The test function conjecture for parahoric local models
Authors:
Thomas J. Haines and Timo Richarz
Journal:
J. Amer. Math. Soc. 34 (2021), 135-218
MSC (2010):
Primary 14G35, 14M15, 20G05
DOI:
https://doi.org/10.1090/jams/955
Published electronically:
December 7, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.
- [AHR]
J. Alper, J. Hall, and D. Rydh,
A Luna étale slice theorem for algebraic stacks,
available at 1504.06467. - [Ana73] Sivaramakrishna Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 5–79. Bull. Soc. Math. France, Mém. 33 (French). MR 0335524, https://doi.org/10.24033/msmf.109
- [ArHa]
E.Arasteh Rad and U.Hartl,
Langlands-Rapoport conjecture over function fields,
1605.01575. - [AB09] Sergey Arkhipov and Roman Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, Israel J. Math. 170 (2009), 135–183. With an appendix by Bezrukavnikov and Ivan Mirković. MR 2506322, https://doi.org/10.1007/s11856-009-0024-y
- [BL95] Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340 (French, with English and French summaries). MR 1320381
- [BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- [BD]
A. Beilinson and V. Drinfeld,
Quantization of Hitchin's integrable system and Hecke eigensheaves,
preprint (1999) available at http://www.math.utexas.edu/users/benzvi/Langlands.html. - [Bo98] Mikhail Borovoi, Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132 (1998), no. 626, viii+50. MR 1401491, https://doi.org/10.1090/memo/0626
- [Br03] Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209–216. MR 1996415, https://doi.org/10.1007/s00031-003-0606-4
- [BG02] A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), no. 2, 287–384. MR 1933587, https://doi.org/10.1007/s00222-002-0237-8
- [BT84] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
- [CGP10] Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010. MR 2723571
- [CN92] Ching-Li Chai and Peter Norman, Singularities of the Γ₀(𝑝)-level structure, J. Algebraic Geom. 1 (1992), no. 2, 251–278. MR 1144439
- [Co14] Brian Conrad, Reductive group schemes, Autour des schémas en groupes. Vol. I, Panor. Synthèses, vol. 42/43, Soc. Math. France, Paris, 2014, pp. 93–444 (English, with English and French summaries). MR 3362641
- [DM82] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
- [dHL] Mark Andrea de Cataldo, Thomas J. Haines, and Li Li, Frobenius semisimplicity for convolution morphisms, Math. Z. 289 (2018), no. 1-2, 119–169. MR 3803785, https://doi.org/10.1007/s00209-017-1946-4
- [dJ93] A. J. de Jong, The moduli spaces of principally polarized abelian varieties with Γ₀(𝑝)-level structure, J. Algebraic Geom. 2 (1993), no. 4, 667–688. MR 1227472
- [DP94] Pierre Deligne and Georgios Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), no. 1, 59–79 (French). MR 1266495
- [DR73] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
- [Dr]
V.G.Drinfeld,
On algebraic spaces with an action of,
available at math/1308.2604. - [DG15] V. Drinfeld and D. Gaitsgory, On a theorem of Braden, Transform. Groups 19 (2014), no. 2, 313–358. MR 3200429, https://doi.org/10.1007/s00031-014-9267-8
- [Fal03] Gerd Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 1, 41–68. MR 1961134, https://doi.org/10.1007/s10097-002-0045-x
- [GGM14] Ofer Gabber, Philippe Gille, and Laurent Moret-Bailly, Fibrés principaux sur les corps valués henséliens, Algebr. Geom. 1 (2014), no. 5, 573–612 (French, with English and French summaries). MR 3296806, https://doi.org/10.14231/AG-2014-025
- [Ga01] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253–280. MR 1826370, https://doi.org/10.1007/s002220100122
- [Gi] V. Ginzburg: Perverse sheaves on a loop group and Langlands' duality, preprint (1995), alg-geom/9511007.
