{℘}, W) ⟶

⟶ D_{st}(W)/(f − 1)D_{st}(W)⟶^{N}D_{st}(W (−1))/(f − 1)D_{st}(W (−1))

Let us apply this machinery to all terms in the exact sequence

0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0

of 3.4. Using the (f, N)-equivariant decomposition, we get a commutative diagram with exact rows and columns:

0  0
↓  ↓
D_{st}(E_{2})^{f=1}  ≃  D_{st}(V)^{f=1}
↓^{(0,N,−i)}  ↓^{(N,−i)}
0 → B ⊕ B ⊗ K_{℘}(1) → B ⊕ D_{st}(V (−1))^{f=1} ⊕ DR(E_{2})/F^{0} → D_{st}(V (−1))^{f=1} ⊕ DR(V)/F^{0} → 0
↓^{≀}  ↓  ↓
0 → H_{st}^{1}(K_{℘}, B(1)) →  H_{st}^{1}(K_{℘}, E_{2})  →  H_{st}^{1}(K_{℘}, V)  → 0
↓  ↓
0  0

As D_{cris}(E_{2})^{f=1} = 0, there are, as in 3.3, isomorphisms

0 ⟶ B ⊗ K_{℘}(1) ⟶ DR(E_{2})/F^{0} ⟶ DR(V)/F^{0} ⟶ 0
↓^{≀exp}  ↓^{≀exp}  ↓^{≀exp}
0 ⟶ H_{f}^{1}(K_{℘}, B(1)) ⟶ H_{f}^{1}(K_{℘}, E_{2}) ⟶ H_{st}^{1}(K_{℘}, V) ⟶ 0

It follows that any splitting of the Hodge filtration

0 ⟶ F^{0}DR(V) ⟶ DR(V) ⟶ DR(V)/F^{0} ⟶ 0

again induces a splitting

DR(V)/F^{0} ⟶ DR(V) ↪ DR(E_{2}) ⟶ DR(E_{2})/F^{0}

of H_{f}^{1}(K_{℘}, E_{2}) ⟶^{⟵} H_{st}^{1}(K_{℘}, V), hence also of H_{st}^{1}(K_{℘}, E_{2}) ⟶^{⟵} H_{st}^{1}(K_{℘}, V).

4. Local height pairings

4.1 Following closely the construction of archimedean height pairings in [Sc 2], we shall decompose the pairing (depending on the choice of splittings)

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

constructed in sec. 2 into a sum of local pairings

h = ∑_{v} h_{v},

the sum extending over all non-archimedean places v of K. The decomposition is not unique and depends on the choice of an auxiliary object, a certain “mixed extension”.

4.2 We retain the assumptions of 2.1. Let there be given two extensions of p-adic representations of G(\overline{K}/K)

\begin{matrix}e_{1} : & 0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0 \\ e_{2} : & 0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0 \end{matrix}

in which the action of G(\overline{K}/K) on A, B is trivial and E_{1,℘}, E_{2,℘} are de Rham representations for all ℘|p (hence crystalline, by 1.14, 1.16).

4.3 Definition. A mixed extension of e_{1}, e_{2} is a p-adic Galois representation E equipped with an increasing weight filtration by subrepresentations

0 = W_{−3}E ⊆ W_{−2}E ⊆ W_{−1}E ⊆ W_{0}E = E

and isomorphisms

W_{−1}E ⟶^{∼} E_{2}, W_{0}E/W_{−2}E ⟶^{∼} E_{1}.

Such an object (called “extension panachée” in [SGA 7/I, IX.9.3]) lives in a commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0
∥  ↓  ↓
0 ⟶ B(1) ⟶ E ⟶ E_{1} ⟶ 0
↓  ↓
A(0) = A(0)
↓  ↓
0  0

There is a natural action of Ext_{K}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H^{1}(K, Q_{p}(1)) on the set of mixed extensions E: for X ∈ Ext_{K}^{1}(A(0), B(1)) the Baer sum of its image in Ext_{K}^{1}(A(0), E_{2}) with

0 ⟶ E_{2} ⟶ E ⟶ A(0) ⟶ 0

is again a mixed extension in a natural way. Similarly, we can take the Baer sum of

0 ⟶ B(1) ⟶ E ⟶ E_{1} ⟶ 0

with the image of X in Ext_{K}^{1}(E_{1}, B(1)), but the resulting mixed extensions will be canonically isomorphic. If E is a mixed extension of e_{1}, e_{2}, then E^{∗}(1) is a mixed extension of

e_{2}^{∗}(1) : 0 ⟶ V^{∗}(1) ⟶ E_{2}^{∗}(1) ⟶ B^{∗}(0) ⟶ 0

e_{1}^{∗}(1) : 0 ⟶ A^{∗}(1) ⟶ E_{1}^{∗}(1) ⟶ V^{∗}(1) ⟶ 0.

Let v be a non-archimedean place of K not dividing p. As H^{1}(K_{v}, V) = H^{1}(K_{v}, V^{∗}(1)) = 0, we have a splitting of the G(\overline{K_{v}}/K_{v})-module E_{v} = V_{v} ⊕ N, where N is an extension

0 ⟶ B(1) ⟶ N ⟶ A(0) ⟶ 0.

We say that E is essentially unramified at v, if N is unramified.

4.4 Proposition. Suppose we are given two extensions e_{1}, e_{2} as in 4.2. Then

(1) The set of isomorphism classes of mixed extensions E of e_{1}, e_{2} is a principal homogeneous space of

Ext_{K}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H^{1}(K, Q_{p}(1))

(under the action described in 4.3)

(2) The set of isomorphism classes of those mixed extensions E for which E_{v} is essentially unramified at all v|̸p and E_{℘} is crystalline for all ℘|p, is a principal homogeneous space of

Ext_{K,f}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H_{f}^{1}(K, Q_{p}(1))

(3) The automorphism group of every mixed extension E is trivial.

(4) If E is any mixed extension of e_{1}, e_{2}, then E_{℘} is de Rham for each ℘|p.

Proof. (1) According to [SGA 7/I, IX.9.3.8], the obstruction for existence of E is given by the class of the Yoneda composition

e_{2} ∘ e_{1} = 0 ⟶ B(1) ⟶ E_{2} ⟶ E_{1} ⟶ A(0) ⟶ 0

in Ext_{K}^{2}(A(0), B(1)) = Hom(A, B) ⊗ H^{2}(K, Q_{p}(1)). This class is, up to a sign, equal to the cup product of

[e_{1}] ∈ A^{∗} ⊗ H_{f}^{1}(K, V) and [e_{2}^{∗}(1)] ∈ B ⊗ H_{f}^{1}(K, V^{∗}(1)).

Since H^{2}(K, Q_{p}(1)) injects into ∏_{v} H^{2}(K_{v}, Q_{p}(1)), this cup product vanishes by 1.14 and 2.1.3. The claim now follows from [SGA 7/I, 9.3.8.b].

(2) There is a commutative diagram with exact rows

0 ⟶ Ext_{K}^{1}(A(0), B(1)) ⟶ Ext_{K}^{1}(A(0), E_{2}) ⟶^{π} Ext_{K}^{1}(A(0), V) ⟶ 0
↑ ∪  ↑ ∪  ↑ ∪
0 ⟶ Ext_{K,f}^{1}(A(0), B(1)) ⟶ Ext_{K,f}^{1}(A(0), E_{2}) ⟶ Ext_{K,f}^{1}(A(0), V) ⟶ 0

(exactness of the bottom row follows from the same argument as in 2.3). The assertion is equivalent to the fact that the fibre π^{−1}([e_{1}]) meets Ext_{K,f}^{1}(A(0), E_{2}) and this is indeed the case, as [e_{1}] ∈ Ext_{K,f}^{1}(A(0), V).

(3) According to [SGA 7/I, 9.3.8.a], the set of automorphisms of any E is equal to Ext_{K}^{0}(A(0), B(1)) = 0.

(4) Follows from (1),(2) and the fact that H^{1}(K_{℘}, Q_{p}(1)) = H_{g}^{1}(K_{℘}, Q_{p}(1)).

4.5 We are now ready to define local pairings h_{v}. Suppose we are given the same data as in Theorem 2.2: local splittings r_{℘} of Hodge filtrations

F^{0}DR(V_{℘}) ⟶^{⟵^{r℘}} DR(V_{℘})

and a non-trivial continuous homomorphism

l : A_{K}^{∗}/K^{∗} ⟶ Q_{p}.

Such a homomorphism can be written as a sum of local terms

l_{v} : K_{v}^{∗} ⟶ Q_{p},

which are continuous and satisfy the reciprocity formula

∑_{v} l_{v}(a) = 0 (∀a ∈ K^{∗}).

Note that for v|̸p the local map l_{v} is unramified, i.e. factors through ord_{v} : K^{∗} ⟶ Z. We shall consider all maps l_{v} as continuous homomorphisms

l_{v} : H^{1}(K_{v}, Q_{p}(1)) ⟶ Q_{p}.

Let there be given extensions e_{1}, e_{2} as in 4.2. We fix one of their mixed extensions E. The data (r_{℘}), E define the same objects for V^{∗}(1): the dual splittings D(r_{℘}) and the mixed extension E^{∗}(1).

4.6 Local pairings at v|̸p Recall from 4.3 the decomposition E_{v} = V_{v} ⊕ N, where N is an extension

0 ⟶ B(1) ⟶ N ⟶ A(0) ⟶ 0.

Its class lies in

[N] ∈ Ext_{K_{v}}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H^{1}(K_{v}, Q_{p}(1)).

Equivalently, cohomology sequences of the first row and column give

A
↓^{[E_{v}]}
ϕ : B ⊗ H^{1}(K_{v}, Q_{p}(1)) ⟶^{∼} H^{1}(K_{v}, E_{2}),

where [E_{v}] ∈ Hom(A, H^{1}(K_{v}, E_{2})) is the extension class of E_{v}, and

ϕ^{−1} ∘ [E_{v}] ∈ Hom(A, B) ⊗ H^{1}(K_{v}, Q_{p}(1))

is equal (by [Bo, X.7.6.5 Cor.1.a]) to [N]. The valuation ord_{v} : K_{v}^{∗} ⟶ Z induces an isomorphism

(ord_{v})_{∗} : H^{1}(K_{v}, Q_{p}(1)) ⟶^{∼} K_{v} ˆ⊗Q_{p} ⟶^{∼} Q_{p}.

Since l_{v} factors through ord_{v}, we may define

h_{v,E} := −l_{v}([N]) ∈ Hom(A, B),

in other words a pairing

h_{v,E} : A ⊗ B^{∗} ⟶ Q_{p}.

We have obvious implications

E is unramified at v ⟹ E is essentially unramified at v ⟹ h_{v,E} = 0.

4.7 Local pairings at ℘|p (crystalline case) Suppose that E_{℘} is crystalline. We shall follow the construction for v|̸p. According to 1.25, the extension class [E_{℘}] lies in

[E_{℘}] ∈ Hom(A, H_{f}^{1}(E_{2,℘}))

(all cohomology groups in 4.7 and 4.8 are over K_{℘}) and the splitting r_{℘} induces by 3.3 a splitting of

0 ⟶ B ⊗ H_{f}^{1}(Q_{p}(1)) ⟶^{⟵} H_{f}^{1}(E_{2,℘}) ⟶ H_{f}^{1}(V_{℘}) ⟶ 0.

Composing [E_{℘}] with this splitting, we obtain an element

(r_{℘})_{∗}[E_{℘}] ∈ Hom(A, B ⊗ H_{f}^{1}(Q_{p}(1))) = Hom(A, B) ⊗ H_{f}^{1}(Q_{p}(1)).

Using

l_{℘} : H_{f}^{1}(Q_{p}(1)) = O_{℘}^{∗} ˆ⊗Q_{p} ⟶ Q_{p}

we define h_{℘,E} ∈ Hom(A, B) by

h_{℘,E} := −l_{℘}((r_{℘})_{∗}[E_{℘}]) ∈ Hom(A, B).

Note that the only data needed to this definition is an extension l′_{℘} : H_{f}^{1}(E_{2,℘}) ⟶ B of l_{℘} : B ⊗ H_{f}^{1}(Q_{p}(1)) ⟶ B. Given such l′_{℘}, we define h_{℘,E} := −l′_{℘} ∘ [E_{℘}]. Such l′_{℘} will arise as “universal norm splittings”, cf. 6.11. 4.8 Local pairings at ℘|p (general case) Denote by π the map H^{1}(E_{2,℘}) ⟶ H^{1}(V_{℘}). There is a commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ B ⊗ H_{f}^{1}(Q_{p}(1)) ⟶^{⟵^{w}} H_{f}^{1}(E_{2,℘}) ⟶ H_{f}^{1}(V_{℘}) ⟶ 0
↓  ↓  ∥
0 ⟶ B ⊗ H^{1}(Q_{p}(1)) ⟶ π^{−1}(H_{f}^{1}(V_{℘})) ⟶ H_{f}^{1}(V_{℘}) ⟶ 0
↓^{1⊗ord_{℘}}  ↓^{ϕ}
1 ⊗ l_{℘}  B  =  B
↓  ↓
B  0  0

equipped with a splitting w of the first row by 3.3. We have again the extension class [E_{℘}] : A ⟶ H^{1}(E_{2,℘}). As E_{1,℘} is crystalline, π(Im([E_{℘}])) lies in H_{f}^{1}(V_{℘}) by 1.25, hence

[E_{℘}] ∈ Hom(A, π^{−1}(H_{f}^{1}(V_{℘}))).

