zbMATH — the first resource for mathematics

Local models of Shimura varieties and a conjecture of Kottwitz. (English) Zbl 1294.14012
Invent. Math. 194, No. 1, 147-254 (2013); erratum ibid. 194, No. 1, 255 (2013).
This paper addresses a problem in arithmetic algebraic geometry. Local models are schemes over a discrete valuation ring, defined as closed subschemes of products of Grassmann schemes by “linear algebra conditions”. They are related to various topics in arithmetic geometry. One of them – maybe the most prominent one – is the theory of integral models of Shimura varieties. This connection, relying on Grothendieck-Messing theory, allows one to define such local models which étale-locally model the singularities of certain integral models of Shimura varieties of PEL-type with parahoric level structure. This theory was developed by de Jong, Deligne, Pappas, Rapoport, Zink and others.
For example, the computation of the trace of Frobenius on the sheaf of nearby cycles, which is required for the standard strategy of computing the local Hasse-Weil zeta function, can be carried out on the local model. In this way, T. Haines and B. C. Ngô [Compos. Math. 133, No. 2, 117–150 (2002; Zbl 1009.11042)] proved for \(\mathrm{GL}_n\) and \(\mathrm{GSp}_{2g}\) a conjecture of Kottwitz describing this trace as a central element in the corresponding parahoric Hecke algebras, partly following previous work of D. Gaitsgory [Invent. Math. 144, No. 2, 253–280 (2001; Zbl 1072.14055)] in the equal characteristic case.
Pappas and Zhu give a group-theoretic definition of “local models” in the context of an arbitrary reductive group. Having a unified approach which is independent of the theory of Shimura varieties is an important step forward in the theory.
Furthermore, they prove the conjecture of Kottwitz in this case, and show that their local model coincides with the one studied previously in most cases of PEL Shimura varieties. Conjecturally, this new construction of local models should also yield étale-local models of suitable integral models of Shimura varieties of Hodge type.
An essential step is the construction of a group scheme \(\mathcal G\) over the \(2\)-dimensional \(\mathbb A^1_{\mathcal O}=\mathrm{Spec} \mathcal O[u]\), where \(\mathcal O\) is a discrete valuation ring, whose base change to \(\text{Quot}(\mathcal O)[[u]]\), and to \((\mathcal O/\varpi)[u]\) respectively, gives Bruhat-Tits schemes for the underlying parahoric group over these discrete valuation rings. If the situation arises from a Shimura variety, then \(\mathcal O\) is of mixed characteristic, and one obtains a tool to “interpolate” between the mixed characteristic \(\mathcal O\) and the equal characteristic \((\mathcal O/\varpi)[[u]]\). It is then possible to define “local” and “global” variants of the affine Grassmannian, à la Beilinson and Drinfeld, and eventually to define the local model as the schematic closure of a certain Schubert cell in the generic fiber, the closure being taken inside a degeneration of a parahoric affine flag variety to the affine Grassmannian. While this idea is already in the work of Gaitsgory and of Haines and Ngô, in the situation at hand it is considerably more complicated to carry it through.
(The erratum only corrects the misprint of two names in the bibliography.)

14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
Full Text: DOI Link
[1] Anantharaman, S.: Schémas en Groupes, Espaces Homogènes et Espaces Algébriques sur Une Base de Dimension 1, sur les groupes algébriques, Soc. Math. France, vol. 33, pp. 5-79. Bull. Soc. Math. France, Paris (1973) · Zbl 0286.14001
[2] Arkhipov, S.; Bezrukavnikov, R., Perverse sheaves on affine flags and Langlands dual group, Isr. J. Math., 170, 135-183, (2009) · Zbl 1214.14011
[3] Artin, M., Versal deformations and algebraic stacks, Invent. Math., 27, 165-189, (1974) · Zbl 0317.14001
[4] Beauville, A., Laszlo, Y.: Un lemme de descente. C. R. Math. Acad. Sci. Paris Sér. I Math. 320(3), 335-340 (1995) · Zbl 0852.13005
[5] Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. Preprint. http://www.math.utexas.edu/users/benzvi/BD/hitchin.pdf · Zbl 0864.14007
[6] Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578. Springer, Berlin (1994) · Zbl 0808.14038
[7] Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Publ. Math. IHÉS 60, 197-376 (1984)
