Multivariable (φ,Γ)(\varphi,\Gamma)-modules and locally analytic vectors

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic means that $\Gamma={\rm Gal}(K_\infty/K)$ where $K_\infty$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that any $(\varphi,\Gamma)$-module is overconvergent. Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\varphi,\Gamma)$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_\infty / K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely the case of a Lubin-Tate extension, most $(\varphi,\Gamma)$-modules fail to be overconvergent. In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $\Gamma$ inside some big modules defined using Fontaine's rings of periods. We show that, in the cyclotomic case, we recover the ususal overconvergent $(\varphi,\Gamma)$-modules. In the Lubin-Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\varphi,\Gamma)$-modules attached to $F$-analytic representations are overconvergent.Comment: v8: final version, to appear in the Duke Math Journal. (In v2, the monodromy conjecture from v1 has been proved. In v3 and then v4, the restriction on ramification has been completely removed. In v5 the introduction has been rewritten. In v6 and v7 there are some improvements

An introduction to the theory of p-adic representations

The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a starting point. It consists mostly of an expanded version of the notes for my two lectures at the "Dwork trimester" in June 2001.Comment: 44 pages, submitted to: Geometric Aspects of Dwork's Theory - A Volume in memory of Bernard Dwor

Bloch and Kato's exponential map: three explicit formulas

The purpose of this article is to give formulas for Bloch-Kato's exponential map and its dual for an absolutely crystalline p-adic representation V, in terms of the (phi,Gamma)-module associated to that representation. As a corollary of these computations, we can give a very simple (and slightly improved) description of Perrin-Riou's exponential map (which interpolates Bloch-Kato's exponentials for V(k)). This new description directly implies Perrin-Riou's reciprocity formula.Comment: 22 pages, in englis

Iterated extensions and relative Lubin-Tate groups

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be a sequence of elements of Q_p^alg such that P(u_{n+1}) = u_n for all n \geq 0. Let K_infty be the field generated over K by all the u_n. If K_infty / K is a Galois extension, then it is abelian, and our main result is that it is generated by the torsion points of a relative Lubin-Tate group (a generalization of the usual Lubin-Tate groups). The proof of this involves generalizing the construction of Coleman power series, constructing some p-adic periods in Fontaine's rings, and using local class field theory.Comment: 15 pages; v2: some edits for clarity, and in the title "iterate" has been changed to "iterated

Multivariable Lubin-Tate (\phi,\Gamma)-modules and filtered \phi-modules

We define some rings of power series in several variables, that are attached to a Lubin-Tate formal module. We then give some examples of (\phi,\Gamma)-modules over those rings. They are the global sections of some reflexive sheaves on the p-adic open unit polydisk, that are constructed from a filtered \phi-module using a modification process. We prove that we obtain every crystalline (\phi,\Gamma)-module over those rings in this way.Comment: This version corrects a mistake from v1: the module M^+(D) is reflexive, but I do not know whether or not it is projective in general. The main theorems and various definitions have been amended as a result. Some other less significant mistakes have been corrected as well. 22 page

Lifting the field of norms

Let K be a finite extension of Q_p. The field of norms of a p-adic Lie extension K_infty/K is a local field of characteristic p which comes equipped with an action of Gal(K_infty/K). When can we lift this action to characteristic 0, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of (phi,Gamma)-modules, and give a condition for the existence of certain types of lifts.Comment: v3: some improvements, to appear in "Journal de l'\'Ecole polytechnique - Math\'ematiques". v2: some minor improvement

Iwasawa theory and FF-analytic Lubin-Tate (φ,Γ)(\varphi,\Gamma)-modules

Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.Comment: v2: final version, to appear in Documenta Mathematica. 31 page

Construction of some families of 2-dimensional crystalline representations

We construct explicitly some analytic families of etale (phi,Gamma)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms.Comment: 13 pages, english and french abstract