#### Vol. 293, No. 2, 2018

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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 $F|{ℚ}_{2}$ be a finite extension. In this paper, we construct an RZ-space ${\mathsc{N}}_{E}$ for split $GU\left(1,1\right)$ over a ramified quadratic extension $E|F$. For this, we first introduce the naive moduli problem ${\mathsc{N}}_{E}^{naive}$ and then define ${\mathsc{N}}_{E}\subseteq {\mathsc{N}}_{E}^{naive}$ as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism between ${\mathsc{N}}_{E}$ 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 ${\mathsc{N}}_{E}^{naive}$. We show the existence of these polarizations in a more general setting over any quadratic extension $E|F$, where $F|{ℚ}_{p}$ 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