Xavier Caruso

An introduction to p-adic period rings

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Introduction

In algebraic geometry, the word period often refers to a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. One of the simplest periods is 2iπ = γ dt t , where γ is the unit circle in the complex plane. Equivalently, a period can be seen as an entry of the matrix (in rational bases) of the de Rham isomorphism:

C ⊗ Q H r sing (X(C), Q) ≃ C ⊗ K H r dR (X) (1)
for an algebraic variety X defined over a number field K. (Here H r sing is the singular cohomology and H r dR denotes the algebraic de Rham cohomology.)

The initial motivation of p-adic Hodge theory is the will to design a relevant p-adic analogue of the notion of periods. To this end, our first need is to find a suitable p-adic generalization of the isomorphism [START_REF] Andreatta | An introduction to perfectoid spaces[END_REF]. In the p-adic setting, the singular cohomology is no longer relevant; it has to be replaced by the étale cohomology. Thus, what we need is a ring B allowing for a canonical isomorphism:

B ⊗ Qp H r ét (X K , Q p ) ≃ B ⊗ K H r dR (X) (2)
when K is now a finite extension of Q p and X is a variety defined over K. Of course, the first natural candidate one thinks at is B = C p , the p-adic completion of an algebraic closure K of K.

Unfortunately, this first period ring does not totally fill our requirements. More precisely, it turns out that C p ⊗ Qp H r ét (X K , Q p ) is isomorphic to the graded module (for the de Rham filtration) of C p ⊗ K H r dR (X) but not to C p ⊗ K H r dR (X) itself. The main objective of this lecture is to detail the construction of two periods rings, namely B crys and B dR , allowing for the isomorphism [START_REF] Ax | Zeros of polynomials over local fields[END_REF] under some additional assumptions on the variety X. The ring B dR (which is the bigger one) is often called the ring of p-adic periods.

Another important aspect of p-adic period rings concerns the Galois structure of H r ét (X K , Q p ). Indeed, we shall see that the mere existence of the isomorphism (2) usually has strong consequences on the Galois module H r ét (X K , Q p ). In order to give depth to this observation, Fontaine developed a general formalism for studying and classifying general Galois representations through the notion of period rings. A large part of this article focuses on the Galois aspects.

Structure of the article. §1 serves as a second long introduction to this article; two results which can be considered as the seeds of p-adic Hodge theory are presented and discussed. The first one is due to Tate and provides a Hodge-like decomposition of the Tate module of a pdivisible group in the spirit of the isomorphism [START_REF] Ax | Zeros of polynomials over local fields[END_REF]. The second result is a classification theorem of p-divisible groups by Fontaine. Fontaine's general formalism for studying Galois representations is also introduced in this section.

In §2, we investigate to what extent C p meets the expected properties of a period ring. We adopt the point of view of Galois representations, which means concretely that we will concentrate on isolating those Galois representations that are susceptible to sit in an isomorphism of the form (2) when B = C p . This study will lead eventually to the notion of Hodge-Tate representations, which is related to the Hodge-like decompositions of cohomology presented in §1.

In §3, we review the construction of the period rings B crys and B dR ; it is the heart of the article but also its most technical part. Finally, in §4, we state several comparison theorems between étale and de Rham cohomologies. We also show how the rings B crys and B dR intervene in the classification of Galois representations, through the notions of crystalline and de Rham representations.

Some advice to the reader. Although we will give frequently reminders, we assume that the reader is familiar with the general theory of local fields as presented in [START_REF] Serre | Corps locaux[END_REF], Chapter 1-4. A minimal knowledge of local class field theory [START_REF] Serre | Corps locaux[END_REF] and of the theory of p-adic analytic functions [START_REF] Lazard | Les zéros des fonctions analytiques d'une variable sur un corps valué complet[END_REF] is also welcome, while not rigourously needed.

To the impatient reader who is afraid by the length of this article and is not interested in the details of the proofs (at least in first reading) but only by a general outline of p-adic Hodge theory, we advise to read §1, then the introduction of §3 until §3.1 and then finally §4.

Acknowledgement. The author is grateful to the editorial board of Panoramas et Synthèses, and especially to Ariane Mézard, who encourages him to finalize this article after many years. He also warmly thanks Olivier Brinon for sharing with him his notes of a master course on p-adic Hodge theory he taught in Bordeaux.

Setting and preliminaries

Let K be a finite extension 1 of Q p . Let v p : K → Q ⊔ {+∞} be the valuation on K normalized by v p (p) = 1. By our assumptions, v p (K × ) is a discrete subgroup of Q containing Z; hence it is equal to 1 e Z for some positive integer e. We recall that this integer e is called the absolute ramification index of K. A uniformizer of K is an element of minimal positive valuation, that is of valuation 1 e . We fix a uniformizer π of K. Let O K be the ring of integers of K, that is the subring of K consisting of elements with nonnegative valuation. We recall that O K is a local ring whose maximal ideal m K consists of elements with positive valuation. The residue field k of K is, by definition, the quotient O K /m K . Under our assumptions, k is a finite field of characteristic p.

Let W (k) denote the ring of Witt vectors with coefficients in k. Set K 0 = Frac W (k). By the general theory of Witt vectors, there exists a canonical embedding K 0 → K. Moreover, through this embedding, K appears as a finite totally ramified extension of K 0 of degree e. Therefore, K 0 is the maximal subextension of K which is unramified over Q p .

The absolute Galois group of K

We choose and fix once for all an algebraic closure K of K. We recall that the valuation v p extends uniquely to K, so that we can talk about the ring of integers O K of K. This ring is a local ring whose maximal ideal will be denoted by m K . The quotient O K /m K is identified with an algebraic closure of k; it will be denoted k in the sequel.

Let G K = Gal( K/K) be the absolute Galois group of K. Any element of G K acts by isometry on K and therefore stabilizes O K and m K . It thus acts on the residue field k. This defines a group homomorphism G K → Gal( k/k), which is surjective. The kernel of this morphism is the inertia subgroup; we shall denote it by I K in the sequel. The subextension of K cut out by I K 1 We could have considered a more general setting where K is a complete discrete valued field of characteristic 0 with perfect residue field of characteristic p. All the results presented in the paper extend to this more general setting. However the case of finite extensions of Qp is the main case of interest and restricting to this case simplifies the exposition at several points. is the maximal unramified extension of K; we will denote it by K ur . Summarizing the above discussion, we find that G K sits in the following exact sequence:

1 -→ I K -→ G K -→ Gal( k/k) → 1.
The structure of Gal( k/k) is also known: if k has cardinality q, Gal( k/k) is the profinite group generated by the Frobenius Frob q : x → x q .

The structure of I K can be further precised. Indeed a simple application of Hensel's lemma shows that any finite extension of K ur whose degree is not divisible by p has the form K ur [ n √ π].

The union of all these extensions is called K tr ; it is the maximal tamely ramified extension of K.

Since K ur contains all n-th roots of unity for n ∤ p (cf the paragraph The cyclotomic extension below for more details), the extension K tr /K ur is Galois and its Galois group is identified with lim ← -n,p∤n Z/nZ ≃ ℓ =p Z ℓ . Moreover, any finite extension of K tr has degree p m for some integer m. On the Galois side, these properties imply that the closed subgroup of I K corresponding to the extension K tr is the unique pro-p-Sylow of I K (which is then a normal subgroup) and that I K sits in the following exact sequence:

1 -→ P K -→ I K -→ lim ← - n,p∤n
Z/nZ → 1

where P K denotes the pro-p-Sylow of I K .

The cyclotomic extension

The cyclotomic extension of K plays a quite important role in p-adic Hodge theory. So we take some time to recall its most important properties. Let µ n ∈ K be a primitive n-th root of unity. We recall that, by definition, the cyclotomic extension of K is the subextension K cycl of K generated by the µ n 's.

The extension K(µ n )/K is Galois and its Galois group canonically embeds into (Z/nZ) × through the map χ n : Gal(K(µ n )/K) → (Z/nZ) × defined by the relation χ n (µ n ) = µ χn(g) n for all g ∈ Gal(K(µ n )/K). We draw the reader's attention to the fact that χ n is in general not surjective although it is for all n when K = Q p .

When n is coprime with p, the extension K(µ n )/K is unramified since the polynomial X n -1 splits over k. In this case, K(µ n ) appears as a subextension of K ur . On the other hand, when n = p r is a power of p, the extension K(µ p r )/K is totally ramified. This dichotomy motivates the introduction of the two following infinite extensions of K:

K p ′ -cycl = n,p∤n K(µ n ) and K p-cycl = r≥0 K(µ p r ).
The first one is actually equal to K ur since, at the level of residue fields, k is obtained by k by adding all p n -th roots of unity for p ∤ n. As for K p-cycl , it is linearly disjoint from K ur . It is sometimes called the p-cyclotomic extension of K. Clearly, the cyclotomic extension of K is the compositum of K ur and K p-cycl .

Let us review briefly the Galois properties of K p-cycl . First of all, we notice that K p-cycl /K is Galois. Its Galois group is equipped with an injective group homomorphism χ p ∞ : Gal(K p-cycl /K) → Z × p which is characterized by the relation gµ p m = µ

χ p ∞ (g) p m
(for all g ∈ Gal(K p-cycl /K) and m ≥ 1). Let χ cycl : G K → Z × p be the homomorphism obtained by precomposing χ p ∞ with the canonical surjection G K → Gal(K p-cycl /K). We shall often see χ cycl as a character and will call it the (p-adic) cyclotomic character. As χ p ∞ , it is determined by the relation:

gµ p m = µ χ cycl (g) p m
for all g ∈ G K and m ≥ 1.

By construction, the extension corresponding to ker χ cycl is K p-cycl and, more generally, for all positive integer r, the extension corresponding to ker(χ cycl mod p r ) is K(µ p r ).

The logarithm defines a group morphism Z × p → Z p where the group structure on the target is given by the addition. It sits in the exact sequence:

1 -→ F × p [•] -→ Z × p log -→ Z p -→ 1 (3)
where [•] denotes the Teichmuller representative function. This sequence is split since a retraction of F × p → Z × p is simply the canonical projection. Therefore Z × p is canonically isomorphic to F × p × Z p . Restricting [START_REF] Berger | Équations différentielles p-adiques et (ϕ, N )-modules filtrés[END_REF] to the image of χ cycl , we find that Gal(K p-cycl /K) sits in another exact sequence which reads as follows:

1 -→ H -→ Gal(K p-cycl /K) log χ cycl -→ p r 0 Z p -→ 1.
Here r 0 is a nonnegative integer and H can be identified as a subgroup of F × p and thus is cyclic of order divisible by p-1. The above sequence splits, so that Gal(K p-cycl /K) is canonically isomorphic to a direct product H × p r 0 Z p ≃ H × Z p .

The subextension of K p-cycl cut out by the factor Z p is nothing but K(µ p ). It is also the maximal tamely ramified subextension of K p-cycl . The Galois group of K p-cycl /K(µ p ) is canonically isomorphic to Z p via the additive character p -r 0 log χ cycl . We say that K p-cycl /K(µ p ) is a Z p -extension. The fact that Gal(K p-cycl /K) splits as a direct product means that this extension descends to K; in particular, K itself admits a Z p -extension.

Characters of G Qp

The representation theory of G K is the main object of interest in this article. Among all representations of G K , the simplest ones are of course characters, which are representations of dimension 1. We have actually already seen an example of such character: the cyclotomic character χ cycl . From χ cycl , we can build the following other character:

ω cycl : G K χ cycl -→ Z × p mod p -→ F × p [•] -→ Z × p
where the last map takes an element to its Teichmüller representative. We observe that ω cycl is a finite order character, whose order divides p-1. When K = Q p , the order of ω cycl is exactly p-1.

Another quite important family of characters are unramified characters, that are those characters which are trivial on the inertia subgroup. Since G K /I K ≃ Gal( k/k) is procyclic, continuous unramified characters are easy to describe: they are all of the form

µ λ : G K -→ G K /I K ≃ Gal( k/k) Frobq →λ -→ Z × p for λ varying in Z ×
p . Using local class field theory (cf [START_REF] Serre | Corps locaux[END_REF]), it is possible to describe explicitely all characters of G K . Indeed such characters all factor through the abelianization of G K , which is closely related to K × through the Artin reciprocity map. When K = Q p , this answer is given by the following proposition.

Proposition 1.1.1. We assume p > 2. Let χ be a character of G K with values in Q × p . Then, there exist unique λ ∈ Z × p , a ∈ Z p and b ∈ Z/(p-1)Z such that χ = µ λ • χ a cycl • ω b cycl .

Proof. We first observe that, by compacity, χ must take its values in Z × p . By the Kronecker-Weber theorem, we know that the maximal abelian extension of Q p is the cyclotomic extension. Therefore χ has to factor through Gal(Q p,cycl /Q p ). In particular, χ |I Qp factors through Gal(Q p,cycl /Q ur p ) which is isomorphic to Z × p by the cyclotomic character.

∈ Z/(p-1)Z such that h(x) = x a • [x mod p] b for x ∈ Z × p . Thus χ I Qp = χ a cycl • ω b cycl . The character χ • χ -a cycl • ω -b
cycl is then unramified. Thus it must be of the form µ λ for some λ ∈ Z × p . The proposition is proved.

Motivations: p-divisible groups and étale cohomology

The starting point of p-adic Hodge theory is Tate's paper of 1966 on p-divisible groups [START_REF] Tate | p-divisible groups[END_REF]. In this seminal article, Tate establishes a Hodge-like decomposition of the Tate module of a pdivisible group on O K . More precisely, let G be a p-divisible group on O K . We define the Tate module of G by

T p G = lim ← -n G[p n ]( K). Observe that T p G is naturally endowed with an action of G K . The algebraic structure of T p G is well-known: it is a free module of finite rank over Z p . Set V p G = Q p ⊗ Zp T p G.
Then, Tate proves the following Hodge-like G K -equivariant decomposition:

C p ⊗ Qp V p G ≃ C p ⊗ O K ω G ∨ ⊕ C p (χ -1 cycl ) ⊗ O K ω ∨ G . (4)
Here G ∨ is the Cartier dual of G, the construction ω -refers to the cotangent space at the origin and C p (χ -1 cycl ) is C p e endowed with the action g(λe) = gλ • χ -1 cycl (g) • e (for g ∈ G K and λ ∈ C p ). Note that the Galois action is trivial over ω G ∨ and ω ∨ G . The isomorphism (4) then reveals the Galois action on the Tate module. Tate's theorem implies in particular that, when A is an abelian variety over K with good reduction, the étale cohomology of A admits the following decomposition:

C p ⊗ Qp H 1 ét (A K , Q p ) ≃ C p ⊗ K H 1 (A, O A ) ⊕ C p (χ -1 cycl ) ⊗ K H 0 (A, Ω A/K ) . (5)
were A K = Spec K × Spec K A, O A is the structural sheaf of A and Ω A/K is the sheaf of Kähler differentials of A over K. We refer to Freixas' lecture in this volume [START_REF] Freixas | Hodge-Tate decomposition[END_REF] for a more detailed discussion-including a sketch of the proof-about Tate's theorem. After Tate's results, p-divisible groups over various bases were studied intensively. In the 1970's, Fontaine [START_REF] Fontaine | Groupes p-divisibles sur les corps locaux[END_REF] obtained a complete classification of p-divisible groups and finite flat group schemes over O K when K/Q p is unramified. The starting point of Fontaine's theorem is the classification of p-divisible groups over perfect fields of characteristic p in terms of Dieudonné modules [START_REF] Demazure | Lectures on p-divisible groups[END_REF]. Let us recall briefly how it works. If G k is a p-divisible group over k, we define

M (G k ) = Hom gr (G k , CW k ) (6)
where CW k is the functor of Witt covectors and the notations Hom gr means that we are considering the set of all natural transformations preserving the group structure. The space M (G k ) is a Dieudonné module. This means that it is a module over W (k) endowed with a Frobenius F (which is a semi-linear endomorphism with respect to the Frobenius on W (k)) and a Verschiebung V (which is a semi-linear endormorphism with respect to the inverse of the Frobenius) with the property that F V = V F = p. One can show that M realizes an anti-equivalence of categories between the category of p-divisible groups over k and that of finite free Dieudonné modules over W (k), the inverse functor being given by the formula

G k (A) = Hom W (k),F,V M, CW (A) for any k-algebra A
which is quite similar to [START_REF] Berthelot | Altérations des variétés algébriques (d'après A.J. de Jong)[END_REF]. Now, if G is a p-divisible over O K with special fibre G k , Fontaine constructs a submodule L(G) ⊂ M (G k ) and demonstrates that it obeys to a certain list of properties. Taking these properties as axioms, Fontaine introduces the notion of Honda systems and proves that the association G → (M (G k ), L(G)) is an anti-equivalence of categories between the category of p-divisible groups over O K and the category of finite free Honda systems over W (k). Moreover, Fontaine establishes a compact formula for the inverse functor. This formula reads:

V p G = Q p ⊗ Zp Hom honda (M (G k ), L(G)), (B, L B ) (7)
where the notation Hom honda means that we are taking the morphisms in the category of Honda system and the target (B, L B ) is a special Honda system2 (the letter B refers to the mathematician Barsotti, who first studied p-divisible groups using this kind of techniques). Moreover, (B, L B ) is endowed with an action of G K , from which we can recover the G K -action on V p G.

Compared to Tate's decomposition formula (4), Fontaine's result is more precise because it describes the Tate module V p G itself, whereas Tate's result only concerns its scalar extension to C p . For many complements about Fontaine's classification results, we refer to [START_REF] Fontaine | Groupes p-divisibles sur les corps locaux[END_REF][START_REF] Conrad | Finite group schemes over bases with low ramification[END_REF]. About ten years later, in 1981, Fontaine came back to Tate's decomposition isomorphism [START_REF] Berthelot | Cohomologie cristalline des schémas de caractéristique p[END_REF]and gave a different proof of it (which is sketched in Freixas' lecture in this volume [START_REF] Freixas | Hodge-Tate decomposition[END_REF]), relaxing at the same time the assumption of good reduction. He also became interested in generalizing Tate's decomposition theorem to higher cohomology group (i.e. H r ét (A K , Q p ) with r > 1) and other types of varieties. Moreover, noticing that the right hand side of ( 5) is the graded module of the de Rham cohomology, one may wonder if one can make the isomorphism (5) more precise and relate the étale cohomology with the de Rham cohomology equipped with its filtration. All these questions had been a strong motivation for the development of p-adic Hodge theory for many years. Nowadays, all of them are solved: it has been proved independently by Faltings [START_REF] Faltings | p-adic Hodge theory[END_REF] and Tsuji [START_REF] Tsuji | p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case[END_REF] that

B dR ⊗ Qp H r ét (X K , Q p ) ≃ B dR ⊗ Qp H r dR (X)
whenever X is a proper smooth variety over K and r is a nonnegative integer. Here B dR is the so-called field of p-adic periods. We will introduce it in this article in §3. Taking the grading in the above isomorphism, we get the following Hodge-like decomposition:

C p ⊗ Qp H r ét (X K , Q p ) ≃ a+b=r C p (χ -a cycl ) ⊗ K H b (X, Ω a X/K ).
We will come back to these results in §4.1.

Finally, it is interesting to confront the two directions of research discussed above, namely classification of p-divisible groups and Hodge-like decomposition theorems. As already mentioned, one important feature of the isomorphism [START_REF] Berthelot | Théorie de Dieudonné cristalline II[END_REF] is the fact that it gives a complete description of the Galois action on the Tate module. On the other hand, it is apparent that Honda systems have important limitations: by design, they can only deal with Tate modules, that is, roughly speaking, with the first cohomology group. Analyzing carefully the situation, Fontaine realized that what is missing to Honda systems is a good tannakian formalism (which is, of course, a key point in the line of Hodge-like decomposition theorems). In more crude terms, the fact that we are limited to the H 1 ét should be understood as a reflection of the fact that we are missing a good notion of tensor product on p-divisible groups. As explained in the introduction of [START_REF] Fontaine | Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate[END_REF], the period ring B crys and the afferent notion of crystalline representations actually emerge when trying to conceal the theory of Honda systems with the tannakian formalism inspired from the Hodge-like decomposition theorems we have presented above.

All the developments we will present in the sequel are stamped by this simple idea that one wants to keep apparent the tannakian structure (i.e. the tensor product) everywhere and, even, to use it as a main tool. The natural framework in which the theory grows is then that of Galois representations, which has a strong tannakian structure.

Notion of semi-linear representations

The Hodge-like decomposition theorems discussed previously motivate the study of representations of the form W = C p ⊗ Qp V where V is a given Q p -representation of G K . Since G K does act on C p , we observe that W is not a C p -linear representation in the usual sense. Instead, it is a so-called semi-linear representation. The aim of this subsection is to introduce and study this notion.

Definitions

In what follows, we let G be a topological group 3 and B be a topological ring equipped with a continuous 4 By our assumptions, B itself (endowed with its G-action) is a B-semi-linear representation of G. Similarly we can turn B n into a B-semi-linear representation by letting G act coordinate by coordinate. The latter representation will be called the trivial representation of dimension n.

If W 1 and W 2 are two B-semi-linear representations of G, a morphism W 1 → W 2 is a Blinear mapping which commutes with the action of G. With this definition, we can form the category of B-semi-linear representations of G (for G and B fixed). In the sequel we will simply denote it Rep B (G). It is easily seen that Rep B (G) is an abelian category. It is moreover endowed with a notion of tensor product and internal hom: if W 1 and W 2 are objects of Rep B (G), then W 1 ⊗ B W 2 (equipped with the action g • (x ⊗ y) = gx ⊗ gy) and Hom B (W 1 , W 2 ) (equipped with the action gϕ : x → gϕ(g -1 x)) are also.

Scalar extension

There is also a natural notion of scalar extension in the framework of semilinear representations. To explain it, let us consider a closed subring C of B, which is stable under the action of G. Then the notion of C-semi-linear representations of G makes sense and there is a canonical functor

Rep C (G) → Rep B (G) taking W to B ⊗ C W .
The latter construction is quite interesting because it allows us to build semi-linear representations from classical representations. Indeed, assume that we are given a field E and we have chosen G and B is such a way that B is an algebra over E and G acts trivially on E. (As an example, B could be a Galois extension of E with G = Gal(B/E).) The scalar extension then defines a functor Rep E (G) → Rep B (G). Moreover, since the action of G on E is trivial, the category Rep E (G) is just the category of E-linear representations of G. In more concrete terms, if V is a classical representation of G defined over E, then W = B ⊗ E V is a B-semi-linear representation. This is actually the prototype of all the semi-linear representations we are going to consider in this article.

Specializing the previous recipe to 1-dimensional representations, we obtain a way to construct semi-linear representations of G from characters of G. Concretely, if χ : G → E × is a multiplicative character, we will denote by B(χ) the 1-dimensional representation generated by a vector e χ on which G acts by ge χ = χ(g) • e χ for all g ∈ G. By semi-linearity, we then have g(a e χ ) = ga • χ(g) e χ for all g ∈ G and a ∈ B.

Recognizing the trivial representation

We keep the setup of the previous subsection: G is a topological group which acts continuously on a topological ring B.

Definition 1.3.2. A B-semi-linear representation of G is trivial if it is isomorphic to the trivial representation B d for some positive integer d.
While it is in general easy to recognize when a linear representation is trivial (it suffices to check that G acts trivially on each vector of the representation), the task becomes more complicated in the context of semi-linear representations. Indeed, coming back to the definition, we see that a B-semi-linear representation of G is trivial if and only if it admits a basis of vectors which are fixed by G. In particular, it is quite possible that a nontrivial semi-linear representation becomes trivial after scalar extension. The latter remark is in fact the starting point of Fontaine's strategy for classifying Galois representations.

We will discuss Fontaine's strategy in much more details in §1.4. Before this, we have to introduce further notations. Given W ∈ Rep B (G), we denote by W G the subset of W consisting of fixed points under G, that is the subset of elements x ∈ W such that gx = x for all g ∈ G.

Clearly W G is a module over B G . Moreover scalar extension provides a canonical morphism in Rep B (G):

α W : B ⊗ B G W G -→ W.
This morphism is useful for recognizing trivial representations. Indeed it is clearly an isomorphism when W is trivial in the sense of Definition 1.3.2 (since (B d ) G = (B G ) d ) and the converse also holds true when W and W G are free of finite rank over B and B G respectively.

