Classification of two-dimensional split trianguline representations ofp-adic fields

Abstract

The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of Gal(Q p /Q p ) for p = 2 by using (ϕ, Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of Gal(K/K) using B-pairs as defined by Berger. Contents

Iwasawa theory of de Rham -modules over the Robba ring

“…For simplicity, we assume that Γ K has a topological generator γ K ; the general case can be proved similarly using the argument of § 2. [23] and by § 2.1 of [25], and we have the following commutative diagram…”

“…Next, we recall the definition of Galois cohomology of B-pairs; see ğ 2 of [25] and the appendix of [26] for details. For a continuous G K -module M, and for each q 0, we denote by…”

Triangulable 𝒪F-analytic (φq,Γ)-modules of rank 2

“…Our computations of dimensions of Ext 1 an match those of Kohlhaase on extensions of locally analytic representations [19]. Nakamura [22] gave a generalization of Colmez's work in another direction. But we think that Nakamura's point of view is probably not the best one for applications to the p-adic local Langlands correspondence.…”

“…Later Nakamura [22] classified 2-dimensional trianguline representations of the Galois group of a p-adic local field that is finite over Q p , generalizing Colmez's work. In this section we classify triangulable O F -analytic (ϕ q , Γ)-modules of rank 2 following Colmez's method [9].…”

Représentations potentiellement triangulines de dimension 2

“…Ses constructions ont été reprises et généralisées par Nakamura dans [Nak09]. La définition de Colmez peut se faire en termes des « (ϕ, Γ)-modules sur l'anneau de Robba » de Fontaine et Kedlaya ou bien en termes de « B-paires ».…”