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Local models in the ramified case. I: The EL-case. (English) Zbl 1063.14029
The subject of the paper under review is to investigate some simply-defined schemes ($$M=M(\Lambda, \text{\textbf{r}}$$, $$\widetilde M$$, $$N$$ etc.) in order to get information about more complicated objects, namely models (denoted by $$X$$) of Shimura varieties of endomorphism and level type over Spec $${\mathcal O}_E$$, where $$E$$ is the completion of the reflex field of $$X$$ at a finite prime. Particularly, information about the phenomenon of possible non-flatness of $$X$$ over Spec $${\mathcal O}_E$$ is obtained. An example of such $$X$$ is a Shimura variety associated to group $$G=\text{Res}_{E/F}\text{GL}(n)$$ where $$E/F$$ is a totally ramified extension of local fields.
$$M=M(\Lambda, \text{\textbf{r}})$$ is called a naive local model for the moduli problem corresponding to $$X$$. $$M$$ is closely related to $$X$$, namely, it contains some closed subschemes (local models) which are locally for the étale topology isomorphic to $$X$$.
Schemes $$M(\Lambda, \text{\textbf{r}})$$ are defined in terms of linear algebra. Namely, fix a totally ramified extension $$F/F_0$$ of local fields, a vector space $$V$$ over $$F$$, an $${\mathcal O}_F$$-lattice $$\Lambda \subset V$$ and a multidimension $$\text{\textbf{r}}=\{r_{\varphi}\}$$ (here the $$\varphi$$’s run over all inclusions of $$F/F_0$$ in $$\bar F_0$$) such that the reflex field of $$\text{\textbf{r}}$$ is $$E$$. $$M$$ represents a functor that (roughly speaking) associates to an extension $$S/F_0$$ the set of sublattices $$\mathcal F$$ of $$\Lambda \otimes_{F_0}S$$ such that the multidimension of $$\mathcal F$$ is $$\text{\textbf{r}}$$. Similarly, $$\widetilde M$$ represents a functor associating to $$S$$ the pair ($$\mathcal F$$, a base of $$\mathcal F$$), and $$N(S)$$ is a set of some $$r\times r$$-matrices over $$S$$ ($$r=\sum_{\varphi}r_{\varphi}$$). There is a diagram $$M\overset{\pi}{\leftarrow} \widetilde M \overset{\phi}{\to}N$$. Further, let $$M^{\text{loc}}$$ be the closure of $$M\otimes_{{\mathcal O}_E}E$$ in $$M$$. The following theorems are proved:
Theorem A. The morphism $$\phi$$ is smooth.
Theorem B. $$M^{\text{loc}}$$ is normal, Cohen-Macaulay, its special fiber is reduced, normal with rational singularities.
The existence of an inclusion of the special fiber of $$M$$ to the affine Grassmannian permits to prove a theorem on its Schubert subvariety. The description of the resolution of the singularities of $$N$$ over the Galois closure of $$F/F_0$$ permits to calculate the complex of nearby cycles on $$M$$ (together with the Gal($$F_0$$)-action) in terms of the constant sheaves on some varieties related to $$M$$; the statement of this result is too large to be given here.
The final result of the discussion about relations between $$X$$ and $$M$$ is the following: while $$X$$ is not always flat over $${\mathcal O}_E$$, it is conjectured that $$M^{\text{loc}}$$ is always flat over $${\mathcal O}_E$$.