Hilbert's theorem 90

As an introduction to Fontaine's strategy, we propose to discuss an easy case where trivial semilinear representations do appear, while they were not expected at first glance. The setting here is the following. We assume that B is a field and, in order to limit confusion, we will call it L. We assume also that G is a finite group, endowed with the discrete topology. Under these assumptions, L G is a subfield of L and the extension L/L G is finite and Galois with Galois group G.

Theorem 1.3.3. We keep the notations and assumptions above. For all W ∈ Rep L (G), the following assertions hold:

1. the morphism α W is surjective, 2. if W is finite dimensional over L, then α W is an isomorphism, i.e. W is trivial.
Proof. Let λ 1 , . . . , λ n be a basis of L over L G . By Artin's linear independence theorem, there exist constants µ 1 , . . . , µ n ∈ L such that n i=1 µ i g(λ i ) is 1 if g is the identity and 0 otherwise. Define the trace function T : W → W by T (x) = g∈G gx. One easily checks that T takes its values in W G . Moreover, for the particular µ i 's we have introduced earlier, we have n i=1 µ i T (λ i x) = x for all x ∈ W . This shows the surjectivity of α W .

We now assume that W is finite dimensional over L. The proof of injectivity is quite similar to the proof of Artin's linear independence theorem. It is enough to check that every finite family of elements of W G which is linearly independent over L G remains linearly independent over L. Let then (x 1 , . . . , x m ) be a linearly independent family over L G with x i ∈ W G for all i. We assume by contradiction that there exists a nontrivial relation of linear dependance of the form:

a 1 x 1 + a 2 x 2 + • • • + a m x m = 0 (8)
with a i ∈ L. We choose such a relation so that the number of nonzero a i 's is minimal. Up to reindexing the a i 's and rescaling the relation, we may assume that a 1 = 1. Let g ∈ G. Applying (gid) to [START_REF] Berthelot | Notes on crystalline cohomology[END_REF], we get the relation (ga 2 -a 2 )x 2 + • • • + (ga m -a m )x m = 0 which is shorter than [START_REF] Berthelot | Notes on crystalline cohomology[END_REF]. From our minimality assumption, we deduce that ga i = a i for all i ≥ 2. Since this is valid for all g ∈ G, we deduce that the linear dependance relation [START_REF] Berthelot | Notes on crystalline cohomology[END_REF]

) → Z × p ⊂ Q × p . We claim that Q p (χ cycl ) is not isomorphic to Q p in the category RepQ p (G Qp ).
Indeed, assume by contraction that there exists a G-equivariant isomorphism Qp ≃ Qp (χ cycl ). Then there should exist an element x ∈ Qp such that gx = χ(g) x for all g ∈ G Qp .

Since x is in Qp , it belongs to a finite extension L of Q p . Let N L/Qp : L → Q p be the norm map from L to Q p . Applying it to (9), we get the relation N L/Qp (x) = χ(g) [L:Qp] • N L/Qp (x).

Since N L/Qp (x) does not vanish, we end up with χ(g)

Fontaine's strategy

We are now ready to explain the general principles of Fontaine's strategy for isolating the most interesting representations of the Galois group of a p-adic field and studying them. The material presented in this subsection comes from [21, Chap. II].

As before, let G be a topological group. Let also E be a fixed topological field. We consider a topological E-algebra B on which G acts and assume that the G-action on E is trivial. Under our assumptions, the category Rep E (G) is the category of E-linear representations of G.

Remark 1.4.1. In fact, in what follows, the topology on B will play no role since all the forthcoming definitions and results will be purely algebraic. Nevertheless, we prefer keeping the datum of the topology on B as it is more natural and all the rings B we shall consider later on will come equipped with a canonical topology.

The following definition is due to Fontaine. Example 1.4.3. Let L be a finite extension of E. Take G = Gal(L/E) and let it act naturally on L. Hilbert's theorem 90 (cf Theorem 1. 3.3) shows that all finite dimension E-representation of G is L-admissible.

A criterium for B-admissibility

The aim of this paragraph is to establish a numerical criterium for recognizing B-admissible representations. In order to do so, we make the following assumptions 5 on the E-algebra B:

(H1) B is a domain, (H2) (Frac B) G = B G , (H3) if b ∈ B, b = 0 and the E-line Eb is stable under G, then b ∈ B × .
It is easily seen that the assumption (H3) implies that B G is a field. Indeed for any b ∈ E, b = 0, the line Eb is clearly stable under G. Thus b has to be invertible in B. Now we conclude by noticing that its inverse is also fixed by G. Moreover, by copying the proof of the second part of Theorem 1.3.3, one shows that the assumptions (H1) and (H2) ensure that the morphism

α W : B ⊗ B G W G → W is injective for all W ∈ Rep B (G) which are free of finite rank over B.
In particular, this property holds true for W of the form B ⊗ E V where V is finite dimensional E-linear representation of G. Proposition 1.4.4. We assume that B satisfies (H1), (H2) and (H3) Let V ∈ Rep E (G) and set W = B ⊗ E V . We assume that V is finite dimensional over E. Then the following assertions are equivalent:

(i) W is trivial, (ii) the morphism α W is an isomorphism, (iii) dim B G W G = dim E V .
Proof. Since B G is a field, the equivalence between (i) and (ii) is obvious. Moreover the fact that (ii) implies (iii) is also clear. We then just have to prove that (iii) implies (ii).

We assume (iii) and denote by d the common dimension of V over E and

W G over B G . The morphism α W : B ⊗ B G W G → B ⊗ E V is a B-
linear morphism between two finite free B-modules of rank d. It is then enough to prove that its determinant is an isomorphism. Let v 1 , . . . , v d be a E-basis of V and let w 1 , . . . , w d be a B G -basis of W G . Let b be the unique element of B such that:

α W (v 1 ) ∧ • • • ∧ α W (v d ) = b • w 1 ∧ • • • ∧ w d . (10)
From the injectivity of α W , we derive b = 0. Let now g ∈ G. Applying g to (10), we get gb = η • b where η is defined by the identity α W (gv

1 ) ∧ • • • ∧ α W (gv d ) = η • α W (v 1 ) ∧ • • • ∧ α W (v d ).
From the fact that the E-span of v 1 , . . . , v d (which is V ) is stable under the action of G, we deduce that η lies in E. Hence gb ∈ Eb. Consequently, the E-line Eb is stable by the action of G. Thanks to hypothesis (H3), we conclude that b ∈ B × as wanted.

Corollary 1.4.5. Under the assumptions (H1), (H2) and (H3), the category Rep B-adm E (G) is stable by subobjects and quotients.

Proof. Assume that we are given an exact sequence 0 → V 1 → V → V 2 → 0 in the category Rep E (G) and assume that V is B-admissible. Tensoring by B and taking the G-invariants, we obtain the exact sequence 0

→ (B ⊗ E V 1 ) G → (B ⊗ E V ) G → (B ⊗ E V 2 )
G from which we derive the inequality:

dim B G (B ⊗ E V ) G ≥ dim B G (B ⊗ E V 1 ) G + dim B G (B ⊗ E V 2 ) G . (11)
Moreover we know that dim

B G (B ⊗ E V i ) G ≤ dim E V i for i ∈ {1, 2}.
Therefore we get:

dim B G (B ⊗ E V ) G ≤ dim E V 1 + dim E V 2 = dim E V. (12)
We know also that dim B G (B ⊗ E V ) G = dim E V thanks to the B-admissibility of V . Combining the inequalities [START_REF] Colmez | Construction des représentations p-adiques semi-stables[END_REF] and [START_REF] Conrad | Finite group schemes over bases with low ramification[END_REF], we find that dim

B G (B ⊗ E V i ) G has to be equal to dim E V i , which proves that V i (for i ∈ {1, 2}) is B-admissible.

What's next?

Until now, we have spent a lot of time at defining a general abstract formalism whose main achievement is the notion of B-admissibility. This is certainly nice but still seems to be quite far from the applications. We devote this subsection to our readers who are impatient to connect the notion of B-admissibility to concrete properties of Galois representations and cohomology of algebraic varieties.

In the sequel, we will often use the locution period rings to refer to various rings B. This terminology is motivation by the role those rings B play in geometry (they often appear in comparison theorem between various cohomologies).

From now, we go back to the setting of §1.1. Precisely, we let K be a finite extension of Q p . We let K denote a fixed algebraic closure of K and we set G K = Gal( K/K). Let C p be the completion of K. The action of G K on K extends to a continuous action of G K on C p . Finally, we recall that χ cycl : G K → Z × p denotes the p-adic cyclotomic character of G K .

C p -admissible representations. The first ring of periods we will consider is C p itself, equipped with the p-adic topology and its natural action of G K . The question of C p -admissibility of representations of G K will be studied in details in §2; we will notably prove the following result (cf Theorem 2.2.1).

C p ⊗ Qp V = C p (χ n 1 cycl ) ⊕ C p (χ n 2 cycl ) ⊕ • • • ⊕ C p (χ n d cycl ) (13)
for some integers n 1 , . . . , n d . The condition actually fits very well in the framework of Badmissibility as introduced above. Indeed, set B HT = C p [t, t -1 ] (HT stands for Hodge-Tate) and let G K act on it by the formula g

• (at i ) = ga • χ cycl (g) i • t i for g ∈ G K , i ∈ Z and a ∈ C p .
One checks that V is Hodge-Tate if and only if it is B HT -admissible. Besides, Theorem 1.4.6 is the starting point for studying Hodge-Tate representations. For example, it implies that the integers n i 's that appeared in [START_REF] Demazure | Lectures on p-divisible groups[END_REF] are uniquely determined up to permutation (cf Proposition 2.2.8). They are called the Hodge-Tate weights of the representation V . Finally, Hodge-like decomposition theorems show that many representations coming from geometry are Hodge-Tate. This class of representations then seems particularly interesting.

De Rham and crystalline representations. Unfortunately, Hodge-Tate representations have several defaults. First, they are actually too numerous and, for this reason, it is difficult to describe them precisely and design tools to work with them efficiently. The second defect of Hodge-Tate representations is of geometric nature. Indeed, tensoring the étale cohomology with C p (or equivalently, with B HT ) captures the graded module of the de Rham cohomology. However, it does not capture the entire complexity of de Rham cohomology, the point being that the de Rham filtration does not split canonically in the p-adic setting.

In order to work around this issues, Fontaine defined other period rings B "finer" than B HT . The most classical period rings introduced by Fontaine are B crys ⊂ B st ⊂ B dR ; the corresponding admissible representations are called crystalline, semi-stable and de Rham respectively. Moreover, B dR is a filtered field whose graded ring can be canonically identified with B HT . This property, together with the aforementionned inclusions, imply the following implications: crystalline =⇒ semi-stable =⇒ de Rham =⇒ Hodge-Tate.

In §3, we will discuss the construction of B dR and B crys , while the arithmetical and geometrical interest of these refined period rings will be presented in §4. Rapidly, let us say here that representations coming from the geometry, i.e. of the form H r ét (X K , Q p ) where X is a smooth projective algebraic variety over Q p , are all de Rham. By definition, this means that the space

B dR ⊗ Qp H r ét (X K , Q p )
G K has the correct dimension. It turns out that this space has a very pleasant cohomological interpretation: it is canonically isomorphic to the de Rham cohomology of X, namely H r dR (X). We thus get an isomorphism:

B dR ⊗ H r dR (X) ∼ -→ B dR ⊗ H r ét (X K , Q p )
which is a the right p-adic analogue of the de Rham comparison theorem. Besides, the de Rham filtration on H r dR (X) can be rebuilt from the filtration on B dR . This is the first apparition of the yoga of additional structures, which actually is ubiquitous in p-adic Hodge theory.

The introduction of B dR resolves elegantly the geometric issue we have pointed out earlier. However, the class of B dR -admissible representations is still rather large and not easy to describe. The ring B crys is a subring of B dR which is equipped with more structures and provides very powerful tools for describing crystalline representations. On the geometric side, crystalline representations correspond to the étale cohomology of varieties with good reduction and the space

B crys ⊗ Qp H r ét (X K , Q p )
G K is related to the crystalline cohomology of (the special fibre of a proper smooth model of) X, equipped with its Frobenius endomorphism. All in all, we will obtain powerful methods for describing the étale cohomology of X with comparatively downto-earth objects.

2 The first period ring: C p After Tate and Fontaine's results on Hodge-like decompositions of cohomology, the first natural period ring to consider is C p itself. In this section, we first study C p -admissibility and prove Theorem 1.4.6. The proof requires some preparation and occupies the first two subsections. The last subsection ( §2.3) is devoted to expose Sen's theory, as developed in [START_REF] Sen | Continuous cohomology and p-adic Galois representations[END_REF], whose aim is to go further than C p -admissibly and classify general C p -semi-linear representations.

Our approach differs a bit from usual presentations in that we will not use the langage of group cohomology but instead will keep working with (semi-linear) representations and vectors throughout the exposition.

Ramification in Z p -extensions

A first important ingredient in the proof of Theorem 1.4.6 is the study of ramification in Galois extensions over K whose Galois group is isomorphic to Z p .

Throughout this section, if L is an algebraic extension of K, we shall denote by O L its ring of integers, by m L the maximal ideal of O L and by k L = O L /m L its residue field. If moreover L/K is finite, we shall denote by v L : L → Z ∪ {+∞} the valuation on L normalized by v L (L × ) = Z. Set e L = v L (p); it is the ramification index of the extension L/Q p . If F and L are two algebraic extensions of K with F ⊂ L and [L : F ] < ∞, we shall use the notation e L/F for the ramification index of L/F and the notation Tr L/F for the trace map of L over F . When F is a finite extension of K, we have e L/F = e L e F . In what follows, it is convenient to allow flexibility and work over a base F which is not necessarily K but a finite extension of it.

Higher ramification groups

We first recall briefly the theory of higher ramification groups as exposed in [START_REF] Serre | Corps locaux[END_REF].

Let L/F be a finite Galois extension with Galois group G. For any nonnegative integer i, we define the i-th higher group of ramification of L/K as the subgroup G i of G consisting of elements g ∈ G such that g acts trivially on the quotient O L /m i+1 L . One easily checks that the G i 's form a nonincreasing sequence of normal subgroups of G and that G 0 is the inertia subgroup of G. Besides, one proves that the quotient G 0 /G 1 naturally embeds into k × F and thus is a cyclic of order prime to p, and, for i > 0, the quotient

G i /G i+1 embeds into m i L /m i+1 L
and thus is a commutative p-group killed by p.

The ramification filtration we have just defined is not compatible with subextensions. However, we can recover some compatibility after a suitable reindexation. In order to do so, we first define the Herbrand function ϕ L/F : R + → R + by:

ϕ L/F (u) = 1 e L/F • u 0 Card G t • dt (14)
where we agree that G t = G ⌈t⌉ when t is a nonnegative real number and ⌈•⌉ is the ceiling function. Clearly ϕ L/F is increasing and defines a bijection from R + to itself. Let ψ L/F be its inverse. For u ∈ R + , we set G u = G ψ L/F (u) . One can then show the following property. If L 1 and L 2 are two finite Galois extensions of K with L 1 ⊂ L 2 , then the canonical projection Gal(L 2 /F ) → Gal(L 1 /F ) maps surjectively Gal(L 2 /F ) u onto Gal(L 1 /F ) u for all u ∈ R + . This compatibility allows us the define ramification subgroups for infinite extensions: if L is a infinite Galois extension of K, we put Gal(L/F ) u = lim ← -F Gal(L ′ /F ) u where L ′ runs over all finite extensions of F included in L.

Moreover the ϕ's and ψ's functions verify very pleasant composition formulae: if F , L 1 and L 2 are extensions of K as above, we have ϕ

L 2 /F = ϕ L 1 /F • ϕ L 2 /L 1 and thus, passing to the inverse, ψ L 2 /F = ψ L 2 /L 1 • ψ L 1 /F .
Finally, we observe that the knowledge of the upper numbering of the ramification filtration is equivalent to that of the lower numbering. Indeed the function ψ L/F can be recovered for the G u 's thanks to the formula:

ψ L/F (t) = e L/F • u 0 du Card G u . ( 15
)
Now, taking the inverse of ψ L/F , we reconstruct the function ϕ L/F and we can finally recover the lower numbering of the filtration ramification by letting

G t = G ϕ L/F (t) .
Relation with the different. We will often use the higher ramification groups in order to compute (or estimate) the different of the extension L/F . Let us first recall that the different

D L/F of L/F is the ideal of O L characterized by the property that Tr L/F (xO L ) ⊂ O F if and only if x ∈ D -1 L/F , i.e. v p (x)+ v p (D L/F ) ≥ 0.
Here we have introduced the valuation of an ideal, which is defined as the minimal valuation of one of its elements (or, equivalently, if it is generated by one element, the valuation of any generator). The following formula relates D L/F with the size of the G i 's:

v L (D L/F ) = i≥0 (Card G i -1).
From this relation, we derive easily the following formulae:

v F (D L/F ) = lim t→∞ ϕ L/F (t) - t e L/F = lim u→∞ u - ψ L/F (u) e L/F . ( 16
)
Ramification and class field theory. When the extension L/F is finite and abelian, local class field theory gives a nice interpretation of the ramification filtration. More precisely, recall first that Artin reciprocity map provides an isomorphism Gal(L/F ) ≃ F × /N L/F (L × ) where N L/F is the norm of L over F . Under this isomorphism, the ramification subgroup Gal(L/F ) u corresponds to the image in F × /N L/F (L × ) of the congruence subgroup:

U u F = x ∈ O × F s.t. x ≡ 1 (mod m u F ) ⊂ F × .
A nontrivial consequence of this result is the Hasse-Arf theorem which states that the jumps of the filtration ramification in upper numbering (i.e. the real numbers u for which G u+ε = G u for all ε > 0) are all integers.

The case of Z p -extensions

We now consider a Galois extension F ∞ of F . We assume that F ∞ /F is ramified and that we are given an isomorphism α : Gal(F ∞ /F ) ≃ Z p . We remark that an extension with these properties always exists; it can be cooked up from the cyclotomic extension of F as discussed in §1.1.2.

For r ≥ 0, let γ r = α -1 (p r ) ∈ Gal(F ∞ /F ) and F r be the finite extension of F cut out by the closed subgroup generated by γ r (that is the subgroup α -1 (p r Z p )). The F r 's then form a tower of extensions, in which each F r+1 /F r is a cyclic extension of order p. More generally, if s ≥ r, the extension F s /F r is cyclic of order p s-r and its Galois group is generated by the class of γ r .

Let also e F be the absolute index of ramification of F defined as e F = v F (p).

Proposition 2.1.1. With the previous notations, there exists a ∈ Z such that, for u large enough,

Gal(F ∞ /F ) u is the closed subgroup generated by γ ⌈ u-a e F ⌉ .
Proof. For u ∈ R + , let ρ(u) be the unique element r of N ∪ {+∞} for which Gal(F ∞ /F ) u is topologically generated by γ r . This definition yields a function ρ : R + → N ∪ {+∞} which is nondecreasing and left-continuous. The fact that F ∞ /F is ramified shows that ρ(0) is finite. By the Hasse-Arf theorem, the points of discontinuity of ρ are all integers. Moreover there must be infinitely many of them since the successive quotients of the ramification filtration are all killed by p. This implies that ρ takes finite values everywhere.

Let s be a positive integer. By local class field theory, we know that Artin's isomorphism Gal(

F s /F ) ≃ F × /N Fr/F (F × r ) maps the subgroup Gal(F s /F ) u onto U u F /(U u F ∩ N Fr/F (F × r )
). Note that the group Gal(F s /F ) u is generated by the class of γ ρ(u) . Its subgroup of p-th powers is then generated by γ ρ(u)+1 . On the other hand, a simple computation shows that the subgroup of p-th powers of U u F is equal to U u+e F F as soon as u > e F p-1 . Comparing the subgroup of p-th powers of both sides, we obtain: whenever u > e F p-1 . Letting s go to infinity, we end up with ρ(u + e F ) = ρ(u) + 1 for u > e F p-1 . This relation, combined with the facts that ρ is nondecreasing, left-continuous and takes integral values, implies that there exists a real constant a such that ρ(u) = ⌈ u-a e F ⌉ for u > e F p-1 . The fact that a is indeed an integer is a consequence of the Hasse-Arf theorem.

min(s, ρ(u) + 1) = min(s, ρ(u + e F )) 1 2 3 4 1 p + 1 p 2 + p + 1 p 3 + p 2 + p + 1
Remark 2.1.2. Using formula [START_REF] Faltings | p-adic Hodge theory[END_REF], one can rephrase Proposition 2.1.1 as follows. For a positive integer r, let ψ r : R + → R + be the function defined by:

ψ r (u) = u if 0 ≤ u < 1 = p(u -1) + 1 if 1 ≤ u < 2 . . . = p r-1 (u -r + 1) + (1 + p + • • • + p r-2 ) if r -1 ≤ u < r = p r (u -r) + (1 + p + • • • + p r-1 )
if u ≥ r (cf Figure 1). Then, there exist u 0 ∈ R + and two constants a and b such that ψ Fr/F (u) = e F • ψ r ( u-a e F ) + b for all integer r and all u ≥ u 0 . Proposition 2.1.1 has several interesting corollaries that we will derive below. We begin with two of them that give information about the behavior of the trace map. The first one (Proposition 2.1.3) concerns extensions living inside F ∞ and shows that traces in such extensions tend to decrease the norm by a large factor. On the contrary, the second one (Proposition 2.1.5) concerns extensions which are "orthogonal" to F ∞ and shows that traces in such extensions have a norm which is close to 1. The conceptual meaning of these results is that the extension F ∞ /F captures almost all the ramification of K/F . Proposition 2.1.3. There exists a constant c 1 (depending only on F and F ∞ ) for which the following property holds: for any positive integers r and s with r ≤ s and for any x ∈ O Fs , we have:

v p Tr Fs/Fr (x) ≥ v p (x) + s -r -c 1 .
Proof. Fix a positive integer r. By the reformulation of Proposition 2.1.1 given in Remark 2.1.2, we have:

ψ Fr/F (u) e F p r = u -r - a e F + b p r e F - p r -1 p r (p -1)
when u is sufficiently large. Thanks to formula [START_REF] Faltings | Crystalline cohomology and p-adic Galois-representations[END_REF], we find:

v p (D Fr/F ) = r + a e F - b p r e F + p r -1 p r (p -1)
.

Given now two integers r and s with r ≤ s, the transitivity property of the different implies that:

v p (D Fs/Fr ) = v p (D Fs/F ) -v p (D Fr/F ) = s -r - b p s e F + b p r e F + p s -1 p s (p -1) - p r -1 p s (p -1)
.

Then there exists c ∈ N, not depending on r and s, such that v p (D Fs/Fr ) ≥ s -r -c. Going back to the definition of the different, we obtain the inclusion Tr Fs/Fr (p r-s+c O Fs ) ⊂ O Fr . Let now x ∈ F s and let v be the integer part of v p (x). Then p -v x falls in O Fs , so that we get Tr Fs/Fr (p r-s+c-v x) ∈ O Fr , i.e. Tr Fs/Fr (x) ∈ p s-r-c+v O Fr . Consequently:

v p (Tr Fs/Fr (x)) ≥ s -r -c + v ≥ v p (x) + s -r -c -1.
We can then take c 1 = c + 1.

Remark 2.1.4. For a fixed integer r, we can glue the Tr Ks/Kr (for s varying) and define a function

R r : K ∞ → K r by R r (x) = p r-s
Tr Ks/Kr (x) for x ∈ K s . We notice that the above definition makes sense because if s ≤ t, the functions p r-s Tr Ks/Kr and p r-t Tr Kt/Kr coincide on K s . Proposition 2.1.3 shows that the function R r obtained this way is uniformly continuous. It then extends (uniquely) to the completion K∞ of K ∞ . The functions R r : K∞ → K r are called the Tate's normalized traces.

