Iwasawa Theory and F-Analytic

Lubin-Tate (φ, Γ)-Modules

Laurent Berger and Lionel Fourquaux

Received: December 20, 2016 Revised: June 6, 2017

Communicated by Otmar Venjakob

Abstract. Let K be a finite extension of Q_{p}. We use the theory of (φ, Γ)-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of V, for certain representations V of Gal(\overline{Q}_{p}/K). If in addition V is crystalline, we describe these classes explicitly using Bloch-Kato’s exponential maps. This allows us to generalize Perrin-Riou’s period map to the Lubin-Tate setting.

2010 Mathematics Subject Classification: 11F; 11S; 14G Keywords and Phrases: p-adic representation; (φ, Γ)-module; Lubin-Tate group; overconvergent representation; p-adic Hodge theory; analytic cohomology; normalized traces; Bloch-Kato exponential; Iwasawa theory; Kummer theory

Contents

1 Lubin-Tate (φ, Γ)-modules 1003 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 1.2 Construction of Lubin-Tate (φ, Γ)-modules . . . . . . . . . . . 1004 1.3 Overconvergent Lubin-Tate (φ, Γ)-modules . . . . . . . . . . . . 1005 1.4 Extensions of (φ, Γ)-modules . . . . . . . . . . . . . . . . . . . 1006

2 Analytic cohomology and Iwasawa theory 1007 2.1 Analytic cohomology . . . . . . . . . . . . . . . . . . . . . . . . 1007 2.2 Cohomology of F -analytic (φ, Γ)-modules . . . . . . . . . . . . 1009 2.3 The space D/(ψ_{q} − 1) . . . . . . . . . . . . . . . . . . . . . . . 1010 2.4 The operator Θ_{b} . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 2.5 Construction of extensions . . . . . . . . . . . . . . . . . . . . . 1015 3 Explicit formulas for crystalline representations 1019

3.1 Crystalline F -analytic representations . . . . . . . . . . . . . . 1019 3.2 Bloch-Kato’s exponentials for analytic representations . . . . . 1020 3.3 Interpolating exponentials and their duals . . . . . . . . . . . . 1021 3.4 Kummer theory and the representation F (χ_{π}) . . . . . . . . . . 1023 3.5 Perrin-Riou’s big exponential map . . . . . . . . . . . . . . . . 1025

Introduction

Let K be a finite extension of Q_{p} and let G_{K} = Gal(\overline{Q}_{p}/K). In this article, we use the theory of (φ, Γ)-modules in the Lubin-Tate setting to construct some classes in H^{1}(K, V), for “F -analytic” representations V of G_{K}. If in addition V is crystalline, we describe these classes explicitly using Bloch and Kato’s exponential maps and generalize Perrin-Riou’s period map to the Lubin-Tate setting. We now describe our constructions in more detail, and introduce some notation which is used throughout this paper. Let F be a finite Galois extension of Q_{p}, with ring of integers O_{F} and maximal ideal 𝔪_{F}, let π be a uniformizer of O_{F} and let k_{F} = O_{F} /π and q = Card(k_{F}). Let LT be the Lubin-Tate formal group [LT65] attached to π. We fix a coordinate T on LT, so that for each a ∈ O_{F} the multiplication-by-a map is given by a power series [a](T) = aT + O(T^{2}) ∈ O_{F} [[T]]. Let log_{LT}(T) denote the attached logarithm and exp_{LT}(T) its inverse for the composition. Let χ_{π} : G_{F} → O_{F}^{×} be the attached Lubin-Tate character. If K is a finite extension of F, let K_{n} = K(LT[π^{n}]) and K_{∞} = ∪_{n⩾1} K_{n} and Γ_{K} = Gal(K_{∞} /K). Let A_{F} denote the set of power series ∑ _{i∈Z} a_{i}T^{i} with a_{i} ∈ O_{F} such that a_{i} → 0 as i → −∞ and let B_{F} = A_{F} [1/π], which is a field. It is endowed with a Frobenius map φ_{q} : f(T) ↦ f([π](T)) and an action of Γ_{F} given by g : f(T) ↦ f([χ_{π}(g)](T)). If K is a finite extension of F, the theory of the field of norms ([FW79a, FW79b] and [Win83]) provides us with a finite unramified extension B_{K} of B_{F}. Recall [Fon90] that a (φ, Γ)-module over B_{K} is a finite dimensional B_{K} -vector space endowed with a compatible Frobenius map φ_{q} and action of Γ_{K}. We say that a (φ, Γ)-module over B_{K} is étale if it has a basis in which Mat(φ_{q}) ∈ GL_{d}(A_{K}). The relevance of these objects is explained by the result below (see [Fon90], [KR09]).

Theorem. There is an equivalence of categories between the category of F - linear representations of G_{K} and the category of étale (φ, Γ)-modules over B_{K}.

Let B_{F}^{†} denote the set of power series f(T) ∈ B_{F} that have a non-empty domain of convergence. The theory of the field of norms again provides us [Mat95] with a finite extension B_{K}^{†} of B_{F}^{†}. We say that a (φ, Γ)-module over B_{K} is overconvergent if it has a basis in which Mat(φ_{q}) ∈ GL_{d}(B_{K}^{†}) and Mat(g) ∈ GL_{d}(B_{K}^{†}) for all g ∈ Γ_{K}. If F = Q_{p}, every étale (φ, Γ)-module over B_{K} is overconvergent [CC98]. If F ≠ Q_{p}, this is no longer the case [FX13].

Iwasawa Theory and Lubin-Tate (φ, Γ)-Modules 1001

Let us say that an F -linear representation V of G_{K} is F -analytic if for all embeddings τ : F → \overline{Q}_{p}, with τ ≠ Id, the representation C_{p} ⊗_{F}^{τ} V is trivial (as a semilinear C_{p}-representation of G_{K}). The following result is known [Ber16].

Theorem. If V is an F -analytic representation of G_{K}, it is overconvergent.

Another source of overconvergent representations of G_{K} is the set of representations that factor through Γ_{K} (see §1.3). Our first result is the following (theorem 1.3.1).

Theorem A. If V is an overconvergent representation of G_{K}, there exists an F -analytic representation X_{an} of G_{K}, a representation Y_{Γ} of G_{K} that factors through Γ_{K}, and a surjective G_{K} -equivariant map X_{an} ⊗_{F} Y_{Γ} → V.

We next focus on F -analytic representations. Let B_{rig,F}^{†} denote the Robba ring, which is the ring of power series f(T) = ∑ _{i∈Z} a_{i}T^{i} with a_{i} ∈ F such that there exists ρ < 1 such that f(T) converges for ρ < |T | < 1. We have B_{F}^{†} ⊂ B_{rig,F}^{†} . The theory of the field of norms again provides us with a finite extension B_{rig,K}^{†} of B_{rig,F}^{†} . If V is an F -linear representation of G_{K}, let D(V) denote the (φ, Γ)-module over B_{K} attached to V. If V is overconvergent, there is a well defined (φ, Γ)-module D^{†}(V) over B_{K}^{†} attached to V, such that D(V) = B_{K} ⊗_{B_{K}^{†}} D^{†}(V). We call D_{rig}^{†}(V) the (φ, Γ)-module over B_{rig,K}^{†} attached to V,

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

The ring B_{rig,K}^{†} is a free φ_{q}(B_{rig,K}^{†} )-module of degree q. This allows us to define [FX13] a map ψ_{q} : B_{rig,K}^{†} → B_{rig,K}^{†} that is a Γ_{K} -equivariant left inverse of φ_{q}, and likewise, if V is an overconvergent representation of G_{K}, a map ψ_{q} : D_{rig}^{†}(V) → D_{rig}^{†}(V) that is a Γ_{K} -equivariant left inverse of φ_{q}. The main result of this article is the construction, for an F -analytic representation V of G_{K}, of a collection of maps

h_{K_{n} ,V}^{1} : D_{rig}^{†}(V)^{ψ_{q}=1} → H^{1}(K_{n}, V),

having a certain number of properties. For example, these maps are compatible with corestriction: cor_{K_{n+1} /K_{n}} ∘ h_{K_{n+1} ,V}^{1} = h_{K_{n} ,V}^{1} if n ⩾ 1. Another property is that if F = Q_{p} and π = p (the cyclotomic case), these maps coïncide with those constructed in [CC99] (and generalized in [Ber03]). If now K = F and V is a crystalline F -analytic representation of G_{F}, we give explicit formulas for h_{F_{n} ,V}^{1} using Bloch and Kato’s exponential maps [BK90]. Let V be as above, let D_{cris} (V) = (B_{cris ,F} ⊗_{F} V)^{G_{F}} (note that because the ⊗ is over F, this is the identity component of the usual D_{cris} ) and let t_{π} = log_{LT}(T). Let {u_{n}}_{n⩾0} be a compatible sequence of primitive π^{n}-torsion points of LT. Let B_{rig,F}^{+} denote the positive part of the Robba ring, namely the ring of power series f(T) = ∑ _{i⩾0} a_{i}T^{i} with a_{i} ∈ F such that f(T) converges for 0 ⩽ |T | < 1. If n ⩾ 0, we have a map φ_{q}^{−n} : B_{rig,F}^{+} → F_{n}[[t_{π}]] given by f(T) ↦ f(u_{n} ⊕ exp_{LT}(t_{π}/π^{n})). Using the results of [KR09], we prove that there is a natural (φ, Γ)-equivariant inclusion D_{rig}^{†}(V)^{ψ_{q}=1} → B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D_{cris} (V). This provides us, by composition, with maps φ_{q}^{−n} : D_{rig}^{†}(V)^{ψ_{q}=1} → F_{n}((t_{π})) ⊗_{F} D_{cris} (V) and ∂_{V} ∘ φ_{q}^{−n} : D_{rig}^{†}(V)^{ψ_{q}=1} → F_{n} ⊗_{F} D_{cris} (V) where ∂_{V} is the “coefficient of t_{π}^{0}” map. Recall finally that we have two maps, Bloch and Kato’s exponential exp_{F_{n} ,V} : F_{n} ⊗_{F} D_{cris} (V) → H^{1}(F_{n}, V) and its dual exp_{F_{n} ,V^{∗} (1)}^{∗} H^{1}(F_{n}, V) → F_{n} ⊗_{F} D_{cris} (V) (the subscript V^{∗}(1) denotes the dual of V twisted by the cyclotomic character, but is merely a notation here). The first result is as follows (theorem 3.3.1).

Theorem B. If V is as above and y ∈ D_{rig}^{†}(V)^{ψ_{q}=1} , then

exp_{F_{n} ,V^{∗} (1)}^{∗} (h_{F_{n} ,V}^{1} (y)) = \{ \begin{matrix}q^{−n} ∂_{V}(φ_{q}^{−n} (y)) & if n ⩾ 1 \\ (1 − q^{−1} φ_{q}^{−1} )∂_{V}(y) & if n = 0. \end{matrix}

Let ∇ = t_{π} · d/dt_{π}, let ∇_{i} = ∇ − i if i ∈ Z and let h ⩾ 1 be such that Fil^{−h} D_{cris} (V) = D_{cris} (V). We prove that if y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} , then ∇_{h−1} ∘ ⋯∘ ∇_{0}(y) ∈ D_{rig}^{†}(V)^{ψ_{q}=1} , and we have the following result (theorem 3.3.2).

Theorem C. If V is as above and y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} , then

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

(−1)^{h−1} (h − 1)! \{ \begin{matrix}exp_{F_{n} ,V} (q^{−n} ∂_{V}(φ_{q}^{−n} (y))) & if n ⩾ 1 \\ exp_{F,V} ((1 − q^{−1} φ_{q}^{−1} )∂_{V}(y)) & if n = 0. \end{matrix}

Using theorems B and C, we give in §3.5 a Lubin-Tate analogue of Perrin-Riou’s “big exponential map” [PR94] using the same method as that of [Ber03] which treats the cyclotomic case. It will be interesting to compare this big exponential map with the “big logarithms” constructed in [Fou05] and [Fou08]. It is also instructive to specialize theorem C to the case V = F (χ_{π}), which corresponds to “Lubin-Tate” Kummer theory. Recall that if L is a finite extension of F, Kummer theory gives us a map δ : LT(𝔪_{L}) → H^{1}(L, F (χ_{π})). When L varies among the F_{n}, these maps are compatible: the diagram

LT(𝔪_{F_{n+1}} ) −−−−→^{δ} H^{1}(F_{n+1} , V)
Tr _{F_{n+1} /Fn}^{LT} ↓  ↓^{cor}^{F_{n+1} /Fn}
LT(𝔪_{F_{n}} ) −−−−→^{δ} H^{1}(F_{n}, V)

commutes. Let S denote the set of sequences {x_{n}}_{n⩾1} with x_{n} ∈ 𝔪_{F_{n}} and such that Tr_{F_{n+1} /F_{n}}^{LT} (x_{n+1} ) = [q/π](x_{n}) for n ⩾ 1. We prove that S is big, in the sense that (if F ≠ Q_{p}) the projection on the n-th coordinate map S ⊗_{O_{F}} F → F_{n} is onto (this would not be the case if we did not have the factor q/π in the definition of S). Furthermore, we prove that if x ∈ S, there exists a power series f(T) ∈ (B_{rig,F}^{+} )^{ψ_{q}=1/π} such that f(u_{n}) = log_{LT}(x_{n}) for n ⩾ 1. We have d/dt_{π}(f(T)) ∈ (B_{rig,F}^{+} )^{ψ_{q}=1} and the following holds (theorem 3.4.5), where u is the basis of F (χ_{π}) corresponding to the choice of {u_{n}}_{n⩾0} .

