Pappas's corrected local model flatness conjecture for orthogonal groups

From papers

Let FF be a field with discriminant algebra A=FDA=F\oplus\mathcal D, let VV be the associated quadratic space of dimension 2n2n, and let MM be the corrected local model obtained from the naive local model after base change to the integral closure OA\mathcal O_A and imposing

n(FΛ)(nΛOA)+OAOS.\wedge^n(\mathcal F_\Lambda)\subset (\wedge^n\Lambda_{\mathcal O_A})_+\otimes_{\mathcal O_A}\mathcal O_S.

The scheme MM is regarded as an OF\mathcal O_F-scheme via its structural morphism.

Corrected local model flatness conjecture. The scheme MM is flat over

Spec(OF).\operatorname{Spec}(\mathcal O_F).

The correction is designed to repair the non-flatness of naive local models while preserving the generic fiber. The source proposes this flatness as a conjecture and gives no resolution in the supplied text.

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Sources & referencesView supporting material

Primary source

G. Pappas and M. Rapoport, “Local Models in the ramified case. III. Unitary groups”, arXiv:math/0702286 (2007).

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