Proposition 2.1.5. Let L be a finite Galois extension of F . For all ε > 0, there exist a positive integer r and an element

x ∈ O L•Fr such that v p (Tr L•Fr/Fr (x)) ≤ ε.
Proof. Up to replacing F by F ∞ ∩ L, we may assume that F ∞ and L are linearly disjoint over F . We set

L ∞ = L•K ∞ and L r = L•F r for all r.
The extension L ∞ /L is then a Z p -extension and the L r 's correspond to the subgroups p r Z p . By formula ( 16) and Remark 2.1.2, there exist u 0 ∈ R + and a, b, a ′ , b ′ ∈ R for which:

ψ L/F (u) = e L/F • u -v F (D L/F ) ψ Lr/Fr (u) = e L/F • u -v Fr (D Lr/Fr ) ψ Fr/F (u) = e F • ψ r u -a e F + b ψ Lr/L (u) = e F • ψ r u -a ′ e L + b ′
for all u ≥ u 0 and all positive integer r. Writing ψ Lr/L • ψ L/F = ψ Lr/Fr • ψ Fr/F , we obtain:

e L • ψ r u e F - a e F + e L/F • b -v Fr (D Lr/Fr ) = e L • ψ r u e F - v F (D L/F ) e F - a ′ e L + b ′
for u ≥ u 0 . When r is sufficiently large, the above identity of functions implies, by comparing slopes, that

a e F = v F (D L/F ) e F + a ′ e L
z y )) < 1 e F p r . Set n = ⌈b-b ′ ⌉ and x = π n Fr z y . We have v p (x) ≥ n-(b-b ′ ) e F p r ≥ 0; hence x ∈ O Lr . Moreover: v p (Tr Lr/Fr (x)) = v p π n Fr Tr Lr/Fr ( z y ) = n e F p r + v p Tr Lr/Fr ( z y ) < n + 1 e F p r < b -b ′ + 2 e F p r .
We conclude the proof by noticing that, when r goes to infinity, the upper bound b-b ′ +2 e F p r converges to 0.

We shall also need the following result which is a refinement of the classical additive Hilbert's theorem 90, allowing in addition some control on the valuation. Proposition 2.1.6. There exists a constant c 2 (depending only on F and F ∞ ) for which the following property holds: for any positive integers r and s with r ≤ s and for any x ∈ O Fs with Tr Fs/Fr (x) = 0, there exists y ∈ F s such that (i) Tr Fs/Fr (y) = 0, (ii) x = γ r y -y and (iii)

v p (y) ≥ v p (x) -c 2 .
Moreover y is uniquely determined by the conditions (i) and (ii).

Proof. We set d = s -r and:

y = - 1 p d • p d -1 i=0 i γ i r (x).
Noticing that γ p d r is the identity on F s , we find that γ r y -y = x + 1 p d Tr Fs/Fr (x) = x. Moreover the assumption on the trace of x implies that Tr Fs/Fr (y) vanishes as well.

Let now c 1 be the constant of Proposition 2.1.3 and set

c 2 = c 1 + 1. For m ∈ {0, . . . , d}, we define x m = 1 p d-m Tr Fs/F r+m (x) and y m = -1 p m • p m -1 i=0 i γ i r (x m ). Obviously x d =
x and y d = y. Moreover, noticing that any integer between 0 and p m -1 can be uniquely written as a + p m-1 b with 0 ≤ a < p m-1 and 0 ≤ b < p, we obtain:

y m-1 -y m = 1 p • p m-1 -1 a=0 p-1 b=0 b γ a+p m-1 b r (x m ). Therefore v p (y m ) ≥ min(v p (x m )-1, v p (y m-1 )) and so v p (y) = v p (y d ) ≥ min 1≤m≤d v p (x m ) -1. By Proposition 2.1.3, we end up with v p (y) ≥ v p (x) -c 2 .
The element y we have constructed then satisfies the requirements (i), (ii) and (iii). It remains to prove unicity. Assume that we have given y 1 and y 2 such that Tr Fs/Fr (y 1 ) = Tr Fs/Fr (y 2 ) and x = γ r y 1 -y 1 = γ r y 2 -y 2 . Set z = y 1 -y 2 . The second condition implies that z is fixed by γ r . Hence z ∈ F r and Tr Fs/Fr (z) = p s-r z. On the other hand, one has Tr Fs/Fr (z) = Tr Fs/Fr (y 1 ) -Tr Fs/Fr (y 2 ) = 0. We conclude that p s-r z = 0 and hence y 1 = y 2 .

Remark 2.1.7. Using Tate's normalized traces (cf Remark 2.1.4), one may extend Proposition 2.1.6 for x ∈ K∞ . The result we obtain reads as follows: for all x ∈ K∞ with R r (x) = 0, there exists a unique y ∈ K r such that x = γ r y -y and R r (y) = 0. Moreover, this element y satisfies v p (y) ≥ v p (x) -c 2 . In other terms, the function (γ rid) is bijective on the kernel of R r and its inverse is continuous.

C p -admissibility

We now come to the study of C p -admissibility of representations of G K . The main objective of this subsection is to prove Theorem 1.4.6 whose statement is recalled below.

Theorem 2.2.1. Let V be a Q p -linear finite dimensional representation of G K . Then V is C p - admissible if and only if the inertia subgroup of G K acts on V through a finite quotient.
As an application, in §2.2.3, we will explain how Theorem 2.2.1 can be used to understand better the internal structure of Hodge-Tate representations.

Preliminaries

Before giving the proof of Theorem 2.2.1, we need some preparation. The first input that we shall use is Ax-Sen-Tate theorem, whose purpose is to compute the fixed subspace of C p under the action of G K . As in the previous subsection, we shall work over a base F which is itself a finite extension of K. For convenience, we set

G F = Gal( K/F ). Theorem 2.2.2 (Ax-Sen-Tate). We have C G F p = F .
We shall prove a "finite" version of Ax-Sen-Tate theorem which is more precise.

Theorem 2.2.3.

There exists a constant c 3 (depending only on F ) for which the following property holds: for all real number v and all x ∈ K such that v p (gx -x) ≥ v for all g ∈ G F , there exists

z ∈ K such that v p (x -z) ≥ v -c 3 .
Proof. Throughout the proof, we fix a Z p -extension F ∞ of F . We recall that such an extension always exists and can be built from the cyclotomic extension of F as discussed in §1.1.2.

Let L be a finite Galois extension of Q p in which x lies. Thanks to Proposition 2.1.5, one can choose an integer r together with an element λ ∈ L•F r with the property that v p (Tr L•Fr/Fr (λ)) ≤ 1. We consider the elements:

y = Tr L•Fr/Fr (λx) Tr L•Fr/Fr (λ) ∈ F r and z = 1 p r • Tr Fr/F (y) ∈ F. The fact that v p (gx -x) ≥ v for all g ∈ G F implies that v p (y -x) ≥ v -1.
Observe that Tr Fr/F (y -z) = p r z -p r z = 0 and (γ 0id)(y -z) = γ 0 y -y. In other words, the element y -z has trace 0 and is an antecedent of γ 0 y -y by the application (γ 0id). By Proposition 2.1.6, it follows that v p (y -z)

≥ v p (γ 0 y -y) -c 2 . Now notice that the combinaison of v p (γ 0 x -x) ≥ v and v p (y -x) ≥ v -1 ensures that v p (γ 0 y -y) ≥ v -1. Therefore, we obtain v p (y -z) ≥ v -(c 2 + 1). Finally v p (x -z) ≥ min(v p (x -y), v p (y -z)) ≥ v -(c 2 + 1) and we can take c 3 = c 2 + 1. Proof of Theorem 2.2.2. Let x ∈ C G F p .
We consider a positive integer n. Since C p is the completion of Qp , one can find an element

x n ∈ Qp such that v p (x -x n ) ≥ n. We then have v p (gx n -x n ) ≥ n for all g ∈ G F . By Theorem 2.2.3, one can find z n ∈ K such that v p (z n -x n ) ≥ n -c 3 .
This implies that v p (z n -x) ≥ n -c 3 as well. The sequence (z n ) n≥1 then converges to x. Since z n ∈ K for all K, we obtain x ∈ K. Remark 2.2.4. As presented above, it seems that the proof of Theorem 2.2.2 uses class field theory (via Proposition 2.1.1). In fact, it is not the case because we have the choice on the Z pextension F ∞ . If we decide to take the Z p -part of the cyclotomic extension, the computation of the ramification filtration of Gal(F ∞ /F ) can be carried out explicitely, so that Proposition 2.1.1 can be proved in this case without making any reference to class field theory.

The proof we have exposed above is essentialy due to Tate [START_REF] Tate | p-divisible groups[END_REF]. A few years later, Ax [START_REF] Ax | Zeros of polynomials over local fields[END_REF] reproves the theorem using a more direct and elementary argument. We presented Tate's proof because we believe that it serves as a very good introduction to the developments we will discuss afterwards, which are all modeled on the same strategy: in order to study the action of G F (on some space), we will always first descend to F ∞ using Proposition 2.1.5 and then to F -or possibly only F r for some finite r-using Proposition 2.1.3 or Proposition 2.1.6.

Ax's proof provides in addition an explicit value for the constant c 3 , namely p (p-1) 2 . Ax asks for the optimality of this constant. In [START_REF] Borgne | Optimisation du théorème d'Ax-Sen-Tate et application à un calcul de cohomologie galoisienne p-adique[END_REF], Le Borgne answers this question and shows that the optimal constant is not p (p-1) 2 , but 1 p-1 . Le Borgne's proof follows Tate's strategy but uses a non Galois extension in place of the cyclotomic extension.

An extension of Hilbert's theorem 90. Another input we shall need is a variant of Hilbert's theorem 90 (cf Theorem 1.3.3) valid for infinite unramified extensions. We recall that K ur denotes the maximal unramified extension of K (inside K). We define Kur as the completion of K ur ; it is a field which naturally embeds into C p and which is equipped with a canonical action of Gal(K ur /K). Proposition 2.2.5. Any finite dimensional Kur -semi-linear representation of Gal(K ur /K) is trivial. Remark 2.2.6. Proposition 2.2.5 implies in particular that unramified representation of G K are Kur -admissible and then a fortiori C p -admissible. It then appears as a first step towards the proof of Theorem 2.2.1.

Proof of Proposition 2.2.5. In order to simplify notations, we denote by O the ring of integers of Kur , and by m its maximal ideal. We recall that the quotient O/m is isomorphic to an algebraic closure k of k. We recall also that Gal(K ur /K) is a procyclic group generated by the Frobenius Frob q : x → x q (where q is the cardinality of k).

Let W be a finite dimensional Kur -semi-linear representation. We fix (v 1,0 , . . . , v d,0 ) a basis of W over Kur . Let O W be a O-span of v 1,0 , . . . , v d,0 . We are going to construct a sequence of tuples (v 1,n , . . . , v d,n ) such that v i,n+1 ≡ v i,n (mod m n ) and Frob q (v i,n ) ≡ v i,n (mod m n ) for all i ∈ {1, . . . , d} and all n ∈ N.

We proceed by induction on n. The case n = 1 reduces to the fact that O W /mO W is trivial as a k-semi-linear representation of Gal(K ur /K) ≃ Gal( k/k). In order to prove this, we remark that, using continuity, O W /mO W descends at finite level: there exist a finite extension ℓ of k and a ℓ-semi-linear representation W ℓ of Gal(ℓ/k) such that k ⊗ ℓ W ℓ = O W /mO W . The property we want to establish then follows from Hilbert's theorem 90 (cf Theorem 1.3.3).

We now assume that (v 1,n , . . . , v d,n ) has been constructed. We look for vectors w 1 , . . . , w n ∈ O W such that Frob q (v i,n + π n w i ) ≡ v i,n + π n w i (mod m n+1 ) for all i. Letting wi be the image of w i in O W /mO W , the system we have to solve can be rewritten Frob q wi -wi = ci (1

≤ i ≤ d)
where ci is defined as the image of

Frobqv i,n -v in π n in O W /mO W .
It is then enough to prove that (Frob qid) is surjective on O W /mO W . This follows directly from the triviality of O W /mO W and the fact that (Frob qid) is surjective on k.

We conclude the proof by remarking that, for any fixed i, the sequence v i,n is Cauchy and hence converges to a vector v i ∈ O W on which Gal(K ur /K) acts trivially. Moreover the family of v i 's is an O-basis of O W (because its reduction modulo m is a basis of O W /mO W ) and then it is also a Kur -basis of W .

Proof of Theorem 2.2.1

We are now ready to prove Theorem 2.2.1.

Write d = dim Qp V . We first assume that the inertia subgroup acts on V through a finite quotient. In other words, there exists a finite extension L of K ur for which Gal( K/L) acts trivially on V . By Hilbert's theorem 90 (cf Theorem 1.3.3), the L-semi-linear representation L ⊗ Qp V admits an L-basis (v 1 , . . . , v d ) on which the action of Gal(L/K ur ) is trivial. Consequently Gal(K ur /K) operates on the Kur -span of v 1 , . . . , v d . By Proposition 2.2.5, this semi-linear representation is trivial. Therefore V is (L• Kur )-admissible. It is then also C p -admissible.

We now focus on the converse. We assume that V is C p -admissible. Then by definition, there exists a C p -basis (w 1 , . . . , w d ) of C p ⊗ Qp V with the property that gw i = w i for all g ∈ G K and all i ∈ {1, . . . , d}. Let (v 1 , . . . , v d ) be a basis of V over Q p and let P ∈ GL d (C p ) be the matrix representating the change of basis between the v i 's and the w i 's. Up to rescaling the v i 's, we may assume without loss of generality that

P ∈ M d (O Cp ).
From the fact that the Q p -span of the v i 's is stable under the action of G K , we derive that the matrix U g = P -1 • gP has coefficients in Q p for all g ∈ G K . Let c 3 be the constant of Theorem 2.2.3 and let v be a positive integer for which p v •P -1 has coefficients in O Cp . By continuity of the action of G K , denoting by I d the identity matrix of size d, there exists an open subgroup H of G K such that v p (U g -I d ) ≥ v + c 3 + 1 for all g ∈ H. Multiplying by P on the left, we get v p (P -gP ) ≥ v + c 3 + 1 for all g ∈ H. Applying now Theorem 2.2.3 (to each entry of P ), we find a matrix P 0 ∈ GL n (L) such that P ≡ P 0 (mod p v+1 ). multiplying by P -1 on the left, we get P -1 P 0 ≡ I d (mod p). Define M = P -1 P 0 . Writing P = P 0 M -1 , we find M • gM -1 = U g for all g ∈ H. Since M ≡ I d (mod p), the matrix N = log M is well defined and satisfies the relation N -gN = log U g for all g ∈ H.

Let F be the extension of K cut out by H. We are going to prove that H ∩ I K operates trivially on N . Let ξ be an entry of N . The relation N -gN = log U g ensures that ξ -gξ ∈ Z p whenever g is in H. Define the function α : H → Z p by α(g) = gξ -ξ. The computation

α(g 1 g 2 ) = g 1 g 2 ξ -ξ = g 1 (g 2 ξ -ξ) + (g 1 ξ -ξ) = (g 2 ξ -ξ) + (g 1 ξ -ξ) = α(g 2 ) + α(g 1 )
shows that α is an additive character. Its kernel defines a Galois extension F ∞ of F whose Galois group embeds into Z p . Moreover, by construction, H ∩ ker α acts trivially on ξ. We then need to prove that F ∞ is unramified over F .

We assume by contraction that the extension F ∞ /F is ramified. In particular, it is not trivial, and hence it is a Z p -extension. Proposition 2.1.3 then applies and ensures that there exists a constant c 1 such that:

v p (Tr Fs/Fr (z)) ≥ v p (z) + s -r -c 1 (17)
whenever s ≥ r and z ∈ F s . Let v be a positive real number and let x be an element of K such that v p (x -ξ) ≥ v. From the equality gξ -ξ = α(g), we derive v p gx -x -α(g) ≥ v for all g ∈ H. In particular, if g ∈ H ∩ ker α, we obtain v p (gx -x) ≥ v. By continuity, this estimation is also correct for g ∈ Gal( K/F s ) for some integer s. Repeating the first part of the proof of Theorem 2.2.3 (with the Z p -extension F ∞ /F ) and possibly enlarging s, we find that there exists y ∈ F s with the property that v p (x -y) ≥ v -1. Thus v p (ξ -y) ≥ v -1 as well. Fix now g ∈ H and set z = gy -y -α(g). By our assumption on ξ, we know that v p (z) ≥ v -1. Using ( 17) with r = 0, we obtain v p (Tr Fs/F 0 (z)) ≥ v + s -c 1 . On the other hand, a direct computations yields Tr Fs/F 0 (z) = -p s α(g). Combining these two inputs, we deduce v p (α(g)) ≥ v -c 1 . Since this estimation holds for all g ∈ H and all v ∈ R + , we end up with α = 0. This means that F ∞ = F and then contradicts our assumption that F ∞ /F was ramified.

Remark 2.2.7. It follows from the proof above (cf in particular the first paragraph of the proof) that a representation is C p -admissible if and only if it is (L• Kur )-admissible for a finite extension L of K ur . We will reuse this property in §4.2 when we will compare C p -representations with de Rham representations.

Application to Hodge-Tate representations

Beyond its obvious own interest, Theorem 2.2.1 can be thought of as a first result towards the study of Hodge-Tate representations. Recall that a finite dimensional

Q p -linear representation V of G K is Hodge-Tate if C p ⊗ Qp V decomposes as: C p ⊗ Qp V = C p (χ n 1 cycl ) ⊕ C p (χ n 2 cycl ) ⊕ • • • ⊕ C p (χ n d cycl ) (18)
for some integers n i 's. Theorem 2.2.1 implies the following unicity result.

Proposition 2.2.8. Let V be a finite dimensional Hodge-Tate representation of G K . Then the integers n i of Eq. ( 18) are uniquely determined up to permutation.

Proof. We have to show that, if

C p (χ n 1 cycl ) ⊕ • • • ⊕ C p (χ n d cycl ) ≃ C p (χ m 1 cycl ) ⊕ • • • ⊕ C p (χ m d ′ cycl
), then d = d ′ and the n i 's agree with the m i 's up to permutation. For this, it is enough to check that, given two integers n and m,

Hom Rep Cp (G K ) C p (χ n cycl ), C p (χ m cycl ) (19)
is a one dimensional K-vector space if n = m, and is zero otherwise.

Let W = Hom Cp C p (χ n cycl ), C p (χ m cycl ) ≃ C p (χ n-m cycl ) (equipped with its Galois action). The space (19) is equal to W G K . When n = m, it is then C G K p
which is indeed equal to K by Ax-Sen-Tate theorem (cf Theorem 2.2.2). If n = m, we need to prove that W is not trivial, which means that the representation V = Q p (χ n-m cycl ) is not C p -admissible. By Theorem 2.2.1, we are reduced to justify that the inertia subgroup of Q p does not act on V through a finite quotient. This is clear because the extension cut out by the kernel of χ n-m cycl is the p-adic cyclotomic extension which is infinitely ramified. Remark 2.2.9. A byproduct of the proof above is that the C p -semi-linear representation C p (χ n cycl ) has no nonzero invariant vector when n = 0. We will reuse repeatedly this property in the sequel.

C p (µ λ • χ a cycl • ω b cycl ) ≃ C p (χ a cycl ).
Hence the character µ λ •χ a cycl •ω b cycl is Hodge-Tate if and only if a ∈ Z. In this case, its Hodge-Tate weight is a.

Example 2.2.11. Let α : G K → Z p be an additive character, e.g. α = log χ cycl . Consider the two dimensional representation V corresponding to the group homomorphism:

G K → GL 2 (Q p ), g → 1 α(g) 0 1
.

In order terms, V = Q 2 p and G K acts to V by g •(u, v) = (u, v + α(g)u).
We have an obvious exact sequence 0 → Q p → V → Q p → 0 where the action of G K on the two copies of Q p is the trivial action. Tensoring this sequence by C p , we get 0 → C p → V → C p → 0. The representation V is Hodge-Tate if and only if the above sequence splits, if and only if V is C p -admissible. By Theorem 2.2.1, this happens if and only if α(I K ) is finite (where I K is the inertia subgroup of G K ). Since α(I K ) is a subgroup of Z p , the previous condition is equivalent to the fact that α(I K ) is reduced to 0. As a conclusion, the representation V is Hodge-Tate if and only if α is unramified. In this case, the Hodge-Tate weights of V are 0 with multiplicity 2.

Hodge-Tate representations and admissibility.

It is important to notice that the class of Hodge-Tate representations fits very well in Fontaine's framework presented in §1.4. Precisely, let us consider the rings B HT = C p [t, t -1 ] and B ′ HT = C p ((t)). We equip them with the Galois action obtained by letting G K act naturally on C p and act on t by gt = χ cycl (g) t for all g ∈ G K . In addition, we define a filtration of

B ′ HT by Fil m B ′ HT = t m C p [[t]] for m varying in Z. The graded ring of B ′
HT is, by definition:

gr B ′ HT = m∈Z Fil m B ′ HT /Fil m B ′ HT .
We observe that it is canonically isomorphic to B HT . Besides, we have a natural G K -equivariant inclusion B HT → B ′ HT . Ax-Sen-Tate theorem, together with the fact that C p (χ n cycl ) has no nonzero invariant vectors as soon as n = 0, implies that (B HT )

G K = (B ′ HT ) G K = K.
Proposition 2.2.12. The rings B HT and B ′ HT satisfy Fontaine's assumptions (H1), (H2) and (H3) (introduced in §1.4.1).

Proof. This is obvious for B ′

HT since it is a field. As for B HT , it is clearly a domain. Moreover since B ′ HT is a field, we have B HT ⊂ Frac B HT ⊂ B ′ HT . Taking the G K -invariants, we obtain (Frac B HT ) G K = K; hence B HT satisfies (H2). Finally, we prove that B HT satisfies (H3). Let x ∈ B HT , x = 0 and assume that the line Q p x is stable by G K . We have to prove that x is invertible in B HT . Up to multiplying x by some power of t, we may assume that x ∈ C p [t]. Write x = a 0 + a 1 t + • • • + a n t n where the a i 's are in C p . Our assumption implies that there exists λ ∈ Q p such that ga i • χ(g) i = λa i for all g ∈ G K and all i ∈ {0, 1, . . . , n}. Let j be an index for which a j = 0 and write µ i = a i a j for all i. We then have gµ i = χ(g) j-i µ i for all g and i. If i = j, this implies that µ i = 0 since C p (χ j-i cycl ) G K = 0. Therefore x has to be equal to a j t j , and so is invertible in B HT . Proposition 2.2.13. Let V be a finite dimensional Q p -linear representation. Then V is Hodge-Tate if and only if it is B HT -admissible, if and only if it is B ′ HT -admissible.

Proof. Write d = dim Qp V . Observe that B HT = m∈Z C p (χ m cycl ) as a C p -semi-linear representation. Therefore:

V ⊗ Qp B HT G K ≃ m∈Z V ⊗ C p (χ m cycl ) G K .
Suppose that V is Hodge-Tate. Let m 1 , . . . , m s be its Hodge-Tate weights and e 1 , . . . , e s be the corresponding multiplicities. The space V ⊗ C p (χ -m i cycl ) G K has then dimension e i . Summing up all these contributions, we find that

(V ⊗ Qp B HT ) G K has dimension d, which means that V is B HT -admissible.
The converse and the case of B ′ HT are proved in a similar fashion and left to the reader.

Complement: Sen's theory

The aim of this subsection is to expose Sen's theory [START_REF] Sen | Continuous cohomology and p-adic Galois representations[END_REF] whose objective is to provide a systematic study of finite dimensional C p -semi-linear representations of G K . In what follows, we choose and fix once for all a Z p -extension K ∞ of K. We let α : Gal(K ∞ /K) → Z p be the attached group isomorphism. As in §2.1.2, we put γ r = α -1 (p r ) and let K r be the subextension of K ∞ corresponding to the closed subgroup α -1 (p r Z p ).

We recall that one possible choice is α = log χ cycl , in which case K ∞ is the Z p -part of the cyclotomic extension of K (cf §1.1.2). Actually, strictly speaking, Sen's theory only concerns this particular choice of α. However the extension of general α's is straightforward. In what follows, we do not restrict ourselves to α = log χ cycl .

Remark 2.3.1. Recently, Berger and Colmez [START_REF] Berger | Théorie de Sen et vecteurs localement analytiques[END_REF] generalized Sen's theory, allowing α to take its values in any p-adic Lie group (possibly noncommutative). Their theory relies on the notion of locally analytic vectors, which is not needed in classical Sen's theory (finite vectors are enough as we shall explain below). We will not expose their generalization in the article and do restrict ourselves to homomorphisms α taking their values in Z p .

We recall that, given a topological group G and a topological ring B on which G acts, we have introduced the notation Rep B (G) for the category of B-semi-linear representations of G. Descend. The first result towards Sen's theory is Proposition 2.3.2 just below, which could understood as an analogue of Hilbert's theorem 90 for the Galois group Gal( K/K ∞ ).

