Topological flatness of orthogonal local models in the split, even case. I.

*(English)*Zbl 1232.14015Local models are schemes which are defined in terms of linear algebra that were introduced by Rapoport and Zink to order to study local structures of integral models of certain Shimura varieties over \(p\)-adic fields. A basic requirement for a good integral model, or equivalently for the corresponding local model is flatness. In this paper the author studies the local model attached to a split even orthogonal group \(GO_{2n}\) with the Iwahori level structure.

It has been known due to G. Pappas [J. Algebr. Geom. 9, No. 3, 577–605 (2000; Zbl 0978.14023)] that the naive local model \(M^{\text{naive}}\) may not be flat in general and a modification is necessary. The desired local model \(M^{\text{loc}}\) may be defined to be the scheme-theoretic closure of the generic fiber in \(M^{\text{naive}}\); however one lacks the good description of this scheme as a functor. G. Pappas and M. Rapoport provided [J. Algebr. Geom. 12, No. 1, 107–145 (2003; Zbl 1063.14029); Duke Math. J. 127, No. 2, 193–250 (2005; Zbl 1126.14028); J. Inst. Math. Jussieu 8, No. 3, 507–564 (2009; Zbl 1185.14018)] various remedies for the non-flatness of the naive local model \(M^{\text{naive}}\). They defined a closed subscheme \(M^{\text{spin}}\subset M^{\text{naive}}\) by adding a so-called spin condition, and showed evidences of their conjecture: \(M^{\text{spin}}=M^{\text{loc}}\). In the paper under review the author shows that \(M^{\text{spin}}\) is topologically flat, that is, the underlying topological spaces of \(M^{\text{spin}}\) and \(M^{\text{loc}}\) are the same.

The method is the same as that in U. Görtz’s papers [Adv. Math. 176, No. 1, 89–115 (2003; Zbl 1051.14027); Math. Z. 250, No. 4, 775–790 (2005; Zbl 1085.14022)]. However, in the case of type \(D\), the closed fiber \(M^{\text{spin}}_k\) has two connected components that correspond to two dominant minuscule coweights \(\mu_1\) and \(\mu_2\) of \(GO_{2n}\). The author then carefully shows the equality of \(\mu\)-admissibility and \(\mu\)-permissibility for each \(\mu\in \{\mu_1,\mu_2\}\) which extends work of R. Kottwitz and M. Rapoport [Manuscr. Math. 102, No.4, 403–428 (2000; Zbl 0981.17003)]. Then it remains to treat a lifting problem for points in maximal cells. This paper is well-written.

It has been known due to G. Pappas [J. Algebr. Geom. 9, No. 3, 577–605 (2000; Zbl 0978.14023)] that the naive local model \(M^{\text{naive}}\) may not be flat in general and a modification is necessary. The desired local model \(M^{\text{loc}}\) may be defined to be the scheme-theoretic closure of the generic fiber in \(M^{\text{naive}}\); however one lacks the good description of this scheme as a functor. G. Pappas and M. Rapoport provided [J. Algebr. Geom. 12, No. 1, 107–145 (2003; Zbl 1063.14029); Duke Math. J. 127, No. 2, 193–250 (2005; Zbl 1126.14028); J. Inst. Math. Jussieu 8, No. 3, 507–564 (2009; Zbl 1185.14018)] various remedies for the non-flatness of the naive local model \(M^{\text{naive}}\). They defined a closed subscheme \(M^{\text{spin}}\subset M^{\text{naive}}\) by adding a so-called spin condition, and showed evidences of their conjecture: \(M^{\text{spin}}=M^{\text{loc}}\). In the paper under review the author shows that \(M^{\text{spin}}\) is topologically flat, that is, the underlying topological spaces of \(M^{\text{spin}}\) and \(M^{\text{loc}}\) are the same.

The method is the same as that in U. Görtz’s papers [Adv. Math. 176, No. 1, 89–115 (2003; Zbl 1051.14027); Math. Z. 250, No. 4, 775–790 (2005; Zbl 1085.14022)]. However, in the case of type \(D\), the closed fiber \(M^{\text{spin}}_k\) has two connected components that correspond to two dominant minuscule coweights \(\mu_1\) and \(\mu_2\) of \(GO_{2n}\). The author then carefully shows the equality of \(\mu\)-admissibility and \(\mu\)-permissibility for each \(\mu\in \{\mu_1,\mu_2\}\) which extends work of R. Kottwitz and M. Rapoport [Manuscr. Math. 102, No.4, 403–428 (2000; Zbl 0981.17003)]. Then it remains to treat a lifting problem for points in maximal cells. This paper is well-written.

Reviewer: Chia-Fu Yu (Taipei)

##### MSC:

14G35 | Modular and Shimura varieties |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

11G18 | Arithmetic aspects of modular and Shimura varieties |

17B20 | Simple, semisimple, reductive (super)algebras |

##### Citations:

Zbl 0978.14023; Zbl 1063.14029; Zbl 1126.14028; Zbl 1185.14018; Zbl 1051.14027; Zbl 1085.14022; Zbl 0981.17003
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\textit{B. D. Smithling}, Math. Ann. 350, No. 2, 381--416 (2011; Zbl 1232.14015)

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##### References:

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