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COMPACTIFICATIONS OF PEL-TYPE SHIMURA VARIETIES IN RAMIFIED CHARACTERISTICS

Published online by Cambridge University Press:  05 January 2016

KAI-WEN LAN*
Affiliation:
University of Minnesota, Minneapolis, MN 55455, USA; kwlan@math.umn.edu

Abstract

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We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2016

References

Arthur, J., Ellwood, D. and Kottwitz, R. (Eds.), Harmonic Analysis, the Trace Formula, and Shimura Varieties. Proceedings of the Clay Mathematics Institute 2003 Summer School, The Fields Institute, Toronto, Canada, June 2–27, 2003, Clay Mathematics Proceedings, 4 (American Mathematical Society, Providence, Rhode Island, 2005), Clay Mathematics Institute, Cambridge, MA.Google Scholar
Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth Compactification of Locally Symmetric Varieties, Lie Groups: History Frontiers and Applications, 4 (Mathematical Science Press, Brookline, MA, 1975).Google Scholar
Baily, W. L. Jr. and Borel, A., ‘Compactification of arithmetic quotients of bounded symmetric domains’, Ann. of Math. (2) 84(3) (1966), 442528.CrossRefGoogle Scholar
Bosch, S., Lütkebohmert, W. and Raybaud, M., Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21 (Springer, Berlin, Heidelberg, New York, 1990).CrossRefGoogle Scholar
Bost, J.-B., Boyer, P., Genestier, A., Lafforgue, L., Lysenko, S., Morel, S. and Ngô, B. C. (Eds.), De la Géometrie Algébrique aux Formes Automorphes (II): Une collection d’articles en l’honneur du soixantième Anniversaire de Gérard Laumon, Astérisque, 370 (Société Mathématique de France, Paris, 2015).Google Scholar
de Jong, A. J., ‘The moduli spaces of polarized abelian varieties’, Math. Ann. 295 (1993), 485503.CrossRefGoogle Scholar
Deligne, P. and Kuyk, W. (Eds.), Modular Functions of One Variable II, Lecture Notes in Mathematics, 349 (Springer, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
Deligne, P. and Mumford, D., ‘The irreducibility of the space of curves of given genus’, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75109.CrossRefGoogle Scholar
Deligne, P. and Rapoport, M., Les schémas de modules de courbes elliptiques, in Deligne and Kuyk [ 7 ], 143–316.CrossRefGoogle Scholar
Faltings, G. and Chai, C.-L., Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 22 (Springer, Berlin, Heidelberg, New York, 1990).CrossRefGoogle Scholar
Farkas, G. and Morrison, I. (Eds.), Handbook of Moduli: Volume III, Advanced Lectures in Mathematics, 24 (International Press, Somerville, MA, 2013), Higher Education Press, Beijing.Google Scholar
Görtz, U., ‘On the flatness of local models for the symplectic group’, Adv. Math. 176 (2003), 89115.CrossRefGoogle Scholar
Grothendieck, A. (Ed.), Revêtements Étales et Groupe Fondamental (SGA 1), Lecture Notes in Mathematics, 224 (Springer, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
Grothendieck, A. and Dieudonné, J., Eléments de Géométrie Algébrique, Publications Mathématiques de Institut des Hautes Etudes Scientifiques, 4, 8, 11, 17, 20, 24, 28, 32 (Institut des Hautes Etudes Scientifiques, Paris, 1960, 1961, 1963, 1964, 1965, 1966, 1967).Google Scholar
Grothendieck, A. and Dieudonné, J., Eléments de Géométrie Algébrique I: Le Langage des Schémas, Grundlehren der Mathematischen Wissenschaften, 166 (Springer, Berlin, Heidelberg, New York, 1971).Google Scholar
Haines, T. J., Introduction to Shimura varieties with bad reductions of parahoric type, in Arthur et al. [ 1 ], 583–658.Google Scholar
Hakim, M., Topos Annelés et Schémas Relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, 64 (Springer, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
Harris, M., ‘Functorial properties of toroidal compactifications of locally symmetric varieties’, Proc. Lond. Math. Soc. (3) 59 (1989), 122.CrossRefGoogle Scholar
Harris, M., Lan, K.-W., Taylor, R. and Thorne, J., On the rigid cohomology of certain Shimura varieties, Preprint, 2013.Google Scholar
Hartwig, P., Kottwitz–Rapoport and $p$ -rank strata in the reduction of Shimura varieties of PEL type, Preprint, 2012.Google Scholar
He, X., ‘Normality and Cohen–Macaulayness of local models of Shimura varieties’, Duke Math. J. 162(13) (2013), 25092523.CrossRefGoogle Scholar
Hochster, M., ‘Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes’, Ann. of Math. (2) 96(2) (1972), 318337.CrossRefGoogle Scholar
Kato, K., ‘Toric singularities’, Amer. J. Math. 116(5) (1994), 10731099.CrossRefGoogle Scholar
Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal Embeddings I, Lecture Notes in Mathematics, 339 (Springer, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
Kisin, M., ‘Integral models for Shimura varieties of abelian type’, J. Amer. Math. Soc. 23(4) (2010), 9671012.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5(2) (1992), 373444.CrossRefGoogle Scholar
Lan, K.-W., ‘Elevators for degenerations of PEL structures’, Math. Res. Lett. 18(5) (2011), 889907.CrossRefGoogle Scholar
Lan, K.-W., ‘Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties’, J. Reine Angew. Math. 664 (2012), 163228. errata available online at the author’s website.Google Scholar
