Abstract
Abstract In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n × n {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.
Highlights
Local models are defined over the ring of integers OE of a local field E
An essential desideratum is that the local model be flat over OE, which involves a set-theoretic condition called topological flatness and a scheme-theoretic condition about the location of embedded primes
In [MWY], we proved a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman about the equations defining affine Grassmannians for SLn
Summary
Shimura varieties serve as a bridge between arithmetic geometry and automorphic forms, and as such they play a central role in the Langlands program. Rapoport and Zink ([RZ96]) introduced the study of local models of Shimura varieties. These local models are intended to capture the behaviour that occurs when considering reduction mod p and often allow one to reduce arithmetic problems to questions about affine Grassmannians and flag varieties. These problems tend to be difficult but often tractible Topological flatness does hold in the cases where, in their notation, the prφqφ differ by at most 1 ([PR03, Proposition 3.2]). We mention results on topological flatness by Görtz and by Smithling ([Gör[05], Smi11b, Smi11a, Smi14])
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