We distinguish two cases:

(a) l_{℘} is unramified: we have 1 ⊗ l_{℘} = a · (1 ⊗ ord_{℘}) for some a ∈ K_{℘}. We then define

h_{℘,E} = −a · (ϕ ∘ [E_{℘}]) ∈ Hom(A, B).

Note that this element does not depend on the splitting w.

(b) l_{℘} is ramified: this means that

l_{℘}(H^{1}(Q_{p}(1))) = l_{℘}(H_{f}^{1}(Q_{p}(1))) = Q_{p},

whence a map

σ : \frac{π^{−1}(H_{f}^{1}(V_{℘}))}{B ⊗ Ker(l_{℘})} ⟶^{∼} \frac{H_{f}^{1}(E_{2,℘})}{B ⊗ Ker(l_{℘,f})} ⟶^{(1⊗l_{℘,f})∘w}B,

where l_{℘,f} is the restriction of l_{℘} to H_{f}^{1}(Q_{p}(1)) and w is the splitting of the first row in the diagram. We then define

h_{℘,E} = −σ ∘ [E_{℘}] ∈ Hom(A, B).

It is easily checked that this definition is compatible with 4.7 for crystalline E_{℘}. Note that for each v, the local height pairing h_{v,E} depends only on the local representation E_{v}, considered as a mixed extension of (e_{1})_{v} and (e_{2})_{v}. It makes sense, therefore, to speak about the local height pairing associated to E_{v}, l_{v} and (if v|p) r_{v}.

4.9 We shall now consider in more detail the local case, i.e. we shall work with Galois representations over K_{℘}. Suppose we are given two extensions of p-adic Galois representations over K_{℘}

\begin{aligned}e_{1} & : 0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0 \\ e_{2} & : 0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0. \end{aligned}

Let E be a crystalline mixed extension of e_{1}, e_{2}. An argument similar to that in the proof of 4.4.2 shows that the set of all such E is a principal homogeneous space of

Ext_{K_{℘},f}^{1}(A(0), B(1)) = Hom(A, B ⊗ K_{℘}(1)).

As in 3.2-3, there is a canonical f-equivariant decomposition

D_{cris}(E) = D_{cris}(V) ⊕ (A ⊗ K_{℘}^{0}) ⊕ (B ⊗ K_{℘}^{0}(1))

and corresponding decompositions

DR(E) = DR(V) ⊕ (A ⊗ K_{℘}) ⊕ (B ⊗ K_{℘}(1))
D_{cris}(E_{1}) = D_{cris}(V) ⊕ (A ⊗ K_{℘}^{0}) ⊕  0
D_{cris}(E_{2}) = D_{cris}(V) ⊕  0  ⊕ (B ⊗ K_{℘}^{0}(1))

and 3.2-3 give us formulas for F^{0}DR(E_{1}), F^{0}DR(E_{2}) in terms of the maps α : A ⊗ K_{℘} ⟶ DR(V)/F^{0}, β : F^{0}DR(V) ⟶ B ⊗ K_{℘}(1), μ : DR(V) ⟶ DR(V)/F^{0}. The projection on the first two factors induces an isomorphism F^{0}DR(E) ⟶^{∼} F^{0}DR(E_{1}), hence

F^{0}DR(E) = {(x, a, ν(x, a)) | μ(x) = α(a)}

for some K_{℘}-linear map ν : F^{0}DR(E_{1}) ⟶ B ⊗ K_{℘}(1). Since F^{0}DR(E_{2}) is the kernel of the projection F^{0}DR(E) ⟶ A ⊗ K_{℘}, the restriction of ν to F^{0}DR(V) must be equal to −β. If we let E vary, the set of possible ν forms a principal homogeneous space of

Hom(A, B ⊗ K_{℘}(1)) = Ext_{K_{℘},f}^{1}(A(0), B(1)),

in agreement with the local version of 4.4.2 alluded to before. Consider now the exact sequence

0 ⟶ B ⊗ K_{℘}(1) ⟶ DR(E_{2})/F^{0}⟶^{π}DR(V)/F^{0} ⟶ 0.

The class of the extension E is a map e = [E] : A ⟶ H_{f}^{1}(K_{℘}, E_{2}) = DR(E_{2})/F^{0} such that π∘e(a) = α(a⊗1).

Let DR(V) ⟶^{⟵^{w}} DR(V)/F^{0} be a splitting of μ. According to 3.3, it defines a splitting DR(E_{2})/F^{0} ⟶^{⟵^{r}} DR(V)/F^{0} given by the formula

r(x) = (w(x), 0, 0) mod F^{0}DR(E_{2}).

Write h = −(1−rπ)∘e : A ⟶ B⊗K_{℘}(1). Then, by definition of the local height pairing in 4.8, h_{℘,E} = l_{℘}∘h. Let us compute h explicitly: for a ∈ A, choose x ∈ DR(V) with μ(x) = α(a ⊗ 1). Then (x, a ⊗ 1, ν(x, a)) ∈ F^{0}DR(E). By 1.20-1,

e(a) ≡ (0, −a ⊗ 1, 0) ≡ (x, 0, ν(x, a)) mod F^{0}DR(E).

Taking x = w ∘ α(a), we get

h(a) = (r ∘ α)(a ⊗ 1) − e(a) = −ν(w ∘ α(a ⊗ 1), a) ∈ B ⊗ K_{℘}(1)

Similarly, the dual mixed extension E^{∗}(1) together with a splitting w′ of DR(V^{∗}(1))⟶^{μ′}DR(V^{∗}(1))/F^{0} defines a map

h′ = −(1 − r′π′) ∘ e′ : B^{∗} ⟶ A^{∗} ⊗ K_{℘}(1).

Let x ∈ DR(V), x′ ∈ DR(V^{∗}(1)), a ∈ A ⊗ K_{℘}, b′ ∈ B^{∗} ⊗ K_{℘} with μ(x) = α(a), μ′(x′) = β′(b′). Since F^{0}DR(E) ⊥ F^{0}DR(E^{∗}(1)), we have

⟨x, x′⟩ + ⟨a, ν′(x′, b′)⟩ + ⟨ν(x, a), b′⟩ = 0.

In particular, for x = w ∘ α(a ⊗ 1), x′ = w′ ∘ β′(b′ ⊗ 1) (a ∈ A, b′ ∈ B^{∗}), we obtain

⟨w ∘ α(a ⊗ 1), w′ ∘ β′(b′ ⊗ 1)⟩ = ⟨a, h′(b′)⟩ + ⟨h(a), b′⟩ ∈ K_{℘}(1).

If the splitting w′ is dual to the projection u = 1 − w ∘ μ, then

⟨w(a), w′(b′)⟩ = ⟨u ∘ w(a), b′) = 0,

hence h′ = −h^{∗}. Applying l_{℘} to both sides, we get

h_{℘,E^{∗}(1)} = −(h_{℘,E})^{∗}.

The formula for h_{℘,E} we have obtained can be summarized by the following commutative diagram, in which both squares are cartesian:

A
↓
F^{0}DR(E_{1,℘}) ⟶^{⟵^{w′}} A ⊗ K_{℘}
ν  ↓  ↓^{α}
0 ⟶ F^{0}DR(V_{℘}) ⟶  DR(V_{℘})  ⟶^{⟵^{w}} DR(V_{℘})/F^{0} ⟶ 0
↓^{β}  ↓
B ⊗ K_{℘}(1) ⟶ DR(E_{2,℘})/F^{0}
↓^{−l_{℘}}
B  ,

where w′ is induced by w and ν controls F^{0}DR(E_{℘}).

4.10 Proposition. Under the hypotheses of 4.2, fix a mixed extension E of e_{1},e_{2} and a non-archimedean place v of K. Then

(1) If E′_{v} is a translation of E_{v} by

x_{v} ∈ Ext_{K_{v}}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H^{1}(K_{v}, Q_{p}(1)),

then

h_{v,E′} = h_{v,E} − (1 ⊗ l_{v})(x_{v}).

(2) If α : A ⟶ A′, β : B′ ⟶ B are homomorphisms and E′ is obtained from E by pushout and pullback

via α, β, then

h_{v,E} = β ∘ h_{v,E′} ∘ α ∈ Hom(A, B).

(3) h_{v,E^{∗}(1)} = −(h_{v,E})^{∗} (with the dual splittings described in 4.5)

Proof. (1) By [Bo, X.7.6.5. Cor.1.b], the extension classes satisfy [E′_{v}] − [E_{v}] = x_{v}, whence the claim.

(2) Follows immediately from the definitions of h_{E,v}.

(3) Supose first that v|̸p. Then we are reduced to prove that for an extension

0 ⟶ Q_{p}(1) ⟶ N ⟶ Q_{p}(0) ⟶ 0,

the class [N^{∗}(1)] of the dual extension is equal to −[N] ∈ H^{1}(K_{v}, Q_{p}(1)). This follows from the relations between extension classes and induced boundary maps [Bo, X.7.6.5. Cor.1]. For v = ℘|p, we first note that the statement is invariant under translations by elements in Ext_{K_{v}}^{1}(A(0), B(1)) (by (1) and the claim used in the proof of the case v|̸p). According to 4.4.2 we can, therefore, assume that E_{v} is crystalline. In this case the statement has been proved in 4.9.

4.11 Theorem. Under the assumptions of 4.2, choose a mixed extension E of e_{1},e_{2}. Then

(1) The sum over all non-archimedean places

h := ∑_{v} h_{v,E} ∈ Hom(A, B)

is convergent and does not depend on the choice of E. If E is unramified outside a finite set of places, then the sum is finite.

(2) Take for e_{1},e_{2} the universal extensions

0 ⟶ V ⟶ E_{1} ⟶ H_{f}^{1}(K, V)(0) ⟶ 0
0 ⟶ H_{f}^{1}(K, V^{∗}(1))^{∗}(1) ⟶ E_{2} ⟶ V ⟶ 0.

Then the pairing

h_{V} = ∑_{v} h_{v,E} : H_{f}^{1}(K, V) × H_{f}^{1}(K, V^{∗}(1)) ⟶ Q_{p}

coincides with the pairing constructed in sec. 2.

(3) This pairing is functorial with respect to isomorphisms: if j : V ⟶^{∼} W is one, then

h_{W} (j(x), y) = h_{V} (x, ĵ(y)), x ∈ H_{f}^{1}(K, V), y ∈ H_{f}^{1}(K, W^{∗}(1))

with ĵ being induced by j^{∗}(1) and splittings compatible via j.

(4) Suppose there exists a skew-symmetric non-degenerate pairing of p-adic Galois representations

[ , ] : V × V ⟶ Q_{p}(1).

Define an isomorphism j : V ⟶^{∼} V^{∗}(1) by [v_{1}, v_{2}] = (j(v_{1}))(v_{2}). The induced pairing

h : H_{f}^{1}(K, V) × H_{f}^{1}(K, V) ⟶ Q_{p}

defined by h(x, y) := h_{V} (x, j(y)) is symmetric, provided the following condition is satisfied at all ℘|p: the image of DR(V_{℘})/F^{0} under the splitting w_{℘} : DR(V_{℘})/F^{0} ⟶ DR(V_{℘}) is isotropic (in fact, Lagrangean) with respect to the symplectic form DR(V_{℘}) × DR(V_{℘}) ⟶ K_{℘}(1) induced by [ , ].

Proof. (1) According to 4.4, E is equal to some translate of E′ by x ∈ H^{1}(K, Q_{p}(1)), where E′ is essentially unramified at all v|̸p and crystalline at all ℘|p. The claim follows from the fact that the sum

l = ∑_{v} l_{v} : K^{∗} ˆ⊗Q_{p} ⟶ Q_{p}

is convergent to zero, which is obvious, since l : A_{K}^{∗} ⟶ Q_{p} extends uniquely to a continuous map l′ : A_{K}^{∗} ˆ⊗Q_{p} ⟶ Q_{p} trivial on K^{∗} ˆ⊗Q_{p}.