Theorem D. We have h_{F_{n} ,F (χ_{π} )}^{1} (d/dt_{π}(f(T)) · u) = (q/π)^{−n} · δ(x_{n}) for all n ⩾ 1.

In the cyclotomic case, there is [Col79] a power series Col_{x}(T) such that Col_{x}(u_{n}) = x_{n} for n ⩾ 1. We then have f(T) = logCol_{x}(T), and theorem D is proved in [CC99]. In the general Lubin-Tate case, we do not know whether there is a “Coleman power series” of which f(T) would be the log_{LT}. This seems like a non-trivial question. It would be interesting to compare our results with those of [SV17]. The authors of [SV17] also construct some classes in H^{1}(K, V), but start from the space D(V (χ_{π} · χ_{cyc}^{−1}))^{ψ_{q}=π/q} . In another direction, is it possible to extend our constructions to representations of the form V ⊗_{F} Y_{Γ} with V F -analytic and Y_{Γ} factoring through Γ_{K}, and in particular recover the explicit reciprocity law of [Tsu04]?

1 Lubin-Tate (φ, Γ)-modules

In this chapter, we recall the theory of Lubin-Tate (φ, Γ)-modules and classify overconvergent representations.

1.1 Notation

Let F be a finite Galois extension of Q_{p} with ring of integers O_{F}, and residue field k_{F}. Let π be a uniformizer of O_{F}. Let d = [F : Q_{p}] and e be the ramification index of F/Q_{p}. Let q = p^{f} be the cardinality of k_{F} and let F_{0} = W (k_{F})[1/p] be the maximal unramified extension of Q_{p} inside F. Let σ denote the absolute Frobenius map on F_{0}. Let LT be the Lubin-Tate formal O_{F} -module attached to π and choose a coordinate T for the formal group law, such that the action of π on LT is given by [π](T) = T^{q} + πT. If a ∈ O_{F}, let [a](T) denote the power series that gives the action of a on LT. Let log_{LT}(T) denote the attached logarithm and exp_{LT}(T) its inverse. If K is a finite extension of F, let K_{n} = K(LT[π^{n}]) and let K_{∞} = ∪_{n⩾1} K_{n}. Let H_{K} = Gal(\overline{Q}_{p}/K_{∞} ) and Γ_{K} = Gal(K_{∞} /K). By Lubin-Tate theory (see [LT65]), Γ_{K} is isomorphic to an open subgroup of O_{F}^{×} via the Lubin-Tate character χ_{π} : Γ_{K} → O_{F}^{×}. Let n(K) ⩾ 1 be such that if n ⩾ n(K), then χ_{π} : Γ_{K_{n}} → 1 + π^{n}O_{F} is an isomorphism, and log_{p} : 1 + π^{n}O_{F} → π^{n}O_{F} is also an isomorphism. Since log_{LT}(T) converges on the open unit disk, it can be seen as an element of B_{rig,F}^{+} and we denote it by t_{π}. Recall that g(t_{π}) = χ_{π}(g) · t_{π} if g ∈ G_{K} and that φ_{q}(t_{π}) = π · t_{π}. Let ∂ = d/dt_{π} so that ∂f(T) = a(T) · df(T)/dT, where a(T) = (d log_{LT}(T)/dT)^{−1} ∈ O_{F} [[T]]^{×} . We have ∂ ∘ g = χ_{π}(g) · g ∘ ∂ if g ∈ Γ_{K} and ∂ ∘ φ_{q} = π · φ_{q} ∘ ∂.

Recall that B_{rig,F}^{†} denotes the Robba ring, the ring of power series f(T) = ∑ _{i∈Z} a_{i}T^{i} with a_{i} ∈ F such that there exists ρ < 1 such that f(T) converges for ρ < |T | < 1. We have B_{F}^{†} ⊂ B_{rig,F}^{†} and by writing a power series as the sum of its plus part and its minus part, we get B_{rig,F}^{†} = B_{rig,F}^{+} + B_{F}^{†}. Each ring R ∈ {B_{rig,F}^{†} , B_{rig,F}^{+} , B_{F}^{†} , B_{F} } is equipped with a Frobenius map φ_{q} : f(T) ↦ f([π](T)) and an action of Γ_{F} given by g : f(T) ↦ f([χ_{π}(g)](T)). Moreover, the ring R is a free φ_{q}(R)-module of rank q, and we define ψ_{q} : R → R by the formula φ_{q}(ψ_{q}(f)) = 1/q·Tr_{R/φ_{q}(R)} (f). The map ψ_{q} has the following properties (see for instance §2A of [FX13] and §1.2.3 of [Col16]): ψ_{q}(x·φ_{q}(y)) = ψ_{q}(x)·y, the map ψ_{q} commutes with the action of Γ_{F}, ∂∘ψ_{q} = π^{−1} ·ψ_{q}∘∂ and if f(T) ∈ B_{rig,F}^{+} then φ_{q}∘ψ_{q}(f) = 1/q·∑ _{z∈LT[π]} f(T ⊕z). If M is a free R-module with a semilinear Frobenius map φ_{q} such that Mat(φ_{q}) is invertible, then any m ∈ M can be written as m = ∑ _{i} r_{i} · φ_{q}(m_{i}) with r_{i} ∈ R and m_{i} ∈ M and the map ψ_{q} : m ↦ ∑ _{i} ψ_{q}(r_{i}) · m_{i} is then well-defined. This applies in particular to the rings B_{rig,K}^{†} , B_{rig,K}^{+} , B_{K}^{†}, B_{K} and to the (φ, Γ)-modules over them.

1.2 Construction of Lubin-Tate (φ, Γ)-modules

A (φ, Γ)-module over B_{K} (or over B_{K}^{†} or over B_{rig,K}^{†} ) is a finite dimensional B_{K} -vector space D (or a finite dimensional B_{K}^{†} -vector space or a free B_{rig,K}^{†} - module of finite rank respectively), along with a semilinear Frobenius map φ_{q} whose matrix (in some basis) is invertible, and a continuous, semilinear action of Γ_{K} that commutes with φ_{q}. We say that a (φ, Γ)-module D over B_{K} is étale if D has a basis in which Mat(φ_{q}) ∈ GL_{d}(A_{K}). Let B be the p-adic completion of ∪_{M/F} B_{M} where M runs through the finite extensions of F. By specializing the constructions of [Fon90], Kisin and Ren prove the following theorem (theorem 1.6 of [KR09]).

Theorem 1.2.1. The functors V ↦ D(V) = (B ⊗_{F} V)^{H_{K}} and D ↦ (B ⊗_{B_{K}} D)^{φ_{q}=1} give rise to mutually inverse equivalences of categories between the category of F -linear representations of G_{K} and the category of étale (φ, Γ)- modules over B_{K}.

We say that a (φ, Γ)-module D is overconvergent if there exists a basis of D in which the matrices of φ_{q} and of all g ∈ Γ_{K} have entries in B_{K}^{†}. This basis then generates a B_{K}^{†} -vector space D^{†} which is canonically attached to D. If V is a padic representation, we say that it is overconvergent if D(V) is overconvergent, and then D^{†}(V) denotes the corresponding (φ, Γ)-module over B_{K}^{†}. The main result of [CC98] states that if F = Q_{p}, then every étale (φ, Γ)-module over B_{K} is overconvergent (the proof is given for π = p, but it is easy to see that it works for any uniformizer). If F ≠ Q_{p}, some simple examples (see [FX13]) show that this is no longer the case. Recall that an F -linear representation of G_{K} is F -analytic if C_{p}⊗_{F}^{τ} V is the trivial C_{p}-semilinear representation of G_{K} for all embeddings τ ≠ Id ∈ Gal(F/Q_{p}).

This definition is the natural generalization of Kisin and Ren’s notion of F - crystalline representation. Kisin and Ren then show that if K ⊂ F_{∞} , and if V is a crystalline F -analytic representation of G_{K}, the (φ, Γ)-module attached to V is overconvergent (see §3.3 of [KR09]; they actually prove a stronger result, namely that the (φ, Γ)-module attached to such a V is of finite height). If D_{rig}^{†} is a (φ, Γ)-module over B_{rig,K}^{†} , and if g ∈ Γ_{K} is close enough to 1, then by standard arguments (see §2.1 of [KR09] or §1C of [FX13]), the series log(g) = log(1 + (g − 1)) gives rise to a differential operator ∇_{g} : D_{rig}^{†} → D_{rig}^{†}. The map v ↦ exp(v) is defined on a neighborhood of 0 in LieΓ_{K}; the map LieΓ_{K} → End(D_{rig}^{†}) arising from v ↦ ∇_{exp (v)} is Q_{p}-linear, and we say that D_{rig}^{†} is F -analytic if this map is F -linear (see §2.1 of [KR09] and §1.3 of [FX13]). If V is an overconvergent representation of G_{K}, we let D_{rig}^{†}(V) = B_{rig,K}^{†} ⊗_{B_{K}^{†}} D^{†}(V). The following is theorem D of [Ber16].

Theorem 1.2.2. The functor V ↦ D_{rig}^{†}(V) gives rise to an equivalence of categories between the category of F -analytic representations of G_{K} and the category of étale F -analytic Lubin-Tate (φ, Γ)-modules over B_{rig,K}^{†} .

In general, representations of G_{K} that are not F -analytic are not overconvergent (see §1.3), and the analogue of theorem 1.2.2 without the F -analyticity condition on both sides does not hold.

1.3 Overconvergent Lubin-Tate (φ, Γ)-modules

By theorem 1.2.2, there is an equivalence of categories between the category of F -analytic representations of G_{K} and the category of étale F -analytic Lubin-Tate (φ, Γ)-modules over B_{rig,K}^{†} . The purpose of this section is to prove a conjecture of Colmez that describes all overconvergent representations of G_{K}. Any representation V of G_{K} that factors through Γ_{K} is overconvergent, since H_{K} acts trivially on V so that D(V) = B_{K} ⊗_{F} V and therefore D(V) has a basis in which Mat(φ_{q}) = Id and Mat(g) ∈ GL_{d}(O_{F}) if g ∈ Γ_{K}. If X is F -analytic and Y factors through Γ_{K}, X ⊗_{F} Y is therefore overconvergent. We prove that any overconvergent representation of G_{K} is a quotient (and therefore also a subobject, by dualizing) of some representation of the form X ⊗_{F} Y as above.

Theorem 1.3.1. If V is an overconvergent representation of G_{K}, there exists an F -analytic representation X of G_{K}, a representation Y of G_{K} that factors through Γ_{K}, and a surjective G_{K} -equivariant map X ⊗_{F} Y → V.

Proof. Recall (see §3 of [Ber16]) that if r > 0, then inside B_{rig,K}^{†} we have the subring B_{rig,K}^{†,r} of elements defined on a fixed annulus whose inner radius depends on r and whose outer raidus is 1, and that (φ, Γ)-modules over B_{rig,K}^{†} can be defined over B_{rig,K}^{†,r} if r is large enough, giving us a module D_{rig}^{†,r} (V).

We also have rings B_{K}^{[r;s]} of elements defined on a closed annulus whose radii depend on r ⩽ s. One can think of an element of B_{rig,K}^{†,r} as a compatible family of elements of {B_{K}^{I} }_{I} where I runs over a set of closed intervals whose union is [r; +∞[. In the rest of the proof, we use this principle of glueing objects defined on closed annuli to get an object on the annulus corresponding to B_{rig,K}^{†,r} .

Choose r > 0 large enough such that D_{rig}^{†,r} (V) is defined, and s ⩾ qr. Let D^{[r;s]}(V) = B_{K}^{[r;s]} ⊗_{B_{rig ,K}^{†,r}} D_{rig}^{†,r} (V). If a ∈ O_{F}, and if val_{p}(a) ⩾ n for n = n(r, s) large enough, the series exp(a·∇) converges in the operator norm to an operator on the Banach space D^{[r;s]}(V). This way, we can define a twisted action of Γ_{K_{n}} on D^{[r;s]}(V), by the formula h ⋆ x = exp(log_{p}(χ_{π}(h)) · ∇)(x). This action is now F -analytic by construction. Since s ⩾ qr, the modules D^{[q^{m} r;q^{m} s]}(V) for m ⩾ 0 are glued together (using the idea explained above) by φ_{q} and we get a new action of Γ_{K_{n}} on D_{rig}^{†,r} (V) = D^{[r;+∞[} (V) and hence on D_{rig}^{†}(V). Since φ_{q} is unchanged, this new (φ, Γ)- module is étale, and therefore corresponds to a representation W of G_{K_{n}} . The representation W is F -analytic by theorem 1.2.2, and its restriction to H_{K} is isomorphic to V. Let X = ind_{G_{Kn}}^{G_{K}} W. By Mackey’s formula, X|_{H_{K}} contains W |_{H_{K}} ≃ V |_{H_{K}} as a direct summand. The space Y = Hom(ind_{G_{Kn}}^{G_{K}} W, V)^{H_{K}} is therefore a nonzero representation of Γ_{K}, and there is an element y ∈ Y whose image is V. The natural map X ⊗_{F} Y → V is therefore surjective. Finally, X is F -analytic since W is F -analytic.

By dualizing, we get the following variant of theorem 1.3.1.

