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COMPACTIFICATIONS OF SUBSCHEMES OF INTEGRAL MODELS OF SHIMURA VARIETIES

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  24 September 2018

KAI-WEN LAN  and
BENOÎT STROH
KAI-WEN LAN
Affiliation:
University of Minnesota, Minneapolis, MN 55455, USA; [email protected]
BENOÎT STROH
Affiliation:
C.N.R.S. and Institut de Mathématiques de Jussieu–Paris Rive Gauche, 75252 Paris Cedex 05, France; [email protected]

Abstract

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We study several kinds of subschemes of mixed characteristic models of Shimura varieties which admit good (partial) toroidal and minimal compactifications, with familiar boundary stratifications and formal local structures, as if they were Shimura varieties in characteristic zero. We also generalize Koecher’s principle and the relative vanishing of subcanonical extensions for coherent sheaves, and Pink’s and Morel’s formulas for étale sheaves, to the context of such subschemes.

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11G15: Complex multiplication and moduli of abelian varieties 11F75: Cohomology of arithmetic groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e18
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s) 2018

References

Andreatta, F., Iovita, A. and Pilloni, V., ‘ p-adic families of Siegel modular cuspforms’, Ann. of Math. (2) 181(2) (2015), 623–697.Google Scholar
Artin, M., ‘Algebraic approximation of structures over complete local rings’, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.Google Scholar
Artin, M., Grothendieck, A. and Verdier, J.-L. (Eds.), Théorie des topos et cohomologie étale des schémas (SGA 4), Tome 3, Lecture Notes in Mathematics, 305 (Springer, Berlin, Heidelberg, New York, 1973).Google Scholar
Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth Compactification of Locally Symmetric Varieties, 2nd edn, Cambridge Mathematical Library (Cambridge University Press, Cambridge, New York, 2010).Google Scholar
Baily, W. L. Jr. and Borel, A., ‘Compactification of arithmetic quotients of bounded symmetric domains’, Ann. of Math. (2) 84(3) (1966), 442–528.Google Scholar
Beilinson, A., Bernstein, J., Deligne, P. and Gabber, O., Faisceaux Pervers, 2nd edn, Astérisque, 100 (Société Mathématique de France, Paris, 2018).Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics, 930 (Springer, Berlin, Heidelberg, New York, 1982).Google Scholar
Borel, A. and Casselman, W. (Eds.), ‘Automorphic forms, representations and L-functions’, inProceedings of Symposia in Pure Mathematics, Vol. 33, Part 2, held at Oregon State University, Corvallis, OR, July 11–August 5, 1977 (American Mathematical Society, Providence, Rhode Island, 1979).Google Scholar
Borel, A. and Ji, L., Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory and Applications (Birkhäuser, Boston, 2006).Google 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).Google 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
Boxer, G. A., ‘Torsion in the coherent cohomology of Shimura varieties and Galois representations’, PhD Thesis, Harvard University, Cambridge, Massachusetts, 2015.Google Scholar
Cartier, P., Illusie, L., Katz, N. M., Laumon, G., Manin, Y. and Ribet, K. A. (Eds.), The Grothendieck festschrift: A Collection of Articles Written in Honer of the 60th Birthday of Alexander Grothendieck, Vol. 2 (Birkhäuser, Boston, 1990).Google Scholar
Deligne, P., Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, in Borel and Casselman [ 8 ], 247–290.Google Scholar