##### MSC:
 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties
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##### References:
 [1] A. Beauville-Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385-419. · Zbl 0815.14015 [2] A. Beilinson-J. Bernstein-P. Deligne: Faisceaux pervers, Astérisque 100 (1982), 3-171. [3] W. Borho-R. MacPherson: Partial resolutions of nilpotent varieties, Astérisque 101-102 (1983), 23-74. [4] A. Braverman-D. Gaitsgory: On Ginzburg’s Lagrangian construction of representations of $$GL_n$$, Math. Res. Lett. 6 (1999), 195-201. · Zbl 0971.22010 [5] C. De Concini-C. Procesi: Symmetric Functions, Conjugacy Classes and the Flag Variety, Invent. Math. 64 (1981), 203-219. · Zbl 0475.14041 [6] P. Deligne: Théorèmes de finitude en cohomologie $$\ell$$-adique, in SGA $$4\frac{1}{2}$$, 233-261, SLN 569, Springer-Verlag 1977. [7] P. Deligne-G. Pappas: Singularités des espaces de modules de Hilbert, en les caractéristiques divisants le discriminant, Compositio Math. 90 (1994), 59-79. [8] D. Eisenbud-D. Saltman: Rank varieties of matrices, Commutative Algebra, MSRI Publications 15 (1989), Hochster, Huneke, Sally ed., Springer Verlag. [9] G. Faltings: Algebraic loop groups and moduli spaces of bundles, preprint Max-Planck-Institut, Bonn 2000, 29 p. [10] E. Frenkel-D. Gaitsgory-D. Kazhdan-K. Vilonen: Geometric realization of Whittaker functions and the Langlands conjecture, J. Amer. Math. Soc. 11 (1998), 451-484. · Zbl 1068.11501 [11] V. Ginzburg: Perverse sheaves on a loop group and Langlands duality, alg-geom/9511007. [12] U. Görtz: On the flatness of models of certain Shimura varieties of PEL type, Math. Ann. 321 (2001), 689-727. · Zbl 1073.14526 [13] U. Görtz: On the flatness of local models for the symplectic group, preprint Köln 2000. [14] T. Haines-B. C. Ngô: Nearby cycles for local models of some Shimura varieties, preprint 1999. [15] R. Hotta-T. Springer: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), 113-127. · Zbl 0389.20037 [16] L. Illusie: Autour du théorème de monodromie locale, Astérisque 223 (1994), 9-58. [17] R. Kottwitz: Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444. · Zbl 0796.14014 [18] R. Kottwitz-M. Rapoport: Minuscule alcoves for $$GL_n$$and $$GSp_{2n}$$, Manuscripta Math. 102 (2000), 403-428. [19] G. Laumon-L. Moret-Bailly. Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 39. Springer-Verlag Berlin Heidelberg 2000. [20] P. Littelmann: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. Journ. AMS 11 3 (1999) pp. 551-567. · Zbl 0915.20022 [21] G. Lusztig: Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178. · Zbl 0473.20029 [22] H. Matsumura: Commutative Algebra. Benjamin/Cummings Publishing Co., Reading, Mass. 1980. · Zbl 0441.13001 [23] O. Mathieu: Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159-160 (1988). [24] V.B. Mehta-W. van der Kallen: A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, Compositio Math. 84 (1992), 211-221. · Zbl 0770.17009 [25] I. Macdonald: Symmetric functions and Hall polynomials, $$2^{\text\textrm{nd}}$$ ed., Oxford 1995. · Zbl 0824.05059 [26] I. Mirkovic-K. Vilonen: Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13-24. [27] B.C. Ngô-P. Polo: Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Alg. Geom. 10 (2001), 514-547. [28] G. Pappas: Local structure of arithmetic moduli for PEL Shimura varieties, J. Alg. Geom. 9 (2000), 577-605. · Zbl 0978.14023 [29] M. Rapoport-Th. Zink: Period spaces for $$p$$-divisible groups. Ann. of Math. Studies, vol. 141, Princeton University Press 1996. · Zbl 0873.14039 [30] N. Spaltenstein: The fixed point set of a unipotent transformation on the flag manifold, Proc. Kon. Ak. v. Wet. 79 (5) (1976), 452-456. · Zbl 0343.20029 [31] E. Strickland: On the variety of projectors, J. Algebra 106 (1987), 135-147. · Zbl 0612.14001 [32] J. Weyman: Two results on equations of nilpotent orbits, J. Alg. Geom., to appear. · Zbl 1009.20052
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