(2) We can perform the construction of sec. 1 also in the universal situation, i.e. considering

0 ⟶ V ⟶ E_{1} ⟶ H_{f}^{1}(K, V)(0) ⟶ 0

instead of an extension of Q_{p}(0). The diagram thus obtained is

0 ⟶ H_{f}^{1}(G_{S}, A^{∗}(1)) ⟶ H_{f}^{1}(G_{S}, E_{1}^{∗}(1)) ⟶^{⟵^{δ}}  B^{∗}  ⟶ 0
↓  ↓^{⊕ρ_{℘}}  ↓
0 ⟶ ⨁_{℘|p} H_{f}^{1}(K_{℘}, A^{∗}(1)) ⟶ ⨁_{℘|p} H_{f}^{1}(K_{℘}, E_{1}^{∗}(1)) ⟶ ⨁_{℘|p} H_{f}^{1}(K_{℘}, V^{∗}(1)) ⟶ 0
↓^{∑ l_{℘}⊗1}
A^{∗}

(with A = H_{f}^{1}(K, V), B^{∗} = H_{f}^{1}(G_{S}, V^{∗}(1))). Any splitting δ is an element of

Hom(B^{∗}, H_{f}^{1}(G_{S}, E_{1}^{∗}(1))) = Ext_{G_{S}}^{1}(B^{∗}, E_{1}^{∗}(1)),

hence defines an extension 0 ⟶ E_{1}^{∗}(1) ⟶ E^{∗}(1) ⟶ B^{∗} ⟶ 0, which makes E^{∗}(1) a mixed extension of the universal extensions

0 ⟶ V^{∗}(1) ⟶ E_{2}^{∗}(1) ⟶ B^{∗}(0) ⟶ 0
0 ⟶ A^{∗}(1) ⟶ E_{1}^{∗}(1) ⟶ V^{∗}(1) ⟶ 0,

essentially unramified at all v|̸p and crystalline at all ℘|p. The given splittings of Hodge filtrations define splittings

H_{f}^{1}(K_{℘}, A^{∗}(1)) ⟶^{⟵^{w℘}} H_{f}^{1}(K_{℘}, E_{1}^{∗}(1))

and the global height pairing of 2.2 is defined as

h_{global} = ∑_{℘}(l_{℘} ⊗ 1) ∘ w_{℘} ∘ ρ_{℘} ∘ δ ∈ Hom(B^{∗}, A^{∗}) = Hom(A, B).

At the same time,

\begin{aligned}h_{v,E^{∗}(1)} & = 0 (v|̸p) \\ h_{℘,E^{∗}(1)} & = −(l_{℘} ⊗ 1) ∘ w_{℘} ∘ ρ_{℘} ∘ δ (℘|p), \end{aligned}

which proves that

h_{global} = − ∑ h_{v,E^{∗}(1)} = ∑ h_{v,E}

by 4.10.2.

(3) is obvious.

(4) The condition on w_{℘} simply means that j carries w_{℘} into the dual splitting D(w_{℘}). Consequently, using ĵ = −j and (3), we have

h(x, y) = h_{V} (x, j(y)) = −h_{V} (x, ĵ(y)) = h_{V^{∗}(1)}(j(x), y) = h_{V} (y, j(x)) = h(y, x).

4.12 One is tempted to speculate about a possible analytic definition of the local pairings at those primes ℘|p where l_{℘} ramifies. Perhaps there is a p-adic Arakelov theory on a regular model X of X over the ring of integers O_{K} in K, i.e. an intersection theory for “enhanced algebraic cycles” on X. We have no candidate for such an object, but the corresponding enhanced Chow group ĈH^{j}(X) (or, rather, its subgroup of homologically trivial enhanced cycles) could fit into an exact sequence

H_{M}^{2j−1}(X, Q_{p}(j)) ⟶ H_{D}^{2j−1}(X, Q_{p}(j)) ⟶ ĈH^{j}(X)_{0} ⊗ Q_{p} ⟶ CH^{j}(X)_{0} ⊗ Q_{p} ⟶ 0

with p-adic Deligne cohomology equal to

H_{D}^{2j−1}(X, Q_{p}(j)) = ⨁_{℘|p} H_{f}^{1}(K_{℘}, H_{et}^{2j−2}(X ⊗ \overline{K})(j))

and “motivic” cohomology H_{M}^{2j−1}(X, Q_{p}(j)) being the j-th graded piece of Im(K′_{1}(X)⊗Q_{p} ⟶ K_{1}(X)⊗Q_{p}) with respect to the Adams filtration.

5. Geometric p-adic height pairings

5.1 Let K be a finite extension of Q and X_{/K} a proper smooth K-scheme, equidimensional of dimension d. Denote by CH^{i}(X_{/K}) the group of algebraic cycles of codimension i on X defined over K, modulo rational equivalence, and by CH^{i}(X_{/K})_{0} its subgroup of homologically trivial cycles (say, in H_{B}^{2i}(X(C), Z(i)) for some embedding K ↪ C). Write \overline{X} for X ⊗ \overline{K}. One can define étale version of the Abel-Jacobi map

Φ_{i} : CH^{i}(X_{/K})_{0} ⊗ Q_{p} ⟶ H^{1}(K, V)

for V = H_{et}^{2i−1}(\overline{X}, Q_{p}(i)) as follows: If Z is a homologically trivial cycle defined over K of codimension i, then the relative cohomology sequence

0 ⟶ H_{et}^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X} − \overline{Z}, Q_{p}(i)) ⟶ H_{|\overline{Z}|}^{2i}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i}(\overline{X}, Q_{p}(i))
∥  ↑ ∪  ↑ ∪  ↑
0 ⟶  V  ⟶  E  ⟶  Q_{p} · cl(\overline{Z})  ⟶  0

defines by pullback an extension of p-adic representations of G(\overline{K}/K)

0 ⟶ V ⟶ E ⟶ Q_{p}(0) ⟶ 0,

whose class in H^{1}(K, V) is Φ_{i}(Z). See [Ja 2,3] for further details.

5.2 Let X_{/O} be a proper flat model of X over the ring of integers of K. Let S_{0} be the set of primes of bad reduction of X. We assume that S_{0} ∩ {℘|p} = ∅, i.e. that X has good reduction at all primes above p. Fix an integer i, 1 ≤ i ≤ d and put V = H_{et}^{2i−1}(\overline{X}, Q_{p}(i)). Then, by Poincaré duality, V^{∗}(1) = H_{et}^{2j−1}(\overline{X}, Q_{p}(j)), where i + j = d + 1.

5.3 Consider now the conditions (A)–(D) of 2.1.2 for V: (A) follows from smooth and proper base change theorems for étale cohomology (B) follows for v ∉ S_{0} from Weil’s conjectures proved by Deligne (C) is proved in [Fa] (D) is proved in [Ka–Me] for smooth projective varieties as a consequence of Weil’s conjectures (B) for v ∈ S_{0} follows from the purity conjecture for the monodromy filtrations on V and V^{∗}(1) (cf.[Ja 1]), which has been proved so far only for certain classes of varieties; these include e.g. abelian varieties and some surfaces (see [Ra–Zi]). It is also known for motives corresponding to elliptic modular forms; they arise as factors of cohomology of non-singular compactifications of fibre products of the universal elliptic curve over a modular curve (see 8.3 for further details).

5.4 Proposition. Assuming that the cohomology groups V, V^{∗}(1) satisfy (B),(D) (in particular if X is projective and the monodromy filtration on V, V^{∗}(1) is pure for each place v of bad reduction), then the Abel-Jacobi maps Φ_{i}, Φ_{j} factor through

\begin{matrix}Φ_{i} : CH^{i}(X_{/K})_{0} ⊗ Q_{p} & ⟶ H_{f}^{1}(K, V) \\ Φ_{j} : CH^{j}(X_{/K})_{0} ⊗ Q_{p} & ⟶ H_{f}^{1}(K, V^{∗}(1)). \end{matrix}

Proof. Let ℘|p. According to [Fa], V_{℘} and V^{∗}(1)_{℘} both satisfy (C). The de Rham conjecture for open varieties 1.11, also proved in [Fa], implies that the representation E defined in 5.1 is de Rham at ℘, hence the ℘- localization of Im(Φ_{i}) lies in H_{g}^{1}(K_{℘}, V). We know that (C),(D) imply D_{cris}(V_{℘})^{f=1} = D_{cris}(V^{∗}(1)_{℘})^{f=1} = 0 and this means that H_{e}^{1} = H_{f}^{1} = H_{g}^{1} for both V_{℘} and V^{∗}(1)_{℘}, by 1.14, 1.16. Thank to (B), H^{1}(K_{v}, V) = 0 for all v|̸p, which means that there is no condition at these primes and Im(Φ_{i}) lies indeed in H_{f}^{1}(K, V). Replacing V by V^{∗}(1), we get the statement for Φ_{j}.

5.5 Theorem. Under the same assumptions as in 5.4, the choice of a non-trivial continuous homomorphism l : A_{K}^{∗}/K^{∗} ⟶ Q_{p} and of Q_{p}-linear splittings of Hodge filtrations F^{i}H_{dR}^{2i−1}(X_{/K_{℘}}) ⟶^{⟵} H_{dR}^{2i−1}(X_{/K_{℘}}) for primes ℘|p ramified with respect to l defines a bilinear pairing

CH^{i}(X_{/K})_{0} ⊗ CH^{j}(X_{/K})_{0} ⟶ Q_{p}

which factors through Abel-Jacobi maps. If i = j = (d + 1)/2 and the condition of 4.11.4 is satisfied at all ℘|p, then the pairing is symmetric.

Proof. Combine 4.11 with 5.4.

5.6 Mixed extensions arise in the geometric situation as follows (cf. [Sc 2, 7.7]): Let A ⊆ H_{f}^{1}(K, V), B^{∗} ⊆ H_{f}^{1}(K, V^{∗}(1)) be Q_{p}-vector subspaces contained in the Im(Φ_{i}) resp. Im(Φ_{j}) (according to general conjectures, p-adic Abel-Jacobi maps should be isomorphisms on H_{f}^{1} in our situation). By the moving lemma, there are disjoint closed subschemes Y, Z of X (of codimensions i resp. j) such that any element of A (resp. B^{∗}) can be represented by a cycle with support in Y (resp. Z). Write Z_{Y}^{i} (X)^{0} for the group of cycles of codimension i with coefficients in Q_{p} which are supported on Y and homologically trivial in H_{et}^{2i}(\overline{X}, Q_{p}(i)), and likewise for Z. Relative cohomology sequences

0 ⟶ H_{et}^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X} − \overline{Y} , Q_{p}(i)) ⟶ H_{|\overline{Y} |}^{2i}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i}(\overline{X}, Q_{p}(i))

0 ⟶ H_{et}^{2i−1}(\overline{X}rel \overline{Z}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X} − \overline{Y} rel \overline{Z}, Q_{p}(i)) ⟶ H_{|\overline{Y} |}^{2i}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i}(\overline{X}rel \overline{Z}, Q_{p}(i))

H_{et}^{2i−2}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i−2}(\overline{Z}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X}rel \overline{Z}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X}, Q_{p}(i)) ⟶ 0

H_{et}^{2i−2}(\overline{X} − \overline{Y} , Q_{p}(i)) ⟶ H_{et}^{2i−2}(\overline{Z}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X} − \overline{Y} rel \overline{Z}, Q_{p}(i)) ⟶ H_{et}^{2i−1}(\overline{X} − \overline{Y} , Q_{p}(i)) ⟶ 0

then define, via maps

Zi ; \overline{Y} (\overline{X})^{0} ⟶^{∼} Ker(H_{|\overline{Y} |}^{2i}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i}(\overline{X}, Q_{p}(i)))

Coker(H_{et}^{2i−2}(\overline{X}, Q_{p}(i)) ⟶ H_{et}^{2i−2}(\overline{Z}, Q_{p}(i))) ⟶^{∼} (Zj ; \overline{Z}(\overline{X})^{0})^{∗}(1)

(given by the cycle class and (−1)× the trace map respectively), a mixed extension

E′ = H_{et}^{2i−1}(\overline{X} − \overline{Y} rel \overline{Z}, Q_{p}(i))

of

0 ⟶ V ⟶ E′_{1} ⟶ Zi ; \overline{Y} (\overline{X})^{0}(0) ⟶ 0

0 ⟶ (Zj ; \overline{Z}(\overline{X})^{0})^{∗}(1) ⟶ E′_{2} ⟶ V ⟶ 0

with

\begin{aligned}E′_{1} & = H_{et}^{2i−1}(\overline{X} − \overline{Y} , Q_{p}(i)) \\ E′_{2} & = H_{et}^{2i−1}(\overline{X}rel \overline{Z}, Q_{p}(i)). \end{aligned}

Choose splittings

\begin{aligned}s_{1} & : A ↪ Z_{Y}^{i} (X)^{0} ⊆ Zi ; \overline{Y} (\overline{X})^{0} \\ s_{2} & : B^{∗} ↪ Z_{Z}^{j}(X)^{0} ⊆ Zj ; \overline{Z}(\overline{X})^{0}. \end{aligned}

Applying to E′ pullback by s_{1} and pushout by s_{2}^{∗}(1), we obtain a mixed extension E of

\begin{matrix}0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0 \\ 0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0. \end{matrix}

This mixed extension is unramified outside a finite set of primes (depending on Y, Z) by the generic base change theorem ([SGA 4\frac{1}{2}, Théorème de finitude]). The extension class of E_{1} resp. E_{2}^{∗}(1) in Hom(A, H^{1}(K, V)) resp. Hom(B^{∗}, H^{1}(K, V^{∗}(1)) is the inclusion map, by [Sc 2,7.5] (this explains why the inverse of the trace map has been used in the definition of E′).