[8] Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. Fr. 112(2), 259-301 (1984) · Zbl 0565.14028
[9] Bruhat, F., Tits, J.: Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 34(3), 671-698 (1987) · Zbl 0657.20040
[10] Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires. Bull. Soc. Math. Fr. 115(2), 141-195 (1987) · Zbl 0636.20027
[11] Chernousov, V.; Gille, P.; Pianzola, A., Torsors over the punctured affine line, Am. J. Math., 134, 1541-1583, (2012) · Zbl 1279.14059
[12] Colliot-Thélène, J.-L.: Résolutions flasques des groupes linéaires connexes. J. Reine Angew. Math. 618, 77-133 (2008) · Zbl 1158.14021
[13] Conrad, B.: Reductive group schemes. Notes for the SGA 3 Summer School, Luminy 2011. http://math.stanford.edu/ conrad/papers/luminysga3.pdf
[14] Deligne, P., Travaux de Shimura, No. 244, 123-165, (1971), Berlin
[15] Edixhoven, B., Néron models and tame ramification, Compos. Math., 81, 291-306, (1992) · Zbl 0759.14033
[16] Faltings, G., Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc., 5, 41-68, (2003) · Zbl 1020.14002
[17] Gaitsgory, D., Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math., 144, 253-280, (2001) · Zbl 1072.14055
[18] Gille, P.: Torseurs sur la droite affine. Transform. Groups 7(3), 231-245 (2002) · Zbl 1062.14061
[19] Görtz, U., On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann., 321, 689-727, (2001) · Zbl 1073.14526
[20] Görtz, U., On the flatness of local models for the symplectic group, Adv. Math., 176, 89-115, (2003) · Zbl 1051.14027
[21] Görtz, U.; Haines, T., The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties, J. Reine Angew. Math., 609, 161-213, (2007) · Zbl 1157.14013
[22] Haines, T., Test functions for Shimura varieties: the Drinfeld case, Duke Math. J., 106, 19-40, (2001) · Zbl 1014.20002
[23] Haines, T., Introduction to Shimura varieties with bad reduction of parahoric type, No. 4, 583-642, (2005), Providence · Zbl 1148.11028
[24] Haines, T., The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J., 149, 569-643, (2009) · Zbl 1194.22019
[25] Haines, T.; Ngô, B.C., Nearby cycles for local models of some Shimura varieties, Compos. Math., 133, 117-150, (2002) · Zbl 1009.11042
[26] Haines, T., Rapoport, M.: On parahoric subgroups (2008). Appendix to [49] · Zbl 1385.11041
[27] Haines, T.; Rostami, S., The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory, 14, 264-284, (2010) · Zbl 1251.22013
[28] He, X.: Normality and Cohen-Macaulayness of local models of Shimura varieties. Preprint (2012). arXiv:1202.4119 · Zbl 1327.14121
[29] Illusie, L., Autour du théorème de monodromie locale, Bures-sur-Yvette, 1988 · Zbl 0837.14013
[30] Jacobson, N., A note on Hermitian forms, Bull. Am. Math. Soc., 46, 264-268, (1940) · Zbl 0024.24503
[31] Kisin, M., Pappas, G.: Integral models for Shimura varieties with parahoric level structure, in preparation · Zbl 1470.14049
[32] Kneser, M.: Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II. Math. Z. 89, 250-272 (1965)
[33] Knutson, D.: Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Springer, Berlin (1971) · Zbl 0221.14001
[34] Kottwitz, R., Shimura varieties and twisted orbital integrals, Math. Ann., 269, 287-300, (1984) · Zbl 0533.14009
[35] Kottwitz, R., Stable trace formula: cuspidal tempered terms, Duke Math. J., 51, 611-650, (1984) · Zbl 0576.22020
[36] Kottwitz, R., Points on some Shimura varieties over finite fields, J. Am. Math. Soc., 5, 373-444, (1992) · Zbl 0796.14014
[37] Kottwitz, R., Isocrystals with additional structure. II, Compos. Math., 109, 255-339, (1997) · Zbl 0966.20022
[38] Kottwitz, R.; Rapoport, M., Minuscule alcoves for \({\rm GL}_{n}\) and \(G{\rm Sp}_{2n}\), Manuscr. Math., 102, 403-428, (2000) · Zbl 0981.17003
[39] Landvogt, E.: A Compactification of the Bruhat-Tits Building. Lecture Notes in Mathematics, vol. 1619. Springer, Berlin (1996) · Zbl 0935.20034
[40] Landvogt, E., Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., 518, 213-241, (2000) · Zbl 0937.20026
[41] Larsen, M., Maximality of Galois actions for compatible systems, Duke Math. J., 80, 601-630, (1995) · Zbl 0912.11026
[42] Laszlo, Y.; Sorger, C., The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections, Ann. Sci. Éc. Norm. Super. (4), 30, 499-525, (1997) · Zbl 0918.14004
[43] Laumon, G., Vanishing cycles over a base of dimension ≥1, Tokyo/Kyoto, 1982, Berlin · Zbl 0567.14003
