Bloch and Kato’s Exponential Map:

Three Explicit Formulas

To Kazuya Kato on the occasion of his fiftieth birthday

Laurent Berger

Received: September 23, 2002 Revised: March 13, 2003

Abstract. The purpose of this article is to give formulas for Bloch-Kato’s exponential map and its dual for an absolutely crystalline padic representation V, in terms of the (φ, Γ)-module associated to V. As a corollary of these computations, we can give a very simple and slightly improved description of Perrin-Riou’s exponential map, which interpolates Bloch-Kato’s exponentials for the twists of V. This new description directly implies Perrin-Riou’s reciprocity formula.

2000 Mathematics Subject Classification: 11F80, 11R23, 11S25, 12H25, 13K05, 14F30, 14G20 Keywords and Phrases: Bloch-Kato’s exponential, Perrin-Riou’s exponential, Iwasawa theory, p-adic representations, Galois cohomology.

Contents

Introduction 100 I. Periods of p-adic representations 102 I.1. p-adic Hodge theory 103 I.2. (φ, Γ)-modules 105 I.3. p-adic representations and differential equations 106 I.4. Construction of cocycles 108 II. Explicit formulas for exponential maps 111 II.1. Preliminaries on some Iwasawa algebras 111 II.2. Bloch-Kato’s exponential map 113 II.3. Bloch-Kato’s dual exponential map 116 II.4. Iwasawa theory for p-adic representations 117 II.5. Perrin-Riou’s exponential map 118 II.6. The explicit reciprocity formula 121 Appendix A. The structure of D(T)^{ψ=1} 124 Appendix B. List of notations 126 Appendix C. Diagram of the rings of periods 127 References 128

Introduction

In his article [Ka93] on L-functions and rings of p-adic periods, K. Kato wrote:

I believe that there exist explicit reciprocity laws for all padic representations of Gal(\overline{K}/K), though I can not formulate them. For a de Rham representation V, this law should be some explicit description of the relationship between D_{dR}(V) and the Galois cohomology of V, or more precisely, some explicit descriptions of the maps exp and exp^{∗} of V.

In this paper, we explain how results of Benois, Cherbonnier-Colmez, Colmez, Fontaine, Kato, Kato-Kurihara-Tsuji, Perrin-Riou, Wach and the author give such an explicit description when V is a crystalline representation of an unramified field.

Let p be a prime number, and let V be a p-adic representation of G_{K} = Gal(\overline{K}/K) where K is a finite extension of Q_{p}. Such objects arise (for example) as the étale cohomology of algebraic varieties, hence their interest in arithmetic algebraic geometry.

Let B_{cris} and B_{dR} be the rings of periods of Fontaine, and let D_{cris}(V) and D_{dR}(V) be the invariants attached to V by Fontaine’s construction. Bloch and Kato have defined in [BK91, §3], for a de Rham representation V, an “exponential” map,

exp_{K,V} : D_{dR}(V)/ Fil^{0} D_{dR}(V) → H^{1}(K, V).

It is obtained by tensoring the so-called fundamental exact sequence:

0 → Q_{p} → B_{cris}^{φ=1} → B_{dR}/B_{dR}^{+} → 0

with V and taking the invariants under the action of G_{K}. The exponential map is then the connecting homomorphism D_{dR}(V)/ Fil^{0} D_{dR}(V) → H^{1}(K, V).

The reason for their terminology is the following (cf. [BK91, 3.10.1]): if G is a formal Lie group of finite height over O_{K}, and V = Q_{p} ⊗_{Z_{p}} T where T is the p-adic Tate module of G, then V is a de Rham representation and D_{dR}(V)/ Fil^{0} D_{dR}(V) is identified with the tangent space tan(G(K)) of G(K). In this case, we have a commutative diagram:

tan(G(K))  ⟶^{exp_{G}} Q ⊗_{Z} G(O_{K})
=↓  ^{δ_{G}}↓
D_{dR}(V)/ Fil^{0} D_{dR}(V) ⟶^{exp_{K,V}} H^{1}(K, V),

where δ_{G} is the Kummer map, the upper exp_{G} is the usual exponential map, and the lower exp_{K,V} is Bloch-Kato’s exponential map.

The cup product ∪ : H^{1}(K, V) × H^{1}(K, V^{∗}(1)) → H^{2}(K, Q_{p}(1)) ≃ Q_{p} defines a perfect pairing, which we can use (by dualizing twice) to define Bloch and Kato’s dual exponential map exp_{K,V^{∗}(1)}^{∗} : H^{1}(K, V) → Fil^{0} D_{dR}(V). Kato has given in [Ka93] a very simple formula for exp_{K,V^{∗}(1)}^{∗}, see proposition II.5 below.

When K is an unramified extension of Q_{p} and V is a crystalline representation of G_{K}, Perrin-Riou has constructed in [Per94] a period map Ω_{V,h} which interpolates the exp_{K,V (k)} as k runs over the positive integers. It is a crucial ingredient in the construction of p-adic L functions, and is a vast generalization of Coleman’s map. Perrin-Riou’s constructions were further generalized by Colmez in [Col98].

Let us recall the main properties of her map. For that purpose we need to introduce some notation which will be useful throughout the article. Let H_{K} = Gal(\overline{K}/K(μ_{p^{∞}})), let Δ_{K} be the torsion subgroup of Γ_{K} = G_{K}/H_{K} = Gal(K(μ_{p^{∞}})/K) and let Γ_{K}^{1} = Gal(K(μ_{p^{∞}})/K(μ_{p})) so that Γ_{K} ≃ Δ_{K} × Γ_{K}^{1}. Let Λ_{K} = Z_{p}[[Γ_{K}]] and H(Γ_{K}) = Q_{p}[Δ_{K}] ⊗_{Q_{p}} H(Γ_{K}^{1}) where H(Γ_{K}^{1}) is the set of f(γ_{1} − 1) with γ_{1} ∈ Γ_{K}^{1} and where f(T) ∈ Q_{p}[[T]] is a power series which converges on the p-adic open unit disk.

Recall that the Iwasawa cohomology groups of V are the projective limits for the corestriction maps of the H^{i}(K_{n}, V) where K_{n} = K(μ_{p^{n}}). More precisely, if T is any lattice of V then H_{Iw}^{i}(K, V) = Q_{p} ⊗_{Z_{p}} H_{Iw}^{i}(K, T) where H_{Iw}^{i}(K, T) = \underleftarrow{lim}_{n} H^{i}(K_{n}, T) so that H_{Iw}^{i}(K, V) has the structure of a Q_{p} ⊗_{Z_{p}} Λ_{K}-module (see §II.4 for more details). Roughly speaking, these cohomology groups are where Euler systems live (at least locally).

The main result of [Per94] is the construction, for a crystalline representation V of G_{K} of a family of maps (parameterized by h ∈ Z):

Ω_{V,h} : H(Γ_{K}) ⊗_{Q_{p}} D_{cris}(V) → H(Γ_{K}) ⊗_{Λ_{K}} H_{Iw}^{1}(K, V)/V^{H_{K}} ,

whose main property is that they interpolate Bloch and Kato’s exponential map. More precisely, if h, j ≫ 0, then the diagram:

(H(Γ_{K}) ⊗_{Q_{p}} D_{cris}(V (j)))^{Δ=0} ⟶^{Ω_{V (j),h}} H(Γ_{K}) ⊗_{Λ_{K}} H_{Iw}^{1}(K, V (j))/V (j)^{H_{K}}
Ξ_{n,V (j)}↓  ^{pr}^{Kn,V (j)}↓
K_{n} ⊗_{K} D_{cris}(V)  ⟶_{exp_{Kn,V (j)}}^{(h+j−1)!×}  H^{1}(K_{n}, V (j))

is commutative where Δ and Ξ_{n,V} are two maps whose definition is rather technical. Let us just say that the image of Δ is finite-dimensional over Q_{p} and that Ξ_{n,V} is a kind of evaluation-at-(ε^{(n)} − 1) map (see §II.5 for a precise definition).

Using the inverse of Perrin-Riou’s map, one can then associate to an Euler system a p-adic L-function (see for example [Per95]). For an enlightening survey of this, see [Col00]. If one starts with V = Q_{p}(1), then Perrin-Riou’s map is the inverse of the Coleman isomorphism and one recovers Kubota-Leopoldt’s p-adic L-functions. It is therefore important to be able to construct the maps Ω_{V,h} as explicitly as possible.

The goal of this article is to give formulas for exp_{K,V}, exp_{K,V^{∗}(1)}^{∗}, and Ω_{V,h} in terms of the (φ, Γ)-module associated to V by Fontaine. As a corollary, we recover a theorem of Colmez which states that Perrin-Riou’s map interpolates the exp_{K,V^{∗}(1−k)}^{∗} as k runs over the negative integers. This is equivalent to Perrin-Riou’s conjectured reciprocity formula (proved by Benois, Colmez and Kato-Kurihara-Tsuji). Our construction of Ω_{V,h} is actually a slight improvement over Perrin-Riou’s (one does not have to kill the Λ_{K}-torsion, see remark II.14). In addition, our construction should generalize to the case of de Rham representations, to families and to settings other than cyclotomic.

We refer the reader to the text itself for a statement of the actual formulas (theorems II.3, II.6 and II.13) which are rather technical.

This article does not really contain any new results, and it is mostly a reinterpretation of formulas of Cherbonnier-Colmez (for the dual exponential map), and of Benois and Colmez and Kato-Kurihara-Tsuji (for Perrin-Riou’s map) in the language of the author’s article [Ber02] on p-adic representations and differential equations.

Acknowledgments. This research was partially conducted for the Clay Mathematical Institute, and I thank them for their support. I would also like to thank P. Colmez and the referee for their careful reading of earlier versions of this article. It is P. Colmez who suggested that I give a formula for Bloch-Kato’s exponential in terms of (φ, Γ)-modules.

Finally, it is a pleasure to dedicate this article to Kazuya Kato on the occasion of his fiftieth birthday.

I. Periods of p-adic representations

Throughout this article, k will denote a finite field of characteristic p > 0, so that if W (k) denotes the ring of Witt vectors over k, then F = W (k)[1/p] is a finite unramified extension of Q_{p}. Let \overline{Q}_{p} be the algebraic closure of Q_{p}, let K be a finite totally ramified extension of F, and let G_{K} = Gal(\overline{Q}_{p}/K) be the absolute Galois group of K. Let μ_{p^{n}} be the group of p^{n}-th roots of unity; for every n, we will choose a generator ε^{(n)} of μ_{p^{n}}, with the additional requirement that (ε^{(n)})^{p} = ε^{(n−1)}. This makes \underleftarrow{lim}_{n} ε^{(n)} into a generator of \underleftarrow{lim}_{n} μ_{p^{n}} ≃ Z_{p}(1). We set K_{n} = K(μ_{p^{n}}) and K_{∞} = ∪_{n=0}^{+∞}K_{n}. Recall that the cyclotomic character χ : G_{K} → Z_{p}^{∗} is defined by the relation: g(ε^{(n)}) = (ε^{(n)})^{χ(g)} for all g ∈ G_{K}. The kernel of the cyclotomic character is H_{K} = Gal(\overline{Q}_{p}/K_{∞}), and χ therefore identifies Γ_{K} = G_{K}/H_{K} with an open subgroup of Z_{p}^{∗}.

A p-adic representation V is a finite dimensional Q_{p}-vector space with a continuous linear action of G_{K}. It is easy to see that there is always a Z_{p}-lattice of V which is stable by the action of G_{K}, and such lattices will be denoted by T. The main strategy (due to Fontaine, see for example [Fo88b]) for studying p-adic representations of a group G is to construct topological Q_{p}-algebras B (rings of periods), endowed with an action of G and some additional structures so that if V is a p-adic representation, then

D_{B}(V) = (B ⊗_{Q_{p}} V)^{G}

is a B^{G}-module which inherits these structures, and so that the functor V ↦ D_{B}(V) gives interesting invariants of V. We say that a p-adic representation V of G is B-admissible if we have B ⊗_{Q_{p}} V ≃ B^{d} as B[G]-modules.

