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COMPACTIFICATIONS OF SUBSCHEMES OF INTEGRAL MODELS OF SHIMURA VARIETIES

  • KAI-WEN LAN (a1) and BENOÎT STROH (a2)
Abstract

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.

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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.
References
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[1] Andreatta, F., Iovita, A. and Pilloni, V., ‘ p-adic families of Siegel modular cuspforms’, Ann. of Math. (2) 181(2) (2015), 623697.
[2] Artin, M., ‘Algebraic approximation of structures over complete local rings’, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 2358.
[3] 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).
[4] 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).
[5] Baily, W. L. Jr. and Borel, A., ‘Compactification of arithmetic quotients of bounded symmetric domains’, Ann. of Math. (2) 84(3) (1966), 442528.
[6] Beilinson, A., Bernstein, J., Deligne, P. and Gabber, O., Faisceaux Pervers, 2nd edn, Astérisque, 100 (Société Mathématique de France, Paris, 2018).
[7] Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics, 930 (Springer, Berlin, Heidelberg, New York, 1982).
[8] 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).
[9] Borel, A. and Ji, L., Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory and Applications (Birkhäuser, Boston, 2006).
[10] Bosch, S., Lütkebohmert, W. and Raybaud, M., Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21 (Springer, Berlin, Heidelberg, New York, 1990).
[11] 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).
[12] Boxer, G. A., ‘Torsion in the coherent cohomology of Shimura varieties and Galois representations’, PhD Thesis, Harvard University, Cambridge, Massachusetts, 2015.
[13] 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).
[14] Deligne, P., Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, in Borel and Casselman [ 8 ], 247–290.
[15] Deligne, P., ‘La conjecture de Weil. II’, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.
[16] 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).
[17] Ekedahl, T., On the Adic Formalism, in Cartier et al. [ 13 ], 197–218.
[18] Faber, C., van der Geer, G. and Oort, F. (Eds.), Moduli of Abelian Varieties, Progress in Mathematics, 195 (Birkhäuser, Boston, 2001).
[19] 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).
[20] 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), 161213.
[21] 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).
[22] Hamacher, P., ‘The p-rank stratification on the Siegel moduli space with Iwahori level structure’, Manuscripta Math. 143(1–2) (2014), 5180.
[23] Hamacher, P., ‘The geometry of Newton strata in the reduction moduli p of Shimura varieties of PEL type’, Duke Math. J. 164(15) (2015), 28092895.
[24] 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.
[25] Hartwig, P., ‘Kottwitz–Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type’, Ann. Inst. Fourier (Grenoble) 65(3) (2015), 10311103.
[26] He, X. and Rapoport, M., ‘Stratifications in the reduction of Shimura varieties’, Manuscripta Math. 152(3–4) (2018), 317343.
[27] Illusie, L., Autour du théorème de monodromie locale, Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (Société Mathématique de France, Paris, 1994), 957.
[28]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.
[29] 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).
[30] Koblitz, N., ‘ p-adic variation of the zeta-function over families of varieties defined over finite fields’, Compos. Math. 31(2) (1975), 119218.
[31] Kottwitz, R. E., ‘Isocrystals with additional structure’, Compos. Math. 56(2) (1985), 201220.
[32] Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5(2) (1992), 373444.
[33] Kottwitz, R. E., ‘Isocrystals with additional structure. II’, Compos. Math. 109 (1997), 255339.
[34] Lan, K.-W., ‘Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties’, J. Reine Angew. Math. 664 (2012), 163228.
[35] Lan, K.-W., ‘Toroidal compactifications of PEL-type Kuga families’, Algebra Number Theory 6(5) (2012), 885966.
[36] 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.
[37] Lan, K.-W., ‘Boundary strata of connected components in positive characteristics’, Algebra Number Theory 9(9) (2015), 20352054, an appendix to the article ‘Families of nearly ordinary Eisenstein series on unitary groups’ by Xin Wan.
[38] Lan, K.-W., ‘Compactifications of PEL-type Shimura varieties in ramified characteristics’, Forum Math. Sigma 4 (2016), e1, 98 pp.
[39] Lan, K.-W., ‘Higher Koecher’s principle’, Math. Res. Lett. 23(1) (2016), 163199.
[40] Lan, K.-W., ‘Vanishing theorems for coherent automorphic cohomology’, Res. Math. Sci. 3 (2016), article no. 39, 43 pp.
[41] Lan, K.-W., ‘Integral models of toroidal compactifications with projective cone decompositions’, Int. Math. Res. Not. IMRN 2017(11) (2017), 32373280.
[42] Lan, K.-W., Compactifications of PEL-type Shimura Varieties and Kuga Families with Ordinary Loci (World Scientific, Singapore, 2018).
[43] Lan, K.-W., ‘Compactifications of splitting models of PEL-type Shimura varieties’, Trans. Amer. Math. Soc. 370(4) (2018), 24632515.
