Nearby Cycles of Automorphic Étale Sheaves, II

  • Kai-Wen LanEmail author
  • Benoît Stroh
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 245)


We review some recent results of ours on the nearby cycles of automorphic étale sheaves, and record some improvements of the arguments.


Integral models of Shimura varieties Compactifications Automorphic étale sheaves Nearby cycles 

2010 Mathematics Subject Classification

Primary 11G18 Secondary 11G15 11F75 



It is our great honor and pleasure to dedicate this article to Joachim Schwermer on the occasion of his 66th birthday. His many works on the cohomology of noncompact locally symmetric spaces have been great sources of information and inspiration for us. We would also like to thank the anonymous referee for a careful reading and helpful suggestions.


  1. 1.
    Arthur, J., Ellwood, D., Kottwitz, R. (eds.), Harmonic analysis, the trace formula, and Shimura varieties. In: Clay Mathematics Proceedings, vol. 4, Proceedings of the Clay Mathematics Institute 2003 Summer School, The Fields Institute, Toronto, Canada, 2–27 June 2003. American Mathematical Society, Providence, Rhode Island, Clay Mathematics Institute, Cambridge, Massachusetts (2005)Google Scholar
  2. 2.
    Artin, M., Grothendieck, A., Verdier, J.-L. (eds.): Théorie des topos et cohomologie étale des schémas SGA 4, Tome 3. Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973)Google Scholar
  3. 3.
    Baily Jr., W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 2 84(3), 442–528 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers, Analyse et topologie sur les espaces singuliers (I) CIRM, 6–10 juillet 1981, Astérisque, vol. 100. Société Mathématique de France, Paris (1982)Google Scholar
  5. 5.
    Borel, A., Casselman, W. (eds.) Automorphic forms, representations and $L$-functions. In: Proceedings of Symposia in Pure Mathematics, vol. 33, Part 2, held at Oregon State University, Corvallis, Oregon, July 11–August 5 1977. American Mathematical Society, Providence, Rhode Island (1979)Google Scholar
  6. 6.
    Breuil, C., Mézard, A.: Multiplicités modulaires et représentations de ${\rm {GL}}_2(\mathbf{Z}_p)$ et de ${\rm {Gal}}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ en $\ell = p$. Duke Math. J. 115(2), 205–310 (2002)Google Scholar
  7. 7.
    Caraiani, A., Scholze, P.: On the generic part of the cohomology of compact unitary Shimura varieties. Ann. Math. (2) 186(3), 649–766 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cartier, P., Illusie, L., Katz, N.M., Laumon, G., Manin, Y., 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
  9. 9.
    Clozel, L., Milne, J.S. (eds.), Automorphic forms, Shimura varieties, and $L$-functions. Volume I In: Perspectives in Mathematics, vol. 10, Proceedings of a Conference held at the University of Michigan, Ann Arbor, 6–16 July 1988. Academic Press Inc., Boston (1990)Google Scholar
  10. 10.
    Deligne, P.: Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, in Borel and Casselman [5], pp. 247–290Google Scholar
  11. 11.
    Deligne, P. (ed.): Cohomologie étale SGA 4$\frac{1}{2}$. Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977)Google Scholar
  12. 12.
    Deligne, P.: La conjecture de Weil. II. Publ. Math. Inst. Hautes Étud. Sci. 52, 137–252 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Deligne, P., Katz, N. (eds.): Groupes de monodromie en géométrie algébriqueSGA 7 II. Lecture Notes in Mathematics, vol. 340. Springer, Berlin (1973)Google Scholar
  14. 14.
    Ekedahl, T.: On the adic formalism, in Cartier et al. [8], pp. 197–218Google Scholar
  15. 15.
    Farkas, G., Morrison, I. (eds.): Handbook of Moduli: Volume II. Advanced Lectures in Mathematics, vol. 25. International Press, Somerville; Higher Education Press, Beijing (2013)Google Scholar
  16. 16.
    Fu, L.: Etale Cohomology Theory. Nankai Tracts in Mathematics, vol. 13. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  17. 17.
    Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991)zbMATHGoogle Scholar
  18. 18.
    Genestier, A., Tilouine, J.: Systèmes de Taylor-Wiles pour GSp$_4$, in Tilouine et al. [59], pp. 177–290Google Scholar
  19. 19.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, Berlin (2009)CrossRefGoogle Scholar
  20. 20.
    Grothendieck, A. (ed.): Revêtements étales et groupe fondamental (SGA 1). Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)Google Scholar
  21. 21.
    Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique. Publications mathématiques de l’I.H.E.S, vol. 4, 8, 11, 17, 20, 24, 28, 32. Institut des Hautes Etudes Scientifiques, Paris (1960, 1961, 1961, 1963, 1964, 1965, 1966, 1967)Google Scholar
  22. 22.
    Haines, T.J.: Introduction to Shimura varieties with bad reductions of parahoric type, in Arthur et al. [1], pp. 583–658Google Scholar
  23. 23.
    Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001)zbMATHGoogle Scholar
  24. 24.
    Helm, D.: Towards a geometric Jacquet–Langlands correspondence for unitary Shimura varieties. Duke Math. J. 155, 483–518 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Illusie, L.: Autour du théorème de monodromie locale, Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque, vol. 223. Société Mathématique de France, Paris, 9–57 (1994)Google Scholar
  26. 26.
    Imai, N., Mieda, Y.: Compactly supported cohomology and nearby cycles of open Shimura varieties of PEL type, preprint (2011)Google Scholar