In the next two paragraphs, we will recall the construction of a number of rings of periods. The relations between these rings are mapped in appendix C.

I.1. p-adic Hodge theory. In this paragraph, we will recall the definitions of Fontaine’s rings of periods. One can find some of these constructions in [Fo88a] and most of what we will need is proved in [Col98, III] to which the reader should refer in case of need. He is also invited to turn to appendix C.

Let C_{p} be the completion of \overline{Q}_{p} for the p-adic topology and let

Ẽ = \underleftarrow{lim}_{x↦x^{p}} C_{p} = \{(x^{(0)}, x^{(1)}, ⋯) | (x^{(i+1)})^{p} = x^{(i)}\},

and let Ẽ^{+} be the set of x ∈ Ẽ such that x^{(0)} ∈ O_{C_{p}}. If x = (x^{(i)}) and y = (y^{(i)}) are two elements of Ẽ, we define their sum x + y and their product xy by:

(x + y)^{(i)} = lim_{j→+∞}(x^{(i+j)} + y^{(i+j)})^{p^{j}} and (xy)^{(i)} = x^{(i)}y^{(i)},

which makes Ẽ into an algebraically closed field of characteristic p. If x = (x^{(n)})_{n≥0} ∈ Ẽ, let v_{E}(x) = v_{p}(x^{(0)}). This is a valuation on Ẽ for which Ẽ is complete; the ring of integers of Ẽ is Ẽ^{+}. Let Ã^{+} be the ring W (Ẽ^{+}) of Witt vectors with coefficients in Ẽ^{+} and let

B̃^{+} = Ã^{+}[1/p] = { ∑_{k≫−∞} p^{k}[x_{k}], x_{k} ∈ Ẽ^{+}}

where [x] ∈ Ã^{+} is the Teichmüller lift of x ∈ Ẽ^{+}. This ring is endowed with a map θ : B̃^{+} → C_{p} defined by the formula

θ ( ∑_{k≫−∞} p^{k}[x_{k}]) = ∑_{k≫−∞} p^{k}x_{k}^{(0)}.

The absolute Frobenius φ : Ẽ^{+} → Ẽ^{+} lifts by functoriality of Witt vectors to a map φ : B̃^{+} → B̃^{+}. It’s easy to see that φ(∑ p^{k}[x_{k}]) = ∑ p^{k}[x_{k}^{p}] and that φ is bijective.

Let ε = (ε^{(i)})_{i≥0} ∈ Ẽ^{+} where ε^{(n)} is defined above, and define π = [ε] − 1, π_{1} = [ε^{1/p}] − 1, ω = π/π_{1} and q = φ(ω) = φ(π)/π. One can easily show that ker(θ : Ã^{+} → O_{C_{p}}) is the principal ideal generated by ω.

The ring B_{dR}^{+} is defined to be the completion of B̃^{+} for the ker(θ)-adic topology:

B_{dR}^{+} = \underleftarrow{lim}_{n≥0} B̃^{+}/(ker(θ)^{n}).

It is a discrete valuation ring, whose maximal ideal is generated by ω; the series which defines log([ε]) converges in B_{dR}^{+} to an element t, which is also a generator of the maximal ideal, so that B_{dR} = B_{dR}^{+}[1/t] is a field, endowed with an action of G_{K} and a filtration defined by Fil^{i}(B_{dR}) = t^{i}B_{dR}^{+} for i ∈ Z.

We say that a representation V of G_{K} is de Rham if it is B_{dR}-admissible which is equivalent to the fact that the filtered K-vector space

D_{dR}(V) = (B_{dR} ⊗_{Q_{p}} V)^{G_{K}}

is of dimension d = dim_{Q_{p}}(V).

Recall that the topology of B̃^{+} is defined by taking the collection of open sets \{([\overline{π}]^{k}, p^{n})Ã^{+}\}_{k,n≥0} as a family of neighborhoods of 0. The ring B_{max}^{+} is defined as being

B_{max}^{+} = \{∑_{n≥0} a_{n} \frac{ω^{n}}{p^{n}} where a_{n} ∈ B̃^{+} is sequence converging to 0\},

and B_{max} = B_{max}^{+}[1/t]. The ring B_{max} was defined in [Col98, III.2] where a number of its properties are established. It is closely related to B_{cris} but tends to be more amenable (loc. cit.). One could replace ω by any generator of ker(θ) in Ã^{+}. The ring B_{max} injects canonically into B_{dR} and, in particular, it is endowed with the induced Galois action and filtration, as well as with a continuous Frobenius φ, extending the map φ : B̃^{+} → B̃^{+}. Let us point out that φ does not extend continuously to B_{dR}. One also sets B̃_{rig}^{+} = ∩_{n=0}^{+∞}φ^{n}(B_{max}^{+}).

We say that a representation V of G_{K} is crystalline if it is B_{max}-admissible or (which is the same) B̃_{rig}^{+}[1/t]-admissible (the periods of crystalline representations live in finite dimensional F -vector subspaces of B_{max}, stable by φ, and so in fact in ∩_{n=0}^{+∞}φ^{n}(B_{max}^{+})[1/t]); this is equivalent to requiring that the F -vector space

D_{cris}(V) = (B_{max} ⊗_{Q_{p}} V)^{G_{K}} = (B̃_{rig}^{+}[1/t] ⊗_{Q_{p}} V)^{G_{K}}

be of dimension d = dim_{Q_{p}}(V). Then D_{cris}(V) is endowed with a Frobenius φ induced by that of B_{max} and (B_{dR} ⊗_{Q_{p}} V)^{G_{K}} = D_{dR}(V) = K ⊗_{F} D_{cris}(V) so that a crystalline representation is also de Rham and K ⊗_{F} D_{cris}(V) is a filtered K-vector space. Note that this definition of D_{cris}(V) is compatible with the “usual” one (via B_{cris}) because ∩_{n=0}^{+∞}φ^{n}(B_{max}^{+}) = ∩_{n=0}^{+∞}φ^{n}(B_{cris}^{+}).

If V is a p-adic representation, we say that V is Hodge-Tate, with Hodge-Tate weights h_{1}, ⋯, h_{d}, if we have a decomposition C_{p} ⊗_{Q_{p}} V ≃ ⊕_{j=1}^{d}C_{p}(h_{j}). We will say that V is positive if its Hodge-Tate weights are negative (the definition of the sign of the Hodge-Tate weights is unfortunate; some people change the sign and talk about geometrical weights). By using the map θ : B_{dR}^{+} → C_{p}, it is easy to see that a de Rham representation is Hodge-Tate and that the Hodge-Tate weights of V are those integers h such that Fil^{−h} D_{dR}(V) ≠ Fil^{−h+1} D_{dR}(V).

To summarize, let us recall that crystalline implies de Rham implies Hodge-Tate. Of course, the significance of these definitions is to be found in geometrical applications. For example, if V is the Tate module of an abelian variety A, then V is de Rham and it is crystalline if and only if A has good reduction.

I.2. (φ, Γ)-modules. The results recalled in this paragraph can be found in [Fo91], and the version which we use here is described in [CC98] and [CC99].

Let Ã be the ring of Witt vectors with coefficients in Ẽ and B̃ = Ã[1/p]. Let A_{F} be the completion of O_{F} [π, π^{−1}] in Ã for this ring’s topology, which is also the completion of O_{F} [[π]][π^{−1}] for the p-adic topology (π being small in Ã). This is a discrete valuation ring whose residue field is k((ε − 1)). Let B be the completion for the p-adic topology of the maximal unramified extension of B_{F} = A_{F} [1/p] in B̃. We then define A = B ∩ Ã, B^{+} = B ∩ B̃^{+} and A^{+} = A ∩ Ã^{+}. These rings are endowed with an action of Galois and a Frobenius deduced from those on Ẽ. We set A_{K} = A^{H_{K}} and B_{K} = A_{K}[1/p]. When K = F, the two definitions are the same. Let B_{F}^{+} = (B^{+})^{H_{F}} as well as A_{F}^{+} = (A^{+})^{H_{F}} (those rings are not so interesting if K ≠ F). One can show that A_{F}^{+} = O_{F} [[π]] and that B_{F}^{+} = A_{F}^{+}[1/p].

If V is a p-adic representation of G_{K}, let D(V) = (B ⊗_{Q_{p}} V)^{H_{K}}. We know by [Fo91] that D(V) is a d-dimensional B_{K}-vector space with a slope 0 Frobenius and a residual action of Γ_{K} which commute (it is an étale (φ, Γ_{K})-module) and that one can recover V by the formula V = (B ⊗_{B_{K}} D(V))^{φ=1}.

If T is a lattice of V, we get analogous statements with A instead of B: D(T) = (A ⊗_{Z_{p}} T)^{H_{K}} is a free A_{K}-module of rank d and T = (A ⊗_{A_{K}} D(T))^{φ=1}.

The field B is a totally ramified extension (because the residual extension is purely inseparable) of degree p of φ(B). The Frobenius map φ : B → B is injective but therefore not surjective, but we can define a left inverse for φ, which will play a major role in the sequel. We set: ψ(x) = φ^{−1}(p^{−1} Tr_{B/φ(B)}(x)).

Let us now set K = F (i.e. we are now working in an unramified extension of Q_{p}). We say that a p-adic representation V of G_{F} is of finite height if D(V) has a basis over B_{F} made up of elements of D^{+}(V) = (B^{+} ⊗_{Q_{p}} V)^{H_{F}}. A result of Fontaine ([Fo91] or [Col99, III.2]) shows that V is of finite height if and only if D(V) has a sub-B_{F}^{+}-module which is free of finite rank d, and stable by φ. Let us recall the main result (due to Colmez, see [Col99, théorème 1] or also [Ber02, théorème 3.10]) regarding crystalline representations of G_{F}:

Theorem I.1. If V is a crystalline representation of G_{F}, then V is of finite height.

If K ≠ F or if V is no longer crystalline, then it is no longer true in general that V is of finite height, but it is still possible to say something about the periods of V. Every element x ∈ B̃ can be written in a unique way as x = ∑_{k≫−∞} p^{k}[x_{k}], with x_{k} ∈ Ẽ. For r > 0, let us set:

B̃^{†,r} = \{x ∈ B̃, lim_{k→+∞} v_{E}(x_{k}) + \frac{pr}{p − 1}k = +∞\} .

This makes B̃^{†,r} into an intermediate ring between B̃^{+} and B̃. Let us set B^{†,r} = B ∩ B̃^{†,r}, B̃^{†} = ∪_{r≥0}B̃^{†,r}, and B^{†} = ∪_{r≥0}B^{†,r}. If R is any of the above rings, then by definition R_{K} = R^{H_{K}}.

We say that a p-adic representation V is overconvergent if D(V) has a basis over B_{K} made up of elements of D^{†}(V) = (B^{†} ⊗_{Q_{p}} V)^{H_{K}}. The main result on the overconvergence of p-adic representations of G_{K} is the following (cf [CC98, corollaire III.5.2]):

Theorem I.2. Every p-adic representation V of G_{K} is overconvergent, that is there exists r = r(V) such that D(V) = B_{K} ⊗_{B_{K}^{†,r}} D^{†,r}(V).

The terminology “overconvergent” can be explained by the following proposition, which describes the rings B_{K}^{†,r}. Let e_{K} be the ramification index of K_{∞}/F_{∞} and let F′ be the maximal unramified extension of F contained in K_{∞} (note that F′ can be larger than F):

Proposition I.3. Let B_{F′}^{α} be the set of power series f(X) = ∑_{k∈Z} a_{k}X^{k} such that a_{k} is a bounded sequence of elements of F′, and such that f(X) is holomorphic on the p-adic annulus {p^{−1/α} ≤ |T | < 1}.

There exist r(K) and π_{K} ∈ B_{K}^{†,r(K)} such that if r ≥ r(K), then the map f ↦ f(π_{K}) from B_{F′}^{e_{K}r} to B_{K}^{†,r} is an isomorphism. If K = F, then F′ = F and one can take π_{F} = π.