Proposition 2.3.2. Let W ∈ Rep f

Cp (G K ). Then there exist an integer r and a

C p -basis v 1 , . . . , v d of W such that gv i = v i for all g ∈ Gal( K/K ∞ ).
Proof. The proof is similar to that of Proposition 2.2.5.

Let O W be any O Cp -lattice in W . As a first step, we are going to construct a C p -basis w 1 , . . . , w d of W with

w i ∈ O W , gw i ≡ w i (mod p 2 O W ) for all g ∈ Gal( K/K ∞ ) and pO W ⊂ O Cp w 1 ⊕ • • • ⊕ O Cp w d .
By continuity of the Galois action, there exists a finite Galois extension L of K such that gw ≡ w (mod p 2 O W ) for all g ∈ Gal( K/L) and all w ∈ W . For a positive integer r, set L r = L•K r . By the proof of Proposition 2.1.3, we know that there exists r for which v p (D Lr/Kr ) < e -1 L/K . We fix such an r. Let λ 1 , . . . , λ m be a O Kr -basis of O Lr and let g 1 , . . . , g m be the elements of Gal(L r /K r ). For i ∈ {1, . . . , m}, choose ĝi ∈ G K a lift of g i . We define the elements:

y i,j = m i ′ =1 ĝi ′ (λ j x i ) = m i ′ =1 g i ′ (λ j ) • ĝi ′ (x i )
for i varying between 1 and d and j varying between 1 and m. It is easily seen that gy i,j ≡ y i,j (mod p 2 O W ) for all g ∈ Gal( K/K r ). Moreover, the determinant of the matrix (g i (λ j )) 1≤i,j≤m is, by definition, the discriminant of L r /K r . Its p-adic valuation is then less than 1 thanks to our assumption on v p (D Lr /Kr ). We deduce that there exist µ 1 , . . . , µ m ∈ O Lr with the property that m j=1 µ j Tr Lr/Kr (λ j ) = p. Hence m j=1 µ j y i,j ≡ px i (mod p 2 O W ) for all i. The O Cp -span of the y i,j 's then contains pO W . Among these vectors, one can select d of them w 1 , . . . , w d whose span still contains pO W . The w i 's satisfy all the announced properties.

The second step of the proof consists in lifting the w i 's by a process of successive approximations. In order to simplify the notations, we redefine O W as the O Cp -span of w 1 , . . . , w d . With the new definition, we have gw i ≡ w i (mod pO W ) for all g ∈ Gal( K/K ∞ ). We will construct by induction on n a sequence of families (v 1,n , . . . , v d,n ) satisfying the following congruences:

v i,n+1 ≡ v i,n (mod p n O W ) and gv i,n ≡ v i,n (mod p n O W )
for all i ∈ {1, . . . , d}, n ∈ N and g ∈ Gal( K/K ∞ ). For n = 1, we set v i,1 = w i . Now we assume that the v i,n 's have been constructed. By continuity there exists a finite Galois extension L of K such that gv i,n ≡ v i,n (mod p n+2 O W ) for all g ∈ Gal( K/L). By Proposition 2.1.3, there exist an integer r and λ ∈ O Lr (with L r = L•K r ) such that v p (Tr Lr/Kr (λ)) ≤ 1. As in the first step, we let g 1 , . . . , g m be the elements of Gal(L r /K r ) and we choose a lifting ĝi ∈ G K of g i . We define:

v i,n+1 = 1 Tr Lr/Kr (λ) • m j=1 ĝj (λv i,n ) = 1 Tr Lr/Kr (λ) • m j=1 g j (λ) • ĝj (v i,n )
and check that the v i,n+1 's satisfy the desired requirements. We conclude the proof by taking the limit with respect to n.

Proposition 2.3.2 tells us that the W is trivial when viewed as a C p -linear representation of Gal( K/K ∞ ). Moreover by the proof of Ax-Sen-Tate theorem, the fixed field C

Gal( K/K∞) p
is the completion of K ∞ , that we shall call K∞ . By general results of trivial semi-linear representations (cf §1.3.2), we then have an isomorphism

C p ⊗ K∞ W Gal( K/K∞) ≃ W
for all C p -semi-linear representation of W . We notice that W Gal( K/K∞) inherits an action of Gal(K ∞ /K).

Finite vectors.

Set Γ = Gal(K ∞ /K). The second step is Sen's theory is the study of K∞semi-linear representations of Γ. To this attempt, Sen defines the subspace of finite vectors as follows.

Definition 2.3.3. Let W ∈ Rep f K∞ (Γ). A vector v ∈ W is finite if the K ∞ -subspace of W generated by the gv for g varying in Γ is finite dimensional over K ∞ .
As an example, the subspace of finite vectors of the semi-linear representation K∞ itself is K ∞ . In general, one easily checks that the subspace of finite vectors is a vector space over K ∞ . Proposition 2.3.4. Let W ∈ Rep f K∞ (Γ). Then, there exist an integer r and a basis (v 1 , . . . , v d ) of W with the property that the K r -span of the v i 's is stable under the Γ-action.

Remark 2.3.5. Obviously, the v i 's of Proposition 2.3.4 are finite in the sense of Definition 2.3.3. Therefore, we deduce that the subspace of finite vectors of W generates W as a K∞ -vector space. Finite vectors are then numerous.

Proof of Proposition 2.3.4. Let c 2 be the constant of Proposition 2.1.6. It is harmless to assume that c 2 is an integer. To simplify notation, we write L = K∞ . Let O L be the ring of integers of L. We choose a O L -lattice O W inside W . By continuity, there exists an integer r such that gw ≡ w (mod p c 2 +1 O W ) for all g ∈ Gal(K ∞ /K r ) and all w ∈ W . We choose and fix such an r. The group Gal(K ∞ /K r ) acts on O W and on all the quotients O W /p n O W for n ∈ N.

We are going to construct, by induction of n, a sequence of families (v 1,n , . . . , v d,n ) n≥1 of elements of O W with the following properties:

(i) for all n, the family v 1,n , . . . , v d,n is an O L -basis of O W , (ii) for all n ≥ 1 and all i, v i,n+1 ≡ v i,n (mod p n O W ) (iii) the O Kr -submodule of O W /p n+c 2 O W generated by the classes of the v i,n 's (1 ≤ i ≤ d) is
stable under the Gal(K ∞ /K r )-action.

For n = 1, we pick an arbitrary

O L -basis v 1,1 , . . . , v d,1 of O W . Since Gal(K ∞ /K r ) acts trivially on O W /p c 2 +1
O W , all the requirements are fulfilled. We now assume that v 1,n , . . . , v d,n have been constructed. By the induction hypothesis, for all i, we can write γ r v i,n = v i,n + d j=1 (λ i,j + ε i,j )v j,n where the λ i,j 's lie in K r and the ε i,j 's have p-adic valuation at least n + c 2 . Moreover, since the action of γ r is trivial modulo p c 2 +1 , we deduce v p (λ i,j ) ≥ c 2 +1. Let R r : L → K r be the Tate's normalized trace defined in Remark 2.1.4. By Proposition 2.1.6 (cf also Remark 2.1.7), for all i and j, there exists µ i,j ∈ L with v p (µ i,j ) ≥ n and ε i,j = R r (ε i,j ) + γ r µ i,j -µ i,j . For all i, define v i,n+1 = v i,n -d j=1 µ i,j v j,n . Since the µ i,j 's have all valuation at least n, the items (i) and (ii) are fulfilled. Besides, a simple computation gives:

γ r v i,n+1 = v i,n+1 + d j=1 λ i,j + R r (ε i,j ) • v j,n + d j=1 γ r µ i,j • v j,n -γ r v j,n .
Since γ r acts trivially modulo p c 2 +1 , the last summand lies in p n+c 2 +1 O W . Noting in addition that v j,n ≡ v j,n+1 (mod p n O W ) and that the λ i,j 's are all divisible by p c 2 +1 , we obtain the congruence:

γ r v i,n+1 ≡ v i,n+1 + d j=1 λ i,j + R r (ε i,j ) • v j,n+1 (mod p n+c 2 +1 O W )
from which the item (iii) follows.

Passing to the limit, we obtain an L-basis v 1 , . . . , v d of W whose K r -span is stable under the action of Gal(K r /K). It remains to prove that it is stable under the whole action of Γ. Let M 0 and M r be the matrices that gives the action of γ 0 and γ r on L respectively, that are:

(γ 0 v 1 • • • γ 0 v d ) = (v 1 • • • v d ) • M 0 (γ r v 1 • • • γ r v d ) = (v 1 • • • v d ) • M r .
We do know that M r has all its entries in K r and we want to prove that the same holds for M 0 . Actually, from our construction of the v i 's, we know further that M r has integral entries and that it is congruent to the identity matrix modulo p c 2 +1 . From the commutation of γ 0 and γ r , we derive the relation

M 0 • γ 0 M r = M r • γ r M 0 . Define C = R r (M r ) -M r
where R r is the Tate's normalized trace. We want to prove that C vanishes. Since R r commutes with γ r , we have the relation

C • γ 0 M r = M r • γ r C, from which we derive γ r C -C = M -1 r • C • γ 0 M r . Set N = M -1 r • C • γ 0 M r
and let v be the smallest valuation of an entry of C. We assume by contradiction that v is finite. The fact that M r ≡ I d (mod p c 2 +1 ) implies that N is divisible by p v+c 2 +1 . By unicity in Proposition 2.1.6 (and Remark 2.1.7), we deduce that C must be divisible by p v+1 . This contradicts the definition of v.

Sen's operator. We now put together the results we have established before. Let W ∈ Rep f Cp (G K ). We define Ŵ∞ = W Gal( K/K∞) and let W ∞ be the subspace of finite vectors of Ŵ∞ . Combining Propositions 2.3.2 and 2.3.4, we find that W admits a C p -basis consisting of elements of W ∞ . In other words, the canonical mapping C p ⊗ K∞ W ∞ → W is an isomorphism. The action of G K on W is then entirely determined by the action of Γ of W ∞ . Using the particularly simple structure of Γ, it is possible to describe its action even more concretely.

More precisely, we consider an integer r and a basis v 1 , . . . , v d of Ŵ∞ such that the K r -vector space

W r = K r v 1 ⊕ • • • ⊕ K r v d is
stable under the action of Γ (or equivalently, G K ). For g ∈ G K , we shall denote by ρ W (g) the endomorphism of W r given by the action of g. Note that ρ W (g) is K r -linear as soon as g ∈ Gal( K/K r ). In particular ρ W (γ s ) is linear whenever s ≥ r. Since γ s converges to the identity in Γ, the logarithm of ρ W (γ s ) is well defined for s sufficiently large. Moreover, we have the relation p • log ρ W (γ s+1 ) = log ρ W (γ s ) as soon as the logarithm ρ W (γ s ) is defined. The sequence p -s log ρ W (γ s ) is then ultimately constant. Sen's operator Φ W is defined as the limit of this sequence:

Φ W = lim s→∞ log ρ W (γ s ) p s .
We extend Φ W to W ∞ by K ∞ -linearity. This extension is canonical in the sense that it does not depend on the choice of r. Besides, the exponential map can be used to reconstruct the representation W we started with, at least on a finite index subgroup of G. Precisely, there exists an integer s such that:

ρ(g) = exp α(g)Φ W for all g ∈ Gal( K/K s ) (20)
where we recall that α : G K → Z p was the character defining the isomorphism between Gal(K ∞ /K) and Z p . From [START_REF] Fontaine | Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux[END_REF], it follows that the action of Gal( K/K s ) on W can be entirely reconstructed by extending the ρ(g)'s to W using semi-linearity.

Example 2.3.6. Consider the representation V given by:

G K → GL 2 (Q p ), g → 1 α(g) 0 1 already discussed in Example 2.2.11. Set W = C p ⊗ Qp V . It is easily checked that W Gal( K/K∞) = K2
∞ and its subspace of finite vectors is K 2 ∞ . Sen's operator Φ V is the nilpotent linear morphism (x, y) → (y, 0).

Sen's operator exhibits very interesting properties. Below, we state the most important ones. Proposition 2.3.7. We keep the above notations. Sen's operator Φ W is defined over K, in the sense that W ∞ admits a basis in which the matrix of Φ W has coefficients in K.

Proof. This follows from the fact that Φ W commutes with the action of Γ.

Let Sen(K, K ∞ ) denote the category of finite dimensional K ∞ -vector spaces equipped with an endomorphism defined over K (in the sense of Proposition 2.3.7). The construction W → (W ∞ , Φ W ) defines a functor S :

Rep f Cp (G K ) → Sen(K, K ∞ ). Indeed a morphism f : W → W ′ in the category Rep f
Cp (G K ) necessarily maps W ∞ to W ′ ∞ and commutes with Sen's operators on both sides because it commutes with the Galois action. The functor S commutes with direct sums, while its behavior under tensor products is governed by the Leibniz rule:

(W ⊗ W ′ ) ∞ = W ∞ ⊗ W ′ ∞ Φ W ⊗W ′ = Φ W ⊗ id W ′ + id W ⊗ Φ W ′ .
Moreover, the functor S is faithful. Indeed, assume that we are given W, W ′ ∈ Rep f Cp (G K ), together with a morphism f : W → W ′ such that S(f ) = 0. Then, by assumption, f vanishes on the subspace W ∞ . Since the latter generates W as a C p -vector spaces, one must have f = 0. In general, S is not full. However, it detects isomorphisms as shown by the next proposition.

Proposition 2.3.8. Let W, W ′ ∈ Rep f Cp (G K ).
We assume that S(W ) and S(W ′ ) are isomorphic in

Sen(K, K ∞ ). Then W and W ′ are isomorphic in Rep f Cp (G K ).
Proof. Let f : W ∞ → W ′ ∞ be an isomorphism commuting with Sen's operators. By C p -linearity, f extends to an isomorphism of C p -vector spaces f : W → W ′ . Moreover, thanks to formula [START_REF] Fontaine | Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux[END_REF], there exists an integer s such that f is Gal( K/K s )-equivariant.

Let V be the space of Gal( K/K s )-equivariant C p -linear morphisms from W to W ′ . It is endowed with a canonical action of Gal(K s /K) and thus appears as an object in the category Rep f Ks (Gal(K s /K)). By Hilbert's theorem 90 (cf Theorem 1.3.3), V admits a basis (f 1 , . . . , f m ) of fixed vectors. In other words the f i 's are G K -equivariant morphisms W → W ′ . It remains to prove that a suitable K-linear combination of the f i 's is invertible. For this, we consider the m-variate polynomial defined by:

P (t 1 , t 2 , . . . , t m ) = det(t 1 f 1 + t 2 f 2 + • • • + t m f m ).
We know that P is not the zero polynomial because the K s -span of the f i 's contains an isomorphism (namely f ). Since K is an infinite field, P cannot vanish everywhere on K m . Hence there exist t 1 , . . . , t m ∈ K such that 3 Two refined period rings: B crys and B dR Previously, we have studied the period rings C p and B HT and discussed the attached notion of Hodge-Tate representations. In the present section, we introduce two new period rings, called B crys and B dR . As we shall see in §4, these rings have a deeper arithmetical and geometrical content that C p and B HT .

t 1 f 1 + • • • + t m f m is
The definition of B crys and B dR is a bit elaborated and occupies all this section. In order to ease the task of the reader, we devote two short paragraphs below to collect the most important properties of B crys and B dR and sketch the main steps of their construction.

Before this, we need to recall and introduce some notations. Throughout this section, K will continue to refer to a finite extension of Q p . Its ring of integers (resp. its residuel field) is denoted by O K (resp. k). We define K 0 = W (k)[ 1 p ]; it is the maximal unramified extension of Q p included in K. We fix an algebraic closure K of K and set G K = Gal( K/K). Observe that K is also an algebraic closure of Q p and hence does not depend on K. We let K ur 0 (resp. K ur ) be the maximal unramified extension of K 0 (resp. of K) inside K. Since K 0 is unramified over Q p , K ur 0 is also the maximal extension of Q p inside K and thus is also independent of K. We let also C p denote the p-adic completion of K. Finally, we choose and fix once for all a uniformizer π of K.

Main properties of B crys and B dR . As discussed in §1.2, the original idea behind the definition of B crys is the wish to design a variant of Barsotti-type spaces (the B of §1.2) which includes the tannakian formalism. On the geometric side, a nice tannakian framework in which p-divisible groups naturally arise is crystalline cohomology. Indeed, in many contexts, crystalline cohomology provides powerful invariants that can be used to classify p-divisible groups (and more generally finite flat group schemes) [START_REF] Berthelot | Théorie de Dieudonné cristalline II[END_REF]. We then expect the ring B crys to have some "crystalline nature" and to be eventually related to crystalline cohomology. Apart from this, recall that another motivation of p-adic Hodge theory is to compare étale cohomology with de Rham cohomology. The period ring making the comparison possible-namely B dR -then needs to be deeply related to de Rham cohomology. The algebraic structure of B crys and B dR is guided by the above general expectations: the ring B crys (resp. B dR ) will carry, as much as possible, the same structures and exhibit similar properties as the crystalline (resp. de Rham) cohomology.

Below, we list the main features of B crys and B dR and, when it is possible, we make the parallel with the corresponding properties of the cohomology. We start with B dR :

• B dR is a discrete valuation field with residue field C p ;

• B dR is an algebra over K• Kur , but not over C p (with a defining morphism preserving the Galois action);

• B dR is equipped with a filtration Fil m B dR (which is nothing but the canonical filtration given by the valuation); this filtration corresponds to the de Rham filtration on the cohomology;

• B dR has a distinguished element t on which Galois acts by multiplication by the cyclotomic character; moreover t is a uniformizer of B dR , so that Fil m B dR = t m B dR ;

• the graded ring of B dR is B HT = C p [t, t -1 ];

• (B dR ) G K = K; this property corresponds to the fact that the de Rham cohomology is a vector space over K.

And now for B crys :

• B crys is an algebra over Kur 0 ; • B crys is equipped with a Frobenius, which is a ring homomorphism ϕ : B crys → B crys ; this structure corresponds to the action of the Frobenius on the crystalline cohomology;

• there is a canonical embedding K ⊗ K 0 B crys ֒→ B dR ; this property corresponds to the fact that the crystalline cohomology defines a canonical K 0 -structure inside the de Rham cohomology (this is Hyodo-Kato isomorphism);

• the distinguished element t of B dR is in B crys ;

• (B crys ) G K = K 0 ; this property corresponds to the fact that the de Rham cohomology is a vector space over K 0 ;

• B crys ∩ Fil 0 B dR ϕ=1 = Q p (the notation "ϕ=1" means that we are taking the fixed points under the Frobenius).

(µ > p-1)

(1 ≤ µ ≤ p-1) B + 1 B + inf B + µ B + crys B + µ B + max B + inf,K B + µ,K B + crys,K B + µ,K B + max,K B + dR B µ B crys B µ B max B µ,K B crys,K B µ,K B max,K B dR
+ inf = A inf [1/p].
One may think of A inf as the universal thickening of O K /pO K ; it is obtained via a general process (detailed in §3.1) involving a perfectization mecanism as a first step and Witt vectors as a second step. Beyond this purely algebraic construction, it is important to notice that the ring B + inf has a strong geometrical interpretation. Indeed as observed first by Colmez and then by Fargues-Fontaine [START_REF] Fargues | Courbes et fibrés vectoriels en théorie de Hodge p-adique[END_REF] and Scholze [START_REF] Scholze | Berkeley lectures on p-adic geometry[END_REF], B + inf appears at a mixed characteristic analogue 6 of the ring of bounded analytic functions on the open unit disc. Moreover B + inf is equipped with a Frobenius (coming from the general theory of Witt vectors) and a distinguished geometric point, which is materalized by a surjective ring homomorphism θ : B + inf → C p . Following the crystalline formalism, we then define the ring B + crys as the completion of the divided powers envelope of B + inf with respect to the ideal ker θ. Unfortunately, B + crys does not have a nice geometrical interpretation, in the sense that it is not the ring of analytic functions on a smaller domain. In order to tackle this difficulty, we introduce (following Colmez) some variants of B + crys . Precisely, given a real parameter µ ≥ 1, one considers the rings B + µ 's of analytic functions defined over some annulus D µ included in the open unit disc, and containing the distinguished point θ. The B + µ 's are closely related to B + crys (we have inclusions in both directions), so that it is often safe to replace the latter by the formers.

Another important feature of B + crys and the B + µ 's is that they contain a period of the cyclotomic character, that is an element t on which G K acts by multiplication by the cyclotomic character. Geometrically, the divisor of t is the orbit of the point θ under the action of the Frobenius, that is the union of all point θ • ϕ n for n varying in Z. The presence of t in B + µ will eventually ensure the admissibility of the representation Q p (χ -1 cycl ). In order to make Q p (χ cycl ) admissible as well (which is of course something we really want to have), we need t to be a unit. So we finally define

B µ = B + µ [ 1 t
] and the construction of B crys is now complete. As for the field B dR , it is defined as the fraction field of the completion of the local field of B + inf (or equivalently, B + µ ) at the special point θ. The filtration on B dR is nothing but the canonical filtration given by the order of the zero (or the pole) at θ.

The diagram presented on Figure 2 summarizes the period rings we will define in this section and the relations between them. We see on this diagram that the B + µ 's and the B µ 's all have a variant denoted by an extra index K. They are defined by extending scalars from K 0 to K. These variants are interesting because, when V is a B µ -admissible representation, the de Rham filtration becomes visible after extending scalars to B µ,K , which is much smaller and sometimes more tractable that B dR .

Preliminaries: the ring B + inf

In this subsection, we introduce the ring B + inf which serves as a common base upon which all the forthcoming constructions will be built.

Perfectization

Let ϕ denote the Frobenius morphism x → x p acting on the quotient O Cp /pO Cp ≃ O K /pO K and observe that ϕ is a ring homomorphism since O Cp /pO Cp is annihilated by p.

Let R be the limit of the projective system 7 :

O Cp /pO Cp ϕ -→ O Cp /pO Cp ϕ -→ O Cp /pO Cp ϕ -→ • • • ϕ -→ O Cp /pO Cp ϕ -→ • • •
Concretely, an element of R is a sequence (ξ n ) n≥0 of elements of O Cp /pO Cp satisfying the following compatibility property: ξ p n+1 = ξ n for all n ≥ 0. Clearly, R is a ring of characteristic p. In a slight abuse of notation, we continue to write ϕ for the Frobenius acting on R. Over this ring, it is an isomorphism, its inverse being given by the shift map (ξ 0 , ξ 1 , ξ 2 , . . .) → (ξ 1 , ξ 2 , ξ 3 , . . .). We sometimes say that R is the perfectization of O Cp /pO Cp . Moreover R is endowed with an action of G K coming from its natural action on O Cp . Some distinguished elements. Choose a primivite p-root of unity in O K and denote it by ε 1 . Similarly, choose a p-th root of ε 1 and denote it by ε 2 ; obviously, ε 2 is a primitive p 2 -th root of unity. Repeating inductively this process, we construct elements ε 3 , ε 4 , . . . ∈ O K such that ε p n+1 = ε n for all n. Let εn ∈ O K /pO K be the image of ε n . The compabitility property ensures that the sequence (1, ε1 , ε2 , . . .) defines an element in R; we shall denote it by ε. We emphasize that ε does depend on the choice of the ε n 's. However, the dependency is easy to write down explicitely: if (ε ′ n ) n≥0 is another compatible sequence of primitive p n -th roots of unity, one can always find an element g ∈ G K such that ε

′ n = gε n = ε χ cycl (g) n
. Hence the element of R defined the ε ′ n 's is ε χ cycl (g) . In what follows, we fix once for all an element ε as above. In a similar fashion, we choose a compatible system (p n ) n≥1 of p n -root of p, i.e. p p 1 = p and p p n+1 = p n for all n ≥ 1. If pn ∈ O K /pO K is the reduction of p n modulo p, the sequence (0, p1 , p2 , . . .) defines an element of R that we will denote by p ♭ . Again, p ♭ depends on the choice of the p n 's but we can check that another choice would finally lead to an element of the form p ♭ •ε a for some a ∈ Z p . The same construction works more generally if we start for any element x ∈ O Cp in place of p; it leads to an element x ♭ ∈ R, which is well defined up to multiplication by ε a with a ∈ Z p . Besides p ♭ , we will fix a choice of π ♭ (where we recall that π is a fixed uniformizer of K) for future use.