Corollary 1.3.2. If V is an overconvergent representation of G_{K}, there exists an F -analytic representation X of G_{K}, a representation Y of G_{K} that factors through Γ_{K}, and an injective G_{K} -equivariant map V → X ⊗_{F} Y.

1.4 Extensions of (φ, Γ)-modules

In this section, we prove that there are no non-trivial extensions between an F -analytic (φ, Γ)-module and the twist of an F -analytic (φ, Γ)-module by a character that is not F -analytic. This is not used in the rest of the paper, but is of independent interest. If δ: Γ_{K} → O_{F}^{×} is a continuous character, and g ∈ Γ_{K}, let w_{δ}(g) = logδ(g)/ logχ_{π}(g). Note that δ is F -analytic if and only if w_{δ}(g) is independent of g ∈ Γ_{K}. We define the first cohomology group H^{1}(D) of a (φ, Γ)-module D as in §4 of [FX13]. Let D be a (φ, Γ)-module over B_{rig,K}^{†} . Let G denote the semigroup

φ_{q}^{Z_{⩾0}} × Γ_{K} and let Z^{1}(D) denote the set of continuous functions f: G → D such that (h − 1)f(g) = (g − 1)f(h) for all g, h ∈ G. Let B^{1}(D) be the subset of Z^{1}(D) consisting of functions of the form g ↦ (g − 1)y, y ∈ D and let H^{1}(D) = Z^{1}(D)/B^{1}(D). If g ∈ G and f ∈ Z^{1}, then [h ↦ (g − 1)f(h)] = [h ↦ (h−1)f(g)] ∈ B^{1}. The natural actions of Γ_{K} and φ_{q} on H^{1} are therefore trivial.

If D_{0} and D_{1} are two (φ, Γ)-modules, then Hom(D_{1}, D_{0}) = Hom_{B_{rig ,K}^{†} -mod} (D_{1}, D_{0}) is a free B_{rig,K}^{†} -module of rank rk(D_{0}) rk(D_{1}) which is

easily seen to be itself a (φ, Γ)-module. The space H^{1}(Hom(D_{1}, D_{0})) classifies the extensions of D_{1} by D_{0}. More precisely, if D is such an extension and if s: D_{1} → D is a B_{rig,K}^{†} -linear map that is a section of the projection D → D_{1}, then g ↦ s − g(s) is a cocycle on G with values in Hom(D_{1}, D_{0}) (the element g(s) ∈ Hom(D_{1}, D) being defined by g(s)(g(x)) = g(s(x)) for all g ∈ G and all x ∈ D_{1}). The class of this cocycle in the quotient H^{1}(Hom(D_{1}, D_{0})) does not depend on the choice of the section s, and every such class defines a unique extension of D_{1} by D_{0} up to isomorphism.

Theorem 1.4.1. If D is an F -analytic (φ, Γ)-module, and if δ: Γ_{K} → O_{F}^{×} is not locally F -analytic, then H^{1}(D(δ)) = {0}.

Proof. If g ∈ Γ_{K} and x(δ) ∈ D(δ) with x ∈ D, we have

∇_{g}(x(δ)) = ∇(x)(δ) + w_{δ}(g) · x(δ).

If g, h ∈ Γ_{K}, this implies that ∇_{g}(x(δ)) − ∇_{h}(x(δ)) = (w_{δ}(g) − w_{δ}(h)) · x(δ). If \overline{f} ∈ H^{1}(D(δ)) and g ∈ Γ_{K}, then g(\overline{f} ) = \overline{f} and therefore ∇_{g}(\overline{f} ) = 0. The formula above shows that if k ∈ Γ_{K}, then ∇_{g}(f(k))−∇_{h}(f(k)) = (w_{δ}(g)−w_{δ}(h))·f(k), so that 0 = (∇_{g} − ∇_{h})(\overline{f} ) = (w_{δ}(g) − w_{δ}(h)) · \overline{f} , and therefore \overline{f} = 0 if δ is not locally analytic.

2 Analytic cohomology and Iwasawa theory

In this chapter, we explain how to construct classes in the cohomology groups of F -analytic (φ, Γ)-modules. This allows us to define our maps h_{K_{n} ,V}^{1} .

2.1 Analytic cohomology

Let G be an F -analytic semigroup and let M be a Fréchet or LF space with a pro-F -analytic (§2 of [Ber16]) action of G. Recall that this means that we can write M = \underrightarrow{lim}_{i} \underleftarrow{lim}_{j} M_{ij} where M_{ij} is a Banach space with a locally analytic action of G. A function f : G → M is said to be pro-F -analytic if its image lies in \underleftarrow{lim}_{j} M_{ij} for some i and if the corresponding function f : G → M_{ij} is locally F -analytic for all j. The analytic cohomology groups H_{an}^{i}(G, M) are defined and studied in §4 of [FX13] and §5 of [Col16]. In particular, we have H_{an}^{0}(G, M) = M^{G} and H_{an}^{1}(G, M) = Z_{an}^{1}(G, M)/B_{an}^{1}(G, M) where Z_{an}^{1}(G, M) is the set of pro-F - analytic functions f : G → M such that (g − 1)f(h) = (h − 1)f(g) for all g, h ∈ G and B_{an}^{1}(G, M) is the set of functions of the form g ↦ (g − 1)m. Let M be a Fréchet space, and write M = \underleftarrow{lim}_{n} M_{n} with M_{n} a Banach space such that the image of M_{n+j} in M_{n} is dense for all j ⩾ 0.

Proposition 2.1.1. We have H_{an}^{1}(G, M) = \underleftarrow{lim}_{n} H_{an}^{1}(G, M_{n}).

Proof. By definition, we have an exact sequence

0 → B_{an}^{1}(G, M_{n}) → Z_{an}^{1}(G, M_{n}) → H_{an}^{1}(G, M_{n}) → 0.

It is clear that B_{an}^{1}(G, M) = \underleftarrow{lim}_{n} B_{an}^{1}(G, M_{n}) and that Z_{an}^{1}(G, M) = \underleftarrow{lim}_{n} Z_{an}^{1}(G, M_{n}), since these spaces are spaces of functions on G satisfying certain compatible conditions. The Banach spaces B_{an}^{1}(G, M_{n}) satisfy the Mittag-Leffler condition: B_{an}^{1}(G, M_{n}) = M_{n}/M_{n}^{G} and the image of M_{n+j} in M_{n} is dense for all j ⩾ 0. This implies that the sequence

0 → \underleftarrow{lim}_{n}B_{an}^{1}(G, M_{n}) → \underleftarrow{lim}_{n}Z_{an}^{1}(G, M_{n}) → \underleftarrow{lim}_{n}H_{an}^{1}(G, M_{n}) → 0

is exact, and the proposition follows.

In this paper, we mainly use the semigroups Γ_{K}, Γ_{K} × Φ where Φ = {φ_{q}^{n}, n ∈ Z_{⩾0}} and Γ_{K} × Ψ where Ψ = {ψ_{q}^{n}, n ∈ Z_{⩾0}}. The semigroups Φ and Ψ are discrete and the F -analytic structure comes from the one on Γ_{K}.

Definition 2.1.2. Let G be a compact group and let H be an open subgroup of G. We have the corestriction map cor : H_{an}^{1}(H, M) → H_{an}^{1}(G, M), which satisfies cor ∘ res = [G : H]. This map has the following equivalent explicit descriptions (see §2.5 of [Ser94] and §II.2 of [CC99]). Let X ⊂ G be a set of representatives of G/H and let f ∈ Z_{an}^{1}(H, M) be a cocycle.

1. By Shapiro’s lemma, H_{an}^{1}(H, M) = H_{an}^{1}(G, ind_{H}^{G} M) and cor is the map induced by i ↦ ∑ _{x∈X} x · i(x^{−1} );

2. if M ⊂ N where N is a G-module and if there exists n ∈ N such that f(h) = (h − 1)(n), then cor(f)(g) = (g − 1)(∑ _{x∈X} xn);

3. if g ∈ G, let τ_{g} : X → X be the permutation defined by τ_{g}(x)H = gxH. We have cor(f)(g) = ∑ _{x∈X} τ_{g}(x) · f(τ_{g}(x)^{−1} gx).

If g ∈ Γ_{K}, let ℓ(g) = log_{p} χ_{π}(g). If M is a Fréchet space with a pro-F -analytic action of Γ_{K} and if g ∈ Γ_{K} is such that χ_{π}(g) ∈ 1 + 2pO_{F}, then lim_{n→∞} (g^{p^{n}} − 1)/(p^{n}ℓ(g)) converges to an operator ∇ on M, which is independent of g thanks to the F -analyticity assumption. If c : Γ_{K} → M is an F -analytic map, let c′(1) denote its derivative at the identity.

Proposition 2.1.3. If M is a Fréchet space with a pro-F -analytic action of Γ_{K}, the map c ↦ c′(1) induces an isomorphism H_{an}^{1}(Γ_{K} , M) = (M/∇M)^{Γ_{K}} , under which cor_{L/K} corresponds to Tr_{L/K} .

Proof. Assume for the time being that M is a Banach space. We first show that the map induced by c ↦ c′(1) is well-defined and lands in (M/∇M)^{Γ_{K}} . The map c ↦ c′(1) from Z_{an}^{1}(Γ_{K} , M) → M is well-defined, and if c(g) = (g − 1)m, then c′(1) = ∇m so that there is a well-defined map H_{an}^{1}(Γ_{K} , M) → M/∇M. If h ∈ Γ_{K} then (h − 1)c′(1) = lim_{g→1} (h − 1)c(g)/ℓ(g) = lim_{g→1} (g − 1)c(h)/ℓ(g) = ∇c(h) so that the image of c ↦ c′(1) lies in (M/∇M)^{Γ_{K}} . The formula for the corestriction follows from the explicit descriptions above: if h ∈ Γ_{L} then τ_{h}(x) = x so that cor(c)(h) = ∑ _{x∈X} x · c(h) and

cor(c)′(1) = lim_{h→1} cor(c)(h)/ℓ(h) = ∑_{x∈X} x · c′(1) = Tr_{L/K} (c′(1)).

We now show that the map is injective. If c′(1) = ∇m, then the derivative of g ↦ c(g) − (g − 1)m at g = 1 is zero and hence c(g) = (g − 1)m on some open subgroup Γ_{L} of Γ_{K} and c = [L : K]^{−1} cor_{L/K} ∘ res_{K/L} (c) = 0. We finally show that the map is surjective. Suppose now that y ∈ (M/∇M)^{Γ_{K}} . The formula g ↦ (exp(ℓ(g)∇) − 1)/∇ · y defines an analytic cocycle c_{L} on some open subgroup Γ_{L} of Γ_{K}. The image of [L : K]^{−1} c_{L} under cor_{L/K} gives a cocyle c ∈ H_{an}^{1}(Γ_{K} , M) such that c′(1) = y. We now let M = \underleftarrow{lim}_{n} M_{n} be a Fréchet space. The map H_{an}^{1}(Γ_{K} , M) → (M/∇M)^{Γ_{K}} induced by c ↦ c′(1) is well-defined, and in the other direction we have the map y ↦ c_{y}:

(M/∇M)^{Γ_{K}} → \underleftarrow{lim}_{n}(M_{n}/∇M_{n})^{Γ_{K}} → \underleftarrow{lim}_{n}H_{an}^{1}(Γ_{K} , M_{n}) → H_{an}^{1}(Γ_{K} , M).

These two maps are inverses of each other.

Remark 2.1.4. Compare with the following theorem (see [Tam15], corollary 21): if G is a compact p-adic Lie group and if M is a locally analytic representation of G, then H_{an}^{i}(G, M) = H^{i}(Lie(G), M)^{G}.

2.2 Cohomology of F -analytic (φ, Γ)-modules

If V is an F -analytic representation, let H_{an}^{1}(K, V) ⊂ H^{1}(K, V) classify the F -analytic extensions of F by V. Let D denote an F -analytic (φ, Γ)-module over B_{rig,K}^{†} , such as D_{rig}^{†}(V).

Proposition 2.2.1. If V is F -analytic, then H_{an}^{1}(K, V) = H_{an}^{1}(Γ_{K} × Φ, D_{rig}^{†}(V)).

Proof. The group H_{an}^{1}(Γ_{K} × Φ, D_{rig}^{†}(V)) classifies the F -analytic extensions of B_{rig,K}^{†} by D_{rig}^{†}(V), which correspond to F -analytic extensions of F by V by theorem 1.2.2.

Theorem 2.2.2. If D is an F -analytic (φ, Γ)-module over B_{rig,K}^{†} and i = 0, 1, then H_{an}^{i}(Γ_{K} , D^{ψ_{q}=0} ) = 0.

Proof. Since B_{rig,F}^{†} ⊂ B_{rig,K}^{†} , the B_{rig,K}^{†} -module D is a free B_{rig,F}^{†} -module of finite rank. Let R_{F} denote B_{rig,F}^{†} and let R_{C_{p}} denote C_{p} ˆ⊗_{F} B_{rig,F}^{†} the Robba ring with coefficients in C_{p}. There is an action of G_{F} on the coefficients of R_{C_{p}} and R_{C_{p}}^{G_{F}} = R_{F}. Theorem 5.5 of [Col16] says that H_{an}^{i}(Γ_{K} , (R_{C_{p}} ⊗_{R_{F}} D)^{ψ_{q}=0} ) = 0. For i = 0, this implies our claim. For i = 1, it says that if c : Γ_{K} → D^{ψ_{q}=0} is an F - analytic cocycle, there exists m ∈ (R_{C_{p}} ⊗_{R_{F}} D)^{ψ_{q}=0} such that c(g) = (g − 1)m for all g ∈ Γ_{K}. If α ∈ G_{F}, then c(g) = (g − 1)α(m) as well, so that α(m) − m ∈ ((R_{C_{p}} ⊗_{R_{F}} D)^{ψ_{q}=0} )^{Γ_{K}} = 0. This shows that m ∈ ((R_{C_{p}} ⊗_{R_{F}} D)^{ψ_{q}=0} )^{G_{F}} = D^{ψ_{q}=0} .