I.3. p-adic representations and differential equations. We shall now recall some of the results of [Ber02], which allow us to recover D_{cris}(V) from the (φ, Γ)-module associated to V. Let H_{F′}^{α} be the set of power series f(X) = ∑_{k∈Z} a_{k}X^{k} such that a_{k} is a sequence (not necessarily bounded) of elements of F′, and such that f(X) is holomorphic on the p-adic annulus {p^{−1/α} ≤ |T | < 1}.

For r ≥ r(K), define B_{rig,K}^{†,r} as the set of f(π_{K}) where f(X) ∈ H_{F′}^{e_{K}r}. Obviously, B_{K}^{†,r} ⊂ B_{rig,K}^{†,r} and the second ring is the completion of the first one for the natural Fréchet topology. If V is a p-adic representation, let

D_{rig}^{†,r}(V) = B_{rig,K}^{†,r} ⊗_{B_{K}^{†,r}} D^{†,r}(V).

One of the main technical tools of [Ber02] is the construction of a large ring B̃_{rig}^{†}, which contains B̃_{rig}^{+} and B̃^{†}. This ring is a bridge between p-adic Hodge theory and the theory of (φ, Γ)-modules.

As a consequence of the two above inclusions, we have:

D_{cris}(V) ⊂ (B̃_{rig}^{†}[1/t] ⊗_{Q_{p}} V)^{G_{K}} and D_{rig}^{†}(V)[1/t] ⊂ (B̃_{rig}^{†}[1/t] ⊗_{Q_{p}} V)^{H_{K}} .

One of the main results of [Ber02] is then (cf. [Ber02, theorem 3.6]):

Theorem I.4. If V is a p-adic representation of G_{K} then: D_{cris}(V) = (D_{rig}^{†}(V)[1/t])^{Γ_{K}}. If V is positive, then D_{cris}(V) = D_{rig}^{†}(V)^{Γ_{K}}.

Note that one does not need to know what B̃_{rig}^{†} looks like in order to state the above theorem. We will not give the rather technical construction of that ring, but recall that B_{rig,K}^{†,r} is the completion of B_{K}^{†,r} for that ring’s natural Fréchet topology and that B_{rig,K}^{†} is the union of the B_{rig,K}^{†,r}. Similarly, there is a natural Fréchet topology on B̃^{†,r}, B̃_{rig}^{†,r} is the completion of B̃^{†,r} for that topology, and B̃_{rig}^{†} = ∪_{r≥0}B̃_{rig}^{†,r}. Actually, one can show that B̃_{rig}^{+} ⊂ B̃_{rig}^{†,r} for any r and there is an exact sequence (see [Ber02, lemme 2.18]):

0 → B̃^{+} → B̃_{rig}^{+} ⊕ B̃^{†,r} → B̃_{rig}^{†,r} → 0,

which the reader can take as providing a definition of B̃_{rig}^{†,r}.

Recall that if n ≥ 0 and r_{n} = p^{n−1}(p − 1), then there is a well-defined injective map φ^{−n} : B̃^{†,r_{n}} → B_{dR}^{+}, and this map extends (see for example [Ber02, §2.2]) to an injective map φ^{−n} : B̃_{rig}^{†,r_{n}} → B_{dR}^{+}.

The reader who feels that he needs to know more about those constructions and theorem I.4 above is invited to read either [Ber02] or the expository paper [Col01] by Colmez. See also appendix C.

Let us now return to the case when K = F and V is a crystalline representation of G_{F}. In this case, Colmez’s theorem tells us that V is of finite height so that one can write D_{rig}^{†,r}(V) = B_{rig,F}^{†,r} ⊗_{B_{F}^{+}} D^{+}(V) and theorem I.4 above therefore

says that D_{cris}(V) = (B_{rig,F}^{†,r} [1/t] ⊗_{B_{F}^{+}} D^{+}(V))^{Γ_{F}}.

One can give a more precise result. Let B_{rig,F}^{+} be the set of f(π) where f(X) = ∑_{k≥0} a_{k}X^{k} with a_{k} ∈ F, and such that f(X) is holomorphic on the p-adic open unit disk. Set D_{rig}^{+}(V) = B_{rig,F}^{+} ⊗_{B_{F}^{+}} D^{+}(V). One can then show (see [Ber03, §II.2]) the following refinement of theorem I.4:

Proposition I.5. We have D_{cris}(V) = (D_{rig}^{+}(V)[1/t])^{Γ_{F}} and if V is positive then D_{cris}(V) = D_{rig}^{+}(V)^{Γ_{F}}.

Indeed if N(V) denotes, in the terminology of [loc. cit.], the Wach module associated to V, then N(V) ⊂ D^{+}(V) when V is positive and it is shown in [loc. cit., §II.2] that under that hypothesis, D_{cris}(V) = (B_{rig,F}^{+} ⊗_{B_{F}^{+}} N(V))^{Γ_{F}}.

I.4. Construction of cocycles. The purpose of this paragraph is to recall the constructions of [CC99, §I.5] and extend them a little bit. In this paragraph, V will be an arbitrary p-adic representation of G_{K}. Recall that in loc. cit., a map h_{K,V}^{1} : D(V)^{ψ=1} → H^{1}(K, V) was constructed, and that (when Γ_{K} is torsion free at least) it gives rise to an exact sequence:

0 ⟶ D(V)_{Γ_{K}}^{ψ=1} ⟶^{h_{K,V}^{1}} H^{1}(K, V) ⟶ (\frac{D(V)}{ψ−1})^{Γ_{K}} ⟶ 0.

We shall extend h_{K,V}^{1} to a map h_{K,V}^{1} : D_{rig}^{†}(V)^{ψ=1} → H^{1}(K, V). We will first need a few facts about the ring of periods B̃_{rig}^{†} and the modules D_{rig}^{†,r}(V).

Lemma I.6. If r is large enough and γ ∈ Γ_{K} then

1 − γ : D_{rig}^{†,r}(V)^{ψ=0} → D_{rig}^{†,r}(V)^{ψ=0}

is an isomorphism.

Proof. We will first show that 1 − γ is injective. By theorem I.4, an element in the kernel of 1 − γ would have to be in D_{cris}(V) and therefore in D_{cris}(V)^{ψ=0} which is obviously 0.

We will now prove surjectivity. Recall that by [CC98, II.6.1], if r is large enough and γ ∈ Γ_{K} then 1 − γ : D^{†,r}(V)^{ψ=0} → D^{†,r}(V)^{ψ=0} is an isomorphism whose inverse is uniformly continuous for the Fréchet topology of D^{†,r}(V).

In order to show the surjectivity of 1 − γ it is therefore enough to show that D^{†,r}(V)^{ψ=0} is dense in D_{rig}^{†,r}(V)^{ψ=0} for the Fréchet topology. For r large enough, D^{†,r}(V) has a basis in φ(D^{†,r/p}(V)) so that

D^{†,r}(V)^{ψ=0} = (B_{K}^{†,r})^{ψ=0} · φ(D^{†,r/p}(V))
D_{rig}^{†,r}(V)^{ψ=0} = (B_{rig,K}^{†,r})^{ψ=0} · φ(D^{†,r/p}(V)).

The fact that D^{†,r}(V)^{ψ=0} is dense in D_{rig}^{†,r}(V)^{ψ=0} for the Fréchet topology will therefore follow from the density of (B_{K}^{†,r})^{ψ=0} in (B_{rig,K}^{†,r})^{ψ=0}. This last statement follows from the facts that by definition B_{K}^{†,r/p} is dense in B_{rig,K}^{†,r/p} and that:

(B_{K}^{†,r})^{ψ=0} = ⊕_{i=1}^{p−1}[ε]^{i}φ(B_{K}^{†,r/p}) and (B_{rig,K}^{†,r})^{ψ=0} = ⊕_{i=1}^{p−1}[ε]^{i}φ(B_{rig,K}^{†,r/p}). □

Lemma I.7. The following maps are all surjective and their kernel is Q_{p}:

1 − φ : B̃^{†} → B̃^{†}, 1 − φ : B̃_{rig}^{+} → B̃_{rig}^{+} and 1 − φ : B̃_{rig}^{†} → B̃_{rig}^{†}.

Proof. We’ll start with the assertion on the kernel of 1 − φ. Since B̃_{rig}^{+} ⊂ B̃_{rig}^{†} and B̃^{†} ⊂ B̃_{rig}^{†} it is enough to show that (B̃_{rig}^{†})^{φ=1} = Q_{p}. If x ∈ (B̃_{rig}^{†})^{φ=1}, then [Ber02, prop 3.2] shows that actually x ∈ (B̃_{rig}^{+})^{φ=1}, and therefore x ∈ (B̃_{rig}^{+})^{φ=1} = (B_{max}^{+})^{φ=1} = Q_{p} by [Col98, III.3].

The surjectivity of 1 − φ : B̃_{rig}^{†} → B̃_{rig}^{†} results from the surjectivity of 1 − φ on the first two spaces since by [Ber02, lemme 2.18], one can write α ∈ B̃_{rig}^{†} as α = α^{+} + α^{−} with α^{+} ∈ B̃_{rig}^{+} and α^{−} ∈ B̃^{†}.

The surjectivity of 1 − φ : B̃_{rig}^{+} → B̃_{rig}^{+} follows from the facts that 1 − φ : B_{max}^{+} → B_{max}^{+} is surjective (see [Col98, III.3]) and that B̃_{rig}^{+} = ∩_{n=0}^{+∞}φ^{n}(B_{max}^{+}).

The surjectivity of 1 − φ : B̃^{†} → B̃^{†} follows from the facts that 1 − φ : B̃ → B̃ is surjective (it is surjective on Ã as can be seen by reducing modulo p and using the fact that Ẽ is algebraically closed) and that if β ∈ B̃ is such that (1 − φ)β ∈ B̃^{†}, then β ∈ B̃^{†} as we shall see presently.

If x = ∑_{i=0}^{+∞} p^{i}[x_{i}] ∈ Ã, let us set w_{k}(x) = inf_{i≤k} v_{E}(x_{i}) ∈ R ∪ {+∞}. The definition of B̃^{†,r} shows that x ∈ B̃^{†,r} if and only if lim_{k→+∞} w_{k}(x) + \frac{pr}{p−1}k = +∞. A short computation also shows that w_{k}(φ(x)) = pw_{k}(x) and that w_{k}(x+ y) ≥ inf(w_{k}(x), w_{k}(y)) with equality if w_{k}(x) ≠ w_{k}(y).

It is then clear that

lim_{k→+∞} w_{k}((1 − φ)x) + \frac{pr}{p − 1}k = +∞ ⟹ lim_{k→+∞} w_{k}(x) + \frac{p(r/p)}{p − 1}k = +∞

and so if x ∈ Ã is such that (1 − φ)x ∈ B̃^{†,r} then x ∈ B̃^{†,r/p} and likewise for x ∈ B̃ by multiplying by a suitable power of p. □

The torsion subgroup of Γ_{K} will be denoted by Δ_{K}. We also set Γ_{K}^{n} = Gal(K_{∞}/K_{n}). When p ≠ 2 and n ≥ 1 (or p = 2 and n ≥ 2), Γ_{K}^{n} is torsion free. If x ∈ 1 + pZ_{p}, then there exists k ≥ 1 such that log_{p}(x) ∈ p^{k}Z_{p}^{∗} and we’ll write log_{p}^{0}(x) = log_{p}(x)/p^{k}.

If K and n are such that Γ_{K}^{n} is torsion-free, then we will construct maps h_{K_{n},V}^{1} such that cor_{K_{n+1}/K_{n}} ∘h_{K_{n+1},V}^{1} = h_{K_{n},V}^{1} . If Γ_{K}^{n} is no longer torsion free, we’ll therefore define h_{K_{n},V}^{1} by the relation h_{K_{n},V}^{1} = cor_{K_{n+1}/K_{n}} ∘h_{K_{n+1},V}^{1} . In the following proposition, we therefore assume that Γ_{K} is torsion free (and therefore procyclic), and we let γ denote a topological generator of Γ_{K}. Recall that if M is a Γ_{K}-module, it is customary to write M_{Γ_{K}} for M/ im(γ − 1).