- [Goe04] Ulrich Görtz, Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for 𝐺𝐿₄ and 𝐺𝐿₅, J. Algebra 278 (2004), no. 1, 148–172. MR 2068071, https://doi.org/10.1016/j.jalgebra.2003.07.002
- [Goe07] Ulrich Görtz, Alcove walks and nearby cycles on affine flag manifolds, J. Algebraic Combin. 26 (2007), no. 4, 415–430. MR 2341858, https://doi.org/10.1007/s10801-007-0063-6
- [Goe08] Ulrich Görtz, Affine Springer fibers and affine Deligne-Lusztig varieties, Affine flag manifolds and principal bundles, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010, pp. 1–50. MR 3013026, https://doi.org/10.1007/978-3-0346-0288-4_1
- [GHKR06] Ulrich Görtz, Thomas J. Haines, Robert E. Kottwitz, and Daniel C. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 467–511 (English, with English and French summaries). MR 2265676, https://doi.org/10.1016/j.ansens.2005.12.004
- [GHKR10] Ulrich Görtz, Thomas J. Haines, Robert E. Kottwitz, and Daniel C. Reuman, Affine Deligne-Lusztig varieties in affine flag varieties, Compos. Math. 146 (2010), no. 5, 1339–1382. MR 2684303, https://doi.org/10.1112/S0010437X10004823
- [GW10] Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. MR 2675155
- [Hai01] Thomas J. Haines, Test functions for Shimura varieties: the Drinfeld case, Duke Math. J. 106 (2001), no. 1, 19–40. MR 1810365, https://doi.org/10.1215/S0012-7094-01-10612-1
- [Hai05] Thomas J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583–642. MR 2192017
- [Hai14] Thomas J. Haines, The stable Bernstein center and test functions for Shimura varieties, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 118–186. MR 3444233
- [Hai15] Thomas J. Haines, On Satake parameters for representations with parahoric fixed vectors, Int. Math. Res. Not. IMRN 20 (2015), 10367–10398. MR 3455870, https://doi.org/10.1093/imrn/rnu254
- [Hai17] Thomas J. Haines, Correction to “On Satake parameters for representations with parahoric fixed vectors” [ MR3455870], Int. Math. Res. Not. IMRN 13 (2017), 4160–4170. MR 3671514, https://doi.org/10.1093/imrn/rnx088
- [Hai18] Thomas J. Haines, Dualities for root systems with automorphisms and applications to non-split groups, Represent. Theory 22 (2018), 1–26. MR 3772644, https://doi.org/10.1090/ert/512
- [HN02] T. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), no. 2, 117–150. MR 1923579, https://doi.org/10.1023/A:1019666710051
- [HR08] G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, https://doi.org/10.1016/j.aim.2008.04.006
- [HR12] Thomas J. Haines and Michael Rapoport, Shimura varieties with Γ₁(𝑝)-level via Hecke algebra isomorphisms: the Drinfeld case, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 5, 719–785 (2013) (English, with English and French summaries). MR 3053008, https://doi.org/10.24033/asens.2177
- [HR10] Thomas J. Haines and Sean Rostami, The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory 14 (2010), 264–284. MR 2602034, https://doi.org/10.1090/S1088-4165-10-00370-5
- [HaRi] Thomas J. Haines and Timo Richarz, The test function conjecture for local models of Weil-restricted groups, Compos. Math. 156 (2020), no. 7, 1348–1404. MR 4120166, https://doi.org/10.1112/s0010437x20007162
- [He10] Jochen Heinloth, Uniformization of 𝒢-bundles, Math. Ann. 347 (2010), no. 3, 499–528. MR 2640041, https://doi.org/10.1007/s00208-009-0443-4
- [He] Jochen Heinloth, Hilbert-Mumford stability on algebraic stacks and applications to \Cal𝐺-bundles on curves, Épijournal Géom. Algébrique 1 (2017), Art. 11, 37. MR 3758902
- [HNY13] Jochen Heinloth, Bao-Châu Ngô, and Zhiwei Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), no. 1, 241–310. MR 2999041, https://doi.org/10.4007/annals.2013.177.1.5
