Shimura varieties with -level via Hecke algebra isomorphisms: the Drinfeld case
Thomas J. Haines; Michael Rapoport
Annales scientifiques de l'École Normale Supérieure (2012)
- Volume: 45, Issue: 5, page 719-785
- ISSN: 0012-9593
Access Full Article
topAbstract
topHow to cite
topHaines, Thomas J., and Rapoport, Michael. "Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case." Annales scientifiques de l'École Normale Supérieure 45.5 (2012): 719-785. <http://eudml.org/doc/272197>.
@article{Haines2012,
abstract = {We study the local factor at $p$ of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at $p$ by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.},
author = {Haines, Thomas J., Rapoport, Michael},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Shimura varieties; Hasse-Weil zeta functions; automorphic $L$-functions},
language = {eng},
number = {5},
pages = {719-785},
publisher = {Société mathématique de France},
title = {Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case},
url = {http://eudml.org/doc/272197},
volume = {45},
year = {2012},
}
TY - JOUR
AU - Haines, Thomas J.
AU - Rapoport, Michael
TI - Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 5
SP - 719
EP - 785
AB - We study the local factor at $p$ of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at $p$ by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.
LA - eng
KW - Shimura varieties; Hasse-Weil zeta functions; automorphic $L$-functions
UR - http://eudml.org/doc/272197
ER -
References
top- [1] J. N. Bernstein, Le “centre” de Bernstein, in Représentations des groupes réductifs sur un corps local (P. Deligne, éd.), Travaux en Cours, Hermann, 1984, 1–32. Zbl0599.22016MR771671
- [2] J. N. Bernstein, Representations of -adic groups, notes taken by K. Rumelhart of lectures by J. Bernstein at Harvard, 1992.
- [3] C. J. Bushnell & P. C. Kutzko, Smooth representations of reductive -adic groups: structure theory via types, Proc. London Math. Soc.77 (1998), 582–634. Zbl0911.22014MR1643417
- [4] W. Casselman, The unramified principal series of -adic groups. I. The spherical function, Compositio Math. 40 (1980), 387–406. Zbl0472.22004MR571057
- [5] L. Clozel, The fundamental lemma for stable base change, Duke Math. J.61 (1990), 255–302. Zbl0731.22011MR1068388
- [6] P. Deligne, Le formalisme des cycles évanescents, in Groupes de monodromie en géométrie algébrique. II, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Math. 340, Springer, 1973, 82–115. Zbl0258.00005
- [7] P. Deligne, Sommes trigonométriques, in Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4, Lecture Notes in Math. 569, Springer, 1977, 168–232. Zbl0349.10031MR463174
- [8] P. Deligne & M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 349, Springer, 1973, 143–316. Zbl0281.14010MR337993
- [9] V. Drinfeld, Elliptic modules, Math. USSR Sb.23 (1974), 561–592. Zbl0321.14014MR384707
- [10] A. Genestier & J. Tilouine, Systèmes de Taylor-Wiles pour , Astérisque302 (2005), 177–290. Zbl1142.11036MR2234862
- [11] D. J. Goldstein, Hecke algebra isomorphisms for tamely ramified characters, Thèse, The University of Chicago, 1990. MR2611915
- [12] U. Görtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann.321 (2001), 689–727. Zbl1073.14526MR1871975
- [13] U. Görtz, Alcove walks and nearby cycles on affine flag manifolds, J. Algebraic Combin.26 (2007), 415–430. Zbl1152.20006MR2341858
- [14] U. Görtz & T. J. Haines, The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties, J. reine angew. Math. 609 (2007), 161–213. Zbl1157.14013MR2350783
- [15] U. Görtz & C.-F. Yu, The supersingular locus in Siegel modular varieties with Iwahori level structure, Math. Annalen353 (2012), 465–498. Zbl1257.14019MR2915544
- [16] T. J. Haines, The combinatorics of Bernstein functions, Trans. Amer. Math. Soc.353 (2001), 1251–1278. Zbl0962.14018MR1804418
- [17] T. J. Haines, Test functions for Shimura varieties: the Drinfeld case, Duke Math. J.106 (2001), 19–40. Zbl1014.20002MR1810365
- [18] T. J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, Amer. Math. Soc., 2005, 583–642. Zbl1148.11028MR2192017
- [19] T. J. Haines, The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J.149 (2009), 569–643. Zbl1194.22019MR2553880
- [20] T. J. Haines, Base change for Bernstein centers of depth zero principal series blocks, Ann. Sci. École Norm. Sup.45 (2012), 681–718. Zbl06155584MR3053007
- [21] T. J. Haines, R. E. Kottwitz & A. Prasad, Iwahori-Hecke algebras, J. Ramanujan Math. Soc.25 (2010), 113–145. Zbl1202.22013MR2642451
- [22] T. J. Haines & B. C. Ngô, Alcoves associated to special fibers of local models, Amer. J. Math.124 (2002), 1125–1152. Zbl1047.20037MR1939783
- [23] T. J. Haines & A. Pettet, Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra, J. Algebra252 (2002), 127–149. Zbl1056.20003MR1922389
- [24] T. J. Haines & M. Rapoport, On parahoric subgroups, Adv. Math. 219 (2008), 188–198, appendix to [48].
