Local models in the ramified case I. The EL-case

By Georgios Pappas, Michael Rapoport. 2000

78 citations recorded by the source corpus.

Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study in detail certain modifications of the local models for G=R_{E/F}GL(n), with E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We show that the new local models are smoothly equivalent to "rank varieties" of matrices, are flat, normal, with rational singularities and that their special fibers contain the expected Schubert strata. A corollary is that Schubert varieties in the affine Grassmannian are smoothly equivalent to nilpotent orbit closures and are normal with rational singularities, even in positive characteristics. We give some applications to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. Finally, in the general EL case, we replace the flatness conjecture of Rapoport-Zink with a conjecture about the modified local models.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9af31f08ff6a825993d779025fea0840f4ef4545

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In Geometry and Topology

1. E. Viehmann
2. Michael Rapoport
3. S. Kudla
4. Xuhua He
5. U. Hartl
6. Xinwen Zhu
7. Georgios Pappas
8. Peter Scholze

In Mathematical Physics

1. E. Viehmann
2. Michael Rapoport
3. S. Kudla
4. Xuhua He
5. U. Hartl
6. Xinwen Zhu
7. Georgios Pappas
8. Peter Scholze

Citations in the corpus

• Special cycles on unitary Shimura varieties I. Unramified local theory (2008, background): “We show that this length coincides with the derivative of a representation density for hermitian forms.”
• The Newton stratification on deformations of local G-shtukas (2008, background): “Since Gr is irreducible, reduced, normal, and Cohen–Macaulay (by [41], Theorem 6.”