- [Il94] Luc Illusie, Autour du théorème de monodromie locale, Astérisque 223 (1994), 9–57 (French). Périodes 𝑝-adiques (Bures-sur-Yvette, 1988). MR 1293970
- [KP] M. Kisin and G. Pappas, Integral models of Shimura varieties with parahoric level structure, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 121–218. MR 3905466, https://doi.org/10.1007/s10240-018-0100-0
- [Ko84] Robert E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), no. 3, 287–300. MR 761308, https://doi.org/10.1007/BF01450697
- [Ko97] Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, https://doi.org/10.1023/A:1000102604688
- [Kr03] N. Krämer, Local models for ramified unitary groups, Abh. Math. Sem. Univ. Hamburg 73 (2003), 67–80. MR 2028507, https://doi.org/10.1007/BF02941269
- [La76] R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 401–418. MR 0437500
- [LS97] Yves Laszlo and Christoph Sorger, The line bundles on the moduli of parabolic 𝐺-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499–525 (English, with English and French summaries). MR 1456243, https://doi.org/10.1016/S0012-9593(97)89929-6
- [LMB00] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
- [LN08] Gérard Laumon and Bao Châu Ngô, Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), no. 2, 477–573 (French, with English summary). MR 2434884, https://doi.org/10.4007/annals.2008.168.477
- [Lev16] Brandon Levin, Local models for Weil-restricted groups, Compos. Math. 152 (2016), no. 12, 2563–2601. MR 3594288, https://doi.org/10.1112/S0010437X1600765X
- [Lou]
João Lourenço,
Grassmanniennes affines tordues sur les entiers,
preprint 2019, 1912.11918. - [Lu81] George Lusztig, Singularities, character formulas, and a 𝑞-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
- [Mar15] Benedictus Margaux, Smoothness of limit functors, Proc. Indian Acad. Sci. Math. Sci. 125 (2015), no. 2, 161–165. MR 3361508, https://doi.org/10.1007/s12044-015-0234-7
- [MV07] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. MR 2342692, https://doi.org/10.4007/annals.2007.166.95
- [Na16] Hiraku Nakajima, Lectures on perverse sheaves on instanton moduli spaces, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2017, pp. 381–436. MR 3752464
- [NP01] B. C. Ngô and P. Polo, Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Algebraic Geom. 10 (2001), no. 3, 515–547 (French, with English summary). MR 1832331
- [PR08] G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, https://doi.org/10.1016/j.aim.2008.04.006
- [PRS13] Georgios Pappas, Michael Rapoport, and Brian Smithling, Local models of Shimura varieties, I. Geometry and combinatorics, Handbook of moduli. Vol. III, Adv. Lect. Math. (ALM), vol. 26, Int. Press, Somerville, MA, 2013, pp. 135–217. MR 3135437
- [PZ13] G. Pappas and X. Zhu, Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194 (2013), no. 1, 147–254. MR 3103258, https://doi.org/10.1007/s00222-012-0442-z
- [Ra90] M. Rapoport, On the bad reduction of Shimura varieties, Automorphic forms, Shimura varieties, and 𝐿-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 253–321. MR 1044832
- [Ra05] Michael Rapoport, A guide to the reduction modulo 𝑝 of Shimura varieties, Astérisque 298 (2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR 2141705
- [RZ96] M. Rapoport and Th. Zink, Period spaces for 𝑝-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439
- [Ri14a] Timo Richarz, A new approach to the geometric Satake equivalence, Doc. Math. 19 (2014), 209–246. MR 3178249
- [Ri14b] T.Richarz: Affine Grassmannians and Geometric Satake Equivalences, Langlands correspondence and constructive Galois theory, Oberwolfach Reports 11, Issue 1 (2014) 287-334.