Deligne, P., ‘La conjecture de Weil. II’, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.Google Scholar
Deligne, P. and Katz, N. (Eds.), Groupes de monodromie en géométrie algébrique (SGA 7 II), Lecture Notes in Mathematics, 340 (Springer, Berlin, Heidelberg, New York, 1973).Google Scholar
Ekedahl, T., On the Adic Formalism, in Cartier et al. [ 13 ], 197–218.Google Scholar
Faber, C., van der Geer, G. and Oort, F. (Eds.), Moduli of Abelian Varieties, Progress in Mathematics, 195 (Birkhäuser, Boston, 2001).Google 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).Google Scholar
Görtz, U. and Haines, T. J., ‘The Jordan–Hölder series for nearby cycles on some Shimura varieties and affine flag varieties’, J. Reine Angew. Math. 609 (2007), 161–213.Google Scholar
Grothendieck, A. and Dieudonné, J., Eléments de géométrie algébrique, Publications Mathématiques de l’I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32 (Institut des Hautes Etudes Scientifiques, Paris, 1960, 1961, 1961, 1963, 1964, 1965, 1966, 1967).Google Scholar
Hamacher, P., ‘The p-rank stratification on the Siegel moduli space with Iwahori level structure’, Manuscripta Math. 143(1–2) (2014), 51–80.Google Scholar
Hamacher, P., ‘The geometry of Newton strata in the reduction moduli p of Shimura varieties of PEL type’, Duke Math. J. 164(15) (2015), 2809–2895.Google Scholar
Harris, M., Lan, K.-W., Taylor, R. and Thorne, J., ‘On the rigid cohomology of certain Shimura varieties’, Res. Math. Sci. 3 (2016), article no. 37, 308 pp.Google Scholar
Hartwig, P., ‘Kottwitz–Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type’, Ann. Inst. Fourier (Grenoble) 65(3) (2015), 1031–1103.Google Scholar
He, X. and Rapoport, M., ‘Stratifications in the reduction of Shimura varieties’, Manuscripta Math. 152(3–4) (2018), 317–343.Google Scholar
Illusie, L., Autour du théorème de monodromie locale, Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (Société Mathématique de France, Paris, 1994), 9–57.Google Scholar
International Congress of Mathematicians, August 19–27, 2010, Hyderabad, India, Proceedings of the International Congress of Mathematicians, 2, Hindustan Book Agency, New Delhi; distributed by World Scientific, Singapore, 2010.Google Scholar
Kiehl, R. and Weissauer, R., Weil Conjectures, Perverse Sheaves, and l’adic Fourier Transform, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 42 (Springer, Berlin, Heidelberg, New York, 2001).Google Scholar
Koblitz, N., ‘ p-adic variation of the zeta-function over families of varieties defined over finite fields’, Compos. Math. 31(2) (1975), 119–218.Google Scholar
Kottwitz, R. E., ‘Isocrystals with additional structure’, Compos. Math. 56(2) (1985), 201–220.Google Scholar
Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5(2) (1992), 373–444.Google Scholar
Kottwitz, R. E., ‘Isocrystals with additional structure. II’, Compos. Math. 109 (1997), 255–339.Google Scholar
Lan, K.-W., ‘Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties’, J. Reine Angew. Math. 664 (2012), 163–228.Google Scholar
Lan, K.-W., ‘Toroidal compactifications of PEL-type Kuga families’, Algebra Number Theory 6(5) (2012), 885–966.Google 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., ‘Boundary strata of connected components in positive characteristics’, Algebra Number Theory 9(9) (2015), 2035–2054, an appendix to the article ‘Families of nearly ordinary Eisenstein series on unitary groups’ by Xin Wan.Google Scholar