Corollary 2.2.3. The groups H_{an}^{i}(Γ_{K} × Φ, D) and H_{an}^{i}(Γ_{K} × Ψ, D) are isomorphic for i = 0, 1.

Proof. If i = 0, then we have an inclusion D^{φ_{q}=1,Γ_{K}} ⊂ D^{ψ_{q}=1,Γ_{K}} . If x ∈ D^{ψ_{q}=1,Γ_{K}} , then x − φ_{q}(x) ∈ D^{ψ_{q}=0,Γ_{K}} = {0} by theorem 2.2.2, so that x = φ_{q}(x) and the above inclusion is an equality. Now let i = 1. If f ∈ Z_{an}^{1}(Γ_{K} × Φ, D), let T f ∈ Z_{an}^{1}(Γ_{K} × Ψ, D) be the function defined by T f(g) = f(g) if g ∈ Γ_{K} and T f(ψ_{q}) = −ψ_{q}(f(φ_{q})). If f ∈ Z_{an}^{1}(Γ_{K} × Ψ, D) and g ∈ Γ_{K}, then (φ_{q}ψ_{q} − 1)f(g) ∈ D^{ψ_{q}=0} and the map g ↦ (φ_{q}ψ_{q}−1)f(g) is an element of Z_{an}^{1}(Γ_{K} , D^{ψ_{q}=0} ). By theorem 2.2.2, applied once for existence and once for unicity, there is a unique m_{f} ∈ D^{ψ_{q}=0} such that (φ_{q}ψ_{q} − 1)f(g) = (g − 1)m_{f} . Let Uf ∈ Z_{an}^{1}(Γ_{K} × Φ, D) be the function defined by Uf(g) = f(g) if g ∈ Γ_{K} and Uf(φ_{q}) = −φ_{q}(f(ψ_{q})) + m_{f} . It is straightforward to check that U and T are inverses of each other (even at the level of the Z_{an}^{1}) and that they descend to the H_{an}^{1}.

Theorem 2.2.4. The map f ↦ f(ψ_{q}) from Z_{an}^{1}(Γ_{K} × Ψ, D) to D gives rise to an exact sequence:

0 → H_{an}^{1}(Γ_{K} , D^{ψ_{q}=1} ) → H_{an}^{1}(Γ_{K} × Ψ, D) → ( \frac{D}{ψ_{q} − 1} ) ^{Γ_{K}}

Proof. If f ∈ Z_{an}^{1}(Γ_{K} × Ψ, D) and g ∈ Γ_{K}, then (g − 1)f(ψ_{q}) = (ψ_{q} − 1)f(g) ∈ (ψ_{q} − 1)D so that the image of f is in (D/(ψ_{q} − 1))^{Γ_{K}} . The other verifications are similar.

2.3 The space D/(ψ_{q} − 1)

By theorem 2.2.4 in the previous section, the cokernel of the map H_{an}^{1}(Γ_{K} , D^{ψ_{q}=1} ) → H_{an}^{1}(Γ_{K} × Ψ, D) injects into (D/(ψ_{q} − 1))^{Γ_{K}} . It can be useful to know that this cokernel is not too large. In this section, we bound D/(ψ_{q} − 1) when D = B_{rig,F}^{†} , with the action of φ_{q} twisted by a^{−1} , for some a ∈ F^{×} .

Theorem 2.3.1. If a ∈ F^{×} , then ψ_{q} − a : B_{rig,F}^{†} → B_{rig,F}^{†} is onto unless a = q^{−1} π^{m} for some m ∈ Z_{⩾1}, in which case B_{rig,F}^{†} /(ψ_{q} − a) is of dimension 1.

In order to prove this theorem, we need some results about the action of ψ_{q} on B_{rig,F}^{†} . Recall that the map ∂ = d/dt_{π} was defined in §1.1.

Lemma 2.3.2. If a ∈ F^{×} , then aφ_{q} − 1 : B_{rig,F}^{+} → B_{rig,F}^{+} is an isomorphism, unless a = π^{−m} for some m ∈ Z_{⩾0}, in which case

ker(aφ_{q} − 1 : B_{rig,F}^{+} → B_{rig,F}^{+} ) = F t_{π}^{m}
im(aφ_{q} − 1 : B_{rig,F}^{+} → B_{rig,F}^{+} ) = {f(T) ∈ B_{rig,F}^{+} | ∂^{m}(f)(0) = 0}.

Proof. This is lemma 5.1 of [FX13].

Lemma 2.3.3. If m ∈ Z_{⩾0}, there is an h(T) ∈ (B_{rig,F}^{+} )^{ψ_{q}=0} such that ∂^{m}(h)(0) ≠ 0.

Proof. We have ψ_{q}(T) = 0 by (the proof of) proposition 2.2 of [FX13]. If there was some m_{0} such that ∂^{m}(T)(0) = 0 for all m ⩾ m_{0}, then T would be a polynomial in t_{π}, which it is not. This implies that there is a sequence {m_{i}}_{i} of integers with m_{i} → +∞, such that ∂^{m_{i}} (T)(0) ≠ 0, and we can take h(T) = ∂^{m_{i} −m} (T) for any m_{i} ⩾ m.

Corollary 2.3.4. If a ∈ F^{×} , then ψ_{q} − a : B_{rig,F}^{+} → B_{rig,F}^{+} is onto.

Proof. If f(T) ∈ B_{rig,F}^{+} and if we can write f = (1 − aφ_{q})g, then f = (ψ_{q} − a)(φ_{q}(g)). If this is not possible, then by lemma 2.3.2 there exists m ⩾ 0 such that a = π^{−m} and ∂^{m}(f)(0) ≠ 0. Let h be the function provided by lemma 2.3.3. The function f − (∂^{m}(f)(0)/∂^{m}(h)(0)) · h is in the image of 1 − aφ_{q} by lemma 2.3.2, and h = (ψ_{q} − a)(−a^{−1} h) since ψ_{q}(h) = 0. This implies that f is in the image of ψ_{q} − a.

Lemma 2.3.5. If a^{−1} ∈ q · O_{F}, then ψ_{q} − a : B_{rig,F}^{†} → B_{rig,F}^{†} is onto.

Proof. We have B_{rig,F}^{†} = B_{rig,F}^{+} +B_{F}^{†} (by writing a power series as the sum of its plus part and of its minus part) and by corollary 2.3.4, ψ_{q}−a : B_{rig,F}^{+} → B_{rig,F}^{+}

is onto. Take f(T) ∈ B_{F}^{†}, choose some r > 0 and let B_{F}^{(0,r]} be the set of f(T) ∈ B_{F}^{†} that converge and are bounded on the annulus 0 < val_{p}(x) ⩽ r. It follows from proposition 1.4 of [Col16] that if n ≫ 0, then ψ_{q}^{n}(f) ∈ B_{F}^{(0,r]} and by proposition 2.4(d) of [FX13], the sequence (q/π · ψ_{q})^{n}(f) is bounded in B_{F}^{(0,r]} . The series ∑ _{n⩾0} a^{−1−n} ψ_{q}^{n}(f) therefore converges in B_{F}^{(0,r]} , and we can write f = (ψ_{q} − a)g where g = a^{−1} (1 − a^{−1} ψ_{q})^{−1} f = ∑ _{n⩾0} a^{−1−n} ψ_{q}^{n}(f).

Let Res : B_{rig,F}^{†} → F be defined by Res(f) = a_{−1} where f(T)dt_{π} = ∑ _{n∈Z} a_{n}T^{n}dT. The following lemma combines propositions 2.12 and 2.13 of [FX13].

Lemma 2.3.6. The sequence 0 → F → B_{rig,F}^{†} ⟶^{∂} B_{rig,F}^{†} −−→^{Res} F → 0 is exact, and Res(ψ_{q}(f)) = π/q · Res(f).

Proof of theorem 2.3.1. Since ∂ ∘ ψ_{q} = π^{−1} ψ_{q} ∘ ∂, the map ∂ induces a map:

\frac{B_{rig,F}^{†}}{ψ_{q} − a} −−−−→^{∂} \frac{B_{rig,F}^{†}}{ψ_{q} − aπ} .  (Der)

Take x ∈ B_{rig,F}^{†} such that Res(x) = 1. We have Res((ψ_{q} − aπ)x) = π/q − aπ. If a ≠ q^{−1} , this is non-zero and if f ∈ B_{rig,F}^{†} , proposition 2.3.6 allows us to write f = ∂g + Res(f)/(π/q − aπ) · (ψ_{q} − aπ)x. This implies that (Der) is onto if a ≠ q^{−1} . Combined with lemma 2.3.5, this implies that B_{rig,F}^{†} /(ψ_{q} − a) = 0 if a is not of the form q^{−1} π^{m} for some m ∈ Z_{⩾1}. When a = q^{−1} , we have an exact sequence

\frac{B_{rig,F}^{†}}{ψ_{q} − q^{−1}} −−−−→^{∂} \frac{B_{rig,F}^{†}}{ψ_{q} − q^{−1} π} −−−−⟶^{Res} F → 0,

which now implies that B_{rig,F}^{†} /(ψ_{q} − q^{−1} π) = F, generated by the class of x. We now assume again that a ≠ q^{−1} and compute the kernel of (Der). If f ∈ B_{rig,F}^{†} is such that ∂f = (ψ_{q} − aπ)g, then Res∂f = Res(ψ_{q} − aπ)g = (π/q − aπ) Res(g), so that Res(g) = 0 and we can write g = ∂h. We have ∂(f − (ψ_{q} − a)h) = 0, so that f = (ψ_{q} − a)h + c, with c ∈ F. By corollary 2.3.4, there exists b ∈ B_{rig,F}^{+} such that (ψ_{q} − a)(b) = c, so that f = (ψ_{q} − a)(h + b) and (Der) is bijective. We then have, by induction on m ⩾ 1, that B_{rig,F}^{†} /(ψ_{q}− q^{−1} π^{m}) = F, generated by the class of ∂^{m}(x).

Remark 2.3.7. More generally, we expect that the following holds: if D is a (φ, Γ)-module over B_{rig,K}^{†} , the F -vector space D/(ψ_{q} − 1) is finite dimensional.

2.4 The operator Θ_{b}

The power series F (X) = X/(exp(X)−1) belongs to Q_{p}[[X]] and has a nonzero radius of convergence. If M is a Banach space with a locally F -analytic action of Γ_{K} and h ∈ Γ_{K} is close enough to 1, then

\frac{∇}{h − 1} = \frac{∇}{exp(ℓ(h)∇) − 1} = ℓ(h)^{−1} F (ℓ(h)∇)

converges to a continuous operator on M. If g ∈ Γ_{K}, we then define

\frac{∇}{1 − g} = \frac{∇}{1 − g^{n}} · \frac{1 − g^{n}}{1 − g} .

This operator is independent of the choice of n such that g^{n} is close enough to 1, and can be seen as an element of the locally F -analytic distribution algebra acting on M.

If M is a Fréchet space, write M = \underleftarrow{lim}_{i} M_{i} and define operators \frac{∇}{1−g} on each M_{i} as above. These operators commute with the maps M_{j} → M_{i} (because n can be taken large enough for both M_{i} and M_{j}). This defines an operator \frac{∇}{1−g} on M itself. The definition of \frac{∇}{1−g} extends to an LF space with a pro-F -analytic action of Γ_{K}. Assume that K contains F_{1} and let r(K) = f + val_{p}([K : F_{1}]). For example, p^{r(F_{n} )} = q^{n} if n ⩾ 1. Assume further that K contains F_{n(K)} , so that χ_{π} : Γ_{K} → O_{F}^{×} is injective and its image is a free Z_{p}-module of rank d. If b = (b_{1}, …, b_{d}) is a basis of Γ_{K} (that is, Γ_{K} = b_{1}^{Z_{p}} ⋯ b_{d}^{Z_{p}} ), then let ℓ^{∗}(b) = ℓ(b_{1}) ⋯ ℓ(b_{d})/p^{r(K)} and

Θ_{b} = ℓ^{∗}(b) · \frac{∇^{d}}{(b_{1} − 1) ⋯ (b_{d} − 1)} .

Lemma 2.4.1. If K = F_{n} and m ⩾ 0 and x ∈ F_{m+n} , then

Θ_{b}(x) = q^{−m−n} · Tr_{F_{m+n} /F_{n}} (x).

Proof. Since ∇ = lim_{k→∞} (b^{p^{k}} − 1)/p^{k}ℓ(b), we have

Θ_{b} = lim_{k→∞} \frac{1}{q^{n}p^{kd}} · \frac{(b_{1}^{p^{k}} − 1) ⋯ (b_{d}^{p^{k}} − 1)}{(b_{1} − 1) ⋯ (b_{d} − 1)} .