6. Universal norms

6.0 In this section we investigate the behavior of universal norms from a Z_{p}-extension for H_{f}^{1} of certain local Galois representations. In some favorable cases, universal norms will define splittings of local cohomology sequences, giving rise to local height pairings (as in the classical approach of Schneider [Sch 2]).

6.1 Suppose that we are in the local situation of 1.2. Let X be a de Rham represantation of G(\overline{K}/K) satisfying F^{0}DR(X) = 0. Choose a G(\overline{K}/K)-stable Z_{p}-lattice T_{X} ⊂ X. Define, for any extension L/K,

X_{n}(L) := H^{1}(L, T_{X} ⊗ Z/p^{n}Z), Y_{n}(L) := H^{0}(L, T_{X} ⊗ Z/p^{n}Z),

X(L) := \underleftarrow{lim}_{n} X_{n}(L), Y (L) := \underleftarrow{lim}_{n} Y_{n}(L).

Of course, Y (L) = H^{0}(L, T_{X}) for all L and X(L) = H^{1}(L, T_{X}) for L of finite degree over K.

6.2 Let K_{∞}/K be a Z_{p}-extension, i.e K_{∞} = ⋃ K_{n} with G(K_{n}/K) = Z/p^{n}Z. Fix an isomorphism l : G(K_{∞}/K) = Γ ⟶^{∼} Z_{p} (i.e. a topological generator γ = l^{−1}(1) of Γ). Via reciprocity map, l defines an “algebraic logarithm” (to be denoted by the same letter) l : K^{∗} ⟶ Z_{p} ↪ Q_{p}. Write

N_{∞}X(K) := ⋂_{n=1}^{∞} cor_{K_{n}/K}X(K_{n}) ⊆ X(K)

for the group of universal norms from K_{∞}/K. In view of 6.3 below, it is also equal to

⋂_{n=1}^{∞} N_{n}X(K_{n}) ⊆ \underrightarrow{lim}_{n} X(K_{n})^{Γ} = X(K),

where N_{n} = res_{K_{n}/K} ∘ cor_{K_{n}/K} is the norm from K_{n}/K. Define N_{∞}H^{1}(K, X) := (N_{∞}X(K)) ⊗ Q – this is independent of the choice of T_{X}.

6.3 Lemma. Let L be an extension of K. Then

(1) If L/K is finite, then H^{0}(L_{ur}, X) = 0.

(2) If L/K is Galois extension and H^{0}(L, X) = 0, then the restriction map X(K) ⟶ X(L)^{G(L/K)} is an isomorphism. For L = K_{∞}, there is an exact sequence

0 ⟶ Y (K_{∞})_{Γ} ⟶ X(K) ⟶ X(K_{∞})^{Γ} ⟶ 0,

in which the first term is finite.

(3) There is a canonical isomorphism

X(K)/N_{∞}X(K) ⟶^{∼} H^{1}(Γ, H^{0}(K_{∞}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}))^{∨}

(where M^{∨} = Hom_{cont}(M, Q_{p}/Z_{p}) for any Z_{p}-module M).

(4) If H^{0}(K, X^{∗}(1)) = 0, then X(K)/N_{∞}X(K) is finite of order equal to

♯(X(K)/N_{∞}X(K)) = ♯H^{0}(K, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p})/♯(Div(H^{0}(K_{∞}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p})))^{Γ}.

Proof. (1) W = H^{0}(L_{ur}, X) is an unramified subrepresentation of X (over L). If nontrivial, it would contribute by F^{0}DR(W_{L}) ≠ 0 to F^{0}DR(X_{L}) = 0.

(2) In the inf–res exact sequence

0 ⟶ H^{1}(L/K, Y_{n}(L)) ⟶ X_{n}(K) ⟶ X_{n}(L)^{G(L/K)} ⟶ H^{2}(L/K, Y_{n}(L))

the groups Y_{n}(L) form a ML-zero system if H^{0}(L, X) = 0 (see [Ja 2] for terminology concerning projective systems). Passing to the projective limit \underleftarrow{lim}_{n} (which is an exact functor on projective systems of finite abelian

groups) we get the isomorphism X(K) ⟶^{∼} X(L)^{G(L/K)}. If L = K_{∞}, then H^{2}(L/K, Y_{n}(K_{∞})) = 0,

H^{1}(L/K, Y_{n}(K_{∞})) ⟶^{∼} Y_{n}(K_{∞})_{Γ}

and we again conclude by passing to the projective limit (the finiteness of Y (K_{∞})_{Γ} follows from Y (K_{∞})^{Γ} = 0).

(3) By local duality, the corestriction cor_{K_{n}/K} : X(K_{n}) ⟶ X(K) is dual to res_{K_{n}/K} : H^{1}(K, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}) ⟶ H^{1}(K_{n}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}), hence

Coker(cor_{K_{n}/K}) ⟶^{∼} Ker(res_{K_{n}/K})^{∨} = H^{1}(K_{n}/K, (T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p})^{Γ_{n}})^{∨}.

Passing to the limit, we get the assertion.

(4) Put M = H^{0}(K_{∞}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}). By assumption, M^{Γ} is finite, hence Div(M)_{Γ} vanishes, being simultaneously divisible and finite (as M has finite corank). Using the multiplicativity of the Herbrand quotient and finiteness of M/Div(M),

♯X(K)/N_{∞}X(K) = ♯H^{1}(Γ, M) = ♯H^{0}(Γ, M)/♯H^{0}(Γ, Div(M))

as claimed.

6.4 Put A := H^{0}(K, X^{∗}(1))^{∗}. There is a canonical surjective morphism of representations X ⟶ A(1). Let T_{A}(1) be the image of T_{X}. The isomorphism of 6.3.3 being canonical, we have a commutative diagram

X(K)/N_{∞}X(K)  ⟶^{∼} (H^{0}(K_{∞}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p})_{Γ})^{∨}
↓  ↓
H^{1}(K, T_{A}(1))/N_{∞}H^{1}(K, T_{A}(1)) ⟶^{∼}  T_{A},

in which the bottom arrow is induced by l ⊗ 1 : K^{∗} ˆ⊗T_{A} ⟶ T_{A}. Tensoring with Q, we get an isomorphism

H^{1}(K, X)/N_{∞}H^{1}(K, X) ⟶^{∼} (H^{0}(K_{∞}, X^{∗}(1))_{Γ})^{∗}.

6.5 Lemma. Suppose that one of the following two conditions is satisfied:

(a) K_{∞}/K is ramified

(b) K_{∞}/K is unramified and the canonical map D_{st}(X^{∗}(1))^{f=1} ⟶ D_{st}(X^{∗}(1))/(f − 1) is an isomorphism. Then the canonical map H^{0}(K_{∞}, X^{∗}(1))^{Γ} ⟶ H^{0}(K_{∞}, X^{∗}(1))_{Γ} is an isomorphism.

Proof. (a) Suppose first that K_{∞}/K is totally ramified. Set E = H^{0}(K_{∞}, X^{∗}(1)) and W = X^{∗}(1)^{I}. Consider the exact sequence

0 ⟶ H^{0}(K_{∞}, W) ⟶ E = H^{0}(K_{∞}, X^{∗}(1)) ⟶ H^{0}(K_{∞}, X^{∗}(1)/W).

By 1.33, the third group has no Γ-invariants, hence

A^{∗} = E^{Γ} = H^{0}(K_{∞}, W)^{Γ} = H^{0}(K, W) = H^{0}(K_{∞}, W)

(the last equality follows from the fact that K_{∞}/K is totally ramified). This implies that (E/E^{Γ})^{Γ} ⊆ H^{0}(K_{∞}, X^{∗}(1)/W)^{Γ} = 0, hence E^{Γ} ⟶^{∼} E_{Γ} as claimed.

In general, K_{∞}/K_{m} is totally ramified for some m < ∞, and by the previous argument E^{Γ_{m}} ⟶^{∼} E_{Γ_{m}}. These two spaces, as well as E, are semisimple as representations of the finite group Γ/Γ_{m}, which proves

that E^{Γ} ⟶^{∼} E_{Γ}.

(b) Put B := X^{∗}(1)^{I}. Then, by 1.28.3-4, D_{st}(X^{∗}(1))^{f=1} = B^{⟨ϕ_{K}⟩}, D_{st}(X^{∗}(1))/(f − 1) = B_{⟨ϕ_{K}⟩}. Since G(K_{ur}/K_{∞}) has (pro) order prime to p, its action on B factors through a finite group, say, H. We have then H^{0}(K_{∞}, X^{∗}(1))^{Γ} = B^{⟨ϕ_{K}⟩}, H^{0}(K_{∞}, X^{∗}(1))_{Γ} = (B^{H})_{Γ} ⟶^{∼} (B_{H})_{Γ} = B_{⟨ϕ_{K}⟩} by the same semisimplicity argument used in (a) above.

6.6 Theorem. Under the assumptions of 6.5, the homomorphism X ⟶ A(1) induces isomorphisms

H^{1}(K, X)/N_{∞}H^{1}(K, X) ⟶^{∼} H^{1}(K, A(1))/N_{∞}H^{1}(K, A(1)) ⟶^{∼}^{l⊗1} A.

If A = 0, then the order of X(K)/N_{∞}X(K) divides ♯H^{0}(K, T_{X}^{∗}(1)⊗Q_{p}/Z_{p}), with equality if H^{0}(K_{∞}, X^{∗}(1)) = 0.

Proof. Combine 6.4, 6.5 and 6.3.4.

6.7 Let V be a p-adic Galois representation over K. We say that V satisfies Pančiškin’s condition (cf. [Pe-Ri 3]), if it sits in an exact sequence of p-adic Galois representations (with X, Y de Rham)

0 ⟶ X ⟶ V ⟶ Y ⟶ 0

such that F^{0}DR(X) = DR(Y)/F^{0} = 0 (V is then also de Rham, by 1.28.1). Note that this exact sequence is uniquely determined by V: if X′, Y′ have the same property, then the induced map α : X ↪ V ⟶ Y′ satisfies DR(Y′) = F^{0}DR(Y′) = F^{0}DR(Coker(α)) ⊆ DR(Coker(α)), hence DR(Y′) = DR(Coker(α)), which implies that α = 0 and X ⊆ X′. By symmetry, X = X′. Choose a G(\overline{K}/K)-invariant Z_{p}-lattice T ⊂ V and put T_{X} = T ∩ X, T_{Y} = T/T_{X}. Basic example the reader should keep in mind is provided by an ordinary representation V, when X = F^{1}V, Y = V/F^{1}V.

6.8 Lemma. Suppose that V is a crystalline representation satisfying the Pančiškin condition and such that D_{cris}(V)^{f=1} = D_{cris}(V^{∗}(1))^{f=1} = 0. Then the following sequence is exact:

0 ⟶ H^{1}(K, T_{X}) ⟶ H_{f}^{1}(K, T) ⟶ H^{0}(K, T_{Y} ⊗ Q_{p}/Z_{p}) ⟶ H^{2}(K, T_{X})_{tors}

Proof. By 1.32.3 and 1.28.4, H_{f}^{1}(K, Y) = 0 and H_{f}^{1}(K, X) = H^{1}(K, X) ⟶^{∼} H_{f}^{1}(K, V). Consider the following commutative diagram with exact rows (the exactness of the bottom row follows from the argument used in 1.25):

H^{1}(K, T_{X}) ⟶ H^{1}(K, T) ⟶^{α} H^{1}(K, T_{Y}) ⟶^{δ} H^{2}(K, T_{X})
∥  ↑ ∪  ↑^{γ}
H_{f}^{1}(K, T_{X}) ⟶ H_{f}^{1}(K, T) ⟶^{β} H_{f}^{1}(K, T_{Y})

The map ϕ : H^{1}(K, T)/H_{f}^{1}(K, T) ⟶ H^{1}(K, T_{Y})/H_{f}^{1}(K, T_{Y}) induced by α has finite kernel, as H_{f}^{1}(K, T_{Y}) = H^{1}(K, T_{Y})_{tors} is finite. On the other hand, H^{1}(K, T)/H_{f}^{1}(K, T) is torsion free, hence Ker(ϕ) = 0. An easy diagram chase then shows that Ker(δγ) = Im(β), which proves the lemma, if we take into account isomorphisms

H_{f}^{1}(K, T_{Y}) = H^{1}(K, T_{Y})_{tors} ⟶^{∼} H^{0}(K, T_{Y} ⊗ Q_{p}/Z_{p})

(the last one follows from [Ta] and the fact that H^{0}(K, Y) = 0, by 1.28.3).

