Construction of a Rapoport–Zink space for $\mathrm{GU}(1,1)$ in the ramified $2$-adic case

Daniel Kirch

Vol. 293 (2018), No. 2, 341–389

DOI: 10.2140/pjm.2018.293.341

Abstract

Let $F|{\mathbb{Q}}_2$ be a finite extension. In this paper, we construct an RZ-space ${\mathcal{N}}_E$ for split $GU\left(1,1\right)$ over a ramified quadratic extension $E|F$. For this, we first introduce the naive moduli problem ${\mathcal{N}}_E^{naive}$ and then define ${\mathcal{N}}_E\subseteq {\mathcal{N}}_E^{naive}$ as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism between ${\mathcal{N}}_E$ and the Drinfeld moduli problem, proving the $2$-adic analogue of a theorem of Kudla and Rapoport. The formulation of the straightening condition uses the existence of certain polarizations on the points of the moduli space ${\mathcal{N}}_E^{naive}$. We show the existence of these polarizations in a more general setting over any quadratic extension $E|F$, where $F|{\mathbb{Q}}_p$ is a finite extension for any prime $p$.

Keywords

Rapoport–Zink spaces, Shimura varieties, bad reduction, unitary group, exceptional isomorphism

Mathematical Subject Classification 2010

Primary: 11G18, 14G35

Milestones

Received: 24 May 2017

Accepted: 6 June 2017

Published: 23 November 2017