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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.

##### MSC:
 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 14M15 Grassmannians, Schubert varieties, flag manifolds
##### Keywords:
Shimura variety; unitary group; parahoric subgroup; local model
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