6.9 Theorem. Suppose that V is a crystalline representation satisfying Pančiškin’s condition, D_{cris}(V)^{f_{m}=1} = D_{cris}(V^{∗}(1))^{f_{m}=1} = 0 for all m < ∞ for which K_{∞}/K is unramified (where f_{m} = f^{p^{m}}) and, if K_{∞}/K is unramified, that X satisfies the condition 6.5.b. Then

(1) H_{f}^{1}(K, T)/N_{∞}H_{f}^{1}(K, T) is finite.

(2) If H^{0}(K_{∞}, X^{∗}(1)) = H^{0}(K_{∞}, Y) = 0, then there is an exact sequence

0 ⟶ H^{1}(K, T_{X})/N_{∞}H^{1}(K, T_{X}) ⟶ H_{f}^{1}(K, T)/N_{∞}H_{f}^{1}(K, T) ⟶ E ⟶ 0,

where E is the image of H_{f}^{1}(K, T) in H^{1}(K, T_{Y})_{tors} ⟶^{∼} H^{0}(K, T_{Y} ⊗ Q_{p}/Z_{p}).

(3) E contains H^{0}(K, T_{Y}^{I} ⊗ Q_{p}/Z_{p}).

Proof. (1) Under our assumptions Lemma 6.8 applies over each K_{n}. Snake lemma then gives an exact sequence

H^{1}(K, T_{X})/N_{∞}H^{1}(K, T_{X}) ⟶ H_{f}^{1}(K, T)/N_{∞}H_{f}^{1}(K, T) ⟶ E ⟶ 0,

where E is a subfactor of the finite group H^{0}(K, T_{Y} ⊗ Q_{p}/Z_{p}). According to 6.6, the first group is finite as well, since A = 0 by 1.28.3.

(2) Applying lemma 6.8 over K_{n}, we get exact sequences

0 ⟶ H^{1}(K_{n}, T_{X}) ⟶  H_{f}^{1}(K_{n}, T)  ⟶ E_{n} ⟶ 0
0 ⟶  E_{n}  ⟶ H^{0}(K_{n}, T_{Y} ⊗ Q_{p}/Z_{p}) ⟶ F_{n} ⟶ 0
0 ⟶  F_{n}  ⟶  H^{2}(K_{n}, T_{X})_{tors}

with some finite groups E_{n}, F_{n}. Write E, F for the corresponding objects over K. Snake lemma implies that E_{n} ⊆ E_{n+1}^{Γ_{n}} and E_{n}^{Γ}/E ⟶^{∼} Ker(F ⟶ F_{n}^{Γ}). As H^{0}(K_{∞}, Y) = 0, E_{∞} = \underrightarrow{lim} E_{n} is finite and the sequence of E_{n} stabilizes. The map F ⟶ F_{n}^{Γ} is induced by the restriction res_{K_{n}/K}, which is dual to cor_{K_{n}/K} : H^{0}(K_{n}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}) ⟶ H^{0}(K, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}). This map is trivial for sufficiently big n, as H^{0}(K_{∞}, T_{X}^{∗}(1) ⊗ Q_{p}/Z_{p}) is finite by our assumptions. For such n, therefore, E_{n}^{Γ}/E = F and E_{n}^{Γ} = H^{0}(K, T_{Y} ⊗ Q_{p}/Z_{p}). We know by 6.3.2 that H^{1}(K, T_{X}) ⟶^{∼} H^{1}(K_{n}, T_{X})^{Γ}. Snake lemma then gives an exact sequence

0 ⟶ H_{f}^{1}(K, T) ⟶ H_{f}^{1}(K_{n}, T)^{Γ} ⟶ Ker(E_{n}^{Γ}/E⟶^{α}H^{1}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X}))) ⟶ 0.

For big n, the composite map

F = E_{n}^{Γ}/E⟶^{α}H^{1}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X})) ↪ H^{2}(K, T_{X})

is the inclusion, hence Ker(α) = 0 and H_{f}^{1}(K, T) ⟶^{∼} H_{f}^{1}(K_{n}, T)^{Γ}. Set

M_{1} = \underrightarrow{lim} H^{1}(K_{n}, T_{X}), M_{2} = \underrightarrow{lim} H_{f}^{1}(K_{n}, T).

We now apply the trick of [Jo 2] to the exact sequence

0 ⟶ M_{1} ⟶ M_{2} ⟶ E_{∞} ⟶ 0,

which generalizes (4) of loc. cit. Its cohomology sequence becomes

0 ⟶ F ⟶ H^{1}(Γ, M_{1}) ⟶ H^{1}(Γ, M_{2}) ⟶ H^{1}(Γ, E_{∞}) ⟶ 0

(apply 6.6 and [Jo 2, 1.1.2]). All of the groups in this sequence are finite: this is trivially true for the first and the fourth group, and

H^{1}(Γ, M_{1}) = \underrightarrow{lim}_{n} H^{1}(Γ/Γ_{n}, M_{1}^{Γ_{n}}) = \underrightarrow{lim}_{n} H^{1}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X})) ⊆ H^{2}(K, T_{X})

is finite as well. Snake lemma gives an exact sequence

\frac{H^{1}(K, T_{X})}{cor_{K_{n}/K}(H^{1}(K_{n}, T_{X}))} ⟶ \frac{H_{f}^{1}(K, T)}{cor_{K_{n}/K}(H_{f}^{1}(K_{n}, T))} ⟶ \frac{E}{cor_{K_{n}/K}(E_{n})} ⟶ 0,

which is for big n nothing else than

Ĥ^{0}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X})) ⟶ Ĥ^{0}(Γ/Γ_{n}, H_{f}^{1}(K_{n}, T)) ⟶ E ⟶ 0.

Multiplicity of the Herbrand quotient then gives

\frac{♯Ĥ^{0}(Γ/Γ_{n}, H_{f}^{1}(K_{n}, T))}{♯Ĥ^{0}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X}))♯E} = \frac{♯H^{1}(Γ/Γ_{n}, H_{f}^{1}(K_{n}, T))}{♯H^{1}(Γ/Γ_{n}, H^{1}(K_{n}, T_{X}))♯E} = \frac{♯H^{1}(Γ, M_{1})}{♯H^{1}(Γ, M_{2})♯E} =

= \frac{♯E_{∞}^{Γ}}{♯E♯F} = 1,

hence the sequence above is exact even with 0 added to the left. Passing to the limit then concludes the proof.

(3) is proved in [Pe-Ri 3, 2.3.2].

6.10 If T = T_{p}(A) is the Tate module of an abelian variety A_{/K} with ordinary good reduction, then T_{X} = F^{1}T, T_{Y} = T/F^{1}T with T_{X}^{∗}(1), T_{Y} unramified. In the exact sequence

0 ⟶ H^{1}(K, F^{1}T) ⟶ H_{f}^{1}(K, T) ⟶ E ⟶ 0,

the first group is equal to the O_{K}-points of the formal group of A, the second is equal to A(K)ˆ⊗Z_{p} and E = H^{0}(K, (T/F^{1}T) ⊗ Q_{p}/Z_{p}) = Ã(k)_{p^{∞}} is the p-primary part of the points of the reduction Ã of A with values in the residue field k of O_{K}. As T^{∗}(1) is the Tate module of the dual abelian variety B, 6.9 gives the well-known formula (under the assumptions of 6.9.2)

♯A(K)/N_{∞}A(K) = p − part of ♯Ã(k)♯B̃(k) = p − part of (♯Ã(k))^{2}.

6.11 Theorem. Suppose V satisfies the assumptions of 6.9 and sits in an exact sequence 0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0 with crystalline E_{2}. Then

(1) The canonical map

B ⟵^{∼} H_{f}^{1}(K, B(1))/N_{∞} ⟶ H_{f}^{1}(K, E_{2})/N_{∞}

is an isomorphism, defining thus an extension l′ : H_{f}^{1}(K, E_{2}) ⟶ B of l : H_{f}^{1}(K, B(1)) ⟶ B.

(2) If 0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0 is an exact sequence with crystalline E_{1} and E a crystalline mixed extension of E_{1}, E_{2} (as Galois representations over K), then the local height pairing h_{E}^{norm} : A ⟶ B defined via l′ coincides with the height pairing h_{E}^{can} : A ⟶ B obtained from the splitting

DR(V)/F^{0} ⟵^{∼} DR(X) ↪ DR(V).

Proof. Using 1.32 and 1.28 applied to V^{∗}(1),

\begin{aligned}Hom_{G(\overline{K}/K)}(X, B(1)) & = B ⊗ H^{0}(K, X^{∗}(1)) = 0 \\ Ext_{K,f}^{1}(X, B(1)) & = B ⊗ H_{f}^{1}(K, X^{∗}(1)) = 0. \end{aligned}

This implies that the pullback of E_{2} via j : X ↪ V admits a unique splitting j^{∗}E_{2} = X ⊕ B(1) as a representation of G(\overline{K}/K). In view of the formulas in 3.3, this decomposition is compatible with the unique f-equivariant splitting D_{cris}(E_{2}) = D_{cris}(V) ⊕ D_{cris}(B(1)). The same is true over all K_{n}. By 1.32 and 1.28, H^{1}(X) ⟶^{∼} H_{f}^{1}(V), H_{f}^{1}(j^{∗}E_{2}) ⟶^{∼} H_{f}^{1}(E_{2}) over all K_{n}. According to 6.6, N_{∞}H^{1}(K, X) = H^{1}(K, X). This means that under the isomorphism

H_{f}^{1}(K, E_{2}) ⟵^{∼} H_{f}^{1}(K, j^{∗}E_{2}) = H^{1}(K, X) ⊕ H_{f}^{1}(K, B(1))

the group of universal norms N_{∞}H_{f}^{1}(K, E_{2}) corresponds to

N_{∞}H^{1}(K, X) ⊕ N_{∞}H_{f}^{1}(K, B(1)) = H^{1}(K, X) ⊕ Ker(1 ⊗ l)

(where 1 ⊗ l : H_{f}^{1}(K, B(1)) = B ˆ⊗O_{K}^{∗} ⟶ B), proving thus (1). Let w : H_{f}^{1}(K, V) ≃ DR(V)/F^{0} ⟶ DR(E_{2})/F^{0} ≃ H_{f}^{1}(K, E_{2}) be the splitting induced by the splitting DR(V)/F^{0} ⟵^{∼} DR(X) ↪ DR(V) (see 3.3). Denote by π : H_{f}^{1}(K, E_{2}) ⟶ H_{f}^{1}(K, V) the canonical projection. Then the map 1 − wπ is equal to the composition

H_{f}^{1}(K, E_{2}) ⟵^{∼} H_{f}^{1}(K, j^{∗}E_{2}) ⟶ H_{f}^{1}(K, j^{∗}E_{2})/H^{1}(K, X) = H_{f}^{1}(K, B(1)).

Consequently, l ∘ (1 − wπ) = l′, which proves (2).

7. Semistable representations

7.1 In this section we extend previous constructions to representations which are no longer crystalline, only semistable at all places over p. We also investigate the behavior of local universal norms for representations satisfying Pančiškin’s condition and define for them an “extended height”, following [Ma–Ta–Te].

7.2 Supose we are in the situation of 2.1, with the condition (C) replaced by (C’) V_{℘} is a semistable representation of G(\overline{K_{℘}}/K_{℘}) for all ℘|p. In particular, we assume that D_{cris}(V_{℘})^{f=1} = D_{cris}(V^{∗}(1)_{℘})^{f=1} = 0. Define for any p-adic representation W of G(\overline{K}/K)

H_{st}^{1}(K, W) = {x ∈ H^{1}(K, W) | x_{℘} ∈ H_{st}^{1}(K_{℘}, W) ∀℘|p}.

We have, as in 2.1, H^{1}(K_{v}, V) = H^{1}(K_{v}, V^{∗}(1)) = 0 for all v|̸p and H_{st}^{1}(K, −) = H_{f}^{1}(K, −) for both V, V^{∗}(1).

7.3 As in 4.4, H_{st}^{1}(K, V) = H_{f}^{1}(K, V) and H_{st}^{1}(K, V^{∗}(1)) = H_{f}^{1}(K, V^{∗}(1)) are orthogonal with respect to the cup product H^{1}(K, V) × H^{1}(K, V^{∗}(1)) ⟶ H^{2}(K, Q_{p}(1)). Consequently, given a pair of extensions of p-adic representations of G(\overline{K}/K)

\begin{matrix}e_{1} : & 0 & ⟶ & V & ⟶ & E_{1} & ⟶ & A(0) & ⟶ & 0 \\ e_{2} : & 0 & ⟶ & B(1) & ⟶ & E_{2} & ⟶ & V & ⟶ & 0 \end{matrix}

with E_{1,℘}, E_{2,℘} de Rham (hence semistable, by 1.27), the Yoneda composition e_{2} ∘ e_{1} = 0 vanishes. This implies that there exists a mixed extension E of e_{1}, e_{2} and that the set of such E is a principal homogeneous space under

Ext_{K,st}^{1}(A(0), B(1)) = Hom(A, B) ⊗ H^{1}(K, Q_{p}(1)).

