Local models in the ramified case. III. Unitary groups.

*(English)*Zbl 1185.14018The 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.

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.

Reviewer: Ulrich Görtz (Essen)

##### MSC:

14G35 | Modular and Shimura varieties |

11G18 | Arithmetic aspects of modular and Shimura varieties |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

##### Citations:

Zbl 0873.14039; Zbl 0978.14023; Zbl 1063.14029; Zbl 1126.14028; Zbl 1159.22010; Zbl 1186.14026
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\textit{G. Pappas} and \textit{M. Rapoport}, J. Inst. Math. Jussieu 8, No. 3, 507--564 (2009; Zbl 1185.14018)

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