Construction of semi-stable \(p\)-adic representations.
(Construction des représentations \(p\)-adiques semi-stables.)

*(French)*Zbl 1010.14004Let \(K\) be a field of characteristic zero complete with respect to some discrete valuation. If \(K_0\) denotes the field of fractions of the associated ring of Witt vectors, then \(K/K_0\) is a finite totally ramified field extension. A \(p\)-adic representation of the Galois group \(G_K=\text{Gal} (\overline K/K)\) is given by a \(\mathbb{Q}_p\)-vector space of finite dimension, together with a continuous linear action of \(G_K\) on it. Among those representations are the so-called semi-stable representations, which are related to étale cohomology theory in algebraic geometry, and for any semi-stable \(p\)-adic representation, there is an associated algebraic object, namely its so-called \((\varphi, N)\)-module \(D\). The correspondence that associates to a representation \(V\) its filtred \((\varphi,N)\)-module \(D\) defines a tensor equivalence between the category of semi-stable \(p\)-adic representations of \(G_K\) and a full subcategory of the category of \((\varphi,N)\)-modules, which is the so-called category of admissible filtred \((\varphi,N)\)-modules. The second author of the present paper, J.-M. Fontaine, has studied this functor in some of his earlier works between 1979 and 1994. However, up to now, the entire theory did not appear to be complete, as no explicit description of the category of admissible filtred \((\varphi,N)\)-modules had been achieved so far.

In the paper under review, the authors prove a fact that had been conjectured in the past. Namely, one can define the category of weakly admissible filtred \((\varphi, N)\)-modules, which seems to be easier to come to grips with, and then the conjecture states that weak admissibility is indeed the same as ordinary admissibility.

The fact that this equivalence holds is a consequence from theorem A proved in the present paper. This principal theorem, which transpires a wealth of important corollaries, with a view to semi-stable \(p\)-adic representations of the Galois group \(G_K\), is proved by a fine and involved analysis of \((\varphi,N)\)-modules using the structure of the extension \(K/K_0\) and previous results of both authors.

The importance of the authors’ main result, theorem A, and its consequences becomes even more evident in the light of the recent fundamental results of J.-P.Wintenberger on \(p\)-adic representations and their relations to \(p\)-adic Hodge theory, which are based on the authors’ now fully established, crucial equivalence theorem. Wintenberger’s results are stated (and referred to) in the paper under review, which helps the reader to grasp the bearing of theorem A.

Also, the utilized earlier results of J.-M. Fontaine’s, which play a prominent rôle in the course of the proof of the central result of the present paper, are carefully restated and reviewed, which makes the entire exposition largely self-contained and easily readable.

In the paper under review, the authors prove a fact that had been conjectured in the past. Namely, one can define the category of weakly admissible filtred \((\varphi, N)\)-modules, which seems to be easier to come to grips with, and then the conjecture states that weak admissibility is indeed the same as ordinary admissibility.

The fact that this equivalence holds is a consequence from theorem A proved in the present paper. This principal theorem, which transpires a wealth of important corollaries, with a view to semi-stable \(p\)-adic representations of the Galois group \(G_K\), is proved by a fine and involved analysis of \((\varphi,N)\)-modules using the structure of the extension \(K/K_0\) and previous results of both authors.

The importance of the authors’ main result, theorem A, and its consequences becomes even more evident in the light of the recent fundamental results of J.-P.Wintenberger on \(p\)-adic representations and their relations to \(p\)-adic Hodge theory, which are based on the authors’ now fully established, crucial equivalence theorem. Wintenberger’s results are stated (and referred to) in the paper under review, which helps the reader to grasp the bearing of theorem A.

Also, the utilized earlier results of J.-M. Fontaine’s, which play a prominent rôle in the course of the proof of the central result of the present paper, are carefully restated and reviewed, which makes the entire exposition largely self-contained and easily readable.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14G20 | Local ground fields in algebraic geometry |

11S31 | Class field theory; \(p\)-adic formal groups |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11R32 | Galois theory |