  27. 27.
    Imai, N., Mieda, Y.: Potentially good reduction loci of Shimura varieties, preprint (2016)Google Scholar
  28. 28.
    Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)zbMATHGoogle Scholar
  29. 29.
    Kisin, M.: Integral models for Shimura varieties of abelian type. J. Am. Math. Soc. 23(4), 967–1012 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kim, W., Madapusi Pera, K.: 2-adic integral canonical models. Forum Math. Sigma 4, e28, 34 pp. (2016)Google Scholar
  31. 31.
    Kisin, M., Pappas, G.: Integral models of Shimura varieties with parahoric level structure, preprint (2015)Google Scholar
  32. 32.
    Kottwitz, R.E.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5(2), 373–444 (1992)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lan, K.-W.: Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties. J. Reine Angew. Math. 664, 163–228 (2012)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lan, K.-W.: Toroidal compactifications of PEL-type Kuga families. Algebra Number Theory 6(5), 885–966 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lan, K.-W.: Arithmetic Compactification of PEL-type Shimura Varieties. London Mathematical Society Monographs, vol. 36. Princeton University Press, Princeton (2013). (errata and revision available online at the author’s website)zbMATHGoogle Scholar
  36. 36.
    Lan, K.-W.: Compactifications of PEL-type Shimura varieties in ramified characteristics. Forum Math. Sigma 4, e1, 98 pp. (2016)Google Scholar
  37. 37.
    Lan, K.-W.: Vanishing theorems for coherent automorphic cohomology. Res. Math. Sci. 3, 43 pp. (2016). (article no. 39)Google Scholar
  38. 38.
    Lan, K.-W.: Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci. World Scientific, Singapore (2018)Google Scholar
  39. 39.
    Lan, K.-W.: Compactifications of splitting models of PEL-type Shimura varieties. Trans. Am. Math. Soc. 370(4), 2463–2515 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Lan, K.-W.: Integral models of toroidal compactifications with projective cone decompositions. Int. Math. Res. Not. IMRN 2017(11), 3237–3280 (2017)MathSciNetGoogle Scholar
  41. 41.
    Lan, K.-W., Stroh, B.: Compactifications of subschemes of integral models of Shimura varieties, preprint (2015)Google Scholar
  42. 42.
    Lan, K.-W., Stroh, B.: Nearby cycles of automorphic étale sheaves. Compos. Math. 154(1), 80–119 (2018)CrossRefGoogle Scholar
  43. 43.
    Lan, K.-W., Suh, J.: Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties. Adv. Math. 242, 228–286 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Madapusi Pera, K.: Toroidal compactifications of integral models of Shimura varieties of Hodge type, preprint (2015)Google Scholar
  45. 45.
    Mantovan, E.: On the cohomology of certain PEL-type Shimura varieties. Duke Math. J. 129(3), 573–610 (2005)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Mantovan, E.: $\ell $-adic étale cohomology of PEL type Shimura varieties with non-trivial coefficients. In: WIN-Woman in Numbers, Fields Institute Communications, vol. 60, pp. 61–83. American Mathematical Society, Providence (2011)Google Scholar
  47. 47.
    Milne, J.S.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)zbMATHGoogle Scholar
  48. 48.
    Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Clozel and Milne [9], pp. 283–414Google Scholar
  49. 49.
    Milne, J.S.: Shimura varieties and moduli, in Farkas and Morrison [15], pp. 467–548Google Scholar
  50. 50.
    Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Oxford University Press, Oxford (1970). (with appendices by C. P. Ramanujam and Yuri Manin)zbMATHGoogle Scholar
  51. 51.
    Mumford, D.: Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42, 239–272 (1977)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Pappas, G., Rapoport, M.: Local models in the ramified case, I. The EL-case. J. Algebr. Geom. 12(1), 107–145 (2003)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Pappas, G., Rapoport, M.: Local models in the ramified case, II. Splitting models. Duke Math. J. 127(2), 193–250 (2005)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Pappas, G., Zhu, X.: Local models of Shimura varieties and a conjecture of Kottwitz. Invent. Math. 194, 147–254 (2013)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Pink, R.: Arithmetic compactification of mixed Shimura varieties, Ph.D. thesis, Rheinischen Friedrich-Wilhelms-Universität, Bonn (1989)Google Scholar
  56. 56.
    Rapoport, M., Zink, T.: Period Spaces for $p$-Divisible Groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)zbMATHGoogle Scholar
  57. 57.
    Scholze, P.: The Langlands–Kottwitz method and deformation spaces of $p$-divisible groups. J. Am. Math. Soc. 26, 227–259 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Skinner, C.: A note on the $p$-adic Galois representations attached to Hilbert modular forms. Doc. Math. 14, 241–258 (2009)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Tilouine, J., Carayol, H., Harris, M., Vignéras, M.-F. (eds.): Formes automorphes (II): Le cas du groupe GSp(4). Astérisque, vol. 302. Société Mathématique de France, Paris (2005)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of MinnesotaMinneapolisUSA
  2. 2.C.N.R.S. and Institut de Mathématiques de Jussieu–Paris Rive GaucheParis Cedex 05France

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