Proposition I.8. If y ∈ D_{rig}^{†}(V)^{ψ=1}, then there exists b ∈ B̃_{rig}^{†} ⊗_{Q_{p}} V such that (γ − 1)(φ − 1)b = (φ − 1)y and the formula

h_{K,V}^{1} (y) = log_{p}^{0}(χ(γ)) [σ ↦ \frac{σ − 1}{γ − 1}y − (σ − 1)b]

then defines a map h_{K,V}^{1} : D_{rig}^{†}(V)_{Γ_{K}}^{ψ=1} → H^{1}(K, V) which does not depend either on the choice of a generator γ of Γ_{K} or on the choice of a particular solution b, and if y ∈ D(V)^{ψ=1} ⊂ D_{rig}^{†}(V)^{ψ=1}, then h_{K,V}^{1} (y) coincides with the cocycle constructed in [CC99, I.5].

Proof. Our construction closely follows [CC99, I.5]; to simplify the notations, we can assume that log_{p}^{0}(χ(γ)) = 1. The fact that if we start from a different γ, then the two h_{K,V}^{1} we get are the same is left as an easy exercise for the reader.

Let us start by showing the existence of b ∈ B̃_{rig}^{†} ⊗_{Q_{p}} V. If y ∈ D_{rig}^{†}(V)^{ψ=1}, then (φ − 1)y ∈ D_{rig}^{†}(V)^{ψ=0}. By lemma I.6, there exists x ∈ D_{rig}^{†}(V)^{ψ=0} such that (γ − 1)x = (φ − 1)y. By lemma I.7, there exists b ∈ B̃_{rig}^{†} ⊗_{Q_{p}} V such that (φ − 1)b = x.

Recall that we define h_{K,V}^{1} (y) ∈ H^{1}(K, V) by the formula:

h_{K,V}^{1} (y)(σ) = \frac{σ − 1}{γ − 1}y − (σ − 1)b.

Notice that, a priori, h_{K,V}^{1} (y) ∈ H^{1}(K, B̃_{rig}^{†} ⊗_{Q_{p}} V), but

(φ − 1)h_{K,V}^{1} (y)(σ) = \frac{σ − 1}{γ − 1}(φ − 1)y − (σ − 1)(φ − 1)b

= \frac{σ − 1}{γ − 1}(γ − 1)x − (σ − 1)x

= 0,

so that h_{K,V}^{1} (y)(σ) ∈ (B_{rig}^{†})^{φ=1} ⊗_{Q_{p}} V = V. In addition, two different choices of b differ by an element of (B̃_{rig}^{†})^{φ=1} ⊗_{Q_{p}} V = V, and therefore give rise to two cohomologous cocycles.

It is clear that if y ∈ D(V)^{ψ=1} ⊂ D_{rig}^{†}(V)^{ψ=1}, then h_{K,V}^{1} (y) coincides with the cocycle constructed in [CC99, I.5], as can be seen by their identical construction, and it is immediate that if y ∈ (γ − 1)D_{rig}^{†}(V), then h_{K,V}^{1} (y) = 0. □

Lemma I.9. We have cor_{K_{n+1}/K_{n}} ∘h_{K_{n+1},V}^{1} = h_{K_{n},V}^{1} .

Proof. The proof is exactly the same as that of [CC99, §II.2] and in any case it is rather easy. □

II. Explicit formulas for exponential maps

Recall that exp_{K,V} : D_{dR}(V)/ Fil^{0} D_{dR}(V) → H^{1}(K, V) is obtained by tensoring the fundamental exact sequence (see [Col98, III.3]):

0 → Q_{p} → B_{max}^{φ=1} → B_{dR}/B_{dR}^{+} → 0

with V and taking the invariants under the action of G_{K} (note once again that B_{cris}^{φ=1} = B_{max}^{φ=1}). The exponential map is then the connecting homomorphism D_{dR}(V)/ Fil^{0} D_{dR}(V) → H^{1}(K, V).

The cup product ∪ : H^{1}(K, V) × H^{1}(K, V^{∗}(1)) → H^{2}(K, Q_{p}(1)) ≃ Q_{p} defines a perfect pairing, which we use (by dualizing twice) to define Bloch and Kato’s dual exponential map exp_{K,V^{∗}(1)}^{∗} : H^{1}(K, V) → Fil^{0} D_{dR}(V).

The goal of this chapter is to give explicit formulas for Bloch-Kato’s maps for a p-adic representation V, in terms of the (φ, Γ)-module D(V) attached to V. Throughout this chapter, V will be assumed to be a crystalline representation of G_{F}.

II.1. Preliminaries on some Iwasawa algebras. Recall that (cf [CC99, III.2] or [Ber02, §2.4] for example) we have maps φ^{−n} : B̃_{rig}^{†,r_{n}} → B_{dR}^{+} whose restriction to B_{rig,F}^{+} satisfy φ^{−n}(B_{rig,F}^{+} ) ⊂ F_{n}[[t]] and which can then characterized by the fact that π maps to ε^{(n)} exp(t/p^{n}) − 1.

If z ∈ F_{n}((t)) ⊗_{F} D_{cris}(V), then the constant coefficient (i.e. the coefficient of t^{0}) of z will be denoted by ∂_{V} (z) ∈ F_{n} ⊗_{F} D_{cris}(V). This notation should not be confused with that for the derivation map ∂ defined below.

We will make frequent use of the following fact:

Lemma II.1. If y ∈ (B_{rig,F}^{+} [1/t] ⊗_{F} D_{cris}(V))^{ψ=1}, then for any m ≥ n ≥ 0, the element p^{−m} Tr_{F_{m}/F_{n}} ∂_{V} (φ^{−m}(y)) ∈ F_{n} ⊗_{F} D_{cris}(V) does not depend on m and we have:

p^{−m} Tr_{F_{m}/F_{n}} ∂_{V} (φ^{−m}(y)) = \begin{cases}p^{−n}∂_{V} (φ^{−n}(y)) & if n ≥ 1 \\ (1 − p^{−1}φ^{−1})∂_{V} (y) & if n = 0. \end{cases}

Proof. Recall that if y = t^{−ℓ} ∑_{k=0}^{+∞} a_{k}π^{k} ∈ B_{rig,F}^{+} [1/t] ⊗_{F} D_{cris}(V), then

φ^{−m}(y) = p^{mℓ}t^{−ℓ} ∑_{k=0}^{+∞} φ^{−m}(a_{k})(ε^{(m)} exp(t/p^{m}) − 1)^{k},

and that by the definition of ψ, ψ(y) = y means that:

φ(y) = \frac{1}{p} ∑_{η^{p}=1} y(η(1 + T) − 1).

The lemma then follows from the fact that if m ≥ 2, then the conjugates of ε^{(m)} under Gal(F_{m}/F_{m−1}) are the ηε^{(m)}, where η^{p} = 1, while if m = 1, then the conjugates of ε^{(1)} under Gal(F_{1}/F) are the η, where η^{p} = 1 but η ≠ 1. □

We will also need some facts about the Iwasawa algebra of Γ_{F} and some differential operators which it contains. Recall that since F is an unramified extension of Q_{p}, Γ_{F} ≃ Z_{p}^{∗} and that Γ_{F}^{n} = Gal(F_{∞}/F_{n}) is the set of elements γ ∈ Γ_{F} such that χ(γ) ∈ 1 + p^{n}Z_{p}.

The completed group algebra of Γ_{F} is Λ_{F} = Z_{p}[[Γ_{F}]] ≃ Z_{p}[Δ_{F}] ⊗_{Z_{p}} Z_{p}[[Γ_{F}^{1} ]], and we set H(Γ_{F}) = Q_{p}[Δ_{F}] ⊗_{Q_{p}} H(Γ_{F}^{1} ) where H(Γ_{F}^{1} ) is the set of f(γ − 1) with γ ∈ Γ_{F}^{1} and where f(X) ∈ Q_{p}[[X]] is convergent on the p-adic open unit disk. Examples of elements of H(Γ_{F}) are the ∇_{i} (which are Perrin-Riou’s ℓ_{i}’s), defined by

∇_{i} = ℓ_{i} = \frac{log(γ)}{log_{p}(χ(γ))} − i.

We will also use the operator ∇_{0}/(γ_{n} − 1), where γ_{n} is a topological generator of Γ_{F}^{n}. It is defined (see [Ber02, §4.1]) by the formula:

\frac{∇_{0}}{γ_{n} − 1} = \frac{log(γ_{n})}{log_{p}(χ(γ_{n}))(γ_{n} − 1)} = \frac{1}{log_{p}(χ(γ_{n}))} ∑_{i≥1} \frac{(1 − γ_{n})^{i−1}}{i} ,

or equivalently by

\frac{∇_{0}}{γ_{n} − 1} = lim_{\begin{matrix}η∈Γ_{F}^{n} \\ η→1 \end{matrix}} \frac{η − 1}{γ_{n} − 1} \frac{1}{log_{p}(χ(η))}.

It is easy to see that ∇_{0}/(γ_{n} − 1) acts on F_{n} by 1/ log_{p}(χ(γ_{n})).

Note that “∇_{0}/(γ_{n} − 1)” is a suggestive notation for this operator but it is not defined as a (meaningless) quotient of two operators.

The algebra H(Γ_{F}) acts on B_{rig,F}^{+} and one can easily check that:

∇_{i} = t\frac{d}{dt} − i = log(1 + π)∂ − i, where ∂ = (1 + π)\frac{d}{dπ} .

In particular, ∇_{0}B_{rig,F}^{+} ⊂ tB_{rig,F}^{+} and if i ≥ 1, then

∇_{i−1} ∘ ⋯ ∘ ∇_{0}B_{rig,F}^{+} ⊂ t^{i}B_{rig,F}^{+} .

Lemma II.2. If n ≥ 1, then ∇_{0}/(γ_{n} − 1)(B_{rig,F}^{+} )^{ψ=0} ⊂ (t/φ^{n}(π))(B_{rig,F}^{+} )^{ψ=0} so that if i ≥ 1, then:

∇_{i−1} ∘ ⋯ ∘ ∇_{1} ∘ \frac{∇_{0}}{γ_{n} − 1}(B_{rig,F}^{+} )^{ψ=0} ⊂ (\frac{t}{φ^{n}(π)})^{i} (B_{rig,F}^{+} )^{ψ=0}.

Proof. Since ∇_{i} = t·d/dt−i, the second claim follows easily from the first one, which we will now show. By the standard properties of p-adic holomorphic functions, what we need to do is to show that if x ∈ (B_{rig,F}^{+} )^{ψ=0}, then

\frac{∇_{0}}{γ_{n} − 1}x(ε^{(m)} − 1) = 0

for all m ≥ n + 1.

On the one hand, up to a scalar factor, one has for m ≥ n + 1:

\frac{∇_{0}}{γ_{n} − 1}x(ε^{(m)} − 1) = Tr_{F_{m}/F_{n}} x(ε^{(m)} − 1)

as can be seen from the fact that

\frac{∇_{0}}{γ_{n} − 1} = lim_{\begin{matrix}η∈Γ_{F}^{n} \\ η→1 \end{matrix}} \frac{η − 1}{γ_{n} − 1} · \frac{1}{log_{p}(χ(η))}.

On the other hand, the fact that ψ(x) = 0 implies that for every m ≥ 2, Tr_{F_{m}/F_{m−1}} x(ε^{(m)} − 1) = 0. This completes the proof. □

Finally, let us point out that the actions of any element of H(Γ_{F}) and of φ commute. Since φ(t) = pt, we also see that ∂ ∘ φ = pφ ∘ ∂.

We will henceforth assume that log_{p}(χ(γ_{n})) = p^{n}, so that log_{p}^{0}(χ(γ_{n})) = 1, and in addition ∇_{0}/(γ_{n} − 1) acts on F_{n} by p^{−n}.

II.2. Bloch-Kato’s exponential map. The goal of this paragraph is to show how to compute Bloch-Kato’s map in terms of the (φ, Γ)-module of V. Let h ≥ 1 be an integer such that Fil^{−h} D_{cris}(V) = D_{cris}(V).

Recall that we have seen that D_{cris}(V) = (D_{rig}^{+}(V)[1/t])^{Γ_{F}} and that by [Ber03, §II.3] there is an isomorphism:

B_{rig,F}^{+} [1/t] ⊗_{F} D_{cris}(V) = B_{rig,F}^{+} [1/t] ⊗_{F} D_{rig}^{+}(V).