The argument of 4.4.4 shows that E_{℘} is de Rham, hence semistable, for all ℘|p. As in 4.3, each E splits as a representation of G(\overline{K_{v}}/K_{v}) at v|̸p: E_{v} = V_{v} ⊕ N for some

0 ⟶ B(1)_{v} ⟶ N ⟶ A(0)_{v} ⟶ 0.

The set of E for which E is essentially unramified (i.e. N is unramified) at all v|̸p forms a pricipal homogeneous space under

Hom(A, B) ⊗ O_{K}[1/p]^{∗}.

7.4 We are now ready to define local height pairings, in analogy to 4.6-7: fix a continuous homomorphism l : A_{K}^{∗}/K^{∗} ⟶ Q_{p} and splittings of the Hodge filtration DR(V_{℘}) ⟶^{⟵} DR(V_{℘})/F^{0} at all ℘|p.

(i) local pairing at v|̸p: The definition of 4.6 works word by word, giving

h_{v,E} = −l_{v}([N]) ∈ Hom(A, B)

(ii) local pairing at ℘|p: The extension class of E_{℘} gives a map [E_{℘}] : A ⟶ H_{st}^{1}(K_{℘}, E_{2}) and the chosen splitting of the Hodge filtration determines, by 3.5, a splitting w_{℘} of

0 ⟶ H_{st}^{1}(K_{℘}, B(1)) ⟶^{⟵^{w℘}} H_{st}^{1}(K_{℘}, E_{2}) ⟶ H_{st}^{1}(K_{℘}, V) ⟶ 0.

We define

h_{℘,E} := −l_{℘} ∘ w_{℘} ∘ [E_{℘}] : A ⟶ B.

7.5 Fix ℘|p. As in 4.9, there is a unique decomposition

D_{st}(E_{℘}) = D_{st}(V_{℘}) ⊕ D_{st}(A(0)_{℘}) ⊕ D_{st}(B(1)_{℘})

which is f-equivariant and “almost N-equivariant”:

N(x, a, b) = (Nx, 0, γ_{0}(a)),

where γ_{0} : D_{st}(A(0)_{℘}) ⟶ D_{st}(B(0)_{℘}) is the K_{℘}^{0}-linear extension of some γ : A ⟶ B. We say that E is essentially crystalline at ℘ if γ = 0. If this is the case, then [E_{℘}] ∈ Hom(A, H_{f}^{1}(K_{℘}, E_{2})) by 1.25 and h_{℘,E} can be defined via splitting of the sequence

0 ⟶ H_{f}^{1}(K_{℘}, B(1)) ⟶ H_{f}^{1}(K_{℘}, E_{2}) ⟶ H_{f}^{1}(K_{℘}, V) ⟶ 0.

If we change E by x ∈ Hom(A, B) ⊗ H^{1}(K, Q_{p}(1)), γ changes by (1 ⊗ v_{℘})(x), where v_{℘} : \overline{K_{℘}}^{∗} ⟶ Q_{p} is the valuation used in the definition of B_{st} for K_{℘}. As

⨁_{℘|p} v_{℘} : K^{∗} ˆ⊗Q_{p} ⟶ ⨁_{℘|p} Q_{p}

is surjective, there exists E which is essentially crystalline at all ℘|p and essentially unramified at all v|̸p (E comes, therefore, from a “mixed motive over O_{K}”, in the terminology of A.J.Scholl [Sc 2]) and the set of such E is a principal homogeneous space under Hom(A, B) ⊗ O_{K}^{∗}.

7.6 In the situation of 7.5, the sum

h = ∑_{v} h_{v,E} ∈ Hom(A, B)

is convergent and does not depend on the choice of E. If E is essentially unramified at all v|̸p and essentially crystalline at all ℘|p, then only terms h_{℘,E} with ℘|p and ramified l_{℘} contribute. Similarly, other statements of 4.11 carry over to the present situation: the universal extensions give rise to

h_{V} : H_{f}^{1}(K, V) = H_{st}^{1}(K, V) ⟶ H_{st}^{1}(K, V^{∗}(1))^{∗} = H_{f}^{1}(K, V^{∗}(1))^{∗}

satisfying h_{V^{∗}(1)} = −(h_{V})^{∗} (indeed, the calculation of 4.9 is valid also for essentially crystalline E_{℘}). In the geometric situation of 5.1, the proof of 5.4 implies that the Abel-Jacobi maps factor through H_{f}^{1}(K, V) resp. H_{f}^{1}(K, V^{∗}(1)), once V = H_{et}^{2i−1}(\overline{X}, Q_{p}(i)) is known to satisfy (B),(C’) and (D). It is expected that examples of such V should be provided by representations corresponding to modular forms of level divisible by p. Despite great recent progress, none of the conditions (C’),(D) seems to have been verified for such V. On the other hand, in an ideal world, general cohomological conjectures (cf. [Fo–Il]) predict that for any proper and smooth X, V should satisfy all of the conditions (A),(B),(C’),(D), possibly after a finite extension of the base field K. If true, then the previous construction covers the most general geometric situation.

7.7 We shall now investigate the behavior of universal norms for H_{st}^{1}(K_{℘}, V). For notational convenience, we shall write K instead of K_{℘} throughout 7.7-11. Let K_{∞}/K be a Γ-extension. Fix an isomorphism Γ = G(K_{∞}/K) ⟶^{∼} Z_{p}. It induces, via reciprocity map, an algebraic logarithm l : H^{1}(K, Q_{p}(1)) = K^{∗} ˆ⊗Q_{p} ⟶ Q_{p}. Suppose that V is a semistable representation of G(\overline{K}/K) satisfying one of the conditions 6.5.a/b and D_{cris}(V)^{f_{m}=1} = D_{cris}(V^{∗}(1))^{f_{m}=1} = 0 whenever K_{m}/K is unramified. We further assume that V satisfies Pančiškin’s condition 6.7, i.e. it is an extension

0 ⟶ X ⟶ V ⟶ Y ⟶ 0

of semistable representations X, Y with F^{0}DR(X) = DR(Y)/F^{0} = 0. Taking pullback (resp. pushout) of V via H^{0}(K, Y)(0) ↪ Y (resp. X ⟶ H^{0}(K, X^{∗}(1))^{∗}(1)), we get an extension

0 ⟶ H^{0}(K, X^{∗}(1))^{∗}(1) ⟶ W ⟶ H^{0}(K, Y)(0) ⟶ 0.

In our situation 1.28.3 implies that

H^{0}(K, Y) = D_{st}(V)^{f=1}, H^{0}(K, X^{∗}(1)) = D_{st}(V^{∗}(1))^{f=1},

thus W is an extension of (D_{st}(V)^{f=1})(0) by (D_{st}(V^{∗}(1))^{f=1})^{∗}(1).

Thank to 1.32.3, we have exact sequences

0 ⟶ H^{0}(K_{n}, Y) ⟶ H^{1}(K_{n}, X) ⟶ H_{st}^{1}(K_{n}, V) ⟶ 0

over all K_{n} (n < ∞). The induced exact sequence

H^{0}(K, Y)/N_{∞} ⟶ H^{1}(K, X)/N_{∞} ⟶ H_{st}^{1}(K, V)/N_{∞} ⟶ 0

is canonically isomorphic, by 6.6, to

H^{0}(K, Y)⟶^{δ}H^{0}(K, X^{∗}(1))^{∗} ⟶ H_{st}^{1}(K, V)/N_{∞} ⟶ 0.

The map δ is the same as for the extension W. Write [W] ∈ Hom(H^{0}(K, Y), H^{0}(K, X^{∗}(1))^{∗})⊗H^{1}(K, Q_{p}(1)) for the extension class of W; then δ = l([W]). Since h^{0}(Y) = h^{0}(X^{∗}(1)) by 1.28.3 and 1.32.2, we see that

H_{st}^{1}(K, V)/N_{∞} = 0 ⟺ l([W]) is surjective ⟺ l([W]) is an isomorphism

7.8 Let 0 ⟶ B(1) ⟶ E_{2} ⟶ V ⟶ 0 be an extension of representations of G(\overline{K}/K) with E_{2} de Rham (hence semistable, by 1.27). Let β : F^{0}DR(V) ⟶ B ⊗ K(1) be the K-linear map defined in 3.4. From 3.5, we get a unique (f, N)-equivariant splitting

D_{st}(E_{2}) = D_{st}(V) ⊕ D_{st}(B(1))

inducing DR(E_{2}) = DR(V)⊕(B ⊗K(1)) with F^{0}DR(E_{2}) = {(x, −β(x)) | x ∈ F^{0}DR(V)}. Write j : X ↪ V for the inclusion map. The induced decomposition

D_{st}(j^{∗}E_{2}) = D_{st}(X) ⊕ D_{st}(B(1))

is in fact a splitting in MF_{K}(f, N), i.e. respects filtrations on both sides of DR(j^{∗}E_{2}) = DR(X)⊕DR(B(1)). It comes, therefore, from an isomorphism j^{∗}E_{2} ⟶^{∼} X ⊕ B(1) of Galois representations (the fact that j^{∗}E_{2}

splits also follows from 1.32.3 applied to V^{∗}(1)). Note that all possible splittings j^{∗}E_{2} ⟶^{⟵} X form a principal homogeneous space under H^{0}(K, X^{∗}(1)) ⊗ B, whereas we have just constructed a canonical one.

Consider the following commutative diagram:

0  0
↑  ↑
0 ⟶ H^{1}(K, B(1)) ⟶ H_{st}^{1}(K, E_{2}) ⟶ H_{st}^{1}(K, V) ⟶ 0
∥  ↑  ↑
0 ⟶ H^{1}(K, B(1)) ⟶ H^{1}(K, j^{∗}E_{2}) ⟶^{⟵} H^{1}(K, X) ⟶ 0

↑

H^{0}(K, Y)

↑

0

Using 3.5, it can be written as

0  0
↑  ↑
B ⊕ B ⊗ K(1) →  B ⊕ DR(E_{2})/F^{0}  →  DR(V)/F^{0}  → 0
∥  ↑  ↑
B ⊕ B ⊗ K(1) → D_{st}(X(−1))^{f=1} ⊕ B ⊕ DR(X) ⊕ B ⊗ K(1) →^{←} D_{st}(X(−1))^{f=1} ⊕ DR(X) → 0

↑

H^{0}(K, Y)

↑

0

(as D_{st}(E_{2}(−1))^{f=1} = D_{st}(V (−1))^{f=1} ⊕ B, D_{st}(V (−1))^{f=1} = D_{st}(X(−1))^{f=1}). The interesting maps in this diagram are given by the following formulas (cf. 3.5 and 1.23):

[a, (i ∘ ε ∘ N_{V}^{−1}(x) + (y, b))mod F^{0}DR(E_{2})] ⟼ can(y) + π ∘ i ∘ ε ∘ N_{V}^{−1}(x)
↑  ↑
(x, y, a, b)  ⟼  (x, y)
(N_{V} (z), −spl ∘ π ∘ i ∘ ε(z))
↑
z

(x ∈ D_{st}(X(−1))^{f=1}, y ∈ DR(X), z ∈ H^{0}(K, Y), a ∈ B, b ∈ B ⊗ K(1)), where

N_{V} : H^{0}(K, Y) = D_{st}(V)^{f=1} = D_{st}(E_{2})^{f=1} ⟶^{∼} D_{st}(V (−1))^{f=1} = D_{st}(X(−1))^{f=1}

is induced by N on D_{st}(V),

ε : H^{0}(K, Y) = D_{st}(E_{2})^{f=1} ↪ D_{st}(E_{2}), i : D_{st}(E_{2}) ⟶ DR(E_{2})

are canonical maps, π is induced by E_{2} ⟶ V, can : DR(X) ⟶^{∼} DR(V)/F^{0} is the canonical isomorphism and spl = can^{−1} : DR(V)/F^{0} ⟶^{∼} DR(X) the canonical splitting. We shall use later on the maps p_{0}, p_{X}, which are given by the first (resp. second) projection in DR(V) ⟶^{∼} F^{0}DR(V) ⊕ DR(X).

7.9 We are now ready to describe N_{∞}H_{st}^{1}(E_{2}). Write ρ : X ⟶ H^{0}(K, X^{∗}(1))^{∗}(1) for the canonical (surjective) map and also for the induced homomorphisms

D_{st}(X(−1))^{f=1} ⟶ H^{0}(K, X^{∗}(1))^{∗}, DR(X) ⟶ H^{0}(K, X^{∗}(1))^{∗} ⊗ K(1).

We have N_{∞}H^{1}(K, B(1)) = B ⊗ Ker(l), if we view l as a homomorphism l : Q_{p} ⊕ K(1) ⟶ Q_{p} (cf. 1.34-5).

Using H^{1}(K, X) ⟶^{∼} D_{st}(X(−1))^{f=1} ⊕ DR(X), the projection

H^{1}(K, X) ⟶ H^{1}(K, X)/N_{∞} ⟶^{∼} H^{0}(K, X^{∗}(1))^{∗}

(the last isomorphism by 6.6) is given by the formula

(x, y) ⟼ l ∘ ρ(y) (x ∈ D_{st}(X(−1))^{f=1}, y ∈ DR(X)).