The set {b_{1}^{a_{1}} ⋯ b_{d}^{a_{d}} } with 0 ⩽ a_{i} ⩽ p^{k} − 1 runs through a set of representatives of Γ_{n}/Γ_{n}^{p^{k}} = Γ_{n}/Γ_{n+ek} so that

\frac{1}{q^{n}p^{kd}} · \frac{(b_{1}^{p^{k}} − 1) ⋯ (b_{d}^{p^{k}} − 1)}{(b_{1} − 1) ⋯ (b_{d} − 1)} = \frac{1}{q^{n}p^{kd}}Tr_{F_{n+ek} /F_{n}} = \frac{1}{q^{n+ek}} · Tr_{F_{n+ek} /F_{n}} .

The lemma follows from taking k large enough so that ek ⩾ m.

For i ∈ Z, let ∇_{i} = ∇ − i.

Lemma 2.4.2. If b is a basis of Γ_{F_{n}} and if f(T) ∈ (B_{rig,F}^{+} )^{ψ_{q}=0} , then Θ_{b}(f(T)) ∈ (t_{π}/φ_{q}^{n}(T)) · B_{rig,F}^{+} , and if h ⩾ 2 then ∇_{h−1} ∘ ⋯ ∘ ∇_{1} ∘ Θ_{b}(f(T)) ∈ (t_{π}/φ_{q}^{n}(T))^{h} · B_{rig,F}^{+} .

Proof. If m ⩾ 1, then by lemma 2.4.1 and using repeatedly the fact (see §1.1) that φ_{q} ∘ ψ_{q}(f) = 1/q · ∑ _{z∈LT[π]} f(T ⊕ z),

Θ_{b}(f(u_{n+m} )) = 1/q^{m+n} · Tr_{F_{m+n} /F_{n}} f(u_{m+n} ) = ψ_{q}^{m}(f)(u_{n}) = 0.

This proves the first claim, since an element f(T) ∈ B_{rig,F}^{+} is divisible by t_{π}/φ_{q}^{n}(T) if and only if f(u_{n+m} ) = 0 for all m ⩾ 1. The second claim follows easily.

Let D be a φ_{q}-module over F. Let φ_{q}^{−n} : B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D → F_{n}((t_{π})) ⊗_{F} D be the map

φ_{q}^{−n} : t_{π}^{−h} f(T) ⊗ x ↦ π^{nh} t_{π}^{−h} f(u_{n} ⊕ exp_{LT}(t_{π}/π^{n})) ⊗ φ_{q}^{−n} (x).

If f(t_{π}) ∈ F_{n}((t_{π})) ⊗_{F} D, let ∂_{D}(f) ∈ F_{n} ⊗_{F} D denote the coefficient of t_{π}^{0}.

Lemma 2.4.3. If y ∈ (B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D)^{ψ_{q}=1} and if m ⩾ n, then

q^{−m} Tr_{F_{m} /F_{n}} ∂_{D}(φ_{q}^{−m} (y)) = \{ \begin{matrix}q^{−n} ∂_{D}(φ_{q}^{−n} (y)) & if n ⩾ 1 \\ (1 − q^{−1} φ_{q}^{−1} )∂_{D}(y) & if n = 0. \end{matrix}

Proof. If y = t_{π}^{−ℓ} ∑ _{k=0}^{+∞} a_{k}T^{k} ∈ B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D, then (by definition of φ_{q}^{−m} )

φ_{q}^{−m} (y) = π^{mℓ}t_{π}^{−ℓ} ∑_{k=0}^{+∞} φ_{q}^{−m} (a_{k})(u_{m} ⊕ exp_{LT}(t_{π}/π^{m}))^{k},

and ψ_{q}(y) = y means that:

φ_{q}(y)(T) = \frac{1}{q} ∑_{[π](ω)=0} y(T ⊕ ω).

If m ⩾ 2, the conjugates of u_{m} under Gal(F_{m}/F_{m−1} ) are the {ω ⊕ u_{m}}_{[π](ω)=0} so that:

Tr_{F_{m} /F_{m−1}} ∂_{D}(φ_{q}^{−m} (y))

= ∂_{D} ( ∑_{[π](ω)=0} π^{mℓ}t_{π}^{−ℓ} ∑_{k=0}^{+∞} φ_{q}^{−m} (a_{k})(ω ⊕ u_{m} ⊕ exp_{LT}(t_{π}/π^{m}))^{k} )

= ∂_{D} ( φ_{q}^{−m} ( ∑_{[π](ω)=0} y(T ⊕ ω)) )

= q∂_{D}(φ_{q}^{−(m−1)} (y)).

For m = 1, the computation is similar, except that the conjugates of u_{1} under Gal(F_{1}/F) are the ω, where [π](ω) = 0 but ω ≠ 0, which results in:

Tr_{F_{1} /F} ∂_{D}(φ_{q}^{−1} (y)) = ∂_{D} ( φ_{q}^{−1} ( ∑_{\begin{matrix}[π](ω)=0 \\ ω≠0 \end{matrix}} y(T ⊕ ω)) ) = ∂_{D}(qy − φ_{q}^{−1} (y)).

2.5 Construction of extensions

Let D be an F -analytic (φ, Γ)-module over B_{rig,K}^{†} . The space D^{ψ_{q}=1} is a closed subspace of D and therefore an LF space. Take K such that K contains F_{n(K)} and let b be a basis of Γ_{K}.

Proposition 2.5.1. If y ∈ D^{ψ_{q}=1} , there is a unique cocycle c_{b}(y) ∈ Z_{an}^{1}(Γ_{K} , D^{ψ_{q}=1} ) such that for all 1 ⩽ j ⩽ d and k ⩾ 0, we have

c_{b}(y)(b_{j}^{k}) = ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (y).

We then have c_{b}(y)′(1) = Θ_{b}(y).

Proof. There is obviously one and only one continuous cocycle satisfying the conditions of the proposition. It is Q_{p}-analytic, and in order to prove that it is F -analytic, we need to check that the directional derivatives are independent of j. We have

lim_{k→0} \frac{c_{b}(y)(b_{j}^{k})}{ℓ(b_{j}^{k})} = ℓ^{∗}(b) · \frac{∇^{d}}{∏ _{i}(b_{i} − 1)} (y) = Θ_{b}(y),

which is indeed independent of j, and thus c_{b}(y)′(1) = Θ_{b}(y).

Lemma 2.5.2. If n ⩾ n(K) and L = K_{n} and M = K_{n+e} and b is a basis of Γ_{L}, then b^{p} is a basis of Γ_{M} and cor_{M/L} c_{b^{p}}(y) = c_{b}(y).

Proof. The Lubin-Tate character maps Γ_{L} to 1+π^{n}O_{F}, and Γ_{M} = Γ_{L}^{p} because (1 + π^{n}O_{F})^{p} = 1 + π^{n+e} O_{F}. Since {b_{1}^{k_{1}} ⋯ b_{d}^{k_{d}} } with 0 ⩽ k_{i} ⩽ p − 1 is a set of representatives for Γ_{L}/Γ_{M}, and since [M : L] = q^{e} = p^{d}, the explicit formula for the corestriction (definition 2.1.2) implies (here and elsewhere ⌈x⌉ is the smallest integer ⩾ x)

cor_{M/L} (c_{b^{p}}(y))(b_{j}^{k})

= ∑_{0⩽k_{1} ,...,k_{d} ⩽p−1} b_{1}^{k_{1}} …b_{d}^{k_{d}} · ℓ^{∗}(b^{p}) · \frac{b_{j}^{p⌈\frac{k−k_{j}}{p} ⌉} − 1}{b_{j}^{p} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i}^{p} − 1)} (y)

= ℓ^{∗}(b) ( ∑_{k_{j} =0}^{p−1} b_{j}^{k_{j}} \frac{b_{j}^{p⌈\frac{k−k_{j}}{p} ⌉} − 1}{b_{j}^{p} − 1} ) · ( ∏_{i≠j} \frac{b_{i}^{p} − 1}{b_{i} − 1} ) · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i}^{p} − 1)} (y)

= ℓ^{∗}(b)\frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (y)

= c_{b}(y)(b_{j}^{k}).

This proves the lemma.

Lemma 2.5.3. If a and b are two bases of Γ_{K}, then c_{a}(y) and c_{b}(y) are cohomologous.

Proof. If α_{1}, …, α_{d} and β_{1}, …, β_{d} are in F^{×} , the Laurent series

\frac{α_{1} ⋯ α_{d} · T^{d−1}}{(exp(α_{1}T) − 1) ⋯ (exp(α_{d}T) − 1)} − \frac{β_{1} ⋯ β_{d} · T^{d−1}}{(exp(β_{1}T) − 1) ⋯ (exp(β_{d}T) − 1)}

is the difference of two Laurent series, each having a simple pole at 0 with equal residues, and therefore belongs to F [[T]]. Let a and b be two bases of Γ_{K} and take y ∈ D^{ψ_{q}=1} . Let N be a Γ_{K} -stable Fréchet subspace of D that contains y and write N = \underleftarrow{lim}M_{j}. Since M = M_{j} is F -analytic, we have g = exp(ℓ(g)∇) on M for g

in some open subgroup of Γ_{K}. Let k ≫ 0 be large enough such that a_{i}^{p^{k}} and b_{i}^{p^{k}} are in this subgroup, and let α_{i} = p^{k}ℓ(a_{i}) and β_{i} = p^{k}ℓ(b_{i}). Taking k large enough (depending on M), we can assume moreover that the power series T/(exp(T) − 1) applied to the operators α_{i}∇ and β_{i}∇ converges on M. The element

w = ( \frac{α_{1} ⋯ α_{d} · ∇^{d−1}}{(exp(α_{1}∇) − 1) ⋯ (exp(α_{d}∇) − 1)}

− \frac{β_{1} ⋯ β_{d} · ∇^{d−1}}{(exp(β_{1}∇) − 1) ⋯ (exp(β_{d}∇) − 1)} ) (y)

of M is well defined. By proposition 2.5.1, we have

c_{a^{p^{k}}} (y)′(1) − c_{b^{p^{k}}} (y)′(1) = Θ_{a^{p^{k}}} (y) − Θ_{b^{p^{k}}} (y) = p^{−r(L)} ∇(w)

where L is the extension of K such that Γ_{L} = Γ_{K}^{p^{k}}. Thus, for g close enough to 1, we have c_{a^{p^{k}}} (y)(g) − c_{b^{p^{k}}} (y)(g) = (g − 1)(p^{−r(L)} w). Lemma 2.5.2 now implies by corestricting that this holds for all g, and, by corestricting again, that c_{a}(y) and c_{b}(y) are cohomologous in M. By varying M, we get the same result in N, which implies the proposition.

Lemma 2.5.4. If L/K is a finite extension contained in K_{∞} , and if b is a basis of Γ_{K} and a is a basis of Γ_{L}, then cor_{L/K} c_{a}(y) = c_{b}(y).

Proof. The groups Γ_{K} and Γ_{L} are both free Z_{p}-modules of rank d, so that by the elementary divisors theorem, we can change the bases a and b in such a way that there exists e_{1}, …, e_{d} with a_{i} = b_{i}^{p^{e_{i}}} . Since {b_{1}^{k_{1}} ⋯ b_{d}^{k_{d}} } with 0 ⩽ k_{i} ⩽ p^{e_{i}} − 1 is a set of representatives for Γ_{K} /Γ_{L}, and since [L : K] = p^{e_{1} +···+e_{d}} , the explicit formula for the corestriction implies

cor_{L/K} (c_{a}(y))(b_{j}^{k})

= ∑_{\begin{matrix}0⩽k_{1} ⩽p^{e_{1}} −1 \\ ... \\ 0⩽k_{d} ⩽p^{e_{d}} −1 \end{matrix}} b_{1}^{k_{1}} …b_{d}^{k_{d}} · ℓ^{∗}(a) · \frac{a_{j}^{⌈ \frac{k−k_{j}}{p^{e_{j}}} ⌉} − 1}{a_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (a_{i} − 1)} (y)

= ℓ^{∗}(b) · ( ∑_{k_{j} =0}^{p^{e_{j}} −1} \frac{a_{j}^{⌈ \frac{k−k_{j}}{p^{e_{j}}} ⌉} − 1}{a_{j} − 1} ) · ( ∏_{i≠j} \frac{a_{i} − 1}{b_{i} − 1} ) · \frac{∇^{d−1}}{∏ _{i≠j} (a_{i} − 1)} (y)

= ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (y)

= c_{b}(y)(b_{j}^{k}).

Definition 2.5.5. Let h_{K,V}^{1} : D_{rig}^{†}(V)^{ψ_{q}=1} → H_{an}^{1}(K, V) denote the map obtained by composing y ↦ \overline{c}_{b}(y) with H_{an}^{1}(Γ_{K} , D_{rig}^{†}(V)^{ψ_{q}=1} ) → H_{an}^{1}(Γ_{K} × Ψ, D_{rig}^{†}(V)) (theorem 2.2.4) and with H_{an}^{1}(Γ_{K} × Ψ, D_{rig}^{†}(V)) ≃ H_{an}^{1}(K, V) (proposition 2.2.1 and corollary 2.2.3).

Proposition 2.5.6. We have cor_{M/L} ∘h_{M,V}^{1} = h_{L,V}^{1} if M/L is a finite extension contained in K_{∞} /K_{n(K)} . In particular, cor_{K_{n+1} /K_{n}} ∘ h_{K_{n+1} ,V}^{1} = h_{K_{n} ,V}^{1} if n ⩾ n(K).