If y ∈ B_{rig,F}^{+} ⊗_{F} D_{cris}(V), then the fact that Fil^{−h} D_{cris}(V) = D_{cris}(V) implies by the results of [Ber03, §II.3] that t^{h}y ∈ D_{rig}^{+}(V), so that if

y = ∑_{i=0}^{d} y_{i} ⊗ d_{i} ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1},

then

∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) = ∑_{i=0}^{d} t^{h}∂^{h}y_{i} ⊗ d_{i} ∈ D_{rig}^{+}(V)^{ψ=1}.

One can then apply the operator h_{F_{n},V}^{1} to ∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y), and the main result of this paragraph is:

Theorem II.3. If y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1}, then

h_{F_{n},V}^{1} (∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y)) =

(−1)^{h−1}(h − 1)! \begin{cases}exp_{F_{n},V} (p^{−n}∂_{V} (φ^{−n}(y))) & if n ≥ 1 \\ exp_{F,V} ((1 − p^{−1}φ^{−1})∂_{V} (y)) & if n = 0. \end{cases}

Proof. Because the diagram

F_{n+1} ⊗_{F} D_{cris}(V) ⟶^{exp_{F_{n+1},V}} H^{1}(F_{n+1}, V)
Tr^{F_{n+1}/Fn} ↓  ^{cor}^{F_{n+1}/Fn} ↓
F_{n} ⊗_{F} D_{cris}(V) ⟶^{exp_{Fn,V}} H^{1}(F_{n}, V)

is commutative, it is enough to prove the theorem under the further assumption that Γ_{F}^{n} is torsion free. Let us then set y_{h} = ∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y). Since we are assuming for simplicity that log_{p}^{0}(χ(γ_{n})) = 1, the cocycle h_{F_{n},V}^{1} (y_{h}) is defined by:

h_{F_{n},V}^{1} (y_{h})(σ) = \frac{σ − 1}{γ_{n} − 1}y_{h} − (σ − 1)b_{n,h}

where b_{n,h} is a solution of the equation (γ_{n} − 1)(φ − 1)b_{n,h} = (φ − 1)y_{h}. In lemma II.2 above, we proved that:

∇_{i−1} ∘ ⋯ ∘ ∇_{1} ∘ \frac{∇_{0}}{γ_{n} − 1}(B_{rig,F}^{+} )^{ψ=0} ⊂ (\frac{t}{φ^{n}(π)})^{i} (B_{rig,F}^{+} )^{ψ=0}.

It is then clear that if one sets

z_{n,h} = ∇_{h−1} ∘ ⋯ ∘ \frac{∇_{0}}{γ_{n} − 1}(φ − 1)y,

then

z_{n,h} ∈ (\frac{t}{φ^{n}(π)})^{h} (B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V)
⊂ φ^{n}(π^{−h})D_{rig}^{+}(V)^{ψ=0}
⊂ D_{rig}^{†}(V)^{ψ=0}.

Recall that q = φ(π)/π. By lemma II.4 (which will be stated and proved below), there exists an element b_{n,h} ∈ φ^{n−1}(π^{−h})B̃_{rig}^{+} ⊗_{Q_{p}} V such that

(φ − φ^{n−1}(q^{h}))(φ^{n−1}(π^{h})b_{n,h}) = φ^{n}(π^{h})z_{n,h},

so that (1 − φ)b_{n,h} = z_{n,h} with b_{n,h} ∈ φ^{n−1}(π^{−h})B̃_{rig}^{+} ⊗_{Q_{p}} V.

If we set

w_{n,h} = ∇_{h−1} ∘ ⋯ ∘ \frac{∇_{0}}{γ_{n} − 1}y,

then w_{n,h} and b_{n,h} ∈ B_{max} ⊗_{Q_{p}} V and the cocycle h_{F_{n},V}^{1} (y_{h}) is then given by the formula h_{F_{n},V}^{1} (y_{h})(σ) = (σ − 1)(w_{n,h} − b_{n,h}). Now (φ − 1)b_{n,h} = z_{n,h} and (φ − 1)w_{n,h} = z_{n,h} as well, so that w_{n,h} − b_{n,h} ∈ B_{max}^{φ=1} ⊗_{Q_{p}} V.

We can also write h_{F_{n},V}^{1} (y_{h})(σ) = (σ − 1)(φ^{−n}(w_{n,h}) − φ^{−n}(b_{n,h})). Since we know that b_{n,h} ∈ φ^{n−1}(π^{−h})B_{max}^{+} ⊗_{Q_{p}} V, we have φ^{−n}(b_{n,h}) ∈ B_{dR}^{+} ⊗_{Q_{p}} V.

The definition of the Bloch-Kato exponential gives rise to the following construction: if x ∈ D_{dR}(V) and x̃ ∈ B_{max}^{φ=1} ⊗_{Q_{p}} V is such that x − x̃ ∈ B_{dR}^{+} ⊗_{Q_{p}} V then exp_{K,V} (x) is the class of the cocyle g ↦ g(x̃) − x̃.

The theorem will therefore follow from the fact that:

φ^{−n}(w_{n,h}) − (−1)^{h−1}(h − 1)!p^{−n}∂_{V} (φ^{−n}(y)) ∈ B_{dR}^{+} ⊗_{Q_{p}} V,

since we already know that φ^{−n}(b_{n,h}) ∈ B_{dR}^{+} ⊗_{Q_{p}} V.

In order to show this, first notice that

φ^{−n}(y) − ∂_{V} (φ^{−n}(y)) ∈ tF_{n}[[t]] ⊗_{F} D_{cris}(V).

We can therefore write

\frac{∇_{0}}{γ_{n} − 1}φ^{−n}(y) = p^{−n}∂_{V} (φ^{−n}(y)) + tz_{1}

and a simple recurrence shows that

∇_{i−1} ∘ ⋯ ∘ \frac{∇_{0}}{1 − γ_{n}} φ^{−n}(y) = (−1)^{i−1}(i − 1)!p^{−n}∂_{V} (φ^{−n}(y)) + t^{i}z_{i},

with z_{i} ∈ F_{n}[[t]] ⊗_{F} D_{cris}(V). By taking i = h, we see that

φ^{−n}(w_{n,h}) − (−1)^{h−1}(h − 1)!p^{−n}∂_{V} (φ^{−n}(y)) ∈ B_{dR}^{+} ⊗_{Q_{p}} V,

since we chose h such that t^{h}D_{cris}(V) ⊂ B_{dR}^{+} ⊗_{Q_{p}} V. □

We will now prove the technical lemma which was used above:

Lemma II.4. If α ∈ B̃_{rig}^{+}, then there exists β ∈ B̃_{rig}^{+} such that

(φ − φ^{n−1}(q^{h}))β = α.

Proof. By [Ber02, prop 2.19] applied to the case r = 0, the ring B̃^{+} is dense in B̃_{rig}^{+} for the Fréchet topology. Hence, if α ∈ B̃_{rig}^{+}, then there exists α_{0} ∈ B̃^{+} such that α − α_{0} = φ^{n}(π^{h})α_{1} with α_{1} ∈ B̃_{rig}^{+} (one may also show this directly; the point is that when one completes all the localizations are the same).

The map φ−φ^{n−1}(q^{h}) : B̃^{+} → B̃^{+} is surjective, because φ−φ^{n−1}(q^{h}) : Ã^{+} → Ã^{+} is surjective, as can be seen by reducing modulo p and using the fact that Ẽ is algebraically closed and that Ẽ^{+} is its ring of integers.

One can therefore write α_{0} = (φ − φ^{n−1}(q^{h}))β_{0}. Finally by lemma I.7, there exists β_{1} ∈ B̃_{rig}^{+} such that α_{1} = (φ − 1)β_{1}, so that φ^{n}(π^{h})α_{1} = (φ − φ^{n−1}(q^{h}))(φ^{n−1}(π^{h})β_{1}). □

II.3. Bloch-Kato’s dual exponential map. In the previous paragraph, we showed how to compute Bloch-Kato’s exponential map for V. We will now do the same for the dual exponential map. The starting point is Kato’s formula [Ka93, §II.1], which we recall below (it is valid for any field K):

Proposition II.5. If V is a de Rham representation, then the map from D_{dR}(V) to H^{1}(K, B_{dR} ⊗_{Q_{p}} V) defined by x ↦ [g ↦ log(χ(\overline{g}))x] is an isomorphism, and the dual exponential map exp_{V^{∗}(1)}^{∗} : H^{1}(K, V) → D_{dR}(V) is equal to the composition of the map H^{1}(K, V) → H^{1}(K, B_{dR} ⊗_{Q_{p}} V) with the inverse of this isomorphism.

Let us point out that the image of exp_{V^{∗}(1)}^{∗} is included in Fil^{0} D_{dR}(V) and that its kernel is H_{g}^{1}(K, V), the subgroup of H^{1}(K, V) corresponding to classes of de Rham extensions of Q_{p} by V.

Let us now return to a crystalline representation V of G_{F}. We then have the following formula, which is proved in much more generality (i.e. for de Rham representations) in [CC99, IV.2.1]:

Theorem II.6. If y ∈ D_{rig}^{†}(V)^{ψ=1} and y ∈ D_{rig}^{+}(V)[1/t] (so that in particular y ∈ (B_{rig,F}^{+} [1/t] ⊗_{F} D_{cris}(V))^{ψ=1}), then

exp_{F_{n},V^{∗}(1)}^{∗}(h_{F_{n},V}^{1} (y)) = \begin{cases}p^{−n}∂_{V} (φ^{−n}(y)) & if n ≥ 1 \\ (1 − p^{−1}φ^{−1})∂_{V} (y) & if n = 0. \end{cases}

Note that by theorem A.3, we know that D^{†}(V)^{ψ=1} ⊂ D_{rig}^{+}(V)[1/t].

Proof. Since the following diagram

H^{1}(F_{n+1}, V) ⟶^{exp_{F_{n+1},V ^{∗}(1)}^{∗}} F_{n+1} ⊗_{F} D_{cris}(V)
cor^{F_{n+1}/Fn} ↓  ^{Tr}^{F_{n+1}/Fn} ↓
H^{1}(F_{n}, V) ⟶^{exp_{Fn,V ^{∗}(1)}^{∗}} F_{n} ⊗_{F} D_{cris}(V)

is commutative, we only need to prove the theorem when Γ_{F}^{n} is torsion free. We then have (bearing in mind that we are assuming that log_{p}^{0}(χ(γ_{n})) = 1 for simplicity):

h_{F_{n},V}^{1} (y)(σ) = \frac{σ − 1}{γ_{n} − 1}y − (σ − 1)b,

where (γ_{n} − 1)(φ − 1)b = (φ − 1)y. Recall that B̃_{rig}^{†} = ∪_{r>0}B̃_{rig}^{†,r}. Since b ∈ B̃_{rig}^{†} ⊗_{Q_{p}} V, there exists m ≫ 0 such that b ∈ B̃_{rig}^{†,r_{m}} ⊗_{Q_{p}} V. Recall also that we have seen in I.3 that the map φ^{−m} embeds B̃_{rig}^{†,r_{m}} into B_{dR}^{+}. We can then write

h^{1}(y)(σ) = \frac{σ − 1}{γ_{n} − 1}φ^{−m}(y) − (σ − 1)φ^{−m}(b),

and φ^{−m}(b) ∈ B_{dR}^{+} ⊗_{Q_{p}} V. In addition, φ^{−m}(y) ∈ F_{m}((t)) ⊗_{F} D_{cris}(V) and γ_{n} − 1 is invertible on t^{k}F_{m} ⊗_{F} D_{cris}(V) for every k ≠ 0. This shows that the cocycle h_{F_{n},V}^{1} (y) is cohomologous in H^{1}(F_{n}, B_{dR} ⊗_{Q_{p}} V) to

σ ↦ \frac{σ − 1}{γ_{n} − 1}(∂_{V} (φ^{−m}(y)))

which is itself cohomologous (since γ_{n} − 1 is invertible on F_{m}^{Tr_{Fm/Fn} =0}) to

σ ↦ \frac{σ − 1}{γ_{n} − 1} (p^{n−m} Tr_{F_{m}/F_{n}} ∂_{V} (φ^{−m}(y)))

= σ ↦ p^{−n} log(χ(\overline{σ}))p^{n−m} Tr_{F_{m}/F_{n}} ∂_{V} (φ^{−m}(y)).