Valuation. The ring R is equipped with a derivation v ♭ that we are going to define now. We start with the following observation: if x is a nonzero element in O Cp /pO Cp , the p-adic valuation of x does not depend on the lifting x of x. The valuation v p then induces a well defined function v p : O Cp /pO Cp → Q ∪ {+∞} where we agree that v p (0) = +∞ as usual. For ξ = (ξ n ) n≥0 in R, we define:

v ♭ (ξ) = lim n→∞ p n v p (ξ n ).
The compatibility condition ξ p n+1 = ξ n implies that the sequence p n v p (ξ n ) n≥0 is ultimately constant; so the limit is well defined. The function v ♭ satisfies the following properties for ξ, ξ ′ ∈ R:

(1) v ♭ (R) = Q ∪ {+∞}, (2) v ♭ (ξ) = ∞ if and only if ξ = 0, (3) v ♭ (ξ) = 0 if and only if ξ is invertible, (4) v ♭ (ξ + ξ ′ ) ≥ min(v ♭ (ξ), v ♭ (ξ ′ )) and equality holds if v ♭ (ξ) = v ♭ (ξ ′ ), (5) v ♭ (ξξ ′ ) = v ♭ (ξ) + v ♭ (ξ ′ ).
Combining ( 2) and ( 5), we find that R is a domain. Indeed if ξ and ξ ′ are nonzero elements of R, then v ♭ (ξ) and v ♭ (ξ ′ ) are finite, and so v ♭ (ξξ ′ ) = v ♭ (ξ) + v ♭ (ξ ′ ) is also finite. The existence of v ♭ implies that R is a local ring with maximal ideal m R consisting of elements of positive valuation. The residue field R/m R is canonically isomorphic to k. We observe in addition that the projection R → k has a canonical splitting defined by:

a → [a] mod p, [a 1/p ] mod p, [a 1/p 2 ] mod p, . . .
where the notation [•] stands for the Teichmuller representative. Besides, the valuation v ♭ equips R with a distance, and hence a topology. The Galois action on R preserves v ♭ ; in particular, it is continuous.

An easy consequence of the existence of a valuation is the following result. Proof. Let f : R → O Cp /pO Cp , (ξ 0 , ξ 1 , . . .) → ξ 0 . The surjectivity of f is a consequence of the fact that C p is algebraically closed. On the other hand, it is clear that the kernel of f consists of elements ξ such that v ♭ (ξ) ≥ 1. Since v ♭ (p ♭ ) = 1, we deduce that ker f is the principal ideal generated by p ♭ . This proves the lemma.

Witt vectors

We set A inf = W (R) (where W (-) stands for the Witt vectors functor) and

B + inf = A inf [ 1 p ]
. For x ∈ R, we let [x] denote its representative Teichmüller in A inf . Since R is perfect, an element of A inf can be written uniquely as a convergent series i≥0 [ξ i ] p i with ξ i ∈ R for all i. A similar decomposition holds for elements in B + inf : each such element x has a unique expansion of the form i≥i 0 [ξ i ] p i (with ξ i ∈ R) where i 0 is a (possibly negative) integer, which depends on x.

The inclusion k → R provides by functoriality a ring morphism W ( k) → A inf . Thus Kur 0 embeds into B + inf . The ring A inf is a local ring whose maximal ideal is the kernel of the composition A inf → W ( k) → k where the first map is induced by the projection R → k and the second map is the reduction modulo p. Concretely, it consists of series i≥0 [ξ i ] p i for which

ξ 0 ∈ m R . We set A inf,K = O K ⊗ W (k) A inf and B + inf,K = K ⊗ K 0 B + inf .
These tensors products make sense because we saw that A inf is an algebra over W (k). The elements of B + inf,K have a canonical expansion of the form i≥i 0 [ξ i ] π i with i 0 ∈ Z and ξ i ∈ R for all i ≥ i 0 . Moreover A inf,K is a local ring and its maximal ideal consists of series as above such that ξ 0 ∈ m R ; its residue field is k.

Additional structures. By definition of the Witt vectors, B +

inf carries an action of a Frobenius, that we shall continue to call ϕ. On the above representation, it is given by the simple formula:

ϕ ∞ i=i 0 [ξ i ] p i = ∞ i=i 0 [ξ p i ] p i (i 0 ∈ Z, ξ i ∈ R). (21)
We emphasize that ϕ does not admit a canonical extension to B + inf,K as there is no canonical Frobenius on K.

The ring B +

inf is also equipped with an action of G K by functorialiy of Witt vectors. Again, this action has a simple expression, namely:

g ∞ i=i 0 [ξ i ] p i = ∞ i=i 0 [gξ i ] p i (i 0 ∈ Z, ξ i ∈ R) (22)
for all g ∈ G K . The G K -action extends to B + inf,K by letting G K act trivially on O K . Finally, we equip A inf and A inf,K with the weak topology, which is the topology defined by the ideal (p, [p ♭ ]) (or equivalently, by the ideal (p, [x]) for any element x ∈ m R ). Concretely, if

x n = ∞ i=0 [ξ i,n ] p i ∈ A inf and x = ∞ i=0 [ξ i ] p i ∈ A inf the sequence (x n ) n≥0 converges to x if ξ i,n → ξ i for each fixed index i ∈ N,
and a similar property holds for A inf,K . The topology on A inf induces a topology on the subset p -v A inf of B + inf for all v. Gluing them, we obtain a topology on B + inf = v≥0 p -v A inf . In concrete terms, a sequence (x n ) n≥0 of elements on B + inf converges to x ∈ B + inf if and only if there exists an integer v such that p v x n ∈ A inf for all n and p v x n tends to p v x in A inf as n goes to infinity. The topology on B + K,inf is defined similarly. From the above descriptions, it follows that the Frobenius acts continuously on A inf and G K acts continuously on A inf and A inf,K .

Newton polygons. In [START_REF] Fargues | Courbes et fibrés vectoriels en théorie de Hodge p-adique[END_REF], Fargues and Fontaine argue that elements of A inf (resp. A inf,K ) should be thought of as analytic functions of the variable p (resp. π); indeed, they share many properties with bounded analytic functions on the open unit disc. Similarly, elements of B + inf and B + inf,K resemble to bounded analytic functions on the punctured open unit disc. In particuler, there is a well-defined notion of Newton polygons for series in B + inf and B + inf,K . Precisely, if x = i≥i 0 [ξ i ] p i ∈ A inf , its Newton polygon is defined as the convex hull in R 2 of the points (i, v ♭ (ξ i )) together with two points at infinity in the direction of the positive xaxis and the direction of the positive y-axis respectively. Similarly, the Newton polygon of x = i≥i 0 [ξ i ] π i ∈ A inf,K is the convex hull of the points ( i e , v ♭ (ξ i )) and the same points at infinity. Using that π e = up for some invertible element u ∈ O K , one easily proves that the above definition coincides with that of Newton polygons on B + inf when x is in B + inf . Let NP inf (x) denote the Newton polygon of x ∈ B + inf,K . Fargues and Fontaine prove that Newton polygons satisfy many excepted properties. For example, they are multiplicative in the sense that NP inf (xy) = NP inf (x) + NP inf (y) where the plus sign on the right hand side denotes the Minkowski sum. Moreover Fargues and Fontaine prove an analogue of the Weierstrass preparation and factorization theorems in this context, showing that Newton polygons serve as a guide for factorization in the rings B + inf and B + inf,K as they do for usual analytic functions. We do not reproduce their proofs here because we will only use Newton polygons for visualizing our forthcoming constructions, and not for proving results. In any case, we refer to [17, §1-3] for many developments in this direction.

We conclude this discussion by examining the action of the additional structures at the level of Newton polygons. Since G K acts on R by isometries, it follows from the formula ( 22) that NP inf (gx) = NP inf (x) whenever g is in G K and x is in B + inf,K . As for Frobenius, formula [START_REF] Fontaine | Périodes p-adiques[END_REF] shows that, for any x ∈ B + inf , we have NP inf (ϕ(x)) = ϕ R 2 (NP inf (x)) where ϕ R 2 : R 2 → R 2 takes (i, v) to (i, pv).

The "sharp" construction

In §3.1, starting with x ∈ O Cp , we have constructed an element x ♭ ∈ R (which was only welldefined up to multiplication by an element of the form ε a with a ∈ Z p ). Let us recall more precisely that the element x ♭ = (x 0 mod p, x 1 mod p, x 2 mod p, . . .) where x 0 = x and x n+1 is a p-th root of x n for n ≥ 0.

It turns out that the datum of x ♭ entirely determines x. Precisely, if we write x ♭ = (ξ 0 , ξ 1 , ξ 2 , . . .) and if we choose a lifting ξn ∈ O Cp of x n for all n, we have x = lim n→∞ ξp n n independently of the choices of the liftings. Indeed, following the definitions, we find that x n ≡ ξn (mod p) and then, raising to the p n -th power, x ≡ ξp n n (mod p n+1 ). This motivates the following definition.

Definition 3.1.2. For ξ = (ξ 0 , ξ 1 , ξ 2 , . . .) ∈ R, we put

ξ ♯ = lim n→∞ ξp n n
where ξn is a lifting of ξ n .

One checks immediately that the function R → O Cp , ξ → ξ ♯ is surjective and multiplicative. Its "kernel" is the closed subgroup of R × generated by ε; it is isomorphic to Z p . Besides, we observe that v p (ξ ♯ ) = v ♭ (ξ) for all ξ ∈ R and that ξ ♯ is the Teichmuller representative of ξ if ξ is in k. By the general properties of Witt vectors, the "sharp" function extends to a surjective homorphism of Kur 0 -algebras θ : B + inf → C p which commutes with the G K -action. Concretely, it is given by: θ :

∞ i=i 0 [ξ i ] p i → ∞ i=i 0 ξ ♯ i p i (i 0 ∈ Z, ξ i ∈ R).
Note that the latter series converges in C p since its i-th summand is a multiple of p i . The morphism θ extends by K-linearity to a surjective G K -equivariant homomorphism of Kur -algebras (ii) Let z ∈ A inf,K be an element such that θ K (z) = 0 and v ♭ (z mod π) = 1 e . Then z generates A inf,K ∩ ker θ K , viewed as an ideal of A inf,K . Remark 3.1.4. In particular, an element z satisfying the condition of the first item (resp. the second item) of Proposition 3.1.3 is a generator of the ideal ker θ (resp. ker θ K ).

θ K : B + inf,K → C p .

Proof of Proposition

3.1.3. Let z ∈ A inf such that θ(z) = 0 and v ♭ (ζ) = 1 with ζ = z mod p. Let x ∈ ker θ ∩ A inf . Write x = i≥0 [ξ i ] p i with ξ i ∈ R. From θ(x) = 0, we derive that v p (ξ ♯ 0 ) ≥ 1 and then v ♭ (ξ 0 ) ≥ 1.
From the assumption v ♭ (ζ) = 1, we find that ζ divides ξ 0 in R. Thus, we can write x = zy 0 + px 1 with y 0 , x 1 ∈ A inf . From this above equality, we derive θ(x 1 ) = 0 and we can then repeat the argument with x 1 , ending up with a writing of the form x = z•(y 0 + py 1 ) + p 2 x 2 with y 1 , x 2 ∈ A inf . Repeating this process again and again, we construct a sequence (y n ) n≥0 of elements of A inf such that:

x ≡ z • (y 0 + py 1 + • • • + p n y n ) (mod p n A inf )
for all n. Passing to the limit we find that x ∈ zA inf , which proves (i).

The statement (ii) is proved similarly.

We remark that there do exist elements in A inf satisfying the condition of Proposition 3.1.3. The simplest one is [p ♭ ]-p, which is then a generator of ker θ. Similarly [π ♭ ]-π ∈ A inf,K satisfies the condition of Proposition 3.1.3 and so is a generator of ker θ K . Another generator of ker θ is E([π ♭ ]) where E is minimal polynomial of π over K 0 . Indeed, on the one hand, we have

θ(E([π ♭ ])) = E(θ([π ♭ ])) = E(π) = 0
and, on the other hand, E([π ♭ ]) reduces modulo p to the constant coefficient of E, which has valuation 1.

The next proposition gives another quite interesting generator of ker θ.

ω = [ε] -1 [ε 1/p ] -1 = [ε 1/p ] + [ε 1/p ] 2 + • • • + [ε 1/p ] p-1
satisfies the condition of Proposition 3.1.3.(i).

Proof. We want to check that θ(ω) = 0 and v ♭ (ω mod p) = 1. The first equality follows from the fact that θ([ε]) = 1 and the fact that θ([ε 1/p ]) is a primitive p-th root of unity. Let us now prove that v ♭ (ω modp) = 1. Reducing modulo p, we find that ω modp = ε-1 ε 1/p -1 . Write ε = (ε 0 , ε 1 , ε 2 , . . .) where ε n is the reduction modulo p of a primitive p n -th root of unity. Coming back to the definition of v ♭ , we find:

v ♭ (ω mod p) = lim n→∞ p n+1 • v p ε n -1 ε n+1 -1 . (23)
By the standard properties of the cyclotomic extension (cf [39, Chap. IV, §4]), we know that the p-adic valuation of ε n -1 is

1 p n (p-1) . Injecting this in (23), we obtain v ♭ (ω mod p) = p p-1 -1 p-1 = 1.
Remark 3.1.6. Since two generators of ker θ differ by multiplication by a unit, they have to share the same Newton polygon up to translation by a horizontal vector. If in addition, they satisfy the conditions of Proposition 3.1.3, the Newton polygons must coincide since they both admit (0, 1) as an extremal point. Clearly, the Newton polygon of [p ♭ ] -p is the convex polygon whose vertices are (0, +∞), (0, 1), (1, 0) and (+∞, 0). The Newton polygon of ω is then the same. Writing

[ε] -1 = ∞ n=0 ϕ -n (ω) (24)
and using the multiplicative properties of the Newton polygons, we find that NP inf ([ε] -1) starts at (0, 1 p-1 ) and then has a segment of length 1 of slope p -n for each nonnegative integer n (cf Figure 3). Proposition 3.1.7. The element [ε] -1 is a generator of the ideal n≥0 ker(θ • ϕ n ). Remark 3.1.8. Proposition 3.1.7 is not surprising after formula [START_REF] Fontaine | Geometric Galois representations[END_REF]. Indeed if x is such that θ • ϕ n (x) = 0 for all n ≥ 0, then x must to divisible by ϕ -n (ω) for all n ≥ 0. It is then reasonable

to expect to [ε] -1 = ∞ n=0 ϕ -n (x) divides
x since the Newton polygon of the factors do not share any common slope (and thus the factors look pairwise coprime). It is possible to turn this vague idea into a rigourous proof. However, we prefer giving below a more direct argument, which is easier to write down.

Proof. Clearly

[ε] -1 ∈ n≥0 ker(θ • ϕ n ).
Repeating the second part of the proof of Proposition 3.1.3, we are reduced to show that any element x ∈ A inf such that θ • ϕ n (x) = 0 verifies v ♭ (x mod p) ≥ p p-1 . From θ(x) = 0, we deduce that x can be written ωx 1 with x 1 ∈ A inf . Since θ • ϕ(ω) = 0, we deduce that θ • ϕ(x 1 ) must vanish. Therefore there exists x 2 ∈ A inf such that x 1 = ϕ -1 (ω)x 2 , i.e. x = ωϕ -1 (ω)x 2 . By induction, we find that x has to be divisible by

x = ωϕ -1 (ω) • • • ϕ -n (ω) in A inf for all n. Reducing modulo p, this implies v ♭ (x mod p) ≥ 1 + 1 p + • • • + 1
p n for all n. Passing to the limit, we find v ♭ (x mod p) ≥ p p-1 as expected.

The ring B crys and some variants

In this subsection, we introduce the ring B crys and its variants B µ 's. The former is interesting because it fits very well in the crystalline framework and therefore is well suited for studying cohomology. Nevertheless, as we shall see, B crys does not behave very well from the purely algebraic point of view. The B µ 's are substitutes to B crys which share its most important features and, in addition, exhibit better algebraic (and analytic) properties, and hence are easier to work with.

Divided powers

Given x ∈ A inf , we denote by A inf x the sub-A inf -algebra of B + inf generated by the elements x n n!

for n varying in N. Obviously if y divides x in A inf , we have A inf x ⊂ A inf y . In particular, A inf x only depends on the principal ideal xA inf . By the proof of Proposition 3.1.3, we know that A inf ∩ ker θ is a principal ideal of A inf . The following definition then makes sense.

Definition 3.2.1. We define A crys as the p-adic completion of A inf z where z is some generator of the ideal A inf ∩ ker θ. We set

B + crys = A crys [ 1 p ].
Rephrasing the definition, we can write:

A crys = A inf [p ♭ ] -p ∧ = A inf ω ∧
where the exponent "∧" means the p-adic completion and the element ω is the one of Proposition 3.1.5.

Lemma 3.2.2. For x, y ∈

A inf with x ≡ y (mod pA inf ), we have A inf x = A inf y .
Proof. By symmetry, it is enough to prove that A inf x ⊂ A inf y , i.e. that x n n! ∈ A inf y for all positive integer n. Writing x = y + pz with z ∈ A inf , we have:

x n n! = (y + pz) n n! = n i=0 p i z i i! • y n-i (n-i)! . (25)
We recall that v p (i!) = i-sp(i)

p-1 where s p (i) denotes the sum of the digits of i in radix p. In particular, we observe that v p (i!) ≤ i, so that the fraction p i i! is in Z p . The formula (25) then presents x n n! as an A inf -linear combination of elements of the form y j j! for j between 0 and n. Therefore, x n n! ∈ A inf y and we are done.

The above lemma shows that A crys = A inf [p ♭ ] ∧ . Since A inf x depends only on the ideal generated by x, we also have Topology and additional structures. Since A crys is defined as a p-adic completion, it is quite natural to endow A crys (and B + crys ) with the p-adic topology. Noticing that we can obviously write

A crys = A inf [ξ] ∧ for any element ξ ∈ R with v ♭ (ξ) = 1.
[p ♭ ] n = n! • [p ♭ ] n
n! , it follows from the defintion of A crys that [p ♭ ] n tends to zero when n goes to infinity (since n! goes to zero for the p-adic topology). In particular the inclusion A inf → A crys is continuous. Inverting p, we find that the inclusion B + inf → B + crys is continuous as well. Besides, we observe that the Frobenius extends canonically to a ring homomorphism ϕ : A crys → A crys . This can be checked by noticing that

A inf [p ♭ ] is stable under the Frobenius since ϕ [p ♭ ] n n! = [p ♭ ] np-n • ϕ [p ♭ ] n n! .
Inverting p, we obtain an extension of the Frobenius to B + crys . We shall continue to denote it by ϕ in the sequel. Similarly, the action of G K extends to B + crys . The embedding W ( k) → A inf → A crys endows A crys with a structure of W ( k)-algebra. Similarly B + crys is an algebra over Kur 0 . It then makes sense to define

A crys,K = O K ⊗ W (k) A crys and B + crys,K = K ⊗ K 0 B + crys = A crys,K [ 1 p ].

Some analytic analogues of A crys

In §3.1, we saw that elements of A inf admitted a nice series expansion, allowing for an analytic interpretation of the ring A inf . To some extent, this point of view is also meaningful for A crys . Indeed, it follows from A crys = A inf [p ♭ ] ∧ , that any element x ∈ A crys has a unique expansion of the form:

x = i∈Z [ξ i ] p i (ξ i ∈ R) with v ♭ (ξ i ) -ν(i) ≥ 0 and lim i→-∞ v ♭ (ξ i ) -ν(i) = +∞ (26)
where, for i ≥ 0, ν(i) = 0 and, for i < 0, ν(i) denotes the smallest integer

n such that v p (n!)+i ≥ 0. From the formula v p (n!) = n-sp(n) p-1
= n p-1 , we derive that, for i ≪ 0, we have the estimation:

-i • (p -1) ≤ ν(i) ≤ -i • (p -1) + O log |i| . (27)
We insist on the fact that the term O(log |i|) is not bounded (it may have order of magnitude (p-1)• log |i| log p ); hence, we cannot replace ν(i) by -i•(p-1) in [START_REF] Grothendieck | Groupes de Barsotti-Tate et cristaux[END_REF]. We will circumvent this difficulty later on. Figure 4 illustrates the convergence conditions discussed above: the grey part is the region on which v ♭ (ξ i ) -ν(i) ≥ 0.

Analytic functions on annuli. The function ν that appeared in the formula (26) has a very erratic behavior. This is unfortunate for two reasons: the ring A crys we defined do not have pleasant algebraic properties (for instance, it is not noetherian), nor a nice analytic interpretation (its elements are not anayltic functions defined on a nice domain). In order to get around these difficulties, we introduce a variant of A crys which does not have these defaults. More precisely, given a positive real number µ, we introduce the ring A µ consisting of series of the form:

x = i∈Z [ξ i ] p i (ξ i ∈ R) with v ♭ (ξ i ) + µi ≥ 0 and lim i→-∞ v ♭ (ξ i ) + µi = +∞. When µ is rational 8 , A µ is the p-adic completion of A inf [ξ] p for any ξ ∈ A inf with v ♭ (ξ) = µ. We let B + µ = A µ [ 1 p ].
The elements of B + µ are series of the form:

x = i∈Z [ξ i ] p i (ξ i ∈ R) with lim i→-∞ v ♭ (ξ i ) + µi = +∞
i.e. the same conditions as for A µ except that the condition of positivity has been dropped. From the analytic point of view, elements of B + µ should be considered as bounded analytic functions (of the variable p) on the annulus

{0 ≤ v ♭ (•) < µ}. It is clear that A µ ⊂ A µ ′ (resp. B + µ ⊂ B + µ ′ )
as soon as µ ≥ µ ′ . However, the reader should be careful that the functions µ → A µ and µ → B + µ are not continuous in the sense that A µ is strictly included in µ ′ <µ A µ ′ , and similarly for the B + µ 's. In the analytic language, a function in µ ′ <µ B + µ ′ is analytic on the annulus {0 ≤ v ♭ (•) < µ} but not necessarily bounded. Similarly µ>0 B + µ is strictly greater than the ring B + inf we have introduced in §3.1; actually, we shall construct soon a quite important element t lying in the former ring but not in the latter. The relation between B + crys and the B + µ 's is also simple to understand. Indeed, the estimation [START_REF] Hyodo | Semi-stable reduction and crystalline cohomology with logarithmic poles[END_REF] shows that B + µ ⊂ B + crys ⊂ B + p-1 for all µ > p-1 (cf also Figure 4). At the integral level, we have

A p ⊂ A crys ⊂ A p-1 .
For µ > 0, we also define

A µ,K = O K ⊗ W (k) A µ and B + µ,K = K ⊗ K 0 B + µ . Elements in B + µ,K are series of the form i∈Z [ξ i ] π i with lim i→-∞ v ♭ (ξ i ) + µi e = +∞.
The subring A µ,K is characterized by the positivity condition v ♭ (ξ i ) + µ i e ≥ 0 for all i ∈ Z. The notion of Newton polygons, which was defined for elements of B + inf (resp. B + inf,K ) in §3.1, admits a straightforward extension to

B + µ (resp. B + µ,K ). Precisely, if x = i∈Z [ξ i ] p i ∈ B + µ (resp. x = i∈Z [ξ i ] π i ∈ B + µ,K
), we define NP µ (x) as the convex hull of the points (i, v ♭ (i)) (resp. ( i e , v ♭ (i))) together with the two points at infinity (0, +∞) and +∞•(-1, µ).