Proof. This follows from the definition and from lemma 2.5.4 above.

Remark 2.5.7. Proposition 2.5.6 allows us to extend the definition of h_{K,V}^{1} to all K, without assuming that K contains F_{n(K)} , by corestricting.

Some of the constructions of this section are summarized in the following theorem. Recall (see §3 of [Ber16]) that there is a ring B̃_{rig}^{†} that contains B_{rig,F}^{†} , is equipped with a Frobenius map φ_{q} and an action of G_{F} and such that V = (B̃_{rig}^{†} ⊗_{B_{rig ,F}^{†}} D_{rig}^{†}(V))^{φ_{q}=1} .

Theorem 2.5.8. If y ∈ D_{rig}^{†}(V)^{ψ_{q}=1} and K contains K_{n(K)} and b is a basis of Γ_{K}, then

1. there is a unique c_{b}(y) ∈ Z_{an}^{1}(Γ_{K} , D_{rig}^{†}(V)^{ψ_{q}=1} ) such that for k ∈ Z_{p},

c_{b}(y)(b_{j}^{k}) = ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (y);

2. there is a unique m_{c} ∈ D_{rig}^{†}(V)^{ψ_{q}=0} such that (φ_{q}−1)c_{b}(y)(g) = (g−1)m_{c} for all g ∈ Γ_{K};

3. the (φ, Γ)-module corresponding to this extension has a basis in which

Mat(g) = ( \begin{matrix}∗ & c_{b}(y)(g) \\ 0 & 1 \end{matrix}) if g ∈ Γ_{K},  and  Mat(φ_{q}) = ( \begin{matrix}∗ & m_{c} \\ 0 & 1 \end{matrix}) ;

4. if z ∈ B̃_{rig}^{†} ⊗_{F} V is such that (φ_{q} − 1)z = m_{c}, then the cocycle

g ↦ c_{b}(y)(g) − (g − 1)z

defined on G_{K} has values in V and represents h_{K,V}^{1} (y) in H_{an}^{1}(K, V).

Proof. Items (1), (2) and (3) are reformulations of the constructions of this chapter. Let us prove (4). Let us write the (φ, Γ)-module corresponding to the extension in (3) as D′ = D_{rig}^{†}(V) ⊕ B_{rig,F}^{†} · e. It is an étale (φ, Γ)-module that comes from the p-adic representation V′ = (B̃_{rig}^{†} ⊗_{B_{rig ,F}^{†}} D′)^{φ_{q}=1} . We have V′ = V ⊕ F · (e − z) as F -vector spaces since φ_{q}(e − z) = e − z. If g ∈ G_{K}, then

g(e − z) = e + c_{b}(y)(g) − g(z) = e − z + c_{b}(y)(g) − (g − 1)z.

This proves (4).

Let F = Q_{p} and π = p = q, and let V be a representation of G_{K}. In §II.1 of [CC99], Cherbonnier and Colmez define a map Log_{V^{∗} (1)}^{∗} : D^{†}(V)^{ψ=1} → H_{Iw}^{1}(K, V), which is an isomorphism (theorem II.1.3 and proposition III.3.2 of [CC99]).

Proposition 2.5.9. If F = Q_{p} and π = p, then the map

D^{†}(V)^{ψ=1} → D_{rig}^{†}(V)^{ψ=1} −−−−−−−−→^{\{h_{Kn ,V}^{1} \}_{n⩾1}} \underleftarrow{lim}_{n}H_{an}^{1}(K_{n}, V) → \underleftarrow{lim}_{n}H^{1}(K_{n}, V)

coincides with the map Log_{V^{∗} (1)}^{∗} : D^{†}(V)^{ψ=1} → H_{Iw}^{1}(K, V) ⊂ \underleftarrow{lim}_{n} H^{1}(K_{n}, V).

Proof. The map Log_{V^{∗} (1)}^{∗} is contructed by mapping x ∈ D^{†}(V)^{ψ=1} to the sequence (…, ι_{ψ,n} (x), …) ∈ \underleftarrow{lim}_{n} H^{1}(K_{n}, V) (see theorem II.1.3 in [CC99] and the paragraph preceding it), where

ι_{ψ,n} (x) = [σ ↦ ℓ_{K_{n}} (γ_{n}) ( \frac{σ − 1}{γ_{n} − 1} x − (σ − 1)b) ]

on G_{K_{n}} and where (see proposition I.4.1, lemma I.5.2 and lemma I.5.5 of ibid.)

1. γ_{n} = γ_{1}^{[K_{n} :K_{1} ]} and γ_{1} is a fixed generator of Γ_{K_{1}} ; 2. ℓ_{K_{n}} (γ_{n}) = \frac{log χ(γ_{n} )}{p^{r(Kn )}} where r(K_{n}) is the integer such that logχ(Γ_{K_{n}} ) =

p^{r(K_{n} )} Z_{p};

3. b ∈ B̃^{†} ⊗_{Q_{p}} V is such that (φ − 1)b = a and a ∈ D^{†}(V)^{ψ=1} is such that (γ_{n} −1)a = (φ−1)x (using the fact that γ_{n} −1 is bijective on D^{†}(V)^{ψ=0} ).

The theorem follows from comparing this with the explicit formula of theorem 2.5.8.

3 Explicit formulas for crystalline representations

In this chapter, we explain how the constructions of the previous chapter are related to p-adic Hodge theory, via Bloch and Kato’s exponential maps. Let B_{dR} be Fontaine’s ring of periods [Fon94] and let B_{max ,F}^{+} be the subring of B_{dR}^{+} that is constructed in §8.5 of [Col02] (recall that B_{max ,F}^{+} = F ⊗_{F_{0}} B_{max}^{+} where F_{0} = F ∩ Q_{p}^{unr} and B_{max}^{+} is a ring that is similar to Fontaine’s B_{cris} ). We assume throughout this chapter that K = F and that the representation V is crystalline and F -analytic.

3.1 Crystalline F -analytic representations

If V is an F -analytic crystalline representation of G_{F}, let D_{cris} (V) = (B_{max ,F} ⊗_{F} V)^{G_{F}} (this is the “component at identity” of the usual D_{cris} ). By corollary 3.3.8 of [KR09], F -analytic crystalline representations of G_{F} are overconvergent. Moreover, if M(D) ⊂ B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D is the object constructed in §2.2 of ibid., then by §2.4 of ibid., M(D_{cris} (V)) contains a basis of D^{†}(V) and D_{rig}^{†}(V) = B_{rig,F}^{†} ⊗_{B_{rig ,F}^{+}} M(D_{cris} (V)). This implies that

D_{rig}^{†}(V) ⊂ B_{rig,F}^{†} [1/t_{π}] ⊗_{F} D_{cris} (V).

Theorem 3.1.1. We have D_{rig}^{†}(V)^{ψ_{q}=1} ⊂ B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D_{cris} (V).

Proof. Take h ⩾ 0 such that the slopes of π^{−h} φ_{q} on D_{cris} (V) are ⩽ −d. Let E be an extension of F such that E contains the eigenvalues of φ_{q} on D_{cris} (V). We show that D_{rig}^{†}(V)^{ψ_{q}=1} ⊂ t_{π}^{−h} E ⊗_{F} B_{rig,F}^{+} ⊗_{F} D_{cris} (V). Let e_{1}, …, e_{n} be a basis of t_{π}^{−h} E ⊗_{F} D_{cris} (V) in which the matrix (p_{i,j} ) of φ_{q} is upper triangular. If y = ∑ _{i=1}^{d} y_{i} ⊗ φ_{q}(e_{i}) with y_{i} ∈ E ⊗_{F} B_{rig,F}^{†} , then ψ_{q}(y) = y if and only if ψ_{q}(y_{k}) = p_{k,k} y_{k} + ∑ _{j>k} p_{k,j} y_{j} for all k. The theorem follows from applying lemma 3.1.2 below to k = n, n − 1, …, 1.

Lemma 3.1.2. Take y ∈ E ⊗_{F} B_{rig,F}^{†} and α ∈ F such that val_{π}(α) ⩽ −d. If ψ_{q}(y) − αy ∈ E ⊗_{F} B_{rig,F}^{+} , then y ∈ E ⊗_{F} B_{rig,F}^{+} .

Proof. This is lemma 5.4 of [FX13].

3.2 Bloch-Kato’s exponentials for analytic representations

We now recall the definition of Bloch-Kato’s exponential map and its dual, and give a similar definition for F -analytic representations.

Lemma 3.2.1. We have an exact sequence

0 → F → (B_{max ,F}^{+} [1/t_{π}])^{φ_{q}=1} → B_{dR}/B_{dR}^{+} → 0.

Proof. This is lemma 9.25 of [Col02].

If V is a de Rham F -linear representation of G_{K}, we can ⊗_{F} the above sequence with V and we get a connecting homomorphism exp_{K,V} : (B_{dR} ⊗_{F} V)^{G_{K}} → H^{1}(K, V). Recall that if W is an F -vector space, there is a natural injective map W ⊗_{F} V → W ⊗_{Q_{p}} V.

Lemma 3.2.2. If V is F -analytic, the map exp_{K,V} : (B_{dR}⊗_{F} V)^{G_{K}} → H^{1}(K, V) defined above coincides with Bloch-Kato’s exponential via the inclusion (B_{dR}⊗_{F} V)^{G_{K}} ⊂ (B_{dR} ⊗_{Q_{p}} V)^{G_{K}} , and its image is in H_{an}^{1}(K, V).

Proof. Bloch and Kato’s exponential is defined as follows (definition 3.10 of [BK90]): if φ_{p} denotes the Frobenius map that lifts x ↦ x^{p} and if x ∈ (B_{dR} ⊗_{Q_{p}} V)^{G_{K}} , there exists x̃ ∈ B_{max ,Q_{p}}^{φ_{p} =1} ⊗_{Q_{p}} V such that x̃ − x ∈ B_{dR}^{+} ⊗_{Q_{p}} V, and exp(x) is represented by the cocyle g ↦ (g − 1)x̃. Lemma 3.2.1 says that we can lift x ∈ (B_{dR} ⊗_{F} V)^{G_{K}} to some x̃ ∈ (B_{max ,F}^{+} [1/t_{π}])^{φ_{q}=1} ⊗_{F} V such that x̃ − x ∈ B_{dR}^{+} ⊗_{F} V ⊂ B_{dR}^{+} ⊗_{Q_{p}} V. In addition, B_{max ,Q_{p}}^{φ_{q}=1} = F_{0} ⊗_{Q_{p}} B_{max ,Q_{p}}^{φ_{p} =1} (see lemma 1.1.11 of [Ber08]) so that

(B_{max ,F}^{+} [1/t_{π}])^{φ_{q}=1} ⊂ F ⊗_{Q_{p}} B_{max ,Q_{p}}^{φ_{p} =1} . We can therefore view x̃ as an element

of B_{max ,Q_{p}}^{φ_{p} =1} ⊗_{Q_{p}} V, and exp_{K,V} (x) = [g ↦ (g − 1)x̃] = exp(x). The construction of exp_{K,V} (x) shows that the cocycle exp_{K,V} (x) is de Rham. At each embedding τ ≠ Id of F, the extension of F by V given by exp_{K,V} (x) is therefore Hodge-Tate with weights 0. This finishes the proof of the lemma.

Recall the following theorem of Kato (see §II.1 of [Kat93]).

Theorem 3.2.3. If V is a de Rham representation, the map from (B_{dR} ⊗_{Q_{p}} V)^{G_{K}} to H^{1}(K, B_{dR} ⊗_{Q_{p}} V) defined by x ↦ [g ↦ log(χ_{cyc}(\overline{g}))x] is an isomorphism, and the dual exponential map exp_{K,V^{∗} (1)}^{∗} : H^{1}(K, V) → (B_{dR} ⊗_{Q_{p}} V)^{G_{K}} 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.

Concretely, if c ∈ Z^{1}(K, B_{dR} ⊗_{Q_{p}} V) is some cocycle, there exists w ∈ B_{dR} ⊗_{Q_{p}} V such that c(g) = log(χ_{cyc}(\overline{g})) · exp_{K,V^{∗} (1)}^{∗} (c) + (g − 1)(w).

Corollary 3.2.4. If c ∈ Z^{1}(K, B_{dR} ⊗_{F} V), and if there exist x ∈ (B_{dR} ⊗_{F} V)^{G_{K}} and w ∈ B_{dR} ⊗_{F} V such that c(g) = ℓ(\overline{g}) · x + (g − 1)(w), then exp_{K,V^{∗} (1)}^{∗} (c) = x.

Proof. This follows from theorem 3.2.3 and from the fact that g ↦ log(χ_{π}(\overline{g})/χ_{cyc}(\overline{g})) is B_{dR}-admissible, since t_{π}/t ∈ (B_{dR}^{+})^{×} so that log(t_{π}/t) ∈ B_{dR}^{+} is well-defined.

3.3 Interpolating exponentials and their duals

Let V be an F -analytic crystalline representation. By theorem 3.1.1, we have D_{rig}^{†}(V)^{ψ_{q}=1} ⊂ B_{rig,F}^{+} [1/t_{π}] ⊗_{F} D_{cris} (V). Let ∂_{V} denote the map ∂_{D} of §2.4 for D = D_{cris} (V).

