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Local models in the ramified case. III. Unitary groups. (English) Zbl 1185.14018
The authors study the reduction at a prime \(p>2\) of Shimura varieties of PEL type with parahoric level structure at \(p\), where the underlying group \(G/\mathbb Q\) is a unitary similitude group for an imaginary quadratic number field \(K/\mathbb Q\) ramified at \(p\). Let \((r,s)\) be the signature of \(G_{\mathbb R}\), \(n=r+s\). An important goal is to find “good” models of the Shimura varieties over the ring of integers of \(E_w\), where \(E\) is the reflex field, and \(w\) is the prime lying over \(p\), and to analyze their étale-local structure.
In their book [Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)], M. Rapoport and T. Zink defined a candidate for such a model in the general framework of PEL data, and explained how to study its local properties in terms of a scheme defined in terms of Grassmannians. This scheme is nowadays called the “naive” local model \(M^{\text{naive}}_I\), because it turned out (see [G. Pappas, J. Algebr. Geom. 9, No. 3, 577–605 (2000; Zbl 0978.14023)] for examples) that \(M^{\text{naive}}_I\) is not in general flat over its base, a serious defect for many purposes. Here \(I\) is an index set which corresponds to the choice of a parahoric subgroup, determined by the level structure imposed at \(p\).
Later, several variants of the naive local model have been defined and investigated, see [G. Pappas and M. Rapoport, J. Algebr. Geom. 12, No. 1, 107–145 (2003; Zbl 1063.14029); Duke Math. J. 127, No. 2, 193–250 (2005; Zbl 1126.14028)]. One approach is to consider the flat closure \(M^{\text{loc}}_I\) of the generic fiber of \(M^{\text{naive}}_I\). This “local model” is flat by definition, and the task then is to describe its geometry, and to give a functorial description, if possible.
As in previously studied cases, an important tool consists of embedding the special fiber \(\overline{M}^{\text{naive}}\) of \(M^{\text{naive}}\), and in particular the special fiber \(\overline{M}^{\text{loc}}\) of \(M^{\text{loc}}\), into an affine flag variety over the residue class field. In the situation at hand, the natural setup is given by the twisted loop groups studied in [G. Pappas and M. Rapoport, Adv. Math. 219, No. 1, 118–198 (2008; Zbl 1159.22010)].
Let us discuss the main results of the paper at hand. Assuming the coherence conjecture (see below), it is shown that \(\bigcup_{w\in\text{Adm}(\mu)} S_w = \overline{M}^{\text{loc}}\) (schematically), where \(\text{Adm}(\mu)\) denotes the so-called admissible set, a finite subset of the extended affine Weyl group attached to the Shimura datum “at \(p\)”, and for each \(w\), \(S_w\) denotes the corresponding Schubert cell in the affine flag variety.
The above-mentioned coherence conjecture is a conjectural explicit formula for the dimension of the space of global sections on \(\bigcup_{w\in\text{Adm}(\mu)} S_w\) of the natural ample line bundle. See op. cit.
Furthermore, the following unconditional results are proved: If \(I = \{0\}\) and \(n\) is odd, or if \(I = \{m\}\) and \(n\) is even, then \(\overline{M}^{\text{loc}}\) is irreducible, normal, Frobenius-split, and has only rational singularities. In these two cases, the underlying parahoric subgroup is special in the sense of Bruhat-Tits theory. The same result in the only remaining case of a special parahoric has been proved in the meantime by K. Arzdorf [Mich. Math. J. 58, No. 3, 683–710 (2009; Zbl 1186.14026)].
In the case of Picard modular surfaces, i.e., if \(n=3\), the authors have similar, and more detailed results for all parahoric subgroups.
There remains the question of giving a modular description of \(M^{\text{loc}}\). To this end, the authors define a closed subscheme of \(M^{\text{naive}}\) which contains \(M^{\text{loc}}\) and is conjecturally equal to the latter. Some computational evidence supports this conjecture.

14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI arXiv
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