Lan, K.-W., ‘Toroidal compactifications of PEL-type Kuga families’, Algebra Number Theory 6(5) (2012), 885966. errata available online at the author’s website.CrossRefGoogle Scholar
Lan, K.-W., Arithmetic Compactification of PEL-type Shimura Varieties, London Mathematical Society Monographs, 36 (Princeton University Press, Princeton, 2013), errata and revision available online at the author’s website.Google Scholar
Lan, K.-W., Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci, Preprint, 2013.CrossRefGoogle Scholar
Langlands, R. P. and Ramakrishnan, D. (Eds.), The Zeta Functions of Picard Modular Surfaces: Based on Lectures Delivered at a CRM Workshop in the Spring of 1988, (Les Publications CRM, Montréal, 1992).Google Scholar
Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 6 (Oxford University Press, Oxford, 2002), translated by Reinie Erné.CrossRefGoogle Scholar
Madapusi Pera, K., Toroidal compactifications of integral models of Shimura varieties of Hodge type, Preprint, 2015.Google Scholar
Matsumura, H., Commutative Algebra, 2nd edn, Mathematics Lecture Note Series (The Benjamin/Cummings Publishing Company, Inc., 1980).Google Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8 (Cambridge University Press, Cambridge, New York, 1986).Google Scholar
Milne, J. S., The points on a Shimura variety modulo a prime of good reduction, in Langlands and Ramakrishnan [ 32 ], 151–253.Google Scholar
Moonen, B., ‘Models of Shimura varieties in mixed characteristic’, in Galois Representations in Arithmetic Algebraic Geometry, [ 53 ], 267–350.CrossRefGoogle Scholar
Moret-Bailly, L., Pinceaux de Variétés Abéliennes, Astérisque, 129 (Société Mathématique de France, Paris, 1985).Google Scholar
Moret-Bailly, L., ‘Un problème de descente’, Bull. Soc. Math. France 124 (1996), 559585.CrossRefGoogle Scholar
Mumford, D., Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5 (Oxford University Press, Oxford, 1970), with appendices by C. P. Ramanujam and Y. Manin.Google Scholar
Mumford, D., ‘An analytic construction of degenerating abelian varieties over complete rings’, Compos. Math. 24(3) (1972), 239272.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, 3rd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34 (Springer, Berlin, Heidelberg, New York, 1994).CrossRefGoogle Scholar
Ngô, B. C. and Genestier, A., ‘Alcôves et p-rang des variétés abéliennes’, Ann. Inst. Fourier. Grenoble 52(6) (2002), 16651680.CrossRefGoogle Scholar
Pappas, G. and Rapoport, M., ‘Local models in the ramified case, I. The EL-case’, J. Algebraic Geom. 12(1) (2003), 107145.CrossRefGoogle Scholar
Pappas, G. and Rapoport, M., ‘Local models in the ramified case, II. Splitting models’, Duke Math. J. 127(2) (2005), 193250.CrossRefGoogle Scholar
Pappas, G. and Zhu, X., ‘Local models of Shimura varieties and a conjecture of Kottwitz’, Invent. Math. 194 (2013), 147254.CrossRefGoogle Scholar
Pappas, G., Rapoport, M. and Smithling, B., ‘Local models of Shimura varieties, I. Geometry and combinatorics’, in Farkas and Morrison [ 11 ], 135–217.Google Scholar
Pink, R., ‘Arithmetic compactification of mixed Shimura varieties’, PhD Thesis, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1989.Google Scholar
Rapoport, M., ‘A guide to the reduction modulo $p$ of Shimura varieties’, in Tilouine et al. [ 58 ], 271–318.Google Scholar
Rapoport, M. and Zink, T., Period Spaces for p-Divisible Groups, Annals of Mathematics Studies, 141 (Princeton University Press, Princeton, 1996).CrossRefGoogle Scholar
Raynaud, M., Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, 119 (Springer, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
Scholl, A. J. and Taylor, R. L., Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Note Series, 254 (Cambridge University Press, Cambridge, New York, 1998).CrossRefGoogle Scholar
Stamm, H., ‘On the reduction of the Hilbert–Blumenthal-moduli scheme with Γ0(p)-level structure’, Forum Math. 9(4) (1997), 405455.CrossRefGoogle Scholar
Stroh, B., ‘Compactification de variétés de Siegel aux places de mauvaise réduction’, Bull. Soc. Math. France 138(2) (2010), 259315.CrossRefGoogle Scholar
Stroh, B., ‘Compactification minimale et mauvaise réduction’, Ann. Inst. Fourier. Grenoble 60(3) (2010), 10351055.CrossRefGoogle Scholar
Stroh, B., ‘Mauvaise réduction au bord’, in Bost et al. [ 5 ], 269–304.Google Scholar
Tilouine, J., Carayol, H., Harris, M. and Vignéras, M.-F. (Eds.), Formes automorphes (I): actes du Semestre du Centre Émile Borel, printemps 2000, Astérisque, 298 (Société Mathématique de France, Paris, 2005).Google Scholar
Vasiu, A., ‘Integral canonical models of Shimura varieties of preabelian type’, Asian J. Math. 3 (1999), 401518.CrossRefGoogle Scholar
Wedhorn, T., ‘Ordinariness in good reductions of Shimura varieties of PEL-type’, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 575618.CrossRefGoogle Scholar
Zarhin, Y. G., ‘Endomorphisms of abelian varieties and points of finite order in characteristic p ’, Math. Notes 21(6) (1977), 415419.CrossRefGoogle Scholar
Zhu, X., ‘On the coherence conjecture of Pappas and Rapoport’, Ann. of Math. (2) 180(1) (2014), 185.CrossRefGoogle Scholar