First covering of Drinfel'd upper half plane and Banach representations of GL₂(Q_p)

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.