[44] Lusztig, G., Singularities, character formulas, and a \(q\)-analog of weight multiplicities, Luminy, 1981, Paris · Zbl 0561.22013
[45] Mirković, I.; Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. Math. (2), 166, 95-143, (2007) · Zbl 1138.22013
[46] Pappas, G., On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebr. Geom., 9, 577-605, (2000) · Zbl 0978.14023
[47] Pappas, G.; Rapoport, M., Local models in the ramified case. I. the EL-case, J. Algebr. Geom., 12, 107-145, (2003) · Zbl 1063.14029
[48] Pappas, G.; Rapoport, M., Local models in the ramified case. II. splitting models, Duke Math. J., 127, 193-250, (2005) · Zbl 1126.14028
[49] Pappas, G.; Rapoport, M., Twisted loop groups and their affine flag varieties, Adv. Math., 219, 118-198, (2008) · Zbl 1159.22010
[50] Haines, T.; Rapoport, M., Local models in the ramified case. III. unitary groups, J. Inst. Math. Jussieu, 8, 507-564, (2009) · Zbl 1185.14018
[51] Gille, P., Rapoport, M., Smithling, B.: Local models of Shimura varieties, I. Geometry and combinatorics. Handbook of Moduli (to appear). arXiv:1011.5551 · Zbl 1157.14013
[52] Philippe, G., Pianzola, A.: Torsors, reductive group schemes and extended affine Lie algebras. Preprint. arXiv:1109.3405 (to appear, Mem. AMS) · Zbl 1354.17017
[53] Prasad, G.; Yu, J.-K., On finite group actions on reductive groups and buildings, Invent. Math., 147, 545-560, (2002) · Zbl 1020.22003
[54] Rapoport, M., A guide to the reduction modulo \(p\) of Shimura varieties, Astérisque, 298, 271-318, (2005) · Zbl 1084.11029
[55] Rapoport, M., Zink, Th.: Period Spaces for \(p\)-Divisible Groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996) · Zbl 0873.14039
[56] Raynaud, M.: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics, vol. 119. Springer, Berlin (1970)
[57] Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1-89 (1971) · Zbl 0227.14010
[58] Richarz, T., Zhu, X.: Appendix to [76] · Zbl 0561.22013
[59] Rostami, S.: Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties. Preprint. arXiv:1112.0074
[60] Scholze, P., The Langlands-Kottwitz method and deformation spaces of \(p\)-divisible groups, J. Am. Math. Soc., 26, 227-259, (2013) · Zbl 1383.11082
[61] Scholze, P.; Shin, S.W., On the cohomology of compact unitary group Shimura varieties at ramified split places, J. Am. Math. Soc., 26, 261-294, (2013) · Zbl 1385.11041
[62] Serre, J.-P.: Groupes de Grothendieck des schémas en groupes réductifs déployés. Publ. Math. Inst. Hautes Études Sci. 34, 37-52 (1968) · Zbl 0195.50802
[63] Serre, J.-P.: Galois Cohomology, 5th edn. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1994)
[64] Seshadri, C.S., Triviality of vector bundles over the affine space \(K\)\^{}{2}, Proc. Natl. Acad. Sci. USA, 44, 456-458, (1958) · Zbl 0081.26603
[65] SGA3: Schémas en groupes. III: In: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, vol. 153. Springer, Berlin (1962/1964)
[66] SGA7I, Groupes de monodromie en géométrie algébrique. I, No. 288, (1972), Berlin
[67] SGA7II, Groupes de monodromie en géométrie algébrique. II, No. 340, (1973), Berlin
[68] Smithling, B., Topological flatness of orthogonal local models in the split, even case. I, Math. Ann., 350, 381-416, (2011) · Zbl 1232.14015
[69] Springer, T.A.: Linear Algebraic Groups, 2nd edn. Modern Birkhäuser Classics. Birkhäuser, Boston (2009) · Zbl 1202.20048
[70] Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of the Am. Math. Soc., vol. 80. Am. Math. Soc., Providence (1968) · Zbl 0164.02902
[71] Thomason, R.W., Equivariant resolution, linearization, and hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., 65, 16-34, (1987) · Zbl 0624.14025
[72] Tits, J., Reductive groups over local fields, Oregon State Univ, Corvallis, Ore, 1977, Providence
[73] Tsukamoto, T., On the local theory of quaternionic anti-Hermitian forms, J. Math. Soc. Jpn., 13, 387-400, (1961) · Zbl 0201.37101
[74] Waterhouse, W.C.; Weisfeiler, B., One-dimensional affine group schemes, J. Algebra, 66, 550-568, (1980) · Zbl 0452.14013
[75] Yu, J.-K.: Smooth models associated to concave functions in Bruhat-Tits theory. Preprint (2002). http://www.math.purdue.edu/ jyu/prep/model.pdf
[76] Zhu, X.: The geometric Satake correspondence for ramified groups. Preprint. arXiv:1107.5762
[77] Zhu, X.: On the coherence conjecture of Pappas and Rapoport. Preprint. arXiv:1012.5979
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.