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

Let F|2 be a finite extension. In this paper, we construct an RZ-space NE for split GU(1,1) over a ramified quadratic extension E|F. For this, we first introduce the naive moduli problem NEnaive and then define NE NEnaive as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism between NE 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 NEnaive. We show the existence of these polarizations in a more general setting over any quadratic extension E|F, where F|p is a finite extension for any prime p.

Rapoport–Zink spaces, Shimura varieties, bad reduction, unitary group, exceptional isomorphism
Mathematical Subject Classification 2010
Primary: 11G18, 14G35
Received: 24 May 2017
Accepted: 6 June 2017
Published: 23 November 2017
Daniel Kirch
Institut de Mathématiques de Jussieu - Paris Rive Gauche