Construction of a Rapoport-Zink space for split GU(1, 1) in the ramified 2-adic case

Abstract

Let F be a finite extension over the field of 2-adic numbers. In this paper, we construct a Rapoport-Zink-space for the split unitary group in two variables over a ramified quadratic extension of F. For this, we first introduce a naive moduli problem and then define the correct Rapoport-Zink-space as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism to the Drinfeld moduli problem, proving the 2-adic analogue of a theorem of Kudla and Rapoport. We also give the definition of a local model as a flat projective scheme over the ring of integers of F, which, locally for the etale topology, models the singularities of the Rapoport-Zink-space. The formulation of the straightening condition uses the existence of certain polarizations on the points of the naive moduli space. We show the existence of these polarizations in a more general setting over any quadratic extension of F, where F is a finite extension of the field of p-adic numbers for any prime p.