Görtz's flatness conjecture for ramified symplectic local models

Let F/F0F/F_0 be totally ramified, and let L{\mathcal L} be a self-dual periodic OF{\mathcal O}_F-lattice chain. Let MG,{μ},LnaiveM^{\mathrm{naive}}_{G,\{\mu\},{\mathcal L}} denote the corresponding naive local model.

Görtz's flatness conjecture. For any self-dual periodic OF{\mathcal O}_F-lattice chain L{\mathcal L}, MG,{μ},LnaiveM^{\mathrm{naive}}_{G,\{\mu\},{\mathcal L}} is flat over

SpecOF0.\operatorname{Spec}{\mathcal O}_{F_0}.

Topological flatness is known in the stated setting, and the naive, vertexwise, and local models agree topologically. The conjecture concerns equality of the schemes themselves and remains open in the supplied source.

Sources & referencesView supporting material

Primary source

G. Pappas, M. Rapoport and B. Smithling, “Local models of Shimura varieties, I. Geometry and combinatorics”, arXiv:1011.5551 (2011).

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