Classification of two dimensional split trianguline representations of $p$-adic fields

TL;DR

This paper extends the classification of two-dimensional split trianguline Galois representations from the rational case to general p-adic fields using B-pairs, broadening the understanding of p-adic Galois representations.

Contribution

It generalizes Colmez's classification of split trianguline representations from Q_p to arbitrary p-adic fields using B-pairs.

Findings

Classified two-dimensional split trianguline representations over general p-adic fields.

Extended the framework from Robba ring to B-pairs for broader applicability.

Provided a comprehensive classification that encompasses previous results as a special case.

Abstract

The aim of this paper 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 $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ by using $(\phi,\Gamma)$-modules over Robba ring. In this paper, we classify two dimensional split trianguline representations of $\mathrm{Gal}(\bar{K}/K)$ for general $p$-adic field $K$ by using $B$-pairs defined by Berger.