It follows that

N_{∞}H^{1}(K, X) = D_{st}(X(−1))^{f=1} ⊕ Ker(l ∘ ρ)

and that the map δ : H^{0}(K, Y) ⟶ H^{0}(K, X^{∗}(1))^{∗} from 7.7 is equal to δ = −l ∘ ρ ∘ spl ∘ π ∘ i ∘ ε:

δ : H^{0}(K, Y)⟶^{π∘i∘ε}DR(V)/F^{0} ⟶^{spl} DR(X)⟶^{ρ}H^{0}(K, X^{∗}(1))^{∗} ⊗ K(1)⟶^{−l}H^{0}(K, X^{∗}(1))^{∗}.

We now distinguish two cases

(i) l is unramified: Then Ker(l) = K(1), N_{∞}H^{1}(K, X) = D_{st}(X(−1))^{f=1} ⊕ DR(X) = H^{1}(K, X), hence H_{st}^{1}(K, V)/N_{∞} = 0, N_{∞}H_{st}^{1}(K, E_{2}) = 0 ⊕ DR(E_{2})/F^{0} and the map

l′ : H_{st}^{1}(K, E_{2}) = B ⊕ DR(E_{2})/F^{0} ⟶ H_{st}^{1}(K, E_{2})/N_{∞} ⟶^{∼} B

(given by the projection to the first factor) extends l : H^{1}(K, B(1)) ⟶ H^{1}(K, B(1))/N_{∞} ⟶^{∼} B. (ii) l is ramified: By 1.34, there is a unique pair of Q_{p}-linear maps Log : K^{∗} ˆ⊗Q_{p} ⟶ K(1), λ : K(1) ⟶ Q_{p} satisfying l = λ ∘ Log. Use this analytic logarithm Log for the embedding B_{st} ↪ B_{dR}. Then, via H^{1}(K, Q_{p}(1)) ⟶^{∼} Q_{p} ⊕ K(1), Ker(l) = Q_{p} ⊕ Ker(λ) (cf. 1.35). This implies that

N_{∞}H_{st}^{1}(K, V) = π ∘ i ∘ ε(H^{0}(K, Y)) + can(Ker(l ∘ ρ)),

hence (as l ∘ ρ : DR(X) ⟶ H^{0}(K, X^{∗}(1))^{∗} is surjective)

H_{st}^{1}(K, V)/N_{∞} = 0 ⟺ DR(X) = spl ∘ π ∘ i ∘ ε(H^{0}(K, Y)) + Ker(l ∘ ρ)
⟺ w_{V} := l ∘ ρ ∘ spl ∘ π ∘ i ∘ ε is surjective
⟺ w_{V} : H^{0}(K, Y) ⟶ H^{0}(K, X^{∗}(1))^{∗} is an isomorphism,

in perfect agreement with 7.7, as w_{V} = −δ = −l([W]) by the formula for δ. Similarly, we have

N_{∞}H_{st}^{1}(K, E_{2}) = B ⊕ \frac{i ∘ ε(H^{0}(K, Y)) + (Ker(λ ∘ ρ) ⊕ B ⊗ Ker(λ))}{F^{0}DR(E_{2})} ⊂ B ⊕ DR(E_{2})/F^{0}.

Suppose that H_{st}^{1}(K, V)/N_{∞} = 0. Then w_{V} : H^{0}(K, Y) ⟶ H^{0}(K, X^{∗}(1))^{∗} is an isomorphism. For x ∈ DR(X), put y := w_{V}^{−1} ∘ λ ∘ ρ(x). Then

x − p_{X} ∘ π ∘ i ∘ ε(y) ∈ Ker(λ ∘ ρ), (π ∘ i ∘ ε(y), 0) ∈ N_{∞}H_{st}^{1}(K, E_{2}),

hence

(x, 0) ≡ (0, −p_{0} ∘ π ∘ i ∘ ε(y), 0) mod N_{∞}H_{st}^{1}(K, E_{2}).

As (z, −β(z)) ∈ F^{0}DR(E_{2}) for all z ∈ F^{0}DR(V),

(x, 0) ≡ (0, −β ∘ p_{0} ∘ π ∘ i ∘ ε(y)) mod N_{∞}H_{st}^{1}(K, E_{2}).

This proves that the map

l′ : [a, (v, b)F^{0}DR(E_{2})] ⟼ l(b + β ∘ p_{0}(1 − π ∘ i ∘ ε ∘ w_{V}^{−1} ∘ λ ∘ ρ ∘ p_{X})v) ∈ B

(a ∈ B, v ∈ DR(V), b ∈ B ⊗ K(1)) defines a homomorphism

l′ : H_{st}^{1}(K, E_{2}) ⟶ H_{st}^{1}(K, E_{2})/N_{∞} ⟶ B

extending l : H^{1}(K, B(1)) ⟶ H^{1}(K, B(1))/N_{∞} ⟶^{∼} B and it is easy to see that it in fact induces an isomorphism H_{st}^{1}(K, E_{2})/N_{∞} ⟶^{∼} B.

The canonical splitting DR(V)/F^{0} ⟵^{∼} DR(X) ↪ DR(V) defines, by 3.5, a splitting H_{st}^{1}(K, E_{2}) ⟶^{⟵^{r}} H_{st}^{1}(K, V), given by

r(v) = [0, (spl(v), 0) F^{0}DR(E_{2})] ∈ B ⊕ DR(E_{2})

(v ∈ DR(V)/F^{0}). It follows from the previous discussion that l′ ∘ r = 0 if l is unramified and

l′ ∘ r(v) = −λ ∘ β ∘ p_{0} ∘ π ∘ i ∘ ε ∘ w_{V}^{−1} ∘ λ ∘ ρ ∘ spl(v)

if l is ramified.

7.10 Suppose we are also given an extension of representations of G(\overline{K}/K)

0 ⟶ V ⟶ E_{1} ⟶ A(0) ⟶ 0

with de Rham (hence semistable) E_{1}. Choose a mixed extension E of E_{1}, E_{2} (it exists, as H_{st}^{1} = H_{f}^{1} for V, V^{∗}(1)). Let α : A ⊗ K ⟶ DR(V)/F^{0} be the map defined in 3.4. The canonical splitting r defines a local height pairing (cf. 7.4)

h_{E}^{can} : A⟶^{[E]}H_{st}^{1}(K, E_{2})⟶^{1−rπ}H^{1}(K, B(1))⟶^{−l}B.

Suppose that H_{st}^{1}(K, V)/N_{∞} = 0. Then the map l′ : H_{st}^{1}(K, E_{2}) ⟶ B defined in 7.9 defines another pairing

h_{E}^{norm} : A⟶^{[E]}H_{st}^{1}(K, E_{2})⟶^{−l′}B

(“norm adapted height”). As π ∘ [E](a) = α(a ⊗ 1), we have

(h_{E}^{norm} − h_{E}^{can})(a) = (l ∘ (1 − rπ) ∘ [E] − l′ ∘ [E])(a) = −l′ ∘ r ∘ π ∘ [E](a) = −l′ ∘ r ∘ α(a ⊗ 1).

Define maps u_{V} , v_{V} as compositions

u_{V} : A ⟶ A ⊗ K⟶^{α}DR(V)/F^{0} ⟶^{spl} DR(X)⟶^{ρ}H^{0}(K, X^{∗}(1))^{∗} ⊗ K(1)⟶^{λ}H^{0}(K, X^{∗}(1))^{∗}

v_{V} : H^{0}(K, Y)⟶^{π∘i∘ε}DR(V)⟶^{p_{0}}F^{0}DR(V)⟶^{β}B ⊗ K(1)⟶^{λ}B

for ramified l and as zero maps for unramified l.

7.11 Proposition. Under the assumptions of 7.10,

(1) h_{E}^{norm} − h_{E}^{can} = v_{V} ∘ w_{V}^{−1} ∘ u_{V}

(2) v_{V^{∗}(1)} = (u_{V})^{∗}

(3) w_{V^{∗}(1)} = −(w_{V})^{∗}

(4) h_{E^{∗}(1)}^{norm} = −(h_{E}^{norm})^{∗}

Proof. (1) Follows from the formulas for l′ ∘ r.

(2) Consider the “K-linear part” of u_{V}:

\overline{u} : A ⊗ K⟶^{α}DR(V)/F^{0} ⟶^{spl} DR(X)⟶^{ρ}H^{0}(K, X^{∗}(1))^{∗} ⊗ K(1).

Applying to \overline{u} the functor D of 2.6 and replacing V by V^{∗}(1), we get

\overline{v} : H^{0}(K, Y) ⊗ K ↪ DR(Y) ⟶^{∼} F^{0}DR(V)⟶^{β}B ⊗ K(1).

The statement now follows from the fact that v_{V} is equal to the composition

H^{0}(K, Y) ⟶ H^{0}(K, Y) ⊗ K⟶^{\overline{v}}B ⊗ K(1)⟶^{λ}B.

(3) As w_{V} = −l([W]), it suffices to recall the fact used in the proof of 4.10.3.

(4) Follows from (1)–(3) and h_{E^{∗}(1)}^{can} = −(h_{E}^{can})^{∗} (cf. 7.6).

7.12 Let us return to the global situation of 7.2. Suppose we are given a global Γ-extension K_{∞}/K. As before, fixing a topological generator γ ∈ Γ defines a global algebraic logarithm l : A_{K}^{∗}/K^{∗} ⟶ G(K_{∞}/K) ⟶^{∼} Z_{p} ↪ Q_{p}. Let P be the set of primes ℘|p for which the induced local logarithm l_{℘} : K_{℘} ˆ⊗Q_{p} ⟶ Q_{p} is non-trivial (i.e. ℘ does not split completely in K_{∞}/K). Suppose, furthermore, that for all ℘ ∈ P the local representation V_{℘} satisfies the assumptions of 7.7. In particular, we have

0 ⟶ X_{℘} ⟶ V_{℘} ⟶ Y_{℘} ⟶ 0

with F^{0}DR(X_{℘}) = DR(Y_{℘})/F^{0} = 0 and also the induced extension

0 ⟶ H^{0}(K_{℘}, X_{℘}^{∗}(1))^{∗}(1) ⟶ W_{℘} ⟶ H^{0}(K_{℘}, Y_{℘})(0) ⟶ 0.

Consider the universal extensions (over K)

\begin{matrix}0 & ⟶ & V & ⟶ & E_{1} & ⟶ & A(0) & ⟶ & 0 \\ 0 & ⟶ & B(1) & ⟶ & E_{2} & ⟶ & V & ⟶ & 0 \end{matrix}

with A = H_{st}^{1}(K, V), B = H_{st}^{1}(K, V^{∗}(1))^{∗} (of course, H_{st}^{1} = H_{f}^{1} for both V, V^{∗}(1)). Restricting to G(\overline{K_{℘}}/K_{℘}), we define, following 7.10, maps

u_{℘,V} : H_{f}^{1}(K, V) = A ⟶ H^{0}(K_{℘}, X_{℘}^{∗}(1))^{∗} = (D_{st}(V^{∗}(1)_{℘})^{f=1})^{∗}
v_{℘,V} : D_{st}(V_{℘})^{f=1} = H^{0}(K_{℘}, Y_{℘}) ⟶ B = H_{f}^{1}(K, V^{∗}(1))^{∗}
w_{℘,V} = −l_{℘}([W_{℘}]) : D_{st}(V_{℘})^{f=1} = H^{0}(K_{℘}, Y_{℘}) ⟶ H^{0}(K_{℘}, X_{℘}^{∗}(1))^{∗} = (D_{st}(V^{∗}(1)_{℘})^{f=1})^{∗}

for all ℘ ∈ P. Choose a mixed extension E of E_{1}, E_{2}, which is essentially unramified at all v|̸p. Canonical splittings

DR(V_{℘})/F^{0} ⟵^{∼} DR(X_{℘}) ↪ DR(V_{℘})

at all ℘ ∈ P then define a global canonical height

h_{V}^{can} = ∑_{℘∈P} h_{℘,E}^{can} : H_{f}^{1}(K, V) ⟶ H_{f}^{1}(K, V^{∗}(1))^{∗}.

If H_{st}^{1}(K_{℘}, V)/N_{∞} = 0 (i.e. w_{℘,V} is an isomorphism) for all ℘ ∈ P, then a global “norm adapted height”

h_{V}^{norm} = ∑_{℘∈P} h_{℘,E}^{norm} : H_{f}^{1}(K, V) ⟶ H_{f}^{1}(K, V^{∗}(1))^{∗}

is defined as well (and does not depend on the choice of E, obviously).

7.13 Theorem. In the notation of 7.12, suppose that V satisfies the hypotheses of 7.7 at all primes ℘ ∈ P. Then

(1) For ℘ ∈ P, H_{st}^{1}(K_{℘}, V)/N_{∞} = 0 iff w_{℘,V} = −l_{℘}([W_{℘}]) is an isomorphism (which is automatically true for l_{℘} unramified).