It follows from this and Kato’s formula (proposition II.5) that

exp_{F_{n},V^{∗}(1)}^{∗}(h_{F_{n},V}^{1} (y)) = p^{−m} Tr_{F_{m}/F_{n}} ∂_{V} (φ^{−m}(y))

= \begin{cases}p^{−n}∂_{V} (φ^{−n}(y)) & if n ≥ 1 \\ (1 − p^{−1}φ^{−1})∂_{V} (y) & if n = 0. \end{cases}

□

II.4. Iwasawa theory for p-adic representations. In this specific paragraph, V can be taken to be an arbitrary representation of G_{K}. Recall that the Iwasawa cohomology groups H_{Iw}^{i}(K, V) are defined by H_{Iw}^{i}(K, V) = Q_{p} ⊗_{Z_{p}} H_{Iw}^{i}(K, T) where T is any G_{K}-stable lattice of V, and where

H_{Iw}^{i}(K, T) = \underleftarrow{lim}_{cor_{K_{n+1}/Kn}} H^{i}(K_{n}, T).

Each of the H^{i}(K_{n}, T) is a Z_{p}[Γ_{K}/Γ_{K}^{n}]-module, and H_{Iw}^{i}(K_{n}, T) is then endowed with the structure of a Λ_{K}-module where

Λ_{K} = Z_{p}[[Γ_{K}]] = Z_{p}[Δ_{K}] ⊗_{Z_{p}} Z_{p}[[Γ_{K}^{1}]].

The H_{Iw}^{i}(K, V) have been studied in detail by Perrin-Riou, who proved the following (see for example [Per94, §3.2]):

Proposition II.7. If V is a p-adic representation of G_{K}, then H_{Iw}^{i}(K, V) = 0 whenever i ≠ 1, 2. In addition:

(1) the torsion sub-module of H_{Iw}^{1}(K, V) is a Q_{p}⊗_{Z_{p}}Λ_{K}-module isomorphic to V^{H_{K}} and H_{Iw}^{1}(K, V)/V^{H_{K}} is a free Q_{p} ⊗_{Z_{p}} Λ_{K}-module whose rank

is [K : Q_{p}]d;
(2) H_{Iw}^{2}(K, V) = (V^{∗}(1)^{H_{K}} )^{∗}.

If y ∈ D(T)^{ψ=1} (where T is still a lattice of V), then the sequence of {h_{F_{n},V}^{1} (y)}_{n} is compatible for the corestriction maps, and therefore defines an element of H_{Iw}^{1}(K, T). The following theorem is due to Fontaine and is proved in [CC99, §II.1]:

Theorem II.8. The map y ↦ \underleftarrow{lim}_{n} h_{K_{n},V}^{1} (y) defines an isomorphism from D(T)^{ψ=1} to H_{Iw}^{1}(K, T) and from D(V)^{ψ=1} to H_{Iw}^{1}(K, V).

Notice that V^{H_{K}} ⊂ D(V)^{ψ=1}, and it is its Q_{p} ⊗_{Z_{p}} Λ_{K}-torsion submodule. In addition, it is shown in [CC99, §II.3] that the modules D(V)/(ψ − 1) and H_{Iw}^{2}(K, V) are naturally isomorphic. One can summarize the above results as follows:

Corollary II.9. The complex of Q_{p} ⊗_{Z_{p}} Λ_{K}-modules

0 ⟶ D(V) ⟶^{1−ψ} D(V) ⟶ 0

computes the Iwasawa cohomology of V.

There is a natural projection map pr_{K_{n},V} : H_{Iw}^{i}(K, V) → H^{i}(K_{n}, V) and when i = 1 it is of course equal to the composition of:

H_{Iw}^{1}(K, V) ⟶ D(V)^{ψ=1} ⟶^{h_{Kn,V}^{1}} H^{1}(K_{n}, V).

II.5. Perrin-Riou’s exponential map. By using the results of the previous paragraphs, we can give a “uniform” formula for the image of an element y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} in H^{1}(F_{n}, V (j)) under the composition of the following maps:

(B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} ⟶^{∇_{h−1}∘···∘∇_{0}} D_{rig}^{†}(V)^{ψ=1} ⟶^{⊗e_{j}}
D_{rig}^{†}(V (j))^{ψ=1} ⟶^{h_{Fn,V (j)}^{1}} H^{1}(F_{n}, V (j)).

Here e_{j} is a basis of Q_{p}(j) such that e_{j+k} = e_{j} ⊗ e_{k} so that if V is a padic representation, then we have compatible isomorphisms of Q_{p}-vector spaces V → V (j) given by v ↦ v ⊗ e_{j}.

Theorem II.10. If y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1}, and h ≥ 1 is such that Fil^{−h} D_{cris}(V) = D_{cris}(V), then for all j with h + j ≥ 1, we have:

h_{F_{n},V (j)}^{1}(∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j}) = (−1)^{h+j−1}(h + j − 1)!

× \begin{cases}exp_{F_{n},V (j)}(p^{−n}∂_{V (j)}(φ^{−n}(∂^{−j}y ⊗ t^{−j}e_{j}))) & if n ≥ 1 \\ exp_{F,V (j)}((1 − p^{−1}φ^{−1})∂_{V (j)}(∂^{−j}y ⊗ t^{−j}e_{j})) & if n = 0, \end{cases}

while if h + j ≤ 0, then we have:

exp_{F_{n},V^{∗}(1−j)}^{∗}(h_{F_{n},V (j)}^{1}(∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j})) =

\frac{1}{(−h − j)!} \begin{cases}p^{−n}∂_{V (j)}(φ^{−n}(∂^{−j}y ⊗ t^{−j}e_{j})) & if n ≥ 1 \\ (1 − p^{−1}φ^{−1})∂_{V (j)}(∂^{−j}y ⊗ t^{−j}e_{j}) & if n = 0. \end{cases}

Proof. If h + j ≥ 1, then the following diagram is commutative:

D_{rig}^{+}(V)^{ψ=1}  ⟶^{⊗e_{j}}  D_{rig}^{+}(V (j))^{ψ=1}
∇_{h−1}∘···∘∇_{0}↑  ^{∇_{h+j−1}∘···∘∇_{0}}↑
(B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} ⟶^{∂^{−j}⊗t^{−j}e_{j}} (B_{rig,F}^{+} ⊗_{F} D_{cris}(V (j)))^{ψ=1} .

and the theorem is then a straightforward consequence of theorem II.3 applied to ∂^{−j}y ⊗ t^{−j}e_{j}, h + j and V (j) (which are the j-th twists of y, h and V).

If on the other hand h + j ≤ 0, and Γ_{F}^{n} is torsion free, then theorem II.6 shows that

exp_{F_{n},V^{∗}(1−j)}^{∗}(h_{F_{n},V (j)}^{1}(∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j})) =

p^{−n}∂_{V (j)}(φ^{−n}(∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j}))

in D_{cris}(V (j)), and a short computation involving Taylor series shows that

p^{−n}∂_{V (j)}(φ^{−n}(∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j})) =

(−h − j)!^{−1}p^{−n}∂_{V (j)}(φ^{−n}(∂^{−j}y ⊗ t^{−j}e_{j})).

Finally, to get the case n = 0, one just needs to use the corresponding statement of theorem II.6 or equivalently to corestrict. □

Remark II.11. The notation ∂^{−j} is somewhat abusive if j ≥ 1 as ∂ is not injective on B_{rig,F}^{+} (it is surjective as can be seen by “integrating” directly a power series) but the reader can check for himself that this leads to no ambiguity in the formulas of theorem II.10 above.

We will now use the above result to give a construction of Perrin-Riou’s exponential map. If f ∈ B_{rig,F}^{+} ⊗_{F} D_{cris}(V), we define Δ(f) to be the image of ⊕_{k=0}^{h}∂^{k}(f)(0) in ⊕_{k=0}^{h}(D_{cris}(V)/(1−p^{k}φ))(k). There is then an exact sequence of Q_{p} ⊗_{Z_{p}} Λ_{F} -modules (see [Per94, §2.2] for a proof):

0 ⟶ ⊕_{k=0}^{h}t^{k}D_{cris}(V)^{φ=p^{−k}} ⟶ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} ⟶^{1−φ}
(B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V) ⟶^{Δ} ⊕_{k=0}^{h} (\frac{D_{cris}(V)}{1−p^{k}φ}) (k) ⟶ 0.

If f ∈ ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0}, then by the above exact sequence there exists

y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1}

such that f = (1 − φ)y, and since ∇_{h−1} ∘ ⋯ ∘ ∇_{0} kills ⊕_{k=0}^{h−1}t^{k}D_{cris}(V)^{φ=p^{−k}} we see that ∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) does not depend upon the choice of such a y unless D_{cris}(V)^{φ=p^{−h}} ≠ 0.

Definition II.12. Let h ≥ 1 be an integer such that Fil^{−h} D_{cris}(V) = D_{cris}(V) and such that D_{cris}(V)^{φ=p^{−h}} = 0. One deduces from the above construction a well-defined map:

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} → D_{rig}^{+}(V)^{ψ=1},

given by Ω_{V,h}(f) = ∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) where y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} is such that f = (1 − φ)y.

If D_{cris}(V)^{φ=p^{−h}} ≠ 0 then we get a map:

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} → D_{rig}^{+}(V)^{ψ=1}/V^{G_{F} =χ^{h}}.

Theorem II.13. If V is a crystalline representation and h ≥ 1 is such that we have Fil^{−h} D_{cris}(V) = D_{cris}(V), then the map

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} → D_{rig}^{+}(V)^{ψ=1}/V^{H_{F}}

which takes f ∈ ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} to ∇_{h−1} ∘ ⋯ ∘ ∇_{0}((1 − φ)^{−1}f) is well-defined and coincides with Perrin-Riou’s exponential map.

Proof. The map Ω_{V,h} is well defined because as we have seen above, the kernel of 1 − φ is killed by ∇_{h−1} ∘ ⋯ ∘ ∇_{0}, except for t^{h}D_{cris}(V)^{φ=p^{−h}}, which is mapped to copies of Q_{p}(h) ⊂ V^{H_{F}}.

The fact that Ω_{V,h} coincides with Perrin-Riou’s exponential map follows directly from theorem II.10 above applied to those j’s for which h+j ≥ 1, and the fact that by Perrin-Riou’s [Per94, théorème 3.2.3], the Ω_{V,h} are uniquely determined by the requirement that they satisfy the following diagram for h, j ≫ 0 (see remark II.17 about the signs however):

(H(Γ_{F}) ⊗_{Q_{p}} D_{cris}(V (j)))^{Δ=0} ⟶^{Ω_{V (j),h}} H(Γ_{F}) ⊗_{Λ_{F}} H_{Iw}^{1}(F, V (j))/V (j)^{H_{F}}
Ξ_{n,V (j)}↓  ^{pr}^{Fn,V (j)}↓
F_{n} ⊗_{F} D_{cris}(V)  ⟶_{exp_{Fn,V (j)}}^{(h+j−1)!×}  H^{1}(F_{n}, V (j)).

Here Ξ_{n,V (j)}(g) = p^{−n}(φ ⊗ φ)^{−n}(f)(ε^{(n)} − 1) where f is such that

(1 − φ)f = g(γ − 1)(1 + π) ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=0}

and the φ on the left of φ ⊗ φ is the Frobenius on B_{rig,F}^{+} while the φ on the right is the Frobenius on D_{cris}(V). Our F_{n} is Perrin-Riou’s H_{n−1}.

