#### Vol. 11, No. 2, 2017

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

### Lue Pan

Vol. 11 (2017), No. 2, 405–503
##### Abstract

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

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Drinfel'd upper half-plane, $p$-adic local Langlands correspondence of $\mathrm{GL}_2(\mathbb{Q}_p)$