First covering of the Drinfel'd upper half-plane and Banach representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Abstract

For an odd prime $p$, we construct some admissible Banach representations of ${GL}_2\left({\mathbb{Q}}_p\right)$ that conjecturally should correspond to some $2$-dimensional tamely ramified, potentially Barsotti–Tate representations of $Gal\left(\overline{{\mathbb{Q}}_p}∕{\mathbb{Q}}_p\right)$ 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.