Local models in the ramified case, II: Splitting models

Abstract

We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" p-adic integral models of these Shimura varieties and study theiŕ etale local structure. In particular, we exhibit a stratification of their (singular) special fibers and give a partial calculation of the sheaf of nearby cycles.

On the coherence conjecture of Pappas and Rapoport

“…cit. (using the results of [Gör01], [Gör03], [PR05]) that the coherence conjecture holds for µ a sum of minuscule coweights. In what follows, we mainly discuss the ramified unitary groups.…”

“…Let us remark that if G is split of type A or C, Theorem 1 is proved in [PR08], using the previous results on the local models of Shimura varieties (cf. [Gör01], [Gör03], [PR05]). However, it seems that Theorem 2 is new even for symplectic groups.…”

“…Therefore, nowadays the (local) models defined in [RZ96] are usually called the naive models. In a series of papers ([PR03], [PR05], [PR09]), Pappas and Rapoport investigated the corrected definition of flat local models. The easiest definition of these local models is by taking the flat closures of the generic fibers in the naive local models.…”

“…Usually, an integral model defined in this way is not useful since the moduli interpretation is lost and therefore it is very difficult to study the special fiber, etc. (In fact, a considerable part [PR03], [PR05], [PR09] is devoted in an attempt to cutting out the correct closed subschemes inside the naive models by strengthening the original moduli problem of [RZ96].) Indeed, most investigations of local models so far used these strengthened moduli problems in a way or another.…”

Unramifiedness of Galois Representations Arising From Hilbert Modular Surfaces

“…We work exclusively with the Pappas-Rapoport splitting model, as indicated in Notation 2.7. The general construction of Iwahori level structure on the splitting model is already explained in [PR05] (in a much more general setup). We here make it more explicit in our particular setup.…”