Theorem 3.3.1. If y ∈ D_{rig}^{†}(V)^{ψ_{q}=1} , then

exp_{F_{n} ,V^{∗} (1)}^{∗} (h_{F_{n} ,V}^{1} (y)) = \{ \begin{matrix}q^{−n} ∂_{V}(φ_{q}^{−n} (y)) & if n ⩾ 1 \\ (1 − q^{−1} φ_{q}^{−1} )∂_{V}(y) & if n = 0. \end{matrix}

Proof. Since the 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 n ⩾ n(F) by lemma 2.4.3 and proposition 2.5.6. By theorem 2.5.8, we have

h_{F_{n} ,V}^{1} (y)(b_{j}^{k}) = ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (y) − (b_{j}^{k} − 1)z,

with z ∈ B̃_{rig}^{†} ⊗_{F} V so that if m ≫ 0, then φ_{q}^{−m} (z) ∈ B_{dR}^{+} ⊗_{F} V (see §3 of [Ber16] and §2.2 of [Ber02]). Moreover, φ_{q}^{−m} (y) ∈ F_{m}((t_{π})) ⊗_{F} D_{cris} (V). Let W = {w ∈ F_{m}((t_{π})) ⊗_{F} D_{cris} (V) such that ∂_{V}(w) = 0}. The operator ∇ is bijective on W, and F_{m}((t_{π})) ⊗_{F} D_{cris} (V) injects into B_{dR} ⊗_{F} V, hence there exists u ∈ B_{dR} ⊗_{F} V such that

h_{F_{n} ,V}^{1} (y)(b_{j}^{k})= ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (∂_{V}(φ_{q}^{−m} (y))) − (b_{j}^{k} − 1)u
= ℓ(b_{j}^{k}) · Θ_{b}(∂_{V}(φ_{q}^{−m} (y))) − (b_{j}^{k} − 1)u
= ℓ(b_{j}^{k}) · q^{−n} ∂_{V}(φ_{q}^{−n} (y))) − (b_{j}^{k} − 1)u,

by lemmas 2.4.1 and 2.4.3. This proves the theorem by corollary 3.2.4.

We now give explicit formulas for exp_{F_{n} ,V} . Take h ⩾ 0 such that Fil^{−h} D_{cris} (V) = D_{cris} (V), so that t_{π}^{h}(B_{rig,F}^{+} ⊗_{F} D_{cris} (V)) ⊂ D_{rig}^{†}(V) (in the notation of §2.2 of [KR09], we have t_{π}^{h}(B_{rig,F}^{+} ⊗_{F} D_{cris} (V)) ⊂ M(D_{cris} (V))). In particular, if y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} , then ∇_{h−1} ∘ ⋯∘ ∇_{0}(y) ∈ D_{rig}^{†}(V)^{ψ_{q}=1} .

Theorem 3.3.2. If y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} , then

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

(−1)^{h−1} (h − 1)! \{ \begin{matrix}exp_{F_{n} ,V} (q^{−n} ∂_{V}(φ_{q}^{−n} (y))) & if n ⩾ 1 \\ exp_{F,V} ((1 − q^{−1} φ_{q}^{−1} )∂_{V}(y)) & if n = 0. \end{matrix}

Proof. Since 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, we only need to prove the theorem when n ⩾ n(F) by lemma 2.4.3 and proposition 2.5.6. By theorem 2.5.8, we have

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

= ℓ^{∗}(b) · \frac{b_{j}^{k} − 1}{b_{j} − 1} · \frac{∇^{d−1}}{∏ _{i≠j} (b_{i} − 1)} (∇_{h−1} ∘ ⋯∘ ∇_{0}(y)) − (b_{j}^{k} − 1)z

= (b_{j}^{k} − 1) · (∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b})(y) − (b_{j}^{k} − 1)z,

so that h_{F_{n} ,V}^{1} (∇_{h−1} ∘⋯ ∘∇_{0}(y))(g) = (g −1)(∇_{h−1} ∘⋯ ∘∇_{1} ∘Θ_{b})(y)−(g −1)z if g ∈ Γ_{K}. By lemma 2.4.2, we have

(∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b})((φ_{q} − 1)y)

∈ (t_{π}/φ_{q}^{n}(T))^{h}(B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=0} ⊂ D_{rig}^{†}(V)^{ψ_{q}=0} ,

so that (in the notation of theorem 2.5.8) m_{c} = (∇_{h−1} ∘⋯ ∘∇_{1}∘Θ_{b})((φ_{q}−1)y). Since (φ_{q} − 1)z = m_{c}, we have (φ_{q} − 1)((∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b})(y) − z) = 0, and therefore

(∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b})(y) − z ∈ (B̃_{rig}^{†} [1/t_{π}])^{φ_{q}=1} ⊗_{F} V

The ring B̃_{rig}^{†} contains B_{max ,F}^{+} and the inclusion (B_{max ,F}^{+} [1/t_{π}])^{φ_{q}=1} ⊂ (B̃_{rig}^{†} [1/t_{π}])^{φ_{q}=1} is an equality (proposition 3.2 of [Ber02]). This implies that

(∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b})(y) − z ⊂ (B_{max ,F}^{+} [1/t_{π}])^{φ_{q}=1} ⊗_{F} V.

Moreover, we have z ∈ B̃_{rig}^{†} ⊗_{F} V so that if m ≫ 0, then φ_{q}^{−m} (z) ∈ B_{dR}^{+}⊗_{F} V. In addition, φ_{q}^{−m} (y) belongs to F_{m}[[t_{π}]]⊗_{F} D_{cris} (V), so that φ_{q}^{−m} (y)−∂_{V}(φ_{q}^{−m} (y)) belongs to t_{π}F_{m}[[t_{π}]] ⊗_{F} D_{cris} (V) and therefore

(∇_{h−1} ∘ ⋯∘ ∇_{1} ∘ Θ_{b}) (φ_{q}^{−m} (y) − ∂_{V}(φ_{q}^{−m} (y))) ∈ t_{π}^{h}F_{m}[[t_{π}]] ⊗_{F} D_{cris} (V)
⊂ B_{dR}^{+} ⊗_{F} V.

We can hence write

h_{F_{n} ,V}^{1} (∇_{h−1} ∘⋯ ∘∇_{0}(y))(g) = (g−1)(∇_{h−1} ∘⋯ ∘∇_{1}∘Θ_{b}∘∂_{V}(φ_{q}^{−m} (y))−(g−1)u,

with u ∈ B_{dR}^{+} ⊗_{F} V. The theorem now follows from the fact that

Θ_{b} ∘ ∂_{V}(φ_{q}^{−m} (y)) = q^{−n} ∂_{V}(φ_{q}^{−n} (y)) ∈ F_{n} ⊗_{F} D_{cris} (V)

by lemmas 2.4.2 and 2.4.3, that ∇_{h−1} ∘ ⋯∘ ∇_{1} = (−1)^{h−1} (h − 1)! on F_{n} ⊗_{F} D_{cris} (V), and from the reminders given in §3.2, in particular the fact that exp_{K,V} is the connecting homomorphism when tensoring the exact sequence of lemma 3.2.1 with V and taking Galois invariants.

3.4 Kummer theory and the representation F (χ_{π})

Throughout this section, V = F (χ_{π}). Let L ⊂ \overline{Q}_{p} be an extension of K. The Kummer map δ : LT(𝔪_{L}) → H^{1}(L, V) is defined as follows. Choose a generator u = (u_{k})_{k⩾0} of T_{π} LT = \underleftarrow{lim}_{k} LT[π^{k}]. If x ∈ LT(𝔪_{L}), let x_{k} ∈ LT(𝔪_{\overline{Q}_{p}}) be such

that [π^{k}](x_{k}) = x. If g ∈ G_{L}, then g(x_{k}) − x_{k} ∈ LT[π^{k}] so that we can write g(x_{k}) − x_{k} = [c_{k}(g)](u_{k}) for some c_{k}(g) ∈ O_{F} /π^{k}. If c(g) = (c_{k}(g))_{k⩾0} ∈ O_{F} then δ(x) = [g ↦ c(g)] ∈ H^{1}(L, V). If x ∈ LT(𝔪_{L}), and L/K is finite Galois, let Tr_{L/K}^{LT} be the map defined by Tr_{L/K}^{LT} (x) = ∑ _{g∈Gal (L/K)}^{LT} g(x) where the superscript LT means that the summation is carried out using the Lubin-Tate addition. If F = Q_{p} and LT = G_{m}, we recover the classical Kummer map, and Tr_{L/K}^{LT} (x) = N_{L/K} (1 + x) − 1.

Lemma 3.4.1. We have the following commutative diagram:

LT(𝔪_{K_{n+1}} ) −−−−→^{δ} H^{1}(K_{n+1} , V)
Tr _{K_{n+1} /Kn}^{LT} ↓  ↓^{cor}^{K_{n+1} /Kn}
LT(𝔪_{K_{n}} ) −−−−→^{δ} H^{1}(K_{n}, V).

Proof. This is a straightforward consequence of the explicit description of the corestriction map.

Recall that φ_{q} ∘ ψ_{q}(f) = \frac{1}{q} ∑ _{ω∈LT[π]} f(T ⊕ ω), so that for n ⩾ 1:

ψ_{q}(f)(u_{n}) = \frac{1}{q} ∑_{ω∈LT[π]} f(u_{n+1} ⊕ ω) = \frac{1}{q}Tr_{F_{n+1} /F_{n}} f(u_{n+1} ).

In particular, if f(T) ∈ B_{rig,F}^{+} is such that ψ_{q}(f(T)) = 1/π · f(T) and y_{n} = f(u_{n}), then Tr_{F_{n+1} /F_{n}} (y_{n+1} ) = q/π · y_{n}.

Proposition 3.4.2. Assume that F ≠ Q_{p}. If {y_{n}}_{n⩾1} is a sequence with y_{n} ∈ F_{n} and Tr_{F_{n+1} /F_{n}} (y_{n+1} ) = q/π · y_{n}, there exists f(T) ∈ B_{rig,F}^{+} such that ψ_{q}(f(T)) = 1/π · f(T) and y_{n} = f(u_{n}) for all n ⩾ 1.

Proof. By [Laz62], there exists a power series g(T) ∈ B_{rig,F}^{+} such that g(u_{n}) = y_{n} for all n ⩾ 1. We also have

ψ_{q}g(0) = \frac{1}{q}g(0) + \frac{1}{q}Tr_{F_{1} /F_{0}} g(u_{1}),

and since q ≠ π (because F ≠ Q_{p}), we can choose g(0) such that

\frac{1}{π}g(0) = \frac{1}{q}g(0) + \frac{1}{q}Tr_{F_{1} /F_{0}} y_{1}.

This implies that (ψ_{q}(g)−1/π·g)(u_{n}) = 0 for all n ⩾ 0, so that ψ_{q}(g)−1/π·g ∈ t_{π} ·B_{rig,F}^{+} . It is therefore enough to prove that ψ_{q}−1/π : t_{π} ·B_{rig,F}^{+} → t_{π} ·B_{rig,F}^{+} is onto. Since ψ_{q}(t_{π}f) = 1/π · t_{π}ψ_{q}(f), this amounts to proving that ψ_{q} − 1 : B_{rig,F}^{+} → B_{rig,F}^{+} is onto, which follows from corollary 2.3.4.

Definition 3.4.3. Let S denote the set of sequences {x_{n}}_{n⩾1} with x_{n} ∈ 𝔪_{F_{n}} and Tr_{F_{n+1} /F_{n}}^{LT} (x_{n+1} ) = [q/π](x_{n}) for n ⩾ 1.

The following proposition says that if F ≠ Q_{p}, then S is quite large: for any k ⩾ 1, the “k-th component” map F ⊗_{O_{F}} S → F_{k} is surjective (if F = Q_{p}, there are restrictions on “universal norms”).

Proposition 3.4.4. Assume that F ≠ Q_{p}. If z ∈ 𝔪_{F_{k}} , there exists ℓ ⩾ 0 and x ∈ S such that x_{k} = [π^{ℓ}](z).

Proof. We claim that Tr_{F_{n+1} /F_{n}} (O_{F_{n+1}} ) = πO_{F_{n}} . Indeed, let D denote the different. We have (see for instance proposition 7.11 of [Iwa86])

val_{p}(D_{F_{n+1} /F_{n}} ) = \frac{1}{e} ( n + 1 − \frac{1}{q − 1} ) − \frac{1}{e} ( n − \frac{1}{q − 1} ) = val_{p}(π).

This implies that Tr_{F_{n+1} /F_{n}} (O_{F_{n+1}} ) = πO_{F_{n}} by proposition 7 of Chapter III of [Ser68]. Since π divides q/π, this shows that given y ∈ O_{F_{k}} , there exists a sequence {y_{n}}_{n⩾1} with x_{n} ∈ O_{F_{n}} such that y_{k} = y, and Tr_{F_{n+1} /F_{n}} (y_{n+1} ) = q/π · y_{n} for n ⩾ 1. Take ℓ_{1}, ℓ_{2} ⩾ 0 such that π^{ℓ_{1}}O_{C_{p}} is in the domain of exp_{LT} and such that π^{ℓ_{2}} log_{LT}(z) ∈ O_{F_{k}} . Let y = π^{ℓ_{2}} log_{LT}(z). Let {y_{n}}_{n⩾1} be a sequence as above, let x_{n} = exp_{LT}(π^{ℓ_{1}}y_{n}) and ℓ = ℓ_{1} + ℓ_{2}. The elements x_{k} ⊖ [π^{ℓ}](z), as well as Tr_{F_{n+1} /F_{n}}^{LT} (x_{n+1} )⊖[q/π](x_{n}) for all n, have their log_{LT} equal to zero and are in a domain in which log_{LT} is injective. This proves the proposition.