Note that by theorem II.8, we have an isomorphism D(V)^{ψ=1} ≃ H_{Iw}^{1}(F, V) and therefore we get a map H(Γ_{F}) ⊗_{Λ_{F}} H_{Iw}^{1}(F, V) → D_{rig}^{†}(V)^{ψ=1}. On the other hand, there is a map

H(Γ_{F}) ⊗_{Q_{p}} D_{cris}(V (j)) → (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=0}

which sends ∑ f_{i}(γ − 1) ⊗ d_{i} to ∑ f_{i}(γ − 1)(1 + π) ⊗ d_{i}. These two maps allow us to compare the diagram above with the formulas given by theorem II.10. □

Remark II.14. By the above remarks, if V is a crystalline representation and h ≥ 1 is such that we have Fil^{−h} D_{cris}(V) = D_{cris}(V) and Q_{p}(h) ⊄ V, then the map

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} → D_{rig}^{+}(V)^{ψ=1}

which takes f ∈ ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} to ∇_{h−1} ∘⋯∘∇_{0}((1−φ)^{−1}f) is well-defined, without having to kill the Λ_{F} -torsion of H_{Iw}^{1}(F, V) which improves upon Perrin-Riou’s construction.

Remark II.15. It is clear from theorem II.10 that we have:

Ω_{V,h}(x) ⊗ e_{j} = Ω_{V (j),h+j}(∂^{−j}x ⊗ t^{−j}e_{j}) and ∇_{h} ∘ Ω_{V,h}(x) = Ω_{V,h+1}(x),

and following Perrin-Riou, one can use these formulas to extend the definition of Ω_{V,h} to all h ∈ Z by tensoring all H(Γ_{F})-modules with the field of fractions of H(Γ_{F}).

II.6. The explicit reciprocity formula. In this paragraph, we shall recall Perrin-Riou’s explicit reciprocity formula and show that it follows easily from theorem II.10 above.

There is a map H(Γ_{F}) → (B_{rig,Q_{p}}^{+})^{ψ=0} which sends f(γ − 1) to f(γ − 1)(1 + π). This map is a bijection and its inverse is the Mellin transform so that if g(π) ∈ (B_{rig,Q_{p}}^{+})^{ψ=0}, then g(π) = Mel(g)(1 + π). See [Per00, B.2.8] for a reference, where Perrin-Riou has also extended Mel to (B_{rig,Q_{p}}^{†})^{ψ=0}. If f, g ∈ (B_{rig,Q_{p}}^{†})^{ψ=0} then we define f∗g by the formula Mel(f∗g) = Mel(f) Mel(g). Let [−1] ∈ Γ_{F} be the element such that χ([−1]) = −1, and let ι be the involution of Γ_{F} which sends γ to γ^{−1}. The operator ∂^{j} on (B_{rig,Q_{p}}^{+})^{ψ=0} corresponds to Tw_{j} on Γ_{F} (Tw_{j} is defined by Tw_{j}(γ) = χ(γ)^{j}γ). For instance, it is a bijection. We will make use of the facts that ι∘∂^{j} = ∂^{−j} ∘ι and that [−1]∘∂^{j} = (−1)^{j}∂^{j} ∘[−1].

If V is a crystalline representation, then the natural maps

D_{cris}(V) ⊗_{F} D_{cris}(V^{∗}(1)) ⟶ D_{cris}(Q_{p}(1)) ⟶^{Tr_{F/Qp}} Q_{p}

allow us to define a perfect pairing [·, ·]_{V} : D_{cris}(V) × D_{cris}(V^{∗}(1)) which we extend by linearity to

[·, ·]_{V} : (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=0} × (B_{rig,F}^{+} ⊗_{F} D_{cris}(V^{∗}(1)))^{ψ=0} → (B_{rig,Q_{p}}^{+})^{ψ=0}

by the formula [f(π) ⊗ d_{1}, g(π) ⊗ d_{2}]_{V} = (f ∗ g)(π)[d_{1}, d_{2}]_{V}.

We can also define a semi-linear (with respect to ι) pairing

⟨·, ·⟩_{V} : D_{rig}^{+}(V)^{ψ=1} × D_{rig}^{+}(V^{∗}(1))^{ψ=1} → (B_{rig,Q_{p}}^{+})^{ψ=0}

by the formula

⟨y_{1}, y_{2}⟩_{V} = \underleftarrow{lim}_{n} ∑_{τ∈Γ_{F} /Γ_{F}^{n}} ⟨τ^{−1}(h_{F_{n},V}^{1} (y_{1})), h_{F_{n},V^{∗}(1)}^{1}(y_{2})⟩_{F_{n},V} · τ(1 + π)

where the pairing ⟨·, ·⟩_{F_{n},V} is given by the cup product:

⟨·, ·⟩_{F_{n},V} : H^{1}(F_{n}, V) × H^{1}(F_{n}, V^{∗}(1)) → H^{2}(F_{n}, Q_{p}(1)) ≃ Q_{p}.

The pairing ⟨·, ·⟩_{V} satisfies the relation ⟨γ_{1}x_{1}, γ_{2}x_{2}⟩_{V} = γ_{1}ι(γ_{2})⟨x_{1}, x_{2}⟩_{V}. Perrin-Riou’s explicit reciprocity formula (proved by Colmez [Col98], Benois [Ben00] and Kato-Kurihara-Tsuji [KKT96]) is then:

Theorem II.16. If x_{1} ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=0} and x_{2} ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V^{∗}(1)))^{ψ=0}, then for every h, we have:

(−1)^{h}⟨Ω_{V,h}(x_{1}), [−1] · Ω_{V^{∗}(1),1−h}(x_{2})⟩_{V} = −[x_{1}, ι(x_{2})]_{V} .

Proof. By the theory of p-adic interpolation, it is enough to prove that if x_{i} = (1 − φ)y_{i} with y_{1} ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V))^{ψ=1} and y_{2} ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris}(V^{∗}(1)))^{ψ=1} then for all j ≫ 0:

(∂^{−j}(−1)^{h}⟨Ω_{V,h}(x_{1}), [−1] · Ω_{V^{∗}(1),1−h}(x_{2})⟩_{V} ) (0) = − (∂^{−j}[x_{1}, ι(x_{2})]_{V} ) (0).

The above formula is equivalent to:

(1) (−1)^{h+j}⟨h_{F,V (j)}^{1}Ω_{V (j),h+j}(∂^{−j}x_{1} ⊗ t^{−j}e_{j}),

h_{F,V^{∗}(1−j)}^{1}Ω_{V^{∗}(1−j),1−h−j}(∂^{j}x_{2} ⊗ t^{j}e_{−j})⟩_{F,V (j)}

= −[∂_{V (j)}(∂^{−j}x_{1} ⊗ t^{−j}e_{j}), ∂_{V^{∗}(1−j)}(∂^{j}x_{2} ⊗ t^{j}e_{−j})]_{V (j)}.

By combining theorems II.10 and II.13 with remark II.15 we see that for j ≫ 0:

h_{F,V (j)}^{1}Ω_{V (j),h+j}(∂^{−j}x_{1} ⊗ t^{−j}e_{j})

= (−1)^{h+j−1} exp_{F,V (j)}((h + j − 1)!(1 − p^{−1}φ^{−1})∂_{V (j)}(∂^{−j}y_{1} ⊗ t^{−j}e_{j})),

and that

h_{F,V^{∗}(1−j)}^{1}Ω_{V^{∗}(1−j),1−h−j}(∂^{j}x_{2} ⊗ t^{j}e_{−j})

= (exp_{F,V^{∗}(1−j)}^{∗})^{−1}(h + j − 1)!^{−1}((1 − p^{−1}φ^{−1})∂_{V^{∗}(1−j)}(∂^{j}y_{2} ⊗ t^{j}e_{−j})).

Using the fact that by definition, if x ∈ D_{cris}(V (j)) and y ∈ H^{1}(F, V (j)) then

[x, exp_{F,V^{∗}(1−j)}^{∗} y]_{V (j)} = ⟨exp_{F,V (j)} x, y⟩_{F,V (j)},

we see that:

(2) ⟨h_{F,V (j)}^{1}Ω_{V (j),h+j}(∂^{−j}x_{1} ⊗ t^{−j}e_{j}),

h_{F,V^{∗}(1−j)}^{1}Ω_{V^{∗}(1−j),1−h−j}(∂^{j}x_{2} ⊗ t^{j}e_{−j})⟩_{F,V (j)}
= (−1)^{h+j−1}[(1 − p^{−1}φ^{−1})∂_{V (j)}(∂^{−j}y_{1} ⊗ t^{−j}e_{j}),

(1 − p^{−1}φ^{−1})∂_{V^{∗}(1−j)}(∂^{j}y_{2} ⊗ t^{j}e_{−j})]_{V (j)}.

It is easy to see that under [·, ·], the adjoint of (1 − p^{−1}φ^{−1}) is 1 − φ, and that if x_{i} = (1 − φ)y_{i}, then

∂_{V (j)}(∂^{−j}x_{1} ⊗ t^{−j}e_{j}) = (1 − φ)∂_{V (j)}(∂^{−j}y_{1} ⊗ t^{−j}e_{j}),
∂_{V^{∗}(1−j)}(∂^{j}x_{2} ⊗ t^{j}e_{−j}) = (1 − φ)∂_{V^{∗}(1−j)}(∂^{j}y_{2} ⊗ t^{j}e_{−j}),

so that (2) implies (1), and this proves the theorem. □

Note that as I. Fesenko pointed out it is better to call the above statement an “explicit reciprocity formula” rather than an “explicit reciprocity law” as the latter terminology is reserved for statements of a more global nature.

Remark II.17. One should be careful with all the signs involved in those formulas. Perrin-Riou has changed the definition of the ℓ_{i} operators from [Per94] to [Per99] (the new ℓ_{i} is minus the old ℓ_{i}). The reciprocity formula which is stated in [Per99, 4.2.3] does not seem (to me) to have the correct sign. On the other hand, the formulas of [Ben00, Col98] do seem to give the correct signs, but one should be careful that [Col98, IX.4.5] uses a different definition for one of the pairings, and that the signs in [CC99, IV.3.1] and [Col98, VII.1.1] disagree. Our definitions of Ω_{V,h} and of the pairing agree with Perrin-Riou’s ones (as they are given in [Per99]).

Appendix A. The structure of D(T)^{ψ=1}

The goal of this paragraph is to prove a theorem which says that for a crystalline representation V, D(V)^{ψ=1} is quite “small”. See theorem A.3 for a precise statement.

Let V be a crystalline representation of G_{F} and let T denote a G_{F} -stable lattice of V. The following proposition, which improves slightly upon the results of N. Wach [Wa96], is proved in detail in [Ber03, §II.1]:

Proposition A.1. If T is a lattice in a positive crystalline representation V, then there exists a unique sub A_{F}^{+}-module N(T) of D^{+}(T), which satisfies the following conditions:

(1) N(T) is an A_{F}^{+}-module free of rank d = dim_{Q_{p}}(V);

(2) the action of Γ_{F} preserves N(T) and is trivial on N(T)/πN(T);

(3) there exists an integer r ≥ 0 such that π^{r}D^{+}(T) ⊂ N(T).

Furthermore, N(T) is stable by φ, and the B_{F}^{+}-module N(V) = B_{F}^{+} ⊗_{A_{F}^{+}} N(T) is the unique sub-B_{F}^{+}-module of D^{+}(V) satisfying the corresponding conditions.

The A_{F}^{+}-module N(T) is called the Wach module associated to T.

Notice that N(T (−1)) = πN(T) ⊗ e_{−1}. When V is no longer positive, we can therefore define N(T) as π^{−h}N(T (−h)) ⊗ e_{h}, for h large enough so that V (−h) is positive. Using the results of [Ber03, §III.4], one can show that:

Proposition A.2. If T is a lattice in a crystalline representation V of G_{F}, whose Hodge-Tate weights are in [a; b], then N(T) is the unique sub-A_{F}^{+}-module of D^{+}(T)[1/π] which is free of rank d, stable by Γ_{F} with the action of Γ_{F} being trivial on N(T)/πN(T), and such that N(T)[1/π] = D^{+}(T)[1/π].

Finally, we have φ(π^{b}N(T)) ⊂ π^{b}N(T) and π^{b}N(T)/φ^{∗}(π^{b}N(T)) is killed by q^{b−a}. The construction T ↦ N(T) gives a bijection between Wach modules over A_{F}^{+} which are lattices in N(V) and Galois lattices T in V.

