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 $p$.