[44] Lan, K.-W. and Stroh, B., ‘Relative cohomology of cuspidal forms on PEL-type Shimura varieties’, Algebra Number Theory 8(8) (2014), 17871799.
[45] Lan, K.-W. and Stroh, B., ‘Nearby cycles of automorphic étale sheaves’, Compos. Math. 154(1) (2018), 80119.
[46] 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.
[47] Laumon, G., ‘Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil’, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 131210.
[48] Lee, D. U., ‘Non-emptiness of Newton strata of Shimura varieties of Hodge type’, Algebra Number Theory 12(2) (2018), 259283.
[49] Liu, Y. and Zheng, W., ‘Enhanced adic formalism and perverse $t$ -structures for higher Artin stacks’. Preprint, 2012.
[50] Madapusi Pera, K., ‘Toroidal compactifications of integral models of Shimura varieties of Hodge type’. Preprint, 2018.
[51] Mantovan, E., ‘On the cohomology of certain PEL-type Shimura varieties’, Duke Math. J. 129(3) (2005), 573610.
[52] Mantovan, E., ‘ -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), 6183.
[53] Mazur, B. and Messing, W., Universal Extensions and One Dimensional Crystalline Cohomology, Lecture Notes in Mathematics, 370 (Springer, Berlin, Heidelberg, New York, 1974).
[54] 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).
[55] Moonen, B., Group Schemes with Additional Structures and Weyl Group Cosets, in Faber et al. [ 18 ], 255–298.
[56] Moonen, B., ‘A dimension formula for Ekedahl–Oort strata’, Ann. Inst. Fourier (Grenoble) 54(3) (2004), 666698.
[57] Moonen, B. and Wedhorn, T., ‘Discrete invariants of varieties in positive characteristic’, Int. Math. Res. Not. IMRN 2004(72) (2004), 38553903.
[58] 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.
[59] Morel, S., ‘Complexes pondérés sur les compactifications de Baily–Borel: le cas des variétés de Siegel’, J. Amer. Math. Soc. 21(1) (2008), 2361.
[60] Morel, S., ‘The intersection complex as a weight truncation and an application to Shimura varieties’, inProceedings of the International Congress of Mathematicians [ 28 ], 312334.
[61] Morel, S., On the Cohomology of Certain Non-compact Shimura Varieties, Annals of Mathematics Studies, 173 (Princeton University Press, Princeton, 2010).
[62] Morel, S., ‘Complexes mixtes sur un schéma de type fini sur $\mathbb{Q}$ ’. Preprint, 2012.
[63] 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.
[64] Ngô, B. C. and Genestier, A., ‘Alcôves et p-rang des variétés abéliennes’, Ann. Inst. Fourier (Grenoble) 52(6) (2002), 16651680.
[65] Oort, F., A Stratification of a Moduli Space of Abelian Varieties, in Faber et al. [ 18 ], 345–416.
[66] Oort, F., ‘Foliations in moduli spaces of abelian varieties’, J. Amer. Math. Soc. 17(2) (2004), 267296.
[67] Pappas, G., ‘On the arithmetic moduli schemes of PEL Shimura varieties’, J. Algebraic Geom. 9(3) (2000), 577605.
[68] Pappas, G. and Rapoport, M., ‘Local models in the ramified case, II. Splitting models’, Duke Math. J. 127(2) (2005), 193250.
[69] Pappas, G. and Zhu, X., ‘Local models of Shimura varieties and a conjecture of Kottwitz’, Invent. Math. 194 (2013), 147254.
[70] Pilloni, V. and Stroh, B., ‘Cohomologie cohérente et représentations Galoisiennes’, Ann. Math. Qué. 40(1) (2016), 167202.
[71] Pink, R., ‘Arithmetic compactification of mixed Shimura varieties’, PhD Thesis, Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1989.
[72] Pink, R., ‘On -adic sheaves on Shimura varieties and their higher direct images in the Baily–Borel compactification’, Math. Ann. 292 (1992), 197240.
[73] Pink, R., Wedhorn, T. and Ziegler, P., ‘Algebraic zip data’, Doc. Math. 16 (2011), 253300.
[74] Rapoport, M. and Richartz, M., ‘On the classification and specialization of F-isocrystals with additional structure’, Compos. Math. 103(2) (1996), 153181.
[75] Rapoport, M. and Zink, T., Period Spaces for p-divisible Groups, Annals of Mathematics Studies, 141 (Princeton University Press, Princeton, 1996).
[76] Springer, T. A., Linear Algebraic Groups, 2nd edn, Progress in Mathematics, 9 (Birkhäuser, Boston, 1998).
[77] Stamm, H., ‘On the reduction of the Hilbert–Blumenthal-moduli scheme with 𝛤0(p)-level structure’, Forum Math. 9(4) (1997), 405455.
[78] Stroh, B., ‘Compactification de variétés de Siegel aux places de mauvaise réduction’, Bull. Soc. Math. France 138(2) (2010), 259315.
[79] Stroh, B., ‘Compactification minimale et mauvaise réduction’, Ann. Inst. Fourier (Grenoble) 60(3) (2010), 10351055.
[80] Stroh, B., ‘Sur une conjecture de Kottwitz au bord’, Ann. Sci. Éc. Norm. Supér. (4) 45(1) (2012), 143165.
[81] Stroh, B., ‘Erratum à ‘sur une conjecture de Kottwitz au bord’’, Ann. Sci. Éc. Norm. Supér. (4) 46(6) (2013), 10231024.
[82] Stroh, B., Mauvaise réduction au bord, in Bost et al. [ 11 ], 269–304.
[83] Viehmann, E. and Wedhorn, T., ‘Ekedahl–Oort and Newton strata for Shimura varieties of PEL type’, Math. Ann. 356(4) (2013), 14931550.
[84] Wedhorn, T., The Dimension of Oort Strata of Shimura Varieties of PEL-Type, in Faber et al. [ 18 ], 441–471.
[85] Zhang, C., ‘Ekedahl–Oort strata for good reductions of Shimura varieties of Hodge type’, Canad. J. Math. 70(2) (2018), 451480.
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