When x ∈ µ>0 B + µ (resp. x ∈ µ>0 B + µ,K ), we define NP inf (x) = µ>0 NP µ (x).
One checks easily that this definition agrees with the definition of NP inf on B + inf (resp. on B + inf,K ) we gave earlier. Finally, we observe that the Galois action and the Frobenius are well-defined on the A µ 's and B + µ 's. Even better, for all µ > 0, the Frobenius induces isomorphisms of rings

A µ → A pµ , B + µ → B + pµ and B + µ,K → B + pµ,K .
As for Newton polygons, they are preserved under the action of G K and we have the following transformation formula under Frobenius:

NP µp (ϕ(x)) = ϕ R 2 (NP µ (x)) where ϕ R 2 : R 2 → R 2 , (i, v) → (i, pv) for x ∈ B + µ,K . Passing to the limit on µ, we find that NP inf (ϕ(x)) = ϕ R 2 (NP inf (x)) for all x ∈ µ>0 B + µ,K . 0 pν p p-1 = ν ν/p ν/p 2 -1 1

The element t

An essential property of A crys is that it contains a period for the cyclotomic character, that is a special element on which Galois acts by multiplication by χ cycl . This distinguished element is:

t = log [ε] = ∞ i=1 (-1) i-1 • ([ε]-1) i i .
Observe that the latter sum converges in A crys since its i-th summand is equal to:

(-1) i-1 • (i-1)! • [ε 1/p ] -1 i • ω i i!
and therefore goes to 0 in A crys , thanks to the factor (i-1)! which converges to 0 for the p-adic topology. A similar computation shows that t actually lies in B + µ for all µ > 0 and in A µ for µ ≥ 1 -1 p . Recall that the Frobenius and the group

G K act on [ε] by ϕ([ε]) = [ε] p and g[ε] = [ε] χ cycl (g) for g ∈ G K .
Taking logarithms, we find ϕ(t) = pt and gt = χ cycl (g) t for all g ∈ G K . The latter relation is what we expected: the element t is a period for the cyclotomic character.

The Newton polygon of t can also be computed 9 . The result we find is displayed on Figure 5; we notice in particular that its slopes are unbounded, reflecting the fact that t is in B + µ for all µ > 0. It also remains unchanged under the transformation (i, v) → (i-1, pv), reflecting the fact that ϕ(t) = pt. In fact, the special shape of NP inf (t) is explained by the existence of a decomposition of t as an infinite convergent product (in all B µ 's), precisely: 9 The computation can be carried out as follows. By Remark 3.1.6, we know that NP inf ([ε] -1)) is the set P + defined as the convex hull of the points An = n, 1 (p-1)p n for n varying in N. By the multiplicativity property of Newton polygons, we find that NP ([ε]-1) i i = τ vp(i) (iP + ) where τu is the translation of vector (0, -u). We now observe that each An (n ∈ Z) belongs to exactly one τ vp(i) (iP + ): when n ≥ 0, we have i = 0 and when n < 0, we have i = p -n . Therefore the Newton polygon of t is the convex hull of the An's for n varying in Z.

t = ∞ n=0 ϕ -n (ω) • ∞ n=1 ϕ n (ω) p (28)
where ω = [ε]-1

[ε] 1/p -1 is the element of Proposition 3.1.5. The factorization [START_REF] De | Smoothness, semi-stability and alterations[END_REF] should be paralleled with [START_REF] Fontaine | Geometric Galois representations[END_REF]. Definition 3.2.3. For µ > 0 or µ = crys, we set

B µ = B + µ [ 1 t ].
The Frobenius and the Galois action extend to B µ without difficulty: for g ∈ G K , x ∈ B + µ and m ∈ N, we put ϕ( x t m ) = ϕ(x) p m t m and g( x t m ) = gx χ cycl (g) m •t m .

The de Rham filtration and the field B dR

We recall that, in §3.1.3, we have constructed a ring homomorphism θ : B + inf → C p , which was given by the explicit formula:

θ : ∞ i=-∞ [ξ i ] p i → ∞ i=-∞ ξ ♯ i p i (ξ i ∈ R), ξ i = 0 for i ≪ 0).
The filtration by the power of the ideal ker θ will play an important role because it will eventually correspond to the de Rham filtration on the cohomology. We devote this subsection to the study of its main properties. This will lead us to the definition of the period ring B dR .

Definition and main properties of the de Rham filtration

First of all, we will need to extend the morphism θ to the rings B µ 's we have introduced earlier.

Actually, just noticing that v p (ξ ♯ i p i ) = v ♭ (ξ i ) + i, we deduce that θ extends readily to a ring homomorphism θ µ : B + µ → C p whenever µ ≥ 1. Extending scalars to K, we obtain a ring homomorphism θ µ,K : B + µ,K → C p for all µ ≥ 1. We observe that θ µ (resp. θ µ,K ) maps the subring A µ (resp. A µ,K ) to O Cp .

The condition µ ≥ 1 for the existence of θ µ suggests that the rings A 1 , A 1,K , B + 1 and B + 1,K will play a particular role. In the literature, they are often denoted by A max , A max,K , B + max and B + max,K respectively; we will also use this notation in the sequel and will set θ max = θ 1 , θ max,K = θ 1,K accordingly. Similarly, we will use the notation θ inf and θ inf,K for θ and θ K respectively.

We recall that A max is the p-adic completion of A inf

[p ♭ ]
p . In particular, we have canonical isomorphisms:

A max /pA max ≃ R/p ♭ R [X] ≃ O Cp /pO Cp [X], [p ♭ ]/p ← X (29)
the second isomorphism coming from Lemma 3.1.1. Similarly, we have:

A max,K /πA max,K ≃ R/π ♭ R [X] ≃ O Cp /πO Cp [X], [π ♭ ]/π ← X. (30)
We can also identify the kernels of θ max and θ max,K (as we did for θ and θ K in Proposition 3.1.3). Proof. We only give the proof for θ max , the case of θ max,K being absolutely similar. We will prove that 1 -[p ♭ ]/p is a generator of the ideal A max ∩ ker θ max of A max . Let θmax : A max /pA max → O Cp /pO Cp be the morphism induced by θ max . Repeating the argument of Proposition 3.1.3, it is enough to show that ker θmax is the principal ideal generated by 1 -[p ♭ ]/p. Under the isomorphism [START_REF] De | Families of curves and alterations[END_REF], θmax acts by the identity on O Cp /pO Cp and takes X to 1. Hence, its kernel is the principal ideal generated by X-1.

We can moreover entirely elucidate the Galois action. Indeed we have the following proposition. Proof. Thanks to the diagram [START_REF] Kisin | The Fontaine-Mazur conjecture for GL 2[END_REF], it is enough to prove the proposition for gr m B + µ . We already know that gr m B + µ is one dimensional over C p . We observe that t lies in ker θ µ since θ µ (t) = log θ µ ([ε]) = log 1 = 0. Thus t m ∈ Fil m B + µ and we are reduced to prove that t m is not zero in gr m+1 B + µ . Noting that t ≡ [ε] -1 (mod Fil 2 B + µ ), we can replace t by [ε] -1. Using again the diagram [START_REF] Kisin | The Fontaine-Mazur conjecture for GL 2[END_REF], it is enough to show that ([ε]-1]) m does not vanish in gr m+1 B + inf . Let ω be the element of Proposition 3.1.5, so that we can write ([ε] -1) m = ω m • ([ε] 1/p -1) m . We know that the class of ω m is a generator of gr m B + inf . It is enough to check θ does not vanish on ([ε] 1/p -1) m . But a direct computation gives θ(([ε] 1/p -1) m ) = (ε 1 -1) m where ε 1 ∈ C p is a primitive p-th root of unity. We conclude by noticing that ε 1 = 1. Remark 3.3.6. We strongly insist on the fact that t m is not a generator of Fil m B + µ (resp. Fil m B + µ,K ) since this is often the source of confusion. Let us clarify this point by examining a bit the case where µ = 1. Then, by Proposition 3.3.1, we know that Fil 1 B + max is generated by the element γ = [p ♭ ]-p. Thus, we can write t = γγ ′ for some γ ′ ∈ B + max . It turns out that γ ′ is not invertible in B + max but is a unit in B + max /Fil 1 B + max (which is isomorphic to C p ), reflecting the fact that t m does not generate Fil m B + max but generates gr m B + max . The situation is quite similar to the following one which is very familiar to the number theorists: pick an odd prime number p, equip Z with the filtration Fil m Z = p m Z and consider the element t = 2p. Then t m is not a generator of Fil m Z but it does generate gr m Z because 2 is invertible modulo p.

It follows from Proposition 3.3.5 that gr m B + µ and gr m B + µ,K are both isomorphic to C p (χ m cycl ) in the category Rep Cp (G K ). Passing to the graduation, we obtain G K -equivariant isomorphisms of rings: gr

B + µ ≃ gr B + µ,K ≃ C p [t] (µ ∈ [1, +∞) ⊔ {inf, crys, max}) (33)
where the letter t on the right hand side is a new variable (corresponding to the special element t ∈ B + µ ) on which Galois acts by multiplication by the cyclotomic character.

Completion with respect to the de Rham filtration

After what we have done previously, it is natural to introduce the completion of the B + µ 's (resp. the B + µ,K 's) with respect to the de Rham filtration. This actually leads to the definition of the period ring B + dR .

Definition 3.3.7. We define B + dR as the completion of B + inf for the (ker θ)-adic topology:

B + dR = lim ← - m B + inf /(ker θ) m = lim ← - m B + inf /Fil m B + inf .
Since each quotient B + inf /Fil m B + inf has a Galois action, B + dR inherits an action of G K . Besides, the algebraic structure of B + dR is very pleasant. Indeed, from the fact that ker θ is a principal ideal of B + inf , we deduce that B + dR is a discrete valuation ring. Its maximal ideal is the ideal generated by ker θ and its residue field is canonically isomorphic to B + inf /Fil 1 B + inf ≃ C p . Therefore, as a ring, B + dR is isomorphic C p ((t)), that is to the ring B ′ HT we introduced in §2.2.3. However, we strongly insist on the fact that there is no such isomorphism preserving the Galois action. The sole connection between B + dR and C p ((t)) is that they share the same graded ring, namely B HT . Observe that, by Proposition 3.3.4, we could have defined alternatively B + dR as the completion of B + µ or B + µ,K , i.e. we have the following canonical identifications:

B + dR = lim ← - m B + µ /Fil m B + µ = lim ← - m B + µ,K /Fil m B + µ,K .
for any µ ∈ [1, +∞) ⊔ {inf, crys, max}. Combining this with the fact that the de Rham filtration is separated (cf Proposition 3.3.3), we deduce that the canonical maps B + µ → B + µ,K → B + dR are injective for all µ as before. In particular t ∈ B + dR and B + dR contains a copy of K. Since the definition of B + dR does not actually depend on K, it follows that B + dR contains (in a coherent way) a copy of any finite extension of Q p , that is a copy of K. Denote by ι : K → B + dR the resulting embedding. It turns out that ι can be understood in more down-to-earth terms. Indeed observe first that K naturally embeds into the residue field of B + dR since the latter is canonically isomorphic to C p . By Hensel lemma, this embedding admits a unique lifting ι : K → B + dR which is a homomorphism of K 0 -algebras: concretely, for x ∈ K whose minimal polynomial over K 0 is denoted by P , ι(x) is the unique root of P that lifts the image of x in C p . In particular, the composite K → B + dR → C p is the natural inclusion. The map θ extends to B + dR easily: we define θ dR as the composite B + dR → B + inf /(ker θ) → C p where the first map is the projection onto the first component and the second map is induced by θ. We set Fil m B + dR = (ker θ dR ) m for m ∈ N. Observe that the kernel of θ dR is nothing but the maximal ideal of B + dR . As a consequence, the de Rham filtration of B + dR coincides with the canonical filtration on the discrete valuation ring B + dR , given by the valuation. Its graded ring is isomorphic to C p [t] (compare with [START_REF] Lazard | Les zéros des fonctions analytiques d'une variable sur un corps valué complet[END_REF] Topology on B + dR . As B + dR is defined as a completion, the first natural topology on B + dR is the (ker θ)-adic topology: a sequence (x n ) n≥0 of elements of B + dR converges to x ∈ B + dR if and only if, for all m, the sequence x n mod Fil m B + dR is eventually constant. This topology is actually not nice because it does not see the p-adic topology: it induces the discrete topology both on the subfield K• Kur and on the residue field C p .

A coarser topology can be defined as follows. Observe that the quotients B + inf /Fil m B + inf have finite length and hence are equipped with a canonical topology. This topology can be described by remarking that the lattice A inf /Fil m A inf defines a valuation v m,inf on B + dR /Fil m B + dR : given

x ∈ B + dR /Fil m B + dR ≃ B + inf /Fil m B + inf ,
we define v m,inf (x) as the largest (possible negative) integer n for which x ∈ p n A inf /Fil m A inf . The valuation v m,inf defines a norm on B + inf /Fil m B + inf , and hence a topology. Remark 3.3.9. Alternatively, instead of B + inf , one could have worked with B + µ for a different µ. We would have ended up this way with a valuation v m,µ on B + dR /Fil m B + dR for which there exists a constant v m,µ with the property that:

v m,inf (x) -v m,µ ≤ v m,µ (x) ≤ v m,inf (x) (34)
for all x ∈ B + dR /Fil m B + dR (see the first part of the proof of Proposition 3.3.4). Therefore the topology induced by v m,µ agrees with that defined by v m,inf for all m.

We extend v m,inf to B + dR by precomposing by the natural projection B + dR → B + dR /Fil m B + dR . When m varies, the v m,inf 's define a family of semi-norms on B + dR , giving it the structure of a Frechet space. The attached topology will be called (in this article) the standard topology on B + dR . Concretely, a sequence (x n ) of elements of B + dR converges to x ∈ B + dR for the standard topology if and only if, for all integer m, the image of x n is B + dR /Fil m B + dR ≃ B + inf /Fil m B + inf converges to the image of x. Clearly, the standard topology induces the usual p-adic topology on the residue field B + dR /Fil 1 B + dR ≃ C p . Colmez proved in [START_REF] Colmez | Une construction de B dR[END_REF] that K is dense in B + dR for the standard topology. We point out that there is no good notion of p-adic topology on B + dR . Indeed, if there were, the inclusion ι : K → B + dR would extend to an inclusion C p → B + dR which would imply that B + dR would be isomorphic to C p ((t)) = B ′ HT and we have already seen that this does not happen. Yet, B + dR admits kinds of lattices, e.g.

A µ,dR = lim ← - m A µ /(A µ ∩ Fil m B + µ ) or A µ,K,dR = lim ← - m A µ,K /(A µ,K ∩ Fil m B + µ,K )
though we have to be careful that A µ,dR [ 1 p ] B dR and similarly for A µ,K,dR . These "lattices" do define topologies on B + dR (which might be considered as sort of p-adic topologies). However, these topologies are all different (and different from the standard topology) and they all have bad properties; for instance, the inclusion ι : K → B + dR is not continuous for any of them. The point behind this is that the constant v m,µ of Eq. ( 34) is not bounded uniformly when m grows. Remark 3.3.10. The situation is quite similar to that Q p [[t]]. The analogue of the standard topology on Q p [[t]] is the standard Fréchet topology on this ring: a sequence (f n ) n≥0 converges to f if and only if f n mod t m converges to f mod t m in Q p [t]/t m for all m ∈ N. This is further equivalent to the fact that, for all fixed m ∈ N, the m-th coefficient of f n converges to the the m-th coefficient of f . Another topology on Q p [[t]] is that defined by the "lattice" Z p [[t]], for which the sequence (f n ) n≥0 converges to f when, for each A ≥ 0, there exists an index n 0 with the property that f n ≡ f (mod p A Z p [[x]]) for all n ≥ n 0 . This notion of convergence is stronger than the previous one because we impose here that the coefficients of f n converge uniformly to that of f (the index n 0 has to the same for all m).

Inverting

m B = B ∩ Fil m B dR (m ∈ Z). (35)
Observe that Fil 0 B is the intersection of two rings and thus is a ring as well. It is easily checked that, when B = B + µ or B + µ,K (for µ ≥ 1 or µ = crys), the above definition leads to the de Rham filtration Fil m B we have defined earlier by different means. Yet, the definition ( 35) is new and interesting for B = B µ and B = B µ,K . The filtrations obtained this way sit in the following diagram (and a similar diagram for B µ,K ):

• • • ⊂ Fil 2 B µ ⊂ Fil 1 B µ ⊂ Fil 0 B µ ⊂ Fil -1 B µ ⊂ Fil -2 B µ ⊂ • • • • • • ⊂ Fil 2 B + µ ⊂ Fil 1 B + µ ⊂ Fil 0 B + µ = B + µ ⊂ ⊂ ⊂
The reader should be very careful that the inclusion B + µ ⊂ Fil 0 B µ is strict. Let us first focus on the case where µ = max. We recall that, in Remark 3. ] in order to avoid the small unpleasantness discussed above. One reason is that the Frobenius does not extend on B + µ [ 1 γ ] because the ideal ker θ µ is not stable under Frobenius. Formula (28) shows that inverting t is very natural if our objective is to keep an action of the Frobenius.

Proposition 3.3.13. For µ ≥ 1 or µ = crys, the inclusions B µ ⊂ B µ,K ⊂ B dR induce G K - equivariants isomorphisms of rings gr B µ ≃ gr B µ,K ≃ gr B dR ≃ B HT .
Proof. The fact that gr B dR is isomorphic to B HT has been already noticed. Now, consider the composite f : gr B + µ → gr B µ → gr B µ,K → gr B dR ≃ gr B + dR in which all maps are injective. Since B + dR is the completing of B + µ with respect to the de Rham filtration, the map f has to be an isomorphism. The proposition follows.

B crys and B dR as period rings

In order to apply Fontaine's general strategy (discussed in §1.4) with the B µ 's (for µ ≥ 1 or µ = crys or µ = dR)-and then "promote" these rings at the level of genuine period rings-a final couple of verifications still need to be done; precisely we need to check that the B µ 's satisfy Fontaine's hypotheses (H1), (H2) and (H3) introduced in §1.4.1, and we need to compute the invariants under the G K -action.

We start with B dR which is easier. First, since it is a field, Fontaine's hypotheses are obviously fulfilled. Concerning the computation of the fixed points, we have the following theorem. Theorem 3.4.1. We have (B dR ) G K = K.

Proof. We have already seen that K embeds into B dR , so that K ⊂ (B dR ) G K . The reverse inclusion follows from the fact that (gr B dR ) G K = (B HT ) G K = K.

We now move to the crystalline setting, that is the ring B crys and its variant B µ with µ ≥ 1.

Theorem 3.4.2. For

µ ≥ 1 or µ = crys, we have (B µ ) G K = K 0 and (B µ,K ) G K = K.
Proof. We have already seen that K 0 ⊂ (B µ ) G K and K ⊂ (B µ,K ) G K . From B µ,K ⊂ B dR , we deduce that (B µ,K ) G K ⊂ (B dR ) G K = K, the latter equality resulting from Theorem 3.4.1. Hence we have proved that (B µ,K ) G K = K. Now remember that, by definition,

B µ,K = K ⊗ K 0 B µ . Taking the G K -invariants, we obtain K = K ⊗ K 0 (B µ ) G K , from which we deduce (B µ ) G K = K 0 .
the coincidence of dimensions cannot be considered as a satisyfing answer to Grothendieck's question.

Hodge-like decomposition theorems discussed in §1.2 (see in particular Eq. ( 5)) constitute a significant process towards Grothendieck's problem. Indeed they show, for some particular X's, that H r ét (X K , Q p ) is isomorphic to the graded module of H r dR (X) after extending scalars to C p . However the de Rham filtration on H r dR (X) is not canonically split in the p-adic setting; therefore some information is lost when passing to the graduation. The point, which was first formulated by Fontaine and Jannsen, is that we can recover this missing information by extending scalars to the larger field B dR . This is the content of the C dR theorem10 : Theorem 4.1.1 (C dR ). Let X be a proper smooth variety over Spec K. For all r, there exists a canonical isomorphism:

γ dR (X) : B dR ⊗ K H r dR (X) ≃ B dR ⊗ Qp H r ét (X K , Q p ) (36)
which respects filtrations and Galois action on both sides. Moreover γ dR (X) is functorial in X.

In the above theorem, the filtration on the source of γ dR (X) is the "convolution" filtration:

Fil m B dR ⊗ H r dR (X) = a+b=m Fil a B dR ⊗ K Fil b H r dR (X)
whereas, on the target, the filtration comes only from that on B dR . In the same way, the Galois action on the source (resp. on the target) of ( 36) is the diagonal action (resp. the action coming from that on B dR ). We observe that Theorem 4.1.1 implies readily that the Q p -linear representation H r ét (X K , Q p ) is de Rham. Moreover, taking G K -invariants on both side of (36), we find a natural isomorphism:

H r dR (X) ≃ B dR ⊗ Qp H r ét (X K , Q p ) G K (37)
which gives a satisfactory answer to Grothendieck's mysterious functor problem. Similarly, passing to the graduation in [START_REF] Scholze | Perfectoid spaces[END_REF], we obtain the following Hodge-like decomposition:

C p ⊗ Qp H r ét (X K , Q p ) ≃ a+b=r C p (χ -a cycl ) ⊗ K H b (X, O X ) (38)
extending Tate's theorem on abelian varieties (cf §1.2). Observe in addition that the above isomorphism gives the Hodge-Tate decomposition of H r ét (X K , Q p ). In particular, we see that all the Hodge-Tate weights of H r ét (X K , Q p ) are in the range [-r, 0].

The Fontaine-Mazur conjecture A classical application of Theorem 4.1.1 is to prove that a representation V does not come from geometry: if we can prove that V is not de Rham (or not Hodge-Tate), it can't arise as the étale cohomology of a proper smooth variety. One may ask for the converse: does any de Rham representation arise as a subquotient of a Tate twist of the étale cohomology of some variety? In the local situation considered up to now, the answer is negative. Nevertheless, a "global" variant of this question is conjectured to admit a positive answer. It is the so-called Fontaine-Mazur conjecture, which first appeared in [START_REF] Fontaine | Geometric Galois representations[END_REF].

Let F be a number field, that is a finite extension of Q. For any prime ideal p in O F (the ring of integers of F ), one can consider the field F p defined as the completion of F with respect to the p-adic topology. If p is the prime number defined by pZ = Z ∩ p, the field F p is a finite extension of Q p . Moreover its absolute Galois group Gal( Qp /F p ) embeds into Gal( Q/F ). This embedding is not unique but it is up to conjugacy by an element of Gal( Q/F ). Therefore, if V is a Q p -representation of Gal( Q/F ), its restrction to Gal( Qp /F p ) is well defined and it makes sense to wonder whether it is de Rham or not. In the same way, a representation of Gal( Q/F ) (with coefficients in any ring) is said to be unramified at p if its restriction to Gal( Qp /F p ) is unramified (i.e. if the inertia subgroup of Gal( Qp /F p ) acts trivially on it). Conjecture 4.1.2 (Fontaine-Mazur). We fix a number field F and a prime number p. Let V be a finite dimensional Q p -representation of Gal( Q/F ). We assume that:

(i) for almost 11 all prime ideals p ∈ O F , the representation V is unramified at p, (ii) for all primes p above p (i.e. such that Z ∩ p = pZ), the representation V |Gal( Qp/Fp) is de Rham. Then V appears as a subquotient of some H r ét (XQ, Q p )(χ m cycl ) where r is a nonnegative integer, X is a proper smooth variety defined over Spec F and m is an integer.

When a representation V satisfies the conclusion of the above conjecture, we usually say that V comes from geometry. From the C dR -theorem, we derive that every representation of the shape H r ét (XQ, Q p )(χ m cycl ) comes from geometry. The Fontaine-Mazur conjecture then appears as a purely algebraic criterium to recognize representations coming from geometry among all representations.

We would like to emphasize that the Fontaine-Mazur conjecture might look surprising at first glance. Indeed it has been known for a long time that the Galois action on the étale cohomology satisfies many additional properties: for instance, the eigenvalues of the Frobenius acting on the étale cohomology have to take very particular values, known as Weyl numbers. However, these properties are not required in Fontaine-Mazur conjecture. It means that, assuming the conjecture to be true, they are implied by the unramified and the de Rham conditions, which is a priori rather unexpected.

Nowadays, the Fontaine-Mazur conjecture is still open. It was recently proved by Emerton [START_REF] Emerton | Local-global compatibility in the p-adic Langlands programme for GL 2/Q[END_REF] and Kisin [START_REF] Kisin | The Fontaine-Mazur conjecture for GL 2[END_REF] for two-dimensional representations of Gal( Q/Q) (satisfying some additional mild conditions) using the most recent developments in algebraic number theory (e.g. modularity lifting theorems, p-adic Langlands program). As far as we know, beyond the dimension 2, nothing is known.

The C crys -theorem We now go back to the local setting and examine the case of the variety X has good reduction. We recall that this means that there exists a proper smooth variety X over Spec O K whose generic fiber is X. We emphasize that the model X is required to be smooth; it is the crucial assumption.