If x ∈ S and y_{n} = log_{LT}(x_{n}), then y_{n} ∈ F_{n} and Tr_{F_{n+1} /F_{n}} (y_{n+1} ) = q/π · y_{n}, so that by proposition 3.4.2, there exists f(T) ∈ B_{rig,F}^{+} such that ψ_{q}(f(T)) = π^{−1} · f(T) and y_{n} = f(u_{n}) for all n ⩾ 1. If f(T) ∈ B_{rig,F}^{+} is such that ψ_{q}(f(T)) = π^{−1} · f(T), then ∂f ∈ (B_{rig,F}^{+} )^{ψ_{q}=1} and ∂f · u can be seen as an element of D_{rig}^{†}(V)^{ψ_{q}=1} .

Theorem 3.4.5. If x ∈ S, and if f(T) ∈ B_{rig,F}^{+} is such that f(u_{n}) = log_{LT}(x_{n}) and ψ_{q}(f(T)) = π^{−1} · f(T), then h_{F_{n} ,V}^{1} (∂f(T) · u) = (q/π)^{−n} · δ(x_{n}) for all n ⩾ 1.

Proof. Let y = f(T) ⊗ t_{π}^{−1} u, so that y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} . By theorem 3.3.2 applied to y with h = 1, we have h_{F_{n} ,V}^{1} (∇(y)) = exp_{F_{n} ,V} (q^{−n} ∂_{V}(φ_{q}^{−n} (y))) if n ⩾ 1. Since φ_{q}^{−n} ∘ ∂ = π^{n} · ∂ ∘ φ_{q}^{−n} , this implies that

h_{F_{n} ,V}^{1} (∂f(T)·u) = exp_{F_{n} ,V} (q^{−n} ∂_{V}(φ_{q}^{−n} (y))) = (q/π)^{−n} ·exp_{F_{n} ,V} (log_{LT}(x_{n})·u).

By example 3.10.1 of [BK90] and lemma 3.2.2, we have δ(x_{n}) = exp_{F_{n} ,V} (log_{LT}(x_{n}) · u). This proves the theorem.

Remark 3.4.6. If F = Q_{p} and π = q = p and x = {x_{n}}_{n⩾1} , this theorem says that Exp_{Q_{p}}^{∗} (δ(x)) = ∂ logCol_{x}(T), which is (iii) of proposition V.3.2 of [CC99] (see theorem II.1.3 of ibid for the definition of the map Exp_{Q_{p}}^{∗} : H_{Iw}^{1}(F, Q_{p}(1)) → D_{rig}^{†}(Q_{p}(1))^{ψ_{q}=1} ).

Remark 3.4.7. If x ∈ S, then by proposition 3.4.2, there is a power series f(T) such that f(u_{n}) = log_{LT}(x_{n}) for n ⩾ 1. Is there a power series g(T) ∈ O_{F} [[T]] such that g(u_{n}) = x_{n}, so that f(T) = logg(T)? If F = Q_{p}, such a power series is the classical Coleman power series [Col79]. If F ≠ Q_{p} and x ∈ S and z is a [q/π]-torsion point, and k ⩾ d − 1 so that z ∈ F_{k}, then the sequence x′ = {x′_{n}}_{n⩾1} defined by x′_{n} = x_{n} if n ≠ k and x′_{k} = x_{k} ⊕ z also belongs to S. This means that we cannot naïvely interpolate x.

3.5 Perrin-Riou’s big exponential map

In this last section, we explain how the explicit formulas of the previous sections can be used to give a Lubin-Tate analogue of Perrin-Riou’s “big exponential map” [PR94]. Take h ⩾ 1 such that Fil^{−h} D_{cris} (V) = D_{cris} (V). If f ∈ B_{rig,F}^{+} ⊗_{F} D_{cris} (V), let Δ(f) be the image of ⊕_{k=0}^{h} ∂^{k}(f)(0) in ⊕_{k=0}^{h} D_{cris} (V)/(1 − π^{k}φ_{q}).

Lemma 3.5.1. There is an exact sequence:

0 → ⊕_{k=0}^{h} t_{π}^{k}D_{cris} (V)^{φ_{q}=π^{−k}} → ( B_{rig,F}^{+} ⊗_{F} D_{cris} (V)) ^{ψ_{q}=1} −−−→^{1−φ_{q}}

(B_{rig,F}^{+} )^{ψ_{q}=0} ⊗_{F} D_{cris} (V) ⟶^{Δ} ⊕_{k=0}^{h} \frac{D_{cris} (V)}{1 − π^{k}φ_{q}} → 0.

Proof. Note that the map φ_{q} acts diagonally on tensor products. It is easy to see that ker(1 − φ_{q}) = ⊕_{k=0}^{h} t_{π}^{k}D_{cris} (V)^{φ_{q}=π^{−k}} , that Δ is surjective, and that im(1 − φ_{q}) ⊂ kerΔ, so we now prove that im(1 − φ_{q}) = kerΔ. If f, g ∈ B_{rig,F}^{+} ⊗_{F} D_{cris} (V) and f = (1 − φ_{q})g, then ψ_{q}(f) = 0 if and only if ψ_{q}(g) = g. It is therefore enough to show that if f ∈ B_{rig,F}^{+} ⊗_{F} D_{cris} (V) is such that Δ(f) = 0, then f = (1 − φ_{q})g for some g ∈ B_{rig,F}^{+} ⊗_{F} D_{cris} (V).

The map 1−φ_{q} : T^{h+1} B_{rig,F}^{+} ⊗_{F} D_{cris} (V) → T^{h+1} B_{rig,F}^{+} ⊗_{F} D_{cris} (V) is bijective because the slopes of φ_{q} on T^{h+1} B_{rig,F}^{+} ⊗_{F} D are > 0. This implies that 1 − φ_{q} induces a sequence

0 → ⊕_{k=0}^{h} t_{π}^{k}D_{cris} (V)^{φ_{q}=π^{−k}} → \frac{B_{rig,F}^{+} ⊗_{F} D_{cris} (V)}{T^{h+1} B_{rig,F}^{+} ⊗_{F} D_{cris} (V)} −−−→^{\overline{1−φ_{q}}}

\frac{B_{rig,F}^{+} ⊗_{F} D_{cris} (V)}{T^{h+1} B_{rig,F}^{+} ⊗_{F} D_{cris} (V)} ⟶^{Δ} ⊕_{k=0}^{h} \frac{D_{cris} (V)}{1 − π^{k}φ_{q}}.

We have ker(\overline{1 − φ_{q}} ) = ⊕_{k=0}^{h} t_{π}^{k}D_{cris} (V)^{φ_{q}=π^{−k}} and by comparing dimensions, we see that coker(\overline{1 − φ_{q}} ) = ⊕_{k=0}^{h} D_{cris} (V)/(1 − π^{k}φ_{q}). This and the bijectivity of 1 − φ_{q} on T^{h+1} B_{rig,F}^{+} ⊗_{F} D_{cris} (V) imply the claim.

If f ∈ ((B_{rig,F}^{+} )^{ψ_{q}=0} ⊗_{F} D_{cris} (V))^{Δ=0} , then by lemma 3.5.1 there exists y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} such that f = (1 − φ_{q})y. Since ∇_{h−1} ∘ ⋯∘ ∇_{0} kills ⊕_{k=0}^{h−1} t_{π}^{k}D_{cris} (V)^{φ_{q}=π^{−k}} we see that ∇_{h−1} ∘ ⋯∘ ∇_{0}(y) does not depend upon the choice of such a y (unless D_{cris} (V)^{φ_{q}=π^{−h}} ≠ 0).

Definition 3.5.2. Let h ⩾ 1 be such that Fil^{−h} D_{cris} (V) = D_{cris} (V) and such that D_{cris} (V)^{φ_{q}=π^{−h}} = 0. We deduce from the above construction a welldefined map:

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

given by Ω_{V,h}(f) = ∇_{h−1} ∘ ⋯∘ ∇_{0}(y) where the element y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} is such that f = (1 − φ_{q})y and is provided by lemma 3.5.1. If D_{cris} (V)^{φ_{q}=π^{−h}} ≠ 0, we get a map

Ω_{V,h} : ((B_{rig,F}^{+} )^{ψ_{q}=0} ⊗_{F} D_{cris} (V))^{Δ=0} → D_{rig}^{†}(V)^{ψ_{q}=1} /V^{G_{F} =χ_{π}^{h}} .

Let u be a basis of F (χ_{π}) as above, and let e_{j} = u^{⊗j} if j ∈ Z.

Theorem 3.5.3. Take y ∈ (B_{rig,F}^{+} ⊗_{F} D_{cris} (V))^{ψ_{q}=1} and let h ⩾ 1 be such that Fil^{−h} D_{cris} (V) = D_{cris} (V). Let f = (1 − φ_{q})y so that f ∈ ((B_{rig,F}^{+} )^{ψ_{q}=0} ⊗_{F} D_{cris} (V))^{Δ=0} . If j ∈ Z and h + j ⩾ 1, then

h_{F_{n} ,V (χ_{π}^{j} )}^{1} (Ω_{V,h}(f) ⊗ e_{j}) = (−1)^{h+j−1} (h + j − 1)!×

\{ \begin{matrix}exp_{F_{n} ,V (χ_{π}^{j} )} (q^{−n} ∂_{V (χ_{π}^{j} )} (φ_{q}^{−n} (∂^{−j} y ⊗ t_{π}^{−j} e_{j}))) & if n ⩾ 1 \\ exp_{F,V (χ_{π}^{j} )} ((1 − q^{−1} φ_{q}^{−1} )∂_{V (χ_{π}^{j} )} (∂^{−j} y ⊗ t_{π}^{−j} e_{j})) & if n = 0. \end{matrix}

If j ∈ Z and h + j ⩽ 0, then

exp_{F_{n} ,V^{∗} (1−j)}^{∗} (h_{F_{n} ,V (χ_{π}^{j} )}^{1} (Ω_{V,h}(f) ⊗ e_{j})) =

\frac{1}{(−h − j)!} \{ \begin{matrix}q^{−n} ∂_{V (χ_{π}^{j} )} (φ_{q}^{−n} (∂^{−j} y ⊗ t_{π}^{−j} e_{j})) & if n ⩾ 1 \\ (1 − q^{−1} φ_{q}^{−1} )∂_{V (χ_{π}^{j} )} (∂^{−j} y ⊗ t_{π}^{−j} e_{j}) & if n = 0. \end{matrix}

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

D_{rig}^{†}(V)^{ψ_{q}=1}  −−−−→^{⊗e_{j}}  D_{rig}^{†}(V (χ_{π}^{j}))^{ψ_{q}=1}
∇_{h−1} ∘···∘∇_{0} ↑  ^{∇_{h+j−1} ∘···∘∇_{0}} ↑
( B_{rig,F}^{+} ⊗_{F} D_{cris} (V)) ^{ψ_{q}=1} −−−−−−−→^{∂^{−j} ⊗t^{−j} e_{j}} ( B_{rig,F}^{+} ⊗_{F} D_{cris} (V (χ_{π}^{j}))) ^{ψ_{q}=1} ,

and the theorem is a straightforward consequence of theorem 3.3.2 applied to ∂^{−j} y ⊗ t^{−j} e_{j}, h + j and V (χ_{π}^{j}) (which are the j-th twists of y, h and V). If h + j ⩽ 0, and Γ_{F_{n}} is torsion free, then theorem 3.3.1 shows that

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

= q^{−n} ∂_{V (χ_{π}^{j} )} (φ_{q}^{−n} (∇_{h−1} ∘ ⋯∘ ∇_{0}(y) ⊗ e_{j}))

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

∂_{V (χ_{π}^{j} )} (φ_{q}^{−n} (∇_{h−1} ∘ ⋯ ∘ ∇_{0}(y) ⊗ e_{j})) = (−h − j)!^{−1} ∂_{V (χ_{π}^{j} )} (φ_{q}^{−n} (∂^{−j} y ⊗ t_{π}^{−j} e_{j})).

To get the other n, we corestrict.

Corollary 3.5.4. We have Ω_{V,h}(x) ⊗ e_{j} = Ω_{V (χ_{π}^{j} ),h+j} (∂^{−j} x ⊗ t_{π}^{−j} e_{j}) and ∇_{h} ∘ Ω_{V,h}(x) = Ω_{V,h+1} (x).

Remark 3.5.5. 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 that this leads to no ambiguity in the formulas of theorem 3.5.3 above.

If F = Q_{p} and π = p, definition 3.5.2 and theorem 3.5.3 are given in §II.5 of [Ber03]. They imply that Ω_{V,h} coïncides with Perrin-Riou’s exponential map (see theorem 3.2.3 of [PR94]) after making suitable identifications (theorem II.13 of [Ber03]). Our definition therefore generalizes Perrin-Riou’s exponential map to the F - analytic setting. We hope to use the results of [Fou05] and [Fou08] to relate our constructions to suitable Iwasawa algebras as in the cyclotomic case.

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Laurent Berger

UMPA, ENS de Lyon UMR 5669 du CNRS Université de Lyon Lionel Fourquaux IRMAR UMR 6625 du CNRS Université Rennes 1