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Topological flatness of local models in the ramified case. (English) Zbl 1085.14022
The author proves the topological flatness of the local model $${\mathbf M}^{\text{loc}}$$ associated to $$\text{Res}_{F/{{\mathbb Q}}_p}{\text{GL}}_n$$ and $$\text{Res}_{F/{\mathbb Q}_p} \text{GSp}_{2n}$$, where $$F$$ is any finite extension of $${\mathbb Q}_p$$. That is, the maximal point of each irreducible component of the special fiber of $${\mathbf M}^{\text{loc}}$$ can be lifted to the generic fiber. In the linear case, the author further proves that $${\mathbf M}^{\text{loc}}$$ is flat over $${\mathbb Z}_p$$, using the method of Frobenius splitting of Schubert varieties in the affine flag variety [U. Görtz, Math. Ann. 321, No. 3, 689–727 (2001; Zbl 1073.14526)].
The local model for the linear case and $$F$$ ramified needs to be modified, as the naive local model $${\mathbf M}^{\text{naive}}$$ defined by M. Rapoport and Th. Zink [Period spaces for $$p$$-divisible groups, Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] is not flat, not even topologically flat [G. Pappas, J. Algebr. Geom. 9, No. 3, 577–605 (2000; Zbl 0978.14023)]. G. Pappas and M. Rapoport [J. Algebr. Geom. 12, No. 1, 107–145 (2003; Zbl 1063.14029)] defined the local model $${\mathbf M}^{\text{loc}}$$ for the maximal parahoric level subgroups $$P_{\{i\}}$$ to be the scheme-theoretic closure of the generic fiber in $${\mathbf M}^{\text{naive}}_{P_{\{i\}}}$$. For general parahoric level subgroups $$P_I$$, the local model $${\mathbf M}^{\text{loc}}_{P_{I}}$$ is defined to be the intersection of ${\mathbf M}^{\text{naive}}_{P_{I}}\times_{{\mathbf M}^{\text{naive}}_{P_{\{i\}}}} {\mathbf M}^{\text{loc}}_{P_{\{i\}}},$ for all $$i\in I$$.
The essential ingredient is to show a combinatorial result which asserts the agreement of the $$\mu$$-admissible set and the $$\mu$$-permissible set in the cases of $$\text{GL}_n$$ and $$\text{GSp}_{2n}$$. This generalizes results of T. J. Haines and B. C. Ngô [Am. J. Math. 124, 1125–1152 (2002; Zbl 1047.20037)] where the case of the Iwahoric level subgroup is proved. The author gives two proofs, using the combinatorial result in different ways. The first one replies on the construction the canonical model $${\mathbf M}^{\text{can}}$$ due to G. Pappas and M. Rapoport [Duke Math. J. 127, No. 2, 193–250 (2005; Zbl 1126.14028)]. An earlier result of the author [Görtz, loc. cit.] then shows the flatness of $${\mathbf M}^{\text{can}}$$. The combinatorial result then shows that the models $${\mathbf M}^{\text{can}}$$ and $${\mathbf M}^{\text{loc}}$$ coincide topologically. The second proof is more direct. It follows from the combinatorial result that the maximal strata correspond to the Weyl translations of $$\mu$$. Then the author constructs an element in the special fiber corresponding to each maximal stratum and constructs a lifting in the generic fiber.
In the symplectic case, the author defines the local model $${\mathbf M}^{\text{loc}}_{\text{GSP}}$$ as the intersection of $${\mathbf M}^{\text{loc}}_{\text{GL}}$$ and $${\mathbf M}^{\text{naive}}_{\text{GSP}}$$ in the flag scheme. Unlike the linear case, the naive local model and local model coincide topologically. Using the same methods in the second (direct) proof of the linear case, the author proves the topological flatness of $${\mathbf M}^{\text{loc}}_{\text{GSP}}$$.
The paper is well-written and friendly to read.

##### MSC:
 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties
##### Keywords:
local model; topological flatness
Full Text:
##### References:
 [1] Görtz, U.: On the flatness of local models for certain Shimura varieties of PEL-type. Math. Ann. 321, 689–727 (2001) · Zbl 1073.14526 [2] Görtz, U.: On the flatness of local models for the symplectic group. Adv. in Math. 176, 89–115 (2003) · Zbl 1051.14027 [3] Haines, T., Ngô, B. C.: Alcoves associated to special fibres of local models. Amer. J. Math. 124(6), 1125–1152 (2002) · Zbl 1047.20037 [4] Kottwitz, R., Rapoport, M.: Minuscule alcoves for GLn and GSp2. manuscripta math. 102, 403–428 (2000) · Zbl 0981.17003 [5] Matsumura, H.: Commutative ring theory. Cambridge University Press, 1986 · Zbl 0603.13001 [6] Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Alg. Geom. 9(3), 577–605 (2000) · Zbl 0978.14023 [7] Pappas, G., Rapoport, M.: Local models in the ramified case. I. The EL-case. J. Alg. Geom. 12, 107–145 (2003) · Zbl 1063.14029 [8] Pappas, G., Rapoport, M.: Local models in the ramified case. II. Splitting Models. math.AG/0205021, to appear in Duke Math. J. · Zbl 1126.14028 [9] Rapoport, M., Zink, T.: Period spaces for p-divisible groups. Ann. of Math. Studies 141, Princeton University Press, 1996 · Zbl 0873.14039
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