First covering of the drinfel’d upper half-plane and banach representations of GL₂.(ℚ_p)

For an odd prime p, we construct some admissible Banach representations of GL₂.(ℚ_p) that conjecturally should correspond to some 2-dimensional tamely ramified, potentially Barsotti-Tate representations of Gal(ℚ_p/ℚ_p) via the p-adic local Langlands correspondence. To achieve this, we generalize Breuil’s work in the semistable case and work on the first covering of the Drinfel;d upper half-plane. Our main tool is an explicit semistable model of the first covering.