We shall now show that D(V)^{ψ=1} is not very far from being included in N(V). Indeed:

Theorem A.3. If V is a crystalline representation of G_{F}, whose Hodge-Tate weights are in [a; b], then D(V)^{ψ=1} ⊂ π^{a−1}N(V).

If in addition V has no quotient isomorphic to Q_{p}(a), then actually D(V)^{ψ=1} ⊂ π^{a}N(V).

Before we prove the above statement, we will need a few results concerning the action of ψ on D(T). In lemmas A.5 through A.7, we will assume that the Hodge-Tate weights of V are ≥ 0. In particular, N(T) ⊂ φ^{∗}N(T) so that ψ(N(T)) ⊂ N(T).

Lemma A.4. If m ≥ 1, then there exists a polynomial Q_{m}(X) ∈ Z_{p}[X] such that ψ(π^{−m}) = π^{−m}(p^{m−1} + πQ_{m}(π)).

Proof. By the definition of ψ, it is enough to show that if m ≥ 1, there exists a polynomial Q_{m}(X) ∈ Z[X] such that

\frac{1}{p} ∑_{η^{p}=1} \frac{1}{(η(1 + X) − 1)^{m}} = \frac{p^{m−1} + ((1 + X)^{p} − 1)Q_{m}((1 + X)^{p} − 1)}{((1 + X)^{p} − 1)^{m}},

which is left as an exercise for the reader (or his students). □

Lemma A.5. If k ≥ 1, then ψ(pD(T) + π^{−(k+1)}N(T)) ⊂ pD(T) + π^{−k}N(T). In addition, ψ(pD(T) + π^{−1}N(T)) ⊂ pD(T) + π^{−1}N(T).

Proof. If x ∈ N(T), then one can write x = ∑ λ_{i}φ(x_{i}) with λ_{i} ∈ A_{F}^{+} and x_{i} ∈ N(T), so that ψ(π^{−(k+1)}x) = ∑ ψ(π^{−(k+1)}λ_{i})x_{i}. By the preceding lemma, ψ(π^{−(k+1)}λ_{i}) ∈ pA_{F} + π^{−k}A_{F}^{+} whenever k ≥ 1. The lemma follows easily, and the second claim is proved in the same way. □

Lemma A.6. If k ≥ 1 and x ∈ D(T) has the property that ψ(x) − x ∈ pD(T) + π^{−k}N(T), then x ∈ pD(T) + π^{−k}N(T).

Proof. Let ℓ be the smallest integer ≥ 0 such that x ∈ pD(T) + π^{−ℓ}N(T). If ℓ ≤ k, then we are done and otherwise lemma A.5 shows that

ψ(x) ∈ pD(T) + π^{−(ℓ−1)}N(T),

so that ψ(x)−x would be in pD(T)+π^{−ℓ}N(T) but not pD(T)+π^{−(ℓ−1)}N(T), a contradiction if ℓ > k. □

Lemma A.7. We have D(T)^{ψ=1} ⊂ π^{−1}N(T).

Proof. We shall prove by induction that D(T)^{ψ=1} ⊂ p^{k}D(T) + π^{−1}N(T) for k ≥ 1. Let us start with the case k = 1. If x ∈ D(T)^{ψ=1}, then there exists some j ≥ 1 such that x ∈ pD(T) + π^{−j}N(T). If j = 1 we are done and otherwise the fact that ψ(x) = x combined with lemma A.5 shows that j can be decreased by 1. This proves our claim for k = 1.

We will now assume our claim to be true for k and prove it for k + 1. If x ∈ D(T)^{ψ=1}, we can therefore write x = p^{k}y + n where y ∈ D(T) and n ∈ π^{−1}N(T). Since ψ(x) = x, we have ψ(n) − n = p^{k}(ψ(y) − y) so that ψ(y) − y ∈ π^{−1}N(T) (this is because p^{k}D(T) ∩ N(T) = p^{k}N(T)). By lemma A.6, this implies that y ∈ pD(T)+π^{−1}N(T), so that we can write x = p^{k}(py′ + n′) + n = p^{k+1}y′ + (p^{k}n′ + n), and this proves our claim.

Finally, it is clear that our claim implies the lemma: if one can write x = p^{k}y_{k} + n_{k}, then the n_{k} will converge for the p-adic topology to a n ∈ π^{−1}N(T) such that x = n. □

Proof of theorem A.3. Clearly, it is enough to show that if T is a G_{F} -stable lattice of V, then D(T)^{ψ=1} ⊂ π^{a−1}N(T). It is also clear that we can twist V as we wish, and we shall now assume that the Hodge-Tate weights of V are in [0; h]. In this case, the theorem says that D(T)^{ψ=1} ⊂ π^{−1}N(T), which is the content of lemma A.7 above.

Let us now prove that if a positive V has no quotient isomorphic to Q_{p}, then actually D(T)^{ψ=1} ⊂ N(T). Recall that N(T) ⊂ φ^{∗}(N(T)), since the Hodge-Tate weights of V are ≥ 0, so that if e_{1}, ⋯, e_{d} is a basis of N(T), then there exists q_{ij} ∈ A_{F}^{+} such that e_{i} = ∑_{j=1}^{d} q_{ij}φ(e_{j}). If ψ(∑_{i=1}^{d} α_{i}e_{i}) = ∑_{i=1}^{d} α_{i}e_{i}, with α_{i} ∈ π^{−1}A_{F}^{+}, then this translates into ψ(∑_{i=1}^{d} α_{i}q_{ij}) = α_{j} for 1 ≤ j ≤ d.

Let α_{i,n} be the coefficient of π^{n} in α_{i}, and likewise for q_{ij,n}. Since ψ(1/π) = 1/π, the equations ψ(∑_{i=1}^{d} α_{i}q_{ij}) = α_{j} then tell us that for 1 ≤ j ≤ d:

∑_{i=1}^{d} α_{i,−1}q_{ij,0} = φ(α_{j,−1}).

Since N(V)/πN(V) ≃ D_{cris}(V) as φ-modules by [Ber03, §III.4], the above equations say that 1 is an eigenvalue of φ on D_{cris}(V). It is easy to see that if a representation has positive Hodge-Tate weights and D_{cris}(V)^{φ=1} ≠ 0, then V has a quotient isomorphic to Q_{p}. □

Remark A.8. It is proved in [Ber03, III.2] that D_{cris}(V) = (B_{rig,F}^{+} ⊗_{B_{F}^{+}}N(V))^{G_{F}}

and that if Fil^{−h} D_{cris}(V) = D_{cris}(V), then

(\frac{t}{π} )^{h} B_{rig,F}^{+} ⊗_{B_{F}^{+}} D_{cris}(V) ⊂ B_{rig,F}^{+} ⊗_{B_{F}^{+}} N(V).

In all the above constructions, one could therefore replace D_{rig}^{+}(V) by B_{rig,F}^{+} ⊗_{B_{F}^{+}} π^{h}N(V). For example, the image of the map Ω_{V,h} is included in (π^{h}B_{rig,F}^{+} ⊗_{B_{F}^{+}} N(V))^{ψ=1} so that we really get a map:

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ=0} ⊗_{F} D_{cris}(V))^{Δ=0} → (π^{h}B_{rig,F}^{+} ⊗_{B_{F}^{+}} N(V))^{ψ=1}.

This slight refinement may be useful in order to prove Perrin-Riou’s δ_{Z_{p}} conjecture.

Appendix B. List of notations

Here is a list of the main notations in the order in which they occur:

Bloch and Kato’s Exponential Map 127

I: p, k, W (k), F, K, G_{K}, μ_{p^{n}}, ε^{(n)}, K_{n}, K_{∞}, H_{K}, Γ_{K}, χ, V, T. I.1: C_{p}, Ẽ, Ẽ^{+}, v_{E}, Ã^{+}, B̃^{+}, θ, φ, ε, π, π_{1}, ω, q, B_{dR}^{+}, B_{dR}, D_{dR}(V), B_{max}^{+}, B_{max}, B_{cris}, B̃_{rig}^{+}, D_{cris}(V), h. I.2: Ã, B̃, A_{F}, B, B_{F}, A, B^{+}, A^{+}, A_{K}, B_{K}, A_{F}^{+}, B_{F}^{+}, D(V), ψ, D^{+}(V), B̃^{†,r}, B^{†,r}, B̃^{†}, B^{†}, D^{†}(V), D^{†,r}(V), e_{K}, F′, π_{K}. I.3: B_{rig,K}^{†,r}, D_{rig}^{†,r}(V), B̃_{rig}^{†}, B̃_{rig}^{†,r}, r_{n}, φ^{−n}, B_{rig,F}^{+} , D_{rig}^{+}(V). I.4: h_{K,V}^{1} , w_{k}, Δ_{K}, Γ_{K}^{n}, log_{p}^{0}, γ, M_{Γ}. II: exp_{K,V}, exp_{K,V^{∗}(1)}^{∗}. II.1: ∂_{V}, Λ_{F}, H(Γ_{F}), ∇_{i}, ∇_{0}/(γ_{n} − 1), ∂. II.4: T, H_{Iw}^{i}(K, V), pr_{K,V}. II.5: e_{j}, Δ, Ω_{V,h}, Ξ_{n,V}. II.6: Mel, Tw_{j}, [−1], ι, [·, ·]_{V}, ⟨·, ·⟩_{V}, ℓ_{i}. A: T, N(V).

Appendix C. Diagram of the rings of periods

The following diagram summarizes the relationships between the different rings of periods. The arrows ending with [figure]are surjective, the dotted arrow [figure]is an inductive limit of maps defined on subrings (φ^{−n} : B̃_{rig}^{†,r_{n}} → B_{dR}^{+}), and all the other ones are injective.

\xymatrix{ &  & B_{max}^{+} \ar[r] & B_{dR}^{+} \ar@{->>}[dd]^{θ} \\  & B̃_{rig}^{†} \ar@{.>}[urr] & B̃_{rig}^{+} \ar[l] \ar[u] \\ B̃ & B̃^{†} \ar[l] \ar[u] & B̃^{+} \ar[l] \ar[u] \ar@{->>}[r]^{θ} & C_{p} \\ Ã \ar[u] \ar@{->>}[d] & Ã^{†} \ar[l] \ar[u] & Ã^{+} \ar[l] \ar[u] \ar@{->>}[d] \ar@{->>}[r]^{θ} & O_{C_{p}} \ar[u] \ar@{->>}[d] \\ Ẽ &  & Ẽ^{+} \ar[ll] \ar@{->>}[r]^{θ} & O_{C_{p}}/p}

Documenta Mathematica · Extra Volume Kato (2003) 99–129

All the rings with tildes also have versions without a tilde: one goes from the latter to the former by making Frobenius invertible and completing.

The three rings in the leftmost column (at least their tilde-free versions) are related to the theory of (φ, Γ)-modules. The two rings on the top line are related to p-adic Hodge theory. To go from one theory to the other, one goes from one place to the other through all the intermediate rings but as the reader will notice, one has to go “upstream”.

Let us finally review the different rings of power series which occur in this article; let C[r; 1[ be the annulus{z ∈ C_{p}, p^{−1/r} ≤ |z|_{p} < 1}. We then have:

\begin{matrix}A_{F}^{+} & O_{F} [[π]] \\ B_{F}^{+} & F ⊗_{O_{F}} O_{F} [[π]] \end{matrix}  \begin{matrix}A_{F} & { \atop O_{F} [[π]][π^{−1}]} \\ B_{F} & F ⊗_{O_{F}} { \atop O_{F} [[π]][π^{−1}]} \end{matrix}

A_{F}^{†,r} Laurent series f(π), convergent on C[r; 1[, and bounded by 1
B_{F}^{†,r}  Laurent series f(π), convergent on C[r; 1[, and bounded
B_{rig,F}^{†,r}  Laurent series f(π), convergent on C[r; 1[
B_{rig,F}^{+} f(π) ∈ F [[π]], f(π) converges on the open unit disk D[0; 1[

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Laurent Berger Harvard University Department of Mathematics One Oxford St Cambridge, MA 02138 USA laurent@math.harvard.edu 130

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