Repr\'esentations potentiellement triangulines de dimension 2

TL;DR

This paper characterizes 2-dimensional potentially trianguline representations of G_Qp, showing they must satisfy specific structural properties, and demonstrates the existence of such representations that are not potentially trianguline.

Contribution

It provides a classification of 2-dimensional potentially trianguline G_Qp-representations and constructs examples that are not potentially trianguline.

Findings

Potentially trianguline representations satisfy one of three properties.

Existence of 2-dimensional G_Qp-representations that are not potentially trianguline.

Classification results for 2-dimensional G_Qp-representations.

Abstract

The two main results of this note are on the one hand that if V is a 2-dimensional potentially trianguline representation of G_Qp then V satisfies at least one of the following properties (1) V is split trianguline (2) V is a direct sum of characters or an induced representation (3) V is a twist of a de Rham representation, and on the other hand that there exists some 2-dimensional representations of G_Qp which are not potentially trianguline.