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

Abstract

Let $F|{\mathbb{Q}}_2$ be a finite extension. In this paper, we construct an RZ-space ${\mathcal{N}}_E$ for split $\mathrm{GU}(1,1)$ over a ramified quadratic extension $E|F$. For this, we first introduce the naive moduli problem ${\mathcal{N}}_E^{\mathrm{naive}}$ and then define ${\mathcal{N}}_E\subseteq {\mathcal{N}}_E^{\mathrm{naive}}$ 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 ${\mathcal{N}}_E^{\mathrm{naive}}$. 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 $p$.

37 pages

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