- [RZ15] Xinwen Zhu, The geometric Satake correspondence for ramified groups, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 409–451 (English, with English and French summaries). MR 3346175, https://doi.org/10.24033/asens.2248
- [Ri16a] Timo Richarz, Affine Grassmannians and geometric Satake equivalences, Int. Math. Res. Not. IMRN 12 (2016), 3717–3767. MR 3544618, https://doi.org/10.1093/imrn/rnv226
- [Ri16b] Timo Richarz, On the Iwahori Weyl group, Bull. Soc. Math. France 144 (2016), no. 1, 117–124. MR 3481263, https://doi.org/10.24033/bsmf.2708
- [Ri19] Timo Richarz, Spaces with 𝔾_{𝕞}-action, hyperbolic localization and nearby cycles, J. Algebraic Geom. 28 (2019), no. 2, 251–289. MR 3912059, https://doi.org/10.1090/S1056-3911-2018-00710-6
- [Ri19b] T.Richarz: Erratum to ``Affine Grassmannians and Geometric Satake Equivalences'', Int. Math. Res. Not., rnz210, https://doi.org/10.1093/imrn/rnz210.
- [Ro15] Sean Rostami, The Bernstein presentation for general connected reductive groups, J. Lond. Math. Soc. (2) 91 (2015), no. 2, 514–536. MR 3355113, https://doi.org/10.1112/jlms/jdu080
- [Ro17] Sean Rostami, Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties, Selecta Math. (N.S.) 23 (2017), no. 1, 643–719. MR 3595903, https://doi.org/10.1007/s00029-016-0252-z
- [SW]
P.Scholze and J.Weinstein,
Berkeley lectures on-adic geometry,
available at http://www.math.uni-bonn.de/people/scholze/. - [SGA1] A. Grothendieck: Séminaire de Géométrie Algébrique du Bois Marie - Revêtements étales et groupe fondamental, Lecture notes in mathematics 224, Springer-Verlag (1971), xxii+447.
- [SGA3] Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458
- [SGA$412$] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA 4\frac{1}2. MR 463174
- [SGA7] Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
- [StaPro]
Stacks Project,
available at http://stacks.math.columbia.edu/. - [Ti69] Jacques Tits, Le problème des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome, 1967/68) Academic Press, London, 1969, pp. 175–185 (French). MR 0254129
- [Xue17]
C.Xue,
Cohomologie cuspidale des champs de Chtoucas,
Université Paris-Saclay, NNT:2017SACLS125, 2017. - [Zhou]
R.Zhou,
Mod-isogeny classes on Shimura varieties with pararhoric level structure,
Preprint available at http://www.math.harvard.edu/~rzhou/. - [Zhu09] Xinwen Zhu, Affine Demazure modules and 𝑇-fixed point subschemes in the affine Grassmannian, Adv. Math. 221 (2009), no. 2, 570–600. MR 2508931, https://doi.org/10.1016/j.aim.2009.01.003
- [Zhu14] Xinwen Zhu, On the coherence conjecture of Pappas and Rapoport, Ann. of Math. (2) 180 (2014), no. 1, 1–85. MR 3194811, https://doi.org/10.4007/annals.2014.180.1.1
- [Zhu15] Xinwen Zhu, The geometric Satake correspondence for ramified groups, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 409–451 (English, with English and French summaries). MR 3346175, https://doi.org/10.24033/asens.2248
- [Zhu] Xinwen Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2017, pp. 59–154. MR 3752460
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Additional Information
Thomas J. Haines
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Email:
tjh@math.umd.edu
Timo Richarz
Affiliation:
Fachbereich Mathematik, TU Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Email:
richarz@mathematik.tu-darmstadt.de
DOI:
https://doi.org/10.1090/jams/955
Received by editor(s):
April 9, 2018
Received by editor(s) in revised form:
February 26, 2020, March 19, 2020, and March 26, 2020
Published electronically:
December 7, 2020
Additional Notes:
The research of the first author was partially supported by NSF DMS-1406787 and by Simons Fellowship 399424.
The research of the second author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 394587809.
Article copyright:
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American Mathematical Society