On the flatness of local models for the symplectic group

By Ulrich Goertz. 2000

80 citations recorded by the source corpus.

Abstract We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p , with parahoric level structure at p . We show that this model is flat, as conjectured by Rapoport and Zink (Ann. of Math. Stud. 141 (1996)), and that its special fibre is reduced. A crucial ingredient is Faltings’ theorem on the normality of Schubert varieties in the affine flag variety.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63afb0a3360edceef0c9c2e034dc4be02dd36957

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In Geometry and Topology

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Citations in the corpus

Local models of Shimura varieties, I. Geometry and combinatorics (2010, methodology): “Then in (i), we have Admf ({µ}) = Permf ({µ}) for any {µ} and any f when R involves only type A; this was proved in the case of minuscule {µ} in [KR, 9.6], and the general case is an immediate consequence of Görtz’s result [Gö4, Cor. 9] (which itself makes crucial use of the cited result in the Iw”
Stratifications in the reduction of Shimura varieties (2015, background): “2 (existence of local models): The local model M KJ coincides in this case with the naive local model defined in [7].”
Integral models of Shimura varieties with parahoric level structure (2015, result): “The corresponding local model M GSp := M loc GSp(V ),{μ0},z for the group GSp(V ), its standard minuscule coweight μ0 and the periodic self dual lattice chain Λ • z that corresponds to z was considered by Görtz in [30]; in this case, this agrees with the corresponding local model of [56] as explaine”
Towards a theory of local Shimura varieties (2014, background): “Something similar holds in the PEL case when B = F is an unramified extension of Qp [25], cf.”
Introduction to Shimura Varieties with Bad Reduction of Parahoric Type (2004, methodology): “We refer to the work of U. G¨ortz for many more complicated calculations of this kind: [Go 1], [Go2], [Go3], [Go4].”
Local models in the ramified case. III Unitary groups (2007, background): “Görtz [10, 11] has shown in this case that the models defined in [29] are indeed flat with reduced special fibre and with irreducible components which are normal and with rational singularities.”