When X has good reduction, the de Rham cohomology of X carries more structures. Indeed, assuming that X has good reduction, one can fix a model X as above and consider its special fiber X . It is a proper smooth scheme defined over Spec k. To X , one can attach a third cohomology group: its crystalline cohomology H r crys ( X ), defined by Berthelot [START_REF] Berthelot | Cohomologie cristalline des schémas de caractéristique p[END_REF]. We refer to [START_REF] Berthelot | Cohomologie cristalline des schémas de caractéristique p[END_REF][START_REF] Berthelot | Notes on crystalline cohomology[END_REF] for a complete exposition of the crystalline theory. For this article, let us just recall very briefly that, for all positive integer r, the crystalline cohomology H r crys ( X ) is a module over W (k) endowed with an endomorphism ϕ : H r crys ( X ) → H r crys ( X ) which is semi-linear with respect to the Frobenius on W (k). In addition, the crystalline cohomology of X is closely related to the de Rham cohomology of X through the Hyodo-Kato isomorphism

K ⊗ W (k) H r crys ( X ) ≃ H r dR (X). Putting K 0 = W (k)[ 1 p ]
as before, we see that Hyodo-Kato isomorphism defines a K 0 -structure in H r dR (X), namely K 0 ⊗ W (k) H r crys ( X ). One can prove that this structure is canonical in the sense that it does not depend on the choice of a proper smooth model X of X. Theorem 4.1.3 (C crys ). Let X be a proper smooth variery over Spec K with good reduction. Let X denote a proper smooth model of X over Spec O K . For all r, there exists a canonical and functorial isomorphism:

γ crys (X) : B crys ⊗ W H r crys ( X ) ≃ B crys ⊗ Qp H r ét (X K , Q p ) (39)
which respects Galois action and Frobenius action on both sides and such that B dR ⊗ γ crys (X) respects filtrations.

Here, the Frobenius action is defined on the source (resp. the target) of γ crys (X) as the diagonal action (resp. the action given by ϕ ⊗ id). Theorem 4.1.3 shows that the representation H r ét (X K , Q p ) is crystalline as soon as X has good reduction. (Remember that we already knew that this representation was de Rham thanks to the C dR -theorem.) More precisely, taking G Kinvariants on both sides of (39), we obtain:

H r crys (X ) ≃ B crys ⊗ Qp H r ét (X K , Q p ) G K (40)
which shows that, when X has good reduction, the étale cohomology of X not only determines its de Rham cohomology but also its canonical K 0 -structure coming from the crystalline cohomology. After the results of §4.3 (to come up), it turns out that the converse also holds true: the crystalline cohomology, equipped with its Frobenius and the de Rham filtration after scalar extension to K, determines the étale cohomology.

A brief history of the C dR -theorem Theorems 4.1.1 and 4.1.3 were first stated as conjecture by Fontaine and Jannsen just after Fontaine introduced the corresponding periods rings B dR and B crys respectively. Fontaine also designed a strategy to prove these conjectures. Very roughly, it can be summarized as follows:

1. prove the C crys -conjecture;

2. extend the C crys -conjecture to the semi-stable case 12 ;

3. derive the C dR -conjecture by reduction to the semi-stable case.

The case of C crys looks easier than that of C dR because the isomorphism (39) can be understood as a kind of Kunneth formula. Indeed, the period ring B crys has a nice cohomological interpretation, that is B crys = H 0 crys O K /pO K . It then becomes plausible that B crys ⊗ K 0 H r crys ( X ) could have something to do with the cohomology of X K . Beyond this remark, it remained to find the way to go back and forth between the crystalline and the étale cohomologies. To this end, Fontaine and Messing proposed to use a third cohomology, the syntomic cohomology, and to compare it to both sides of the isomorphism [START_REF] Scholze | Perfectoid spaces[END_REF]. Using these ideas, they managed to prove the C crys -theorem under the additional assumption that X has dimension at most p-1 2 [START_REF] Fontaine | p-adic periods and p-adic étale cohomology[END_REF]. Regarding the second step, Fontaine and Illusie introduced and proposed to develop log geometry. The main feature of log geometry is that it sees a normal crossing divisor as a log-smooth scheme. It then should be the right framework to perform local computations in the semi-stable case and then, hopefully, to extend the proof by Fontaine and Messing to all varieties admitting semi-stable reduction. The development of log geometry was achieved by the Japanese school [START_REF] Kato | Logarithmic structure of Fontaine-Illusie, Algebraic, Analysis, Geometry and Number Theory[END_REF][START_REF] Hyodo | Semi-stable reduction and crystalline cohomology with logarithmic poles[END_REF], who defined an analogue of the crystalline cohomology in this setting -the so-called log-crystalline cohomology -and related it to the de Rham cohomology via a log-analogue of the Hyodo-Kato isomorphism.

The initial idea for the third step was to prove that every proper smooth variety over Spec K admits semi-stable reduction after a finite extension. Unfortunately, this problem turns out to be quite difficult and is still open nowadays. Nevertheless, de Jong [START_REF] De | Smoothness, semi-stability and alterations[END_REF][START_REF] De | Families of curves and alterations[END_REF][START_REF] Berthelot | Altérations des variétés algébriques (d'après A.J. de Jong)[END_REF] proved a weaker result which was enough to complete the last step of Fontaine's strategy. In the very long paper [START_REF] Tsuji | p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case[END_REF], Tsuji gathered all these inputs and finally came up with a complete proof of the C dR -theorem. The main ingredients of the proof will be presented in Yamashita's lecture [43, §2] in this volume.

In the meanwhile, Faltings published another proof of the C crys and C dR -theorem [START_REF] Faltings | Crystalline cohomology and p-adic Galois-representations[END_REF] (but did not state a semi-stable version). Faltings' strategy is quite different from Fontaine's one and relies on almost mathematics, a theory specifically developped by Faltings for this application, which can be thought of as a wild generalization of Tate-Sen's methods presented in §2. The common idea which unifies these two proofs is, roughly speaking, to develop advanced methods to control extensions obtained by extracting p-th roots: in Fontaine's approach, it is achieved by the syntomic topology 13 whereas Faltings' initial idea is to work over infinite extensions obtained by extracting successive p-th roots and to use almost mathematics as the main tool to study the cohomology of varieties defined over such extensions.

More recently, Scholze designed a very powerful framework to do geometry over many "very ramified" bases including those obtained from usual Z p -schemes by adjoining iterated p-th roots: it is the theory of perfectoid spaces [START_REF] Scholze | Perfectoid spaces[END_REF]. Based on this, he obtained in [START_REF] Scholze | p-adic Hodge theory for rigid-analytic varieties[END_REF] a new proof of the C dR -theorem which extends readily to analytic varieties (without any hypothesis of type Kähler)! This proof will be sketched in the article of Andreatta and al.'s in this volume [START_REF] Andreatta | An introduction to perfectoid spaces[END_REF].

More on de Rham representations

Now we have seen the relevance of crystalline and de Rham representations, it looks important to study systematically their properties. We start with the de Rham case. Conversely, let us assume that V |G L is de Rham. Without loss of generality, we may assume that the extension L/K is Galois (if not, replace L by its Galois closure). Define D = (B dR ⊗ Qp V ) G L , so that we have dim L D = dim Qp V . Moreover D inherits a semi-linear action of Gal(L/K). By Hilbert's theorem 90 (cf Theorem 1.3.3), D is spanned by a basis of fixed vectors. In other words, dim K D Gal(L/K) = dim L D = dim Qp V . Since D Gal(L/K) = (B dR ⊗ Qp V ) G K , we have proved that V is de Rham. h must be an isomorphism. We conclude the proof by remarking that B HT ⊗ h = f • g where g is the canonical mapping: g : B HT ⊗ Qp gr D dR (V ) -→ gr B dR ⊗ Qp D dR (V ) .

By definition of the filtration on B dR ⊗ Qp D dR (V ), g is surjective. Since h is a bijection, we deduce, first, that g is an isomorphism and, then, that f is an isomorphism as well.

As a byproduct of the above proof, we obtain the following quite interesting corollary. Corollary 4.2.6. Let V de a de Rham representation of G K . Then V is Hodge-Tate and its Hodge-Tate weights are the integers m for which gr -m D dR (V ) = 0, the multiplicity of m being equal to dim K gr -m D dR (V ).

Proof. The corollary follows from the isomorphism B HT ⊗ Qp V ≃ B HT ⊗ K gr D dR (V ), which was established in the proof of Proposition 4.2.5.

For one dimensional representations, the converse of Corollary 4.2.6 holds. Indeed, if χ :

G K → Q ×
p is a Hodge-Tate character, then there exists some integer m for which χ • χ m cycl is C p -admissible. By Theorem 4.2.1, we deduce that χ • χ m cycl is de Rham. Hence χ is de Rham as well. However for higher dimensional representation, there do exist Hodge-Tate representations which are not de Rham.

B dR -representations. After what we have achieved so far, it is quite tempting to study B dRsemi-linear representations on their own in the spirit of Sen's theory (presented in §2.3). This work was achieved by Fontaine in [START_REF] Fontaine | Arithmétique des représentations galoisiennes p-adiques[END_REF]. Let us give rapidly a few details on Fontaine's results. Let K ∞ denote the p-adic cyclotomic extension of K. Generalizing Sen's arguments, Fontaine first shows that any B dR -semi-linear representation of G K descends to K ∞ ((t)). We are then reduced to study the K ∞ ((t))-semi-linear representations of Γ = Gal(K ∞ /K). Fontaine then defines an analogue of the Sen's operator which is no longer a linear map, but instead a derivation. More precisely, given a K ∞ ((t))-semi-linear representation W of Γ, Fontaine shows that, for γ ∈ Γ sufficiently closed to the identity, the formula log γ log χ cycl (γ) defines a K ∞ -linear mapping ∇ W : W → W which satisfies the Leibniz rule, i.e.

∇ W (f w) = df dt • w + f • ∇ W (w) (f ∈ K ∞ ((t)), w ∈ W ).
Moreover, as in Sen's theory, this construction is functorial and the datum of ∇ W caracterizes the representation W . For much more details, we refer to Fontaine's original paper [START_REF] Fontaine | Arithmétique des représentations galoisiennes p-adiques[END_REF].

More on crystalline representations

Let Rep crys Qp (G K ) be the category of Q p -linear crystalline representations of G K . It is a abelian subcategory of Rep dR Qp (G K ), which is stable by direct sums, duals, tensor products, subobjects and quotients. Unlike the de Rham case, the fact that a representation is crystalline cannot be detected on the restriction to an open subgroup in general. Nevertheless, we have a weaker result in this direction. (i) if V is unramified (i.e. the inertia subgroup acts trivially on V ), then V is crystalline, (ii) if there exists a finite unramified extension L of K such that V |G L is crystalline, then V is crystalline.

Proof. By Proposition 2.2.5, if V is unramified then it is Kur -admissible. Since Kur ⊂ B crys , it is then a fortioti crystalline. This proves (i).

We now assume that V |G L is crystalline for some finite unramified extension L of K. Without loss of generality, we may assume that L/K is Galois. We let L 0 be the maximal unramified extension of Q p inside L. Then Gal(L/K) ≃ Gal(L 0 /K 0 ). Set D = (B crys ⊗ Qp V ) G L ; it is a L 0 -vector space endowed with a semi-linear action of Gal(L/K) ≃ Gal(L 0 /K 0 ). By Hilbert's theorem 90, we have dim K 0 D Gal(L/K) = dim L 0 D. Moreover since V |G L is crystalline, we know that dim L 0 D = dim Qp V . Consequently dim K 0 D Gal(L/K) = dim Qp V , which proves that V is crystalline because D Gal(L/K) = (B crys ⊗ Qp V ) G K .

We insist again on the fact that the assumption that L/K is unramified is crucial in Proposition 4.3.1.(ii). For example, one can prove (using Proposition 4.3.2 below for example) that a character is crystalline if and only if it is the product of an unramified character by a power of the cyclotomic character. In particular the finite order character ω cycl = [χ cycl mod p] of G Qp is not crystalline.

A finite dimensional Q p -linear representation that becomes crystalline over a finite extension (non necessarily ramified) is called potentially crystalline. Combining Theorems 2.2.1, 4. Proof of Proposition 4.3.2. Let V be a crystalline C p -admissible representation. From Remark 2.2.7, we derive that there exists a finite extension L of K such that V is (L• Kur 0 )-admissible. Let f : V ⋆ → B dR be a G K -equivariant Q p -linear morphism. Since V is crystalline, we know that f (V ⋆ ) ⊂ B crys . Similarly, using that V is (L• Kur 0 )-admissible, we find f (V ⋆ ) ⊂ (L• Kur 0 ). On the other hand, we know that L ⊗ L 0 B crys embebs into B dR . The canonical morphism (L• Kur 0 ) ⊗ Kur 0 B crys → B dR is then injective. As a consequence (L• Kur 0 ) ∩ B crys = Kur 0 and we deduce that f takes its values in Kur 0 . As a conclusion, Hom Qp[G K ] (V ⋆ , B dR ) = Hom Qp[G K ] (V ⋆ , Kur 0 ). Since V is de Rham, we deduce from the above equality that V is Kur 0 -admissible. In particular V embeds into a direct sum of copies of Kur 0 . Since the inertia subgroup acts trivially on Kur 0 , it acts trivially on V as well.

Example 4.3.4. We give an example of a two dimensional representation which is de Rham but not crystalline. For any positive integer n, let ε n ∈ K be a primitive p n -th root of unity. Similarly, let ̟ n ∈ K be a p n -root of p. For any g ∈ G Qp , there exists a unique element c(g) ∈ Z p such that g̟ n = ε c(g) n ̟ n for all n. In the language of §3, the previous equation reads:

gp ♭ = ε c(g) • p ♭ (g ∈ G Qp ) (43)
where p ♭ = (p, p1 , p2 , . . .) and ε = (1, ε1 , ε2 , . . .) are the elements of R defined in §3.1. A direct computation shows that c(gh) = c(g) + χ cycl (g) • c(h) (we say that c is a cocycle). From this

  action of G, which is compatible with the ring structure, i.e. g • (a + b) = ga + gb and g • (ab) = ga gb for all g ∈ G and a, b ∈ B. Definition 1.3.1. A B-semi-linear representation of G is the datum of a B-module W equipped with a continuous action of G such that: g • (x + y) = gx + gy and g • (ax) = ga • gx for all g ∈ G, a ∈ B and x, y ∈ W . Clearly, if G acts trivially on B, the notion of B-semi-linear representation of G agrees with the usual notion of B-linear representation of G.

Figure 1 :

 1 Figure 1: The graph of the function ψ r (r ≥ 4)

  and b -v Fr (D Lr /Fr ) = b ′ . From the latter equality, we derive v p (D Lr/Fr ) = b-b ′ e F p r . Let π Fr be a uniformizer of F r . Let y be an element of D Lr/Fr with v p (y) = b-b ′ e F p r . By definition of the different, there exists z ∈ O Lr such that Tr Lr/Fr ( z π Fr y ) ∈ O Fr . In other words, v p (Tr Lr/Fr (

Example 2 . 2 . 10 .

 2210 Recall that, assuming p > 2, we have classified the characters of G Qp in Proposition 1.1.1: they are all of the form µ λ • χ a cycl • ω b cycl with a ∈ Z p and b ∈ Z/(p-1)Z. Here µ λ denotes the unramified character taking the Frobenius Frob q to λ and ω cycl = [χ cycl mod p]. Since the representations C p (µ λ ) and C p (ω b cycl ) are C p -admissible, we obtain:

  Let Rep f B (G) denote the full subcategory of Rep B (G) consisting of representations which are finitely generated as a B-module. When B is the field, Rep f B (G) is then the category of finite dimensional B-semi-linear representations of G.

Figure 2 :

 2 Figure 2: Diagram of period rings; all arrows are injective

Lemma 3 . 1 . 1 .

 311 The projection onto the first coordinate R → O Cp /pO Cp induces an isomorphism R/p ♭ R ≃ O Cp /pO Cp .

Proposition 3 . 1 . 3 .

 313 (i) Let z ∈ A inf be an element such that θ(z) = 0 and v ♭ (z mod p) = 1. Then z generates A inf ∩ ker θ, viewed as an ideal of A inf .

Figure 3 : 1 Proposition 3 . 1 . 5 .

 31315 Figure 3: The Newton polygon of [ε] -1

Figure 4 :

 4 Figure 4: Convergence conditions for elements in A crys

2 s l o p e : -p s l o p e : - 1 sFigure 5 :

 215 Figure 5: The Newton polygon of t

Proposition 3 . 3 . 1 .

 331 The ideal ker θ max (resp. ker θ max,K ) is the principal ideal generated by the element [p ♭ ] -p (resp. [π ♭ ] -π).

Definition 3 . 3 . 2 .

 332 For µ ≥ 1, µ = crys or µ = inf and for m ∈ N, we define: Fil m B + µ = (ker θ µ ) m and Fil m B + µ,K = (ker θ µ,K ) m . Proposition 3.3.4 implies that for all m, all the maps of the commutative square below are isomorphisms of C p -semi-linear representations: gr m B +

Proposition 3 . 3 . 5 .

 335 For µ ≥ 1 or µ = crys and for m ∈ N, the spaces gr m B + µ and gr m B + µ,K are generated by the class of t m .

Proposition 4 . 3 . 1 .

 431 Let V be a finite dimensional Q p -linear representation of G K . Then:

  2.1, 4.2.2 and Proposition 4.3.1, we obtain the following diagram of implications: unramified =⇒ =⇒ crystalline =⇒ C p -admissible =⇒ pot. crys. =⇒ de Rham =⇒ Hodge-Tate Proposition 4.3.2. A representation which is at the same time crystalline and C p -admissible is unramified. Remark 4.3.3. Recall that, for a Hodge-Tate representation, C p -admissibility means that all Hodge-Tate weights are 0. Proposition 4.3.2 then says that any crystalline representation with Hodge-Tate weights 0 is unramified.

  denotes the Teichmuller representative of a. From this description, we derive that there exist a ∈ Z p and b

Consequently, χ |I Qp = h • χ cycl for some group homomorphism h : Z × p → Z × p . Moreover, when p > 2, we have an isomorphism Z × p ≃ F × p × Z p , x → (x mod p, log x), the inverse being given by (a, b) → [a]• exp b where [a]

  has coefficients in L G . This is a contradiction since we have assumed that the family (x 1 , . . . , x m ) is linearly independent over L G . is an infinite extension and G is equipped with its natural profinite topology. As an example, take G = G Qp = Gal( Qp /Q p ) and let it act on L = Qp . The fixed subfield L G is Q p . Consider the semi-linear representation Q p (χ cycl ) where we recall that χ cycl denotes the cyclotomic character Gal( Qp /Q p

	Remark 1.3.4. Theorem 1.3.3 is often referred to as Hilbert's theorem 90. The reason is that
	it can be rephrased in the language of group cohomology, then asserting that H 1 (G, GL d (L)) is
	reduced to one element. This latter statement is an extension of the classical Hilbert's theorem
	90 to higher d.
	Example 1.3.5. We emphasize that Theorem 1.3.3 does not hold in general when G = Gal(L/K)
	where L/K

Definition 1.4.2. Let

  V ∈ Rep E (G) be finite dimensional over E. We say that V is B-admissible if the B-semi-linear representation B ⊗ E V is trivial. following sense: any B 1 -admissible representation is automatically B 2 -admissible as soon as B 2 appears as an algebra over B 1 .

	We denote by Rep B-adm E representations of E which are B-admissible. It is easy to check that Rep B-adm (G) the full subcategory of Rep E (G) consisting of finite dimensional (G) is stable by E E (G) is increas-direct sums, tensor products, and duals. Moreover the association B → Rep B-adm ing in the

Theorem 1.4.6. Let V be a Q p -linear finite dimensional representation of G K . Then V is C p - admissible if and only if the inertia subgroup of G K acts on V through a finite quotient.

	In other words, C p -admissibility detects those representations which are potentially unramified.
	This notion has then a strong arithmetical meaning.
	Hodge-Tate representations. Theorem 1.4.6 shows that the notion of C p -admissibility is too
	strong and does not capture all interesting representations; for instance, the cyclotomic charac-
	ter is not C p -admissible. A larger class of representations is given by the notion of Hodge-Tate
	representations. By definition, a Q p -linear representation of G K is Hodge-Tate if C p ⊗ Qp V decomposes as:

  an isomorphism. It suffices to apply Proposition 2.3.8 with W ′ = C dim W p . We conclude our exposition of Sen's theory by noticing that Sen's operator is closely related to the notion of Hodge-Tate representations. Precisely, a representation V ∈ Rep f Qp (G K ) is Hodge-Tate if and only if the operator Φ Cp⊗ Qp V is semi-stable with eigenvalues in Z, these eigenvalues being the Hodge-Tate weights of V . (Combine Examples 2.2.11 and 2.3.6 for an illustration of this property.) Given a general W ∈ Rep f Cp (G K ), the eigenvalues of Φ W are sometimes called the generalized Hodge-Tate weights of W .

	Corollary 2.3.9. A representation W ∈ Rep f Cp (G

K ) is trivial if and only if Sen's operation Φ W vanishes.

Proof.

  Remark 3.3.8. Continuing Remark 3.3.6 (and importing notations from there), we observe that the element γ ′ ∈ B + max ⊂ B + dR is invertible in B + dR since it is nonzero in the residue field; thus t = γγ ′ is a generator of ker θ dR as γ is. This contrasts with the fact that t did not generate ker θ max because γ ′ was not invertible in B + max .

	). Moreover, any generator of B + µ or B + µ,K (for dR , i.e. a uniformizer of B + µ ∈ {inf, max}) is a generator of B + dR . Even better, by completeness, an element of B + dR is a uniformizer if and only if it does not belong to Fil 1 B + dR or, equivalently, it does not vanish if gr 1 B + dR . In particular, the special element t is a uniformizer of B + dR by
	Proposition 3.3.5.

  t. Recall that we have defined B µ and B µ,K as B + µ [ 1 t ] and B + µ,K [ 1 t ] respectively for µ ≥ 1 or µ = crys (recall that this definition does not make sense for µ = inf because t ∈ B + inf ). Similarly we set B dR = B + dR [ 1 t ]. Since B + dR is a discrete valuation ring with uniformizer t, B dR is also the fraction field of B + dR ; in particular, it is a field. Moreover since localization is exact, the rings B µ and B µ,K appear as subrings of B dR . The de Rham filtration extends readily to B dR by letting Fil m B dR = t m B + dR for m ∈ Z. The graded ring of B dR is then canonically isomorphic to C p [t, t -1 ] = B HT . If B is any subring of B dR , we define: Fil

  3.6, we have set γ = [p ♭ ] -p ∈ B + max and noticed that t = γγ ′ for some γ ′ ∈ B + max . The element γ ′ is not invertible in B + max but we have seen in Remark 3.3.8 that it is invertible in B + dR . Besides, since γ ′ is a divisor of t, it is invertible in B max . Now consider 1 γ ′ ∈ B max . It does not lie in B + max . However, its image in B dR falls in B + dR , so that 1 γ ′ ∈ Fil 0 B max . Actually, one can (easily) prove that Fil 0 B max = B + max [ 1 γ ′ ]. A similar description is also possible for a general µ. Precisely let S be the multiplicative part consisting of all divisors in B + µ of some power of t. Then Fil 0 B µ = B + µ [S -1 ]. Remark 3.3.11. As discussed in Remark 3.3.6, what happens here is very similar to the following very classical situation: assume that Z is endowed with the filtration Fil m Z = p m Z, which induces the usual valuation filtration on Z p and Q p after completion. Now consider the localization Z[ 1 2p ]; it is a subring of Q p and then inherits the valuation filtration. For this filtration, we have Fil 0 Z[ 1 2p ] = Z[ 1 2 ]. Remark 3.3.12. The reader may wonder why we defined B µ as B + µ [ 1 t ] and not B + µ [ 1 γ

  Let Rep dR Qp (G K ) denote the category of Q p -linear de Rham representations of G K . By Fontaine's general formalism, we know that Rep dR Qp (G K ) is a full abelian subcategory of Rep Qp (G K ). It is moreover stable under direct sums, duals, tensor products, subobjects and quotients. Any finite dimensional Cp -admissible representation of G K is de Rham. Proof. Let V be a finite dimensional C p -admissible representation of G K . By Remark 2.2.7, there exists a finite extension L of K ur such that V is (L• Kur )-admissible. Since L• Kur ⊂ B dR , we conclude that V is de Rham.Another interesting result is that de Rham representations can be detected by looking at the restriction to open subgroups. Precisely, we have the following theorem. Let L be a finite extension of K and let V be a finite dimensional Q p -linear representation of G K . Then V is de Rham if and only if V |G L is de Rham. Remark 4.2.3. In other terms, Theorem 4.2.2 says that the following diagram is cartesian.Proof of Theorem 4.2.2. By definition, if V is de Rham, the B dR -semi-linear representation B dR ⊗ Qp V is trivial as a G K -representation. It is then a fortiori trivial as a G L -representation, which means that V |G L is de Rham.

