First covering of the Drinfel’d upper half-plane and Banach representations of GL2(ℚp)

Pan, Lue

doi:10.2140/ant.2017.11.405

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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

DOI: 10.2140/ant.2017.11.405

Abstract

For an odd prime $p$ , 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 $Gal (\bar{ℚ_{p}} ∕ ℚ_{p})$ 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 11S37

Secondary: 22E50, 11F85, 11G25

Milestones

Received: 12 October 2015

Revised: 17 June 2016

Accepted: 18 November 2016

Published: 15 April 2017

Authors

Lue Pan
Department of Mathematics Princeton University Fine Hall Washington Road Princeton, NJ 08540 United States

Vol. 11, No. 2, 2017