(2) If H_{st}^{1}(K_{℘}, V)/N_{∞} = 0 for all ℘ ∈ P, then

h_{V}^{norm} : H_{f}^{1}(K, V) ⟶ H_{f}^{1}(K, V^{∗}(1))^{∗}

is defined and is equal to

h_{V}^{norm} = h_{V}^{can} + ∑_{℘∈P^{ram}} v_{℘,V} ∘ w_{℘,V}^{−1} ∘ u_{℘,V} ,

where the summation extends over those ℘|p for which l_{℘} ramifies.

(3) Define extended height

h_{V}^{ext} : H_{f}^{1}(K, V) ⊕ ⨁_{℘∈P} D_{st}(V_{℘})^{f=1} ⟶ H_{f}^{1}(K, V^{∗}(1))^{∗} ⊕ ⨁_{℘∈P}(D_{st}(V^{∗}(1)_{℘})^{f=1})^{∗}

by the matrix

h_{V}^{ext} = ( \begin{matrix}h_{V}^{can} & ⊕ − v_{℘,V} \\ ⊕u_{℘,V} & ⊕w_{℘,V} \end{matrix} )

Then h_{V}^{ext} is an isomorphism iff h_{V}^{norm} is and
det(h_{V}^{ext}) = det(h_{V}^{norm}) ∏_{℘∈P} det(−l_{℘}([W_{℘}]))

in det(H_{f}^{1}(K, V^{∗}(1)))^{∗} ⊗ det(H_{f}^{1}(K, V))^{∗} ⊗ ⨂_{℘∈P} det(D_{st}(V^{∗}(1)_{℘})^{f=1})^{∗} ⊗ det(D_{st}(V_{℘})^{f=1})^{∗}

Proof. (1),(2) In view of 7.9 and 7.11, we only have to show that our assumptions imply that w_{℘,V} = −l_{℘}([W_{℘}]) is an isomorphism for unramified (and non-trivial) l_{℘}. In this case, w_{℘,V} is proportional to the monodromy operator on W_{℘}, which sits in a commutative diagram

N_{V_{℘}} : D_{st}(Y_{℘})^{f=1} ⟶^{∼}  D_{st}(X_{℘}(−1))^{f=1}
∥  ↓^{α}
N_{W_{℘}} : H^{0}(K_{℘}, Y_{℘}) ⟶ D_{st}(X_{℘}(−1))/(f − 1) = H^{0}(K_{℘}, X_{℘}^{∗}(1))^{∗}

By our assumptions, α is an isomorphism, hence N_{W_{℘}} is, too.

(3) Follows from the formula

( \begin{matrix}A & B \\ C & D \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & D \end{matrix} ) ( \begin{matrix}A − BD^{−1}C & B \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ D^{−1}C & 1 \end{matrix} ) .

7.14 Suppose that we are given a symplectic form on V, i.e. an isomorphism j : V ⟶^{∼} V^{∗}(1) satisfying j^{∗}(1) = −j. It induces, then, isomorphisms H_{f}^{1}(K, V) ⟶^{∼} H_{f}^{1}(K, V^{∗}(1)), D_{st}(V_{℘})^{f=1} ⟶^{∼} D_{st}(V^{∗}(1)_{℘})^{f=1}, to be denoted by the same letter. Writing ⟨., .⟩ for the canonical pairing between a vector space and its dual, we may define various pairings (depending on the choice of j) h^{?}(x, x) = ⟨j(x), h_{V}^{?} (x)⟩ (for ? = can, norm, ext) and similarly w_{℘}(y, y) = ⟨j(y), w_{℘,V} (y)⟩ (for y ∈ D_{st}(V_{℘})^{f=1} = H^{0}(K_{℘}, Y_{℘})). In our situation, the condition of 4.11.4 is satisfied, hence all of the pairings are symmetric and

det(h^{ext}) = det(h^{norm}) ∏_{℘∈P} det(w_{℘}).

It is instructive to give explicit formulas for elliptic curves: suppose that V = T_{p}(E) ⊗ Q, where E is an elliptic curve over K. We further assume that E has ordinary reduction at all ℘ ∈ P. Let P_{T} ⊆ P be the subset of primes in which E has split multiplicative reduction. For ℘ ∈ P_{T}, we have X_{℘} = Q_{p}(1), Y_{℘} = Q_{p}(0) and the extension class of W_{℘} = V_{℘} is equal to the Tate parameter q_{℘} ⊗ 1 ∈ H^{1}(K_{℘}, Q_{p}(1)), hence w_{℘,V} = −l_{℘}(q_{℘}). The map u_{℘,V} : H_{f}^{1}(K, V) ⟶ Q_{p} is given as follows: for an element x ∈ H_{f}^{1}(K, V), its local component x_{℘} lies in

E(K_{℘}) ⊗ Q_{p} = (K_{℘}^{∗}/q_{℘}^{Z})ˆ⊗Q_{p} ⟶^{∼} O_{℘}^{∗} ˆ⊗Q_{p}.

Taking the unit representative t_{℘}(x) ∈ O_{℘}^{∗} ˆ⊗Q_{p} of x_{℘}, we have u_{℘,V} (x) = l_{℘}(t_{℘}(x)). There is a canonical pair of isomorphisms j, which differ by a sign, given by Weil’s pairing. The corresponding pair of symmetrized extended heights

h^{ext} : [H_{f}^{1}(K, V) ⊕ ⨁_{℘∈P_{T}} Q_{p}] × [H_{f}^{1}(K, V) ⊕ ⨁_{℘∈P_{T}} Q_{p}] ⟶ Q_{p}

is given by the following formula:

±h^{ext}([x, (a_{℘})], [y, (b_{℘})]) = h^{can}(x, y) + ∑_{℘∈P_{T}} (a_{℘}l_{℘}(t_{℘}(y)) + b_{℘}l_{℘}(t_{℘}(x)) + l_{℘}(q_{℘})a_{℘}b_{℘})

In matrix form,

±h^{ext} = ( \begin{matrix}h^{can} & ⊕l_{℘} ∘ t_{℘} \\ ⊕l_{℘} ∘ t_{℘} & ⊕l_{℘}(q_{℘}) \end{matrix} )

and the norm adapted height, defined iff l_{℘}(q_{℘}) ≠ 0 for all ℘ ∈ P_{T}, is equal to (cf. 7.13.2)

h^{norm}(x, y) = h^{can}(x, y) − ∑_{℘∈P_{T}} \frac{l_{℘}(t_{℘}(x))l_{℘}(t_{℘}(y))}{l_{℘}(q_{℘})}.

The formula given in [Ma–Ta–Te, II.6] (without a proof) is different. In fact, Mazur-Tate-Teitelbaum give an ad hoc definition of h^{ext} (which is not to say that our definition in 7.13 is any more conceptual than theirs) which contains an extra term ord_{℘}(q_{℘}) in the denominator. There is no justification for such a term in our framework, as the computation of universal norms in 7.9 is independent on the choice of any particular valuation v_{℘} (and hence of the normalization of the monodromy operator N at ℘). One can, therefore, expect that the full L-invariant of [Ma–Ta–Te], equal to the product of l_{℘}(q_{℘})/ord_{℘}(q_{℘}), appears only in a more detailed study of the arithmetic of the whole situation (cf. [Jo 1]).

8. Comparisons and open questions

8.1 Suppose that X_{/K} is a proper smooth variety over a number field K. Let A = Pic^{0}(X_{/K}) be its Picard variety. Then V = H^{1}(\overline{X}, Q_{p}(1)) is equal to T_{p}(A) ⊗ Q, where T_{p}(A) is the Tate module of the abelian variety A and the Abel-Jacobi map

Φ_{1} : CH^{1}(X_{/K})_{0} ⟶ H^{1}(K, V)

is, essentially, nothing else then the Kummer map A(K) ⟶ H^{1}(K, T_{p}(A) ⊗ Q) (cf. 1.34). The subgroup H_{f}^{1}(K, V) ⊆ H^{1}(K, V) is equal to the classical Selmer group (more precisely, to the projective limit of Selmer groups of levels p^{n}, tensored with Q), hence equal to A(K) ⊗ Q_{p} if the p-primary part of the Tate-Šafarevič group of A_{/K} is finite. Bearing this in mind, then the construction of sec. 2 for this particular V is easily seen to be identical to that in [Za]. Similarly, our h_{V}^{norm}, if defined, coincides with Schneider’s height [Sch 2], hence also with the canonical height of Mazur–Tate [Ma–Ta] in the case of good ordinary reduction at all primes dividing p. By the same token, for general V satisfying the assumptions of 6.11, h_{V}^{norm} coincides with the height defined by Perrin-Riou [Pe-Ri 3].

8.2 It seems very likely that the canonical height of Mazur–Tate coincides with our h_{V}^{can} even for abelian varieties with bad ordinary reduction. It would also be interesting to compare our construction to [Co–Gr], where the authors define p-adic height pairings on a curve, which also depend on the choice of splittings of Hodge filtrations. Their approach is analytic and uses integration of p-adic Green currents. Is it possible to give a similar analytic interpretation of our construction in the higher-dimensional geometric case as well?

8.3 Suppose that f ∈ S_{2r}^{new}(Γ_{0}(N)) is a normalized newform of weight 2r ≥ 4 with rational coefficients. Let Y be the (2r − 1)-dimensional Kuga-Sato variety over Y (N), i.e. the (2r − 2)-fold fibre product of the universal elliptic curve over the open modular curve Y (N) with itself. There is a canonical nonsingular compactification X of Y (see [Sc 1]) and a ‘motive’ M(f) of rank 2 over Q associated to f, sitting inside the ‘motive’ of X ([Ja 3], [Sc 1]). Its p-adic realization V = M(f)_{p} is a factor of the cohomology group H_{et}^{2r−1}(\overline{X}, Q_{p}(r)). In particular, we have, over any number field K, the Abel-Jacobi map

Φ_{f,K} : CH^{r}(X_{/K})_{0} ⟶ H^{1}(K, H_{et}^{2r−1}(\overline{X}, Q_{p}(r))) ⟶ H^{1}(K, V).

For every quadratic field K in which all primes dividing N split completely, it is possible to define a ‘Heegner cycle’

y_{K} ∈ CH^{r}(X_{/K})_{0} ⊗ Q.

Write y = Φ_{f,K}(y_{K}) ∈ H^{1}(K, V). Using a Kolyvagin-type descent, it was proved in [Ne] (for p not dividing 2(2r − 2)!Nφ(N)) that Im(Φ_{f,K}) ⊗ Q_{p} is equal to Q_{p} · y, provided y ≠ 0. Unfortunately, there does not seem to be any known criterion for deciding whether y vanishes or not in a given situation. Note, however, that V satisfies all conditions (A)–(D) of 2.1 (see [Ca], [Sc 1]; in fact, monodromy conjecture is automatic in this case and does not require deep results of [Ca]: it follows from the fact that V is two-dimensional and its determinant is equal to Q_{p}(1), as was pointed out to me by A.J.Scholl) and therefore it can be plugged into our height machinery. In particular, if p is ordinary for f, then there is a canonical symmetric height pairing

h : H_{f}^{1}(K, V) × H_{f}^{1}(K, V) ⟶ Q_{p}

and it is quite natural to expect a formula of Gross-Zagier type

h(y, y) = Ω_{f⊗K,p}L′_{p}(f ⊗ K, r)

for the height of y as a multiple of the central value of the derivative of the p-adic L-function associated to f (over K). If true, it would provide a sought for criterion for non-vanishing of y.

8.4 Let us finally make a few remarks about height pairings for representations which do not satisfy Pančiškin’s condition. If ℘ is unramified in K/Q and V_{℘} is a crystalline representation of G(\overline{K_{℘}}/K_{℘}), then there is a canonical splitting of the Hodge filtration

0 ⟶ F^{0}DR(V_{℘}) ⟶ DR(V_{℘}) ⟶^{⟵} DR(V_{℘})/F^{0} ⟶ 0

constructed in [Wt] (provided the span of the filtration on DR(V_{℘}) is less than p). Consequently, if V satisfies (A)–(D) of 2.1 and p is unramified in K/Q, these splittings define a height pairing

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

which one may be tempted to call canonical. Some degree of caution might not be quite out of place at this point: B.Perrin-Riou gave in [Pe-Ri 2] another construction of p-adic heights for supersingular elliptic curves, which is based on a generalized notion of universal norms, first proposed by P.Schneider. Although Wintenberger’s splitting appears in her construction, the height pairing also depends on the choice of one of the roots of the local Euler factor at p. The referee informs me, however, that such a choice is artifical and should be ultimately eliminated, as the recent progress in the theory of p-adic L-functions suggests.

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Mathematical Institute Charles University Sokolovská 83 186 00 Praha 8 Czechoslovakia

Department of Mathematics^{1} University of California Berkeley, CA 94720 U.S.A.

1 Miller Fellow