	Theorem 4.2.1. Theorem 4.2.2. Rep dR Qp (G K )

/ / restriction Rep Qp (G K ) restriction Rep dR Qp (G L ) / / Rep Qp (G L )

Its construction is subtle and we will not give it here. However, we would like to encourage the reader to look at it in Fontaine's paper [18, Chap. V, §1] because it is instructive to realize that it is actually quite close to the construction of the periods B dR and Bcrys we shall detail in §3.

In the application we have in mind, G will be the absolute Galois group of a p-adic field. However, for now, it is better to allow more flexibility and let G be an arbitrary topological group.

By continuous, we mean that the map G × B → B, (g, x) → gx is continuous.

Our assumptions are a bit stronger than Fontaine's ones. We chose these stronger hypothesis because they simplify the exposition and are sufficient for the applications we want to discuss here.

This analogy has been placed in the framework of Huber geometry by Scholze in[START_REF] Scholze | Berkeley lectures on p-adic geometry[END_REF] and then takes a very substantial meaning. However, for this article, it will be sufficient to keep in mind that elements of B + inf behave like analytic functions over a nonarchimedian base.

This definition is s special case of a general construction (the tilt) in the theory of perfectoid spaces[START_REF] Scholze | Perfectoid spaces[END_REF]. We refer to Andreatta and al. lecture[START_REF] Andreatta | An introduction to perfectoid spaces[END_REF] in this volume for an introduction to perfectoid spaces. The notations p ♭ , π ♭ and v ♭ that we will introduce later comes from the language of perfectoid spaces.

Otherwise, an element ξ with the required properties does not exist.

This result is sometimes referred to as the C dR -conjecture (even if it is now proved) since it has been a conjecture for a long time. The letter "C" in C dR stands for "comparison" or "conjecture".

"almost all" means "all expect possibly a finite number of them"

We say that a variety X over K has semi-stable reduction if it has a proper model X over Spec OK whose generic fibre is a divisor with normal crossings.

A morphism of schemes obtained by extraction of a p-th root of some function turns out to be a covering for the syntomic cohomology.

We will use the notation gr to refer to the graded ring of a filtered ring: if m is an integer and A is a filtered ring, we put gr m A = Fil m A/Fil m+1 A and gr A = m≥0 gr m A. We recall that gr 0 A is a ring and that gr m A is a module over gr 0 A for all m ≥ 0. As for gr A, is it a graded algebra over gr 0 A. In our case, we have gr 0 B + µ = gr 0 B + µ,K = C p (since θ µ and θ µ,K are surjective). Hence gr m B + µ and gr m B + µ,K have a natural structure of C p -vector space. They moreover inherit a Galois action, so that they are actually C p -semi-linear representations of G K , i.e. objects of the category Rep Cp (G K ). From Proposition 3.1.3 and Proposition 3.3.1, we deduce that gr m B + inf , gr m B + inf,K , gr m B + max and gr m B + max,K are all one dimensional over C p . As we shall see below (cf Proposition 3.3.4), this property also holds for gr m B + µ and gr m B + µ,K for any µ. The next proposition shows that the de Rham filtration separated. Proof. Since B + µ and B + µ,K contain B + max,K , it is enough to prove the proposition for θ max,K . After Proposition 3.3.1, we are reduced to check that if x ∈ B + max,K is divisible by 1

π m for all m, then x = 0. Multiplying x by the adequate power of π, we may assume that x ∈ A max,K and in addition, if x = 0, that x ∈ πA max . Using isomorphism [START_REF] De | Families of curves and alterations[END_REF], we find that x vanishes in A max,K /πA max,K , i.e. x ∈ πA max,K . By our assumption, this implies that x = 0. Proposition 3.3.4. For µ ≥ 1 or µ = crys and for m ∈ N, the inclusion

Proof. In the case µ = 1, the proposition follows by combining Propositions 3.1.3 and 3.3.1. Before moving to the case of a general µ, we will prove an additional continuity property of the isomorphism B + inf /Fil m B + inf ≃ B + max /Fil m B + max , which will be useful later. Precisely, we claim that, for all m ≥ 0, there exists a nonnegative integer v m such that:

We prove the claim by induction on m. For m = 0, there is nothing to prove (we can take v 0 = 0). We now assume that (31) is proved for m. We consider the following commutative diagram with exact rows:

From the fact that gr m B + inf → gr m B + max is a C p -linear mapping between two one-dimensional C p -vector spaces, we deduce that there exists an integer v such that p v •gr m A max ⊂ gr m A inf . A diagram chase then shows that [START_REF] Kisin | Crystalline representations and F -crystals, Algebraic geometry and number theory[END_REF]

We now go back to the proof of the proposition. We pick µ ∈ (1, +∞)⊔{crys} and m ∈ N. Let f : B + inf /(ker θ) m → B + µ /(ker θ µ ) m be the morphism of the proposition. Let A ′ µ be the sub-A infalgebra of B + inf generated by all the elements of the form [ξ] p i (ξ ∈ R, i ∈ N), which belong to A µ . Then A ′ µ ⊂ B + inf and A µ appears as the p-adic completion of A ′ µ . The former property implies that we have a morphism g ′ : A ′ µ /(ker θ µ ) m → B + inf /(ker θ) m . We claim that g ′ is continuous. Indeed, by [START_REF] Kisin | Crystalline representations and F -crystals, Algebraic geometry and number theory[END_REF], we have A max ⊂ p -vm A inf + (ker θ max ) m . Since A ′ µ ⊂ A max , we deduce that g ′ maps A ′ µ /(ker θ µ ) m to p -vm A inf /(ker θ) m , which implies its continuity. Now, passing to the p-adic completion and inverting p, we find that g ′ induces a ring morphism g :

Hence B µ and B µ,K satisfy hypothesis (H2).

Let us now prove that B µ satisfies Fontaine's hypothesis (H3). Let x ∈ B µ , x = 0 and assume that the line Q p x is stable under the action of G K . We have to prove that x is invertible in B µ . In what follows, we will consider x as an element on B dR . Replacing possibly x by t n x for some integer n (which is safe since t is invertible in B µ ), we may assume that x ∈ B + dR and x ∈ Fil 1 B + dR . The morphism θ dR then induces a G K -equivariant embedding Q p x ֒→ C p . Thus the representation Q p x is C p -admissible. By Theorem 2.2.1, the inertia subgroup I K of G K acts on x through a finite quotient. Therefore there exists a positive integer n such that I K acts trivially on y = x n . The line Q p y then inherits an action of G K /I K = Gal(K ur /K) ≃ Gal(K ur 0 /K 0 ). Applying Proposition 2.2.5 to the Gal(K ur 0 /K 0 )-representation Kur 0 y (recall that Kur 0 ⊂ B + inf ⊂ B dR ), we find that there exists λ ∈ Kur 0 such that λy is fixed by G K . By Theorem 3.4.2, we obtain λy ∈ K 0 and then y ∈ Kur 0 . We deduce that y is invertible in B µ , and so also is x. The fact that B µ,K satisfies (H3) is proved in a similar fashion.

We conclude this section by stating another important property of the rings B µ .

Proof. We first prove the proposition when µ = µ 0 = p p-1 . In this case, it is easily checked that

By definition of B µ 0 , we can write x = t -m y with m ∈ N and y ∈ B + µ 0 . We choose m minimal with this property. We assume by contradiction that m > 0. By the first paragraph of the proof, we can write y = a + tb with a ∈ B + inf and b ∈ B + µ 0 . Besides, for any nonnegative integer n, we have ϕ n (y) = p nm y and then:

the last equality coming from the fact that y = t m x ∈ Fil m B µ 0 ⊂ Fil 1 B µ 0 . By Proposition 3.1.7, we find that [ε] -1 divides a in B + inf . On the other hand, from the definition of t, we have:

from what we derive that t and [ε] p -1 differ by a unit in B + µ 0 . From the divisibility observed above, we deduce that

µ 0 , which contradicts the minimality of m. As a conclusion, we find m = 0, i.e.

We now move to the general case. Let x ∈ (Fil 0 B µ ) ϕ=1 . In particular x ∈ Fil 0 B max and therefore x = ϕ(x) ∈ Fil 0 B p ⊂ Fil 0 B µ 0 . The conclusion now follows by the first part of the proof.

Remark 3.4.5. Proposition 3.4.4 can be written in the shorter form:

where the exponent "ϕ=1" means that we are taking the subspace of fixed points under ϕ.

The reader should be aware that restricting to Fil 0 is essential: B ϕ=1 µ is much bigger than Q p . Precisely, we have the so-called fundamental exact sequence:

where the map B ϕ=1 µ → B dR /B + dR is induced by the natural inclusion B µ ֒→ B dR .

Crystalline and de Rham representations

We keep the general notations of the previous section: the letter K denotes a finite extension of Q p , G K is its Galois group, etc. So far, we have defined the periods rings B crys and B dR (together with some variants). By Fontaine's formalism (cf §1.4), these rings cut out full subcategories of Rep Qp (G K ). The objective of this section is to study these categories and to demonstrate that they are relevant for geometric purpose. We begin with a definition.

Rephrasing the definition of B-admissibilty and using Theorems 3.4.1 and 3.4.2, we have:

Moreover, B crys is a subring of B dR , any crystalline representation is de Rham.

Comparison theorems: statements

We start by discussing the geometric relevance of the notion of crystalline and de Rham representations. Our ambition is only to state the relevant theorems in this direction and definitely not to prove them. The most important ingredients of the proofs will be presented and discussed in Yamashita's lecture [START_REF] Yamashita | An introduction to p-adic Hodge theory for open varieties via syntomic cohomology[END_REF] and Andreatta and al.'s lecture [START_REF] Andreatta | An introduction to perfectoid spaces[END_REF] in this volume. From now on, we fix a proper smooth variety X defined over SpecK. (At least) two different cohomology theories taking coefficients in Q p can be naturally attached to X, namely:

• the (algebraic) de Rham cohomology

vector space endowed with a continuous action of Gal( K/K).

In the early 1970's, Grothendieck [START_REF] Grothendieck | Groupes de Barsotti-Tate et cristaux[END_REF] wondered whether one can compare these cohomology groups. More precisely, he raised the so-called problem of the mysterious functor, asking for the existence of a purely algebraic recipe to recover H r dR (X) from H r ét (X K , Q p ). When X C is a complex variety, the problem of the "mysterious" functor has been solved for a long time; indeed, the de Rham comparison theorem ensures that H r dR (X C ) is isomorphic to the singular cohomology of X C (C) with coefficients in C (which plays the role of the étale cohomology). As we shall see, the p-adic case is more subtle.

Using standard arguments, one proves that H r dR (X) and H r ét (X K , Q p ) have the same dimension for all r. Thus K ⊗ Qp H r ét (X K , Q p ) has to be isomorphic to H r dR (X) as abstract K-vector spaces. However there does not exist any functorial isomorphism between them. Therefore

The function D dR . If V is a de Rham representation of G K , we define:

where Hom Qp[G K ] refers to the set of Q p -linear G K -equivariant morphisms and V ⋆ is the dual representation of V . Fontaine's formalism shows that we have a canonical isomorphism:

Remark 4.2.4. When V is the étale cohomology of a proper smooth variety X over Spec K, the isomorphism ( 42) is the isomorphism (36) of the C dR -theorem. Notably, we have H r dR (X) = D dR (H r ét (X K , Q p )) for all integer r. Formula (41) defines a functor D dR : Rep dR Qp (G K ) → Vect K where Vect K is the category of finite dimensional vector spaces over K. One can actually be more precise and endow D dR (V ) with a filtration coming from the filtration on B dR . Precisely, for an integer m ∈ Z, we define:

Clearly Fil m D dR (V ) is sub-K-vector space of D dR (V ) and Fil m+1 D dR (V ) ⊂ Fil m D dR (V ) for all m. Moreover observe that:

the second equality coming from the fact that D dR (V ) has finite dimension over K. Since again D dR (V ) is finite dimensional, we deduce that Fil m D dR (V ) = 0 for m ≫ 0 and Fil m D dR (V ) = D dR (V ) for m ≪ 0; we say that the filtration of D dR (V ) is separated and exhaustive.

With the above construction, we have promoted D dR to a functor D dR : Rep dR Qp (G K ) → MF K where MF K denotes the category of finite dimension K-vector spaces equipped with a nonincreasing separated and exhaustive filtration by sub-K-vector spaces. This functor has an extra remarkable property given by the next proposition.

Proposition 4.2.5. For any V ∈ Rep dR

Qp (G K ) and any integer m, the isomorphism (42) identifies Fil m (B dR ⊗ Qp V ) with Fil m (B dR ⊗ K D dR (V )) where, by definition:

induced by the inverse of ( 42) is an isomorphism. For this, we consider the exact sequence

Tensoring it by V and taking the G K -invariants, we obtain an injective morphism h m :

Taking the direct sum of the h m 's, we end up with an injective K-linear mapping h : gr

by the general Fontaine's formalism. As a consequence, observation, we deduce that the function:

is a group homomorphism and then defines a two dimensional Q p -linear representation V of G Qp . We are going to compute the space D = Hom Qp[G Qp ] (V, B dR ). By the general theory, we know that:

(i) D is a K-vector space of dimension at most 2, To this end, we observe that it can be defined as follows:

(here we have chosen the convention that log p = 0). Note that the series converges in Fil

. The space D is two dimensional and spanned by (0, 1) and (t, log[p ♭ ]). Hence V ⋆ is de Rham. The fact that log[p ♭ ] ∈ B crys , i.e. that V ⋆ is not crystalline can be checked as follows. Assume by contradiction that log[p ♭ ] ∈ B crys . Then, it would lies in Fil 1 B crys , so that a = log[p ♭ ] t ∈ Fil 0 B crys . Moreover, we would have ϕ(a) = a since the Frobenius takes [p ♭ ] to [p ♭ ] p . By Proposition 3.4.4, this would implies that a ∈ Q p . Applying Galois to the relation log[p ♭ ] = at, we would obtain a + c(g) = χ cycl (g) for all g ∈ G K , which is obviously not true. Finally, we deduce that V ⋆ is not crystalline. Remark 4.3.5. The representation V of the previous example is the prototype of semi-stable representations. On the geometric side, it corresponds to the Tate curve, which is the prototype of elliptic curve without good reduction. Semi-stable representations will be introduced and widely discussed in Brinon's lecture. In particular, it will be proved in [9, Proposition 2.7] is actually transcendantal over Frac B crys . About B µ -admissibility. Recall that, in §3, we have introduced a whole family of rings B µ 's (where µ ≥ 1 is a real paramater); there rings serve as variants of B crys , which have the advantage of exhibiting more pleasant properties from the algebraic and analytic point of view. The next theorem shows that changing B crys by B µ does not affect the notion of crystalline representation.

Theorem 4.3.6. Let µ ≥ 1 and let V be a finite dimension Q p -linear representation of G K . Then V is crystalline if and only if it is B µ -admissible.

Proof. Since the B µ 's form a decreasing sequence of rings and B µ ⊂ B crys ⊂ B p-1 for each µ < p-1, it is enough to show that B µ -admissibility implies B pµ -admissibility for all µ ≥ 1. But the latter assertion follows from the fact that the Frobenius induces a Galois equivariant ring isomorphism B µ ∼ → B pµ and therefore an isomorphism

The functor D crys . When V is a crystalline representation of G K , we set:

for µ ≥ 1, µ = crys or µ = max (which is, we recall, a redundant notation for µ = 1). By Theorem 4.3.6, D µ (V ) is a vector space over K 0 of dimension dim Qp V . Observe in addition that D µ (V ) is equipped with a Frobenius map ϕ : D µ (V ) → D µ (V ) which is semi-linear with respect to the Frobenius on K 0 . Moreover, one checks easily that:

Therefore K ⊗ K 0 D µ (V ) comes equipped with a filtration, namely the de Rham filtration.

The inclusion B µ ⊂ B max induces an injective K 0 -linear mapping f µ : D µ (V ) → D max (V ), which commutes with all additional structures. Since the source and the target of f µ are both K 0 -vector spaces of dimension dim Qp V , we conclude that f µ is an isomorphism. In other words, the D µ does not depend on the choice of µ; in what follows, we will prefer the notation D crys (in order to make apparent the fact that we are considering the crystalline case) but the reader should keep in mind that D crys = D µ for all µ.

The above constructions motivate the following definition.

Definition 4.3.7.

A filtered ϕ-module over K is a K 0 -vector space D equipped with a semi-linear endomorphism ϕ : D → D and a nonincreasing, exhaustive and separated filtration on K ⊗ K 0 D.

We denote by MF K (ϕ) the category of filtered ϕ-modules over K (the morphisms are the K 0 -linear mappings commuting with ϕ and preserving the filtration after scalar extension to K). We have a natural functor MF K (ϕ) → MF K taking D to K ⊗ K 0 D equipped with its filtration. Besides, the previous constructions give rise to a functor Proof. The fact that D crys is exact follows directly by a dimension argument.

Let V ∈ Rep crys Qp (G K ) and set D = D crys (V ). We then have a canonical isomorphism B crys ⊗ Qp V ≃ B crys ⊗ K 0 V which commutes with Frobenius and respects the filtration after extending scalars to B dR . Taking the Fil 0 and the fixed points under the Frobenius and using Proposition 3.4.4, we obtain:

(where the supscript "ϕ=1" means that we are taking fixed points under the Frobenius). Formula (45) defines a functor V crys : MF K (ϕ) → Rep Qp (G K ) and we have just proved that V crys • D crys is the identity. This is enough to ensure that D crys is fully faithful.

Remark 4.3.9. In the proof of Theorem 4.3.8, instead of B dR , we could have used the smaller ring B crys,K . Similarly, we could have replaced everywhere the subscript "crys" by µ for any real number µ ≥ 1.

Remark 4.3.10. Let A is an abelian variety over K with good reduction and let A[p ∞ ] be the p-divisible groups of its points of p ∞ -torsion. The étale cohomology (resp. the crystalline cohomology) of A is then identified with the Tate module (resp. the Dieudonné module) of

The fact that D crys is fully faithful then reflects the fact that Dieudonné modules (equipped with the de Rham filtration) classify p-divisible groups.

Admissibility for ϕ-modules. We say that a filtered ϕ-module over K is admissible if it belongs to the essential image of D crys , and we denote by MF adm K (ϕ) the full subcategory of MF K (ϕ) consisting of admissible filtered ϕ-modules. Theorem 4.3.8 indicates that D crys induces an equivalence of categories Rep crys Qp (G K ) ≃ MF adm K (ϕ). This result provides a very concrete description of crystalline representations as soon as we are able to recognize admissible filtered ϕ-modules among all filtered ϕ-modules. This is actually possible: there exists an easy numerical criterium that caracterizes admissibility. We would like to conclude this article by stating it (without proof). Let D ∈ MF K (ϕ) and set d = dim K 0 D. The maximal exterior product det D = d D has a natural structure of filtered ϕ-module: the Frobenius on it is d ϕ (where the latter ϕ is the Frobenius acting to D) and:

where we have set D K = K ⊗ K 0 D. Since det D is one dimensional, there exists a unique integer m for which Fil m (K ⊗ K 0 det D) = K ⊗ K 0 det D and Fil m+1 (K ⊗ K 0 det D) = 0. This integer is called the Hodge number of D and is usually denoted by t H (D). It is an easy exercise to check that we have the following alternative formula for t H (D):

Similarly, we can assign an integer to D measuring the action of the Frobenius. Precisely, if v is nonzero element of det D, we have d ϕ(v) = λv for some λ ∈ K 0 . One checks easily that v p (λ) does not depend on the choice of V . We call it the Newton number of D and denote by t N (D). (ii) for all sub-K 0 -vector space D ′ ⊂ D stable by the Frobenius, we have t H (D ′ ) ≤ t N (D ′ ), where D ′ is endowed with the induced filtration defined by:

Theorem 4.3.11 was first conjectured by Fontaine in [START_REF] Fontaine | Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate[END_REF]. It has been proved first by Colmez and Fontaine in [START_REF] Colmez | Construction des représentations p-adiques semi-stables[END_REF] about twenty years later. Today, other proofs are been proposed by different authors [START_REF] Berger | Équations différentielles p-adiques et (ϕ, N )-modules filtrés[END_REF][START_REF] Kisin | Crystalline representations and F -crystals, Algebraic geometry and number theory[END_REF][START_REF] Fargues | Courbes et fibrés vectoriels en théorie de Hodge p-adique[END_REF], but Theorem 4.3.11 remains a difficult result in all cases. Kisin's proof [START_REF] Kisin | Crystalline representations and F -crystals, Algebraic geometry and number theory[END_REF] will be sketched in Brinon's lecture in this volume [START_REF] Brinon | Filtered (ϕ, N )-modules and semi-stable representations[END_REF] (in the more general framework of filtered (ϕ, N )-modules).

Example 4.3.12. As an easy example, let us give a complete classification of filtered ϕ-modules

Let e be a basis of D. Then, there exists λ ∈ K 0 such that ϕ(e) = λe. Observe that if e is changed to ue (with u ∈ K 0 ), λ becomes λ • ϕ(u) u . By Hilbert's theorem 90, the elements of the form ϕ(u) u are exactly the elements of norm 1 over Q p . Therefore N K 0 /Qp (λ) does not depend on a choice of e and is a complete invariant classying the possible ϕ's on D. Concerning the filtration, there exists a unique integer r such that Fil m D K = D K if m ≤ r and Fil m D K = 0 otherwise.

One sees immediately that the D is admissible if and only if v p (λ) = r. Moreover, when admissibility holds, an easy computation shows that the attached Galois representation V crys (D) is given by the character χ -r cycl • µ -1 α with α = N K 0 /Qp (p -r λ). Example 4.3.13. We now investigate the admissible filtered ϕ-modules of dimension 2 over Q p . We then consider D ∈ MF adm Qp (ϕ) with dim Qp D = 2. The filtration on D is easy to describe: there exist two integers r and s with r ≤ s and a line

it follows from Proposition 4.3.2 that the crystalline representation associated to D (if D is admissible) has the form V (χ -r cycl ) for an unramified representation V . We leave this case to the reader and assume now that r < s. Then L is uniquely determined.

We want to describe the action of the Frobenius ϕ : D → D. Let us first notice that ϕ is a linear mapping because the Frobenius acts trivially on Q p . We first assume that L is stable by the Frobenius. We let α ∈ Q p be the scalar by which ϕ acts on L and we let β be the second eigenvalue of ϕ. From the admissibility condition, we deduce v p (α)+v p (β) = r+s and v p (α) ≥ s. Therefore v p (β) ≤ r < s. Hence α = β and ϕ is diagonalizable. If L ′ denotes the eigenspace associated to β, we have t N (L ′ ) = v p (β) and t H (L ′ ) = r. By the admissibility condition, this implies that v p (β) = r and then v p (α) = s. Then L and L ′ are themselves admissible filtered ϕ-modules of dimension 1 and D splits as a direct sum D = L ⊕ L ′ . The attached Galois representation is then a direct sum of two crystalline characters.

We now assume that L is not stable under ϕ. Let e 1 be a nonzero vector in L. Define p , ϕ has two eigenvalues of valuation r and s respectively. Let L r and L s be the corresponding eigenspaces. Since e 1 is not an eigenvector, we have t H (L r ) = t H (L s ) = r. Hence, L r is admissible and we have the exact sequence 0 → L r → D → D/L r → 0 is MF adm Qp (ϕ). Passing to Galois representations, we find that V crys (D) is a non split extension of Q p (χ -s cycl µ α ) by Q p (χ -r cycl µ β ) with α, β ∈ Z × p . On the contrary, when v p (b) > 0, D is admissible and irreducible in the category MF adm Qp (ϕ). It then gives rise to an irreducible crystalline representation of dimension 2 of G Qp , whose Hodge-Tate weights are r and s.