2020/07/10 by Stephen S. Kudla, Kudla, Stephen, Michael Rapoport +3
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2007.05211
openalex publication_date 2020/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove p-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces V with respect to an arbitrary CM field K with maximal totally real subfield F. For a place v|p of F that is not split in K and for which Vv is anisotropic, let ν be an extension of v to the reflex field E. We define an integral model of the corresponding Shimura curve over \rm Spec OE, (ν) by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to p. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The first two conditions are conditions on the Lie algebra of the abelian varieties; the last condition is a condition on the Riemann form of the polarization. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane for Fv. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal OFv-modules with a Rapoport-Zink space of p-divisible groups which arise from the moduli problem, where the OFv-action is usually not strict when Fv≠ \mathbb Qp. Our main tool is the theory of displays, in particular the Ahsendorf functor.