First covering of Drinfel'd upper half plane and Banach representations of GL_2(Q_p)

TL;DR

This paper constructs admissible Banach representations of GL_2(Q_p) linked to certain Galois representations, extending Breuil's work through explicit models of the Drinfel'd upper half plane's first covering.

Contribution

It generalizes Breuil's semi-stable case work by constructing explicit semi-stable models of the first covering of Drinfel'd upper half plane for p-adic representation theory.

Findings

Constructed admissible Banach representations of GL_2(Q_p).

Linked these representations to tamely ramified Galois representations.

Extended the semi-stable case to more general settings.

Abstract

We construct some admissible Banach representations of GL_2(Q_p) that conjecturally should correspond to some 2-dimensional tamely ramified, potentially Barsotti-Tate representations of G_{Q_p} via the p-adic local Langlands correspondence. To achieve this, we generalize Breuil's work in the semi-stable case and work on the first covering of Drinfel'd upper half plane. Our main tool is an explicit semi-stable model of the first covering.