Lan, K.-W., ‘Compactifications of PEL-type Shimura varieties in ramified characteristics’, Forum Math. Sigma 4 (2016), e1, 98 pp.Google Scholar
Lan, K.-W., ‘Higher Koecher’s principle’, Math. Res. Lett. 23(1) (2016), 163–199.Google Scholar
Lan, K.-W., ‘Vanishing theorems for coherent automorphic cohomology’, Res. Math. Sci. 3 (2016), article no. 39, 43 pp.Google Scholar
Lan, K.-W., ‘Integral models of toroidal compactifications with projective cone decompositions’, Int. Math. Res. Not. IMRN 2017(11) (2017), 3237–3280.Google Scholar
Lan, K.-W., Compactifications of PEL-type Shimura Varieties and Kuga Families with Ordinary Loci (World Scientific, Singapore, 2018).Google Scholar
Lan, K.-W., ‘Compactifications of splitting models of PEL-type Shimura varieties’, Trans. Amer. Math. Soc. 370(4) (2018), 2463–2515.Google Scholar
Lan, K.-W. and Stroh, B., ‘Relative cohomology of cuspidal forms on PEL-type Shimura varieties’, Algebra Number Theory 8(8) (2014), 1787–1799.Google Scholar
Lan, K.-W. and Stroh, B., ‘Nearby cycles of automorphic étale sheaves’, Compos. Math. 154(1) (2018), 80–119.Google Scholar
Lan, K.-W. and Stroh, B., Nearby Cycles of Automorphic étale Sheaves, II, Cohomology of Arithmetic Groups: On the Occasion of Joachim Schwermer’s 66th Birthday, Bonn, Germany, June 2016 (eds. J. Cogdell, G. Harder, S. Kudla and F. Shahidi), Springer Proceedings in Mathematics & Statistics, 245 (Springer International Publishing, 2018), 83–106.Google Scholar
Laumon, G., ‘Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil’, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 131–210.Google Scholar
Lee, D. U., ‘Non-emptiness of Newton strata of Shimura varieties of Hodge type’, Algebra Number Theory 12(2) (2018), 259–283.Google Scholar
Liu, Y. and Zheng, W., ‘Enhanced adic formalism and perverse $t$ -structures for higher Artin stacks’. Preprint, 2012.Google Scholar
Madapusi Pera, K., ‘Toroidal compactifications of integral models of Shimura varieties of Hodge type’. Preprint, 2018.Google Scholar
Mantovan, E., ‘On the cohomology of certain PEL-type Shimura varieties’, Duke Math. J. 129(3) (2005), 573–610.Google Scholar
Mantovan, E., ‘ ℓ-adic étale cohomology of PEL type Shimura varieties with non-trivial coefficients’, inWIN–Woman in Numbers, Fields Institute Communications, 60 (American Mathematical Society, Providence, Rhode Island, 2011), 61–83.Google Scholar
Mazur, B. and Messing, W., Universal Extensions and One Dimensional Crystalline Cohomology, Lecture Notes in Mathematics, 370 (Springer, Berlin, Heidelberg, New York, 1974).Google Scholar
Messing, W., The Crystals Associated to Barsotti–Tate Groups: With Applications to Abelian Schemes, Lecture Notes in Mathematics, 264 (Springer, Berlin, Heidelberg, New York, 1972).Google Scholar
Moonen, B., Group Schemes with Additional Structures and Weyl Group Cosets, in Faber et al. [ 18 ], 255–298.Google Scholar
Moonen, B., ‘A dimension formula for Ekedahl–Oort strata’, Ann. Inst. Fourier (Grenoble) 54(3) (2004), 666–698.Google Scholar
Moonen, B. and Wedhorn, T., ‘Discrete invariants of varieties in positive characteristic’, Int. Math. Res. Not. IMRN 2004(72) (2004), 3855–3903.Google Scholar
Morel, S., ‘Complexes d’intersection des compactifications de Baily–Borel. Le cas des groupes unitaires sur $\mathbb{Q}$ ’, PhD Thesis, Université Paris-Sud, Orsay, France, 2005.Google Scholar
Morel, S., ‘Complexes pondérés sur les compactifications de Baily–Borel: le cas des variétés de Siegel’, J. Amer. Math. Soc. 21(1) (2008), 23–61.Google Scholar