- [25] M. Harris & R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies 151, Princeton Univ. Press, 2001. Zbl1036.11027MR1876802
- [26] M. Harris & R. Taylor, Regular models of certain Shimura varieties, Asian J. Math.6 (2002), 61–94. Zbl1008.11022MR1902647
- [27] R. B. Howlett & G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math.58 (1980), 37–64. Zbl0435.20023MR570873
- [28] L. Illusie, Autour du théorème de monodromie locale, Astérisque223 (1994), 9–57. Zbl0837.14013MR1293970
- [29] T. Ito, Hasse invariants for some unitary Shimura varieties, in Algebraische Zahlentheorie. Abstracts from the workshop held June 17–23, Oberwolfach, 4, 2007, 1565–1568.
- [30] H. Jacquet, Principal -functions of the linear group, in Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., 1979, 63–86. Zbl0413.12007MR546609
- [31] N. M. Katz & B. Mazur, Arithmetic moduli of elliptic curves, Annals of Math. Studies 108, Princeton Univ. Press, 1985. Zbl0576.14026MR772569
- [32] R. E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J.49 (1982), 785–806. Zbl0506.20017MR683003
- [33] R. E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann.269 (1984), 287–300. Zbl0533.14009MR761308
- [34] R. E. Kottwitz, Base change for unit elements of Hecke algebras, Compositio Math.60 (1986), 237–250. MR868140
- [35] R. E. Kottwitz, Shimura varieties and -adic representations, in Automorphic forms, Shimura varieties, and -functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, 1990, 161–209. Zbl0743.14019MR1044820
- [36] R. E. Kottwitz, On the -adic representations associated to some simple Shimura varieties, Invent. Math.108 (1992), 653–665. Zbl0765.22011MR1163241
- [37] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc.5 (1992), 373–444. Zbl0796.14014MR1124982
- [38] R. E. Kottwitz & M. Rapoport, Minuscule alcoves for and , Manuscripta Math.102 (2000), 403–428. Zbl0981.17003MR1785323
- [39] S. Kudla, Letter to Rapoport, April 18, 2010.
- [40] S. Kudla & M. Rapoport, Special cycles on unitary Shimura varieties II: Global theory, preprint arXiv:0912.3758.
- [41] J.-P. Labesse, Fonctions élémentaires et lemme fondamental pour le changement de base stable, Duke Math. J.61 (1990), 519–530. Zbl0731.22012MR1074306
- [42] K. W. Lan, Arithmetic compactifications of PEL Shimura varieties, preprint http://www.math.princeton.edu/~klan/articles/cpt-PEL-type-thesis- revision-20100208.pdf, 2010. Zbl1284.14004
- [43] R. P. Langlands, Modular forms and -adic representations, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 349, Springer, 1973, 361–500. Zbl0279.14007MR354617
- [44] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc.2 (1989), 599–635. Zbl0715.22020MR991016
- [45] L. Morris, Tamely ramified intertwining algebras, Invent. Math.114 (1993), 1–54. Zbl0854.22022MR1235019
- [46] B. C. Ngô & A. Genestier, Alcôves et -rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble) 52 (2002), 1665–1680. Zbl1046.14023MR1952527
- [47] G. Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom.9 (2000), 577–605. Zbl0978.14023MR1752014
- [48] G. Pappas & M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math.219 (2008), 118–198. Zbl1159.22010MR2435422
- [49] A. Ram, Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Pure Appl. Math. Q. 2 (2006), 963–1013. Zbl1127.20005MR2282411
- [50] M. Rapoport, On the bad reduction of Shimura varieties, in Automorphic forms, Shimura varieties, and -functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math. 11, Academic Press, 1990, 253–321. Zbl0716.14010MR1044832
- [51] M. Rapoport, A guide to the reduction modulo of Shimura varieties, Astérisque298 (2005), 271–318. Zbl1084.11029MR2141705
- [52] M. Rapoport & T. Zink, Period spaces for -divisible groups, Annals of Math. Studies 141, Princeton Univ. Press, 1996. Zbl0873.14039MR1393439
- [53] A. Roche, Types and Hecke algebras for principal series representations of split reductive -adic groups, Ann. Sci. École Norm. Sup.31 (1998), 361–413. Zbl0903.22009MR1621409
- [54] F. Rodier, Représentations de où est un corps -adique, in Bourbaki Seminar, Vol. 1981/1982, Astérisque 92, Soc. Math. France, 1982, 201–218. Zbl0506.22019MR689531
- [55] P. Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not.2011 (2011), 3368–3425. Zbl1309.14019MR2822177
- [56] P. Scholze, The Langlands-Kottwitz approach for some simple Shimura varieties, to appear in Invent. Math. Zbl1309.14020MR3049931
- [57] M. Strauch, Galois actions on torsion points of one-dimensional formal modules, J. Number Theory130 (2010), 528–533. Zbl1221.11227MR2584836
- [58] J. Tate & F. Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup.3 (1970), 1–21. Zbl0195.50801MR265368
- [59] D. A. J. Vogan, The local Langlands conjecture, in Representation theory of groups and algebras, Contemp. Math. 145, Amer. Math. Soc., 1993, 305–379. Zbl0802.22005MR1216197
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.