Morel, S., ‘The intersection complex as a weight truncation and an application to Shimura varieties’, inProceedings of the International Congress of Mathematicians [ 28 ], 312–334.Google Scholar
Morel, S., On the Cohomology of Certain Non-compact Shimura Varieties, Annals of Mathematics Studies, 173 (Princeton University Press, Princeton, 2010).Google Scholar
Morel, S., ‘Complexes mixtes sur un schéma de type fini sur $\mathbb{Q}$ ’. Preprint, 2012.Google 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 Yuri Manin.Google Scholar
Ngô, B. C. and Genestier, A., ‘Alcôves et p-rang des variétés abéliennes’, Ann. Inst. Fourier (Grenoble) 52(6) (2002), 1665–1680.Google Scholar
Oort, F., A Stratification of a Moduli Space of Abelian Varieties, in Faber et al. [ 18 ], 345–416.Google Scholar
Oort, F., ‘Foliations in moduli spaces of abelian varieties’, J. Amer. Math. Soc. 17(2) (2004), 267–296.Google Scholar
Pappas, G., ‘On the arithmetic moduli schemes of PEL Shimura varieties’, J. Algebraic Geom. 9(3) (2000), 577–605.Google Scholar
Pappas, G. and Rapoport, M., ‘Local models in the ramified case, II. Splitting models’, Duke Math. J. 127(2) (2005), 193–250.Google Scholar
Pappas, G. and Zhu, X., ‘Local models of Shimura varieties and a conjecture of Kottwitz’, Invent. Math. 194 (2013), 147–254.Google Scholar
Pilloni, V. and Stroh, B., ‘Cohomologie cohérente et représentations Galoisiennes’, Ann. Math. Qué. 40(1) (2016), 167–202.Google Scholar
Pink, R., ‘Arithmetic compactification of mixed Shimura varieties’, PhD Thesis, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1989.Google Scholar
Pink, R., ‘On ℓ-adic sheaves on Shimura varieties and their higher direct images in the Baily–Borel compactification’, Math. Ann. 292 (1992), 197–240.Google Scholar
Pink, R., Wedhorn, T. and Ziegler, P., ‘Algebraic zip data’, Doc. Math. 16 (2011), 253–300.Google Scholar
Rapoport, M. and Richartz, M., ‘On the classification and specialization of F-isocrystals with additional structure’, Compos. Math. 103(2) (1996), 153–181.Google Scholar
Rapoport, M. and Zink, T., Period Spaces for p-divisible Groups, Annals of Mathematics Studies, 141 (Princeton University Press, Princeton, 1996).Google Scholar
Springer, T. A., Linear Algebraic Groups, 2nd edn, Progress in Mathematics, 9 (Birkhäuser, Boston, 1998).Google Scholar
Stamm, H., ‘On the reduction of the Hilbert–Blumenthal-moduli scheme with 𝛤0(p)-level structure’, Forum Math. 9(4) (1997), 405–455.Google Scholar
Stroh, B., ‘Compactification de variétés de Siegel aux places de mauvaise réduction’, Bull. Soc. Math. France 138(2) (2010), 259–315.Google Scholar
Stroh, B., ‘Compactification minimale et mauvaise réduction’, Ann. Inst. Fourier (Grenoble) 60(3) (2010), 1035–1055.Google Scholar
Stroh, B., ‘Sur une conjecture de Kottwitz au bord’, Ann. Sci. Éc. Norm. Supér. (4) 45(1) (2012), 143–165.Google Scholar
Stroh, B., ‘Erratum à ‘sur une conjecture de Kottwitz au bord’’, Ann. Sci. Éc. Norm. Supér. (4) 46(6) (2013), 1023–1024.Google Scholar
Stroh, B., Mauvaise réduction au bord, in Bost et al. [ 11 ], 269–304.Google Scholar
Viehmann, E. and Wedhorn, T., ‘Ekedahl–Oort and Newton strata for Shimura varieties of PEL type’, Math. Ann. 356(4) (2013), 1493–1550.Google Scholar
Wedhorn, T., The Dimension of Oort Strata of Shimura Varieties of PEL-Type, in Faber et al. [ 18 ], 441–471.Google Scholar
Zhang, C., ‘Ekedahl–Oort strata for good reductions of Shimura varieties of Hodge type’, Canad. J. Math. 70(2) (2018), 451–480.Google Scholar