Shimura varieties with Γ 1 (p)-level via Hecke algebra isomorphisms: the Drinfeld case
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 5, p. 719-785

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.

On étudie le facteur local en p de la fonction zêta semi-simple d’une variété de Shimura du type de Drinfeld, où la structure de niveau en p est donnée par le radical pro-unipotent d’un sous-groupe d’Iwahori. La méthode suivie est une adaptation à ce cas de la méthode de comptage de Langlands-Kottwitz. On détermine de façon explicite la fonction test dans l’algèbre de Hecke correspondante  ; puis on démontre que c’est un élément central en déterminant ses images sous des isomorphismes d’algèbres de Hecke dus à Goldstein, Morris et Roche.

DOI : https://doi.org/10.24033/asens.2177
Classification:  14G35,  11F70
Keywords: Shimura varieties, Hasse-Weil zeta functions, automorphic L-functions
@article{ASENS_2012_4_45_5_719_0,
     author = {Haines, Thomas J. and Rapoport, Michael},
     title = {Shimura varieties with $\Gamma \_1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {5},
     year = {2012},
     pages = {719-785},
     doi = {10.24033/asens.2177},
     mrnumber = {3053008},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_5_719_0}
}
Haines, Thomas J.; Rapoport, Michael. Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 5, pp. 719-785. doi : 10.24033/asens.2177. http://www.numdam.org/item/ASENS_2012_4_45_5_719_0/

[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. | MR 771671 | Zbl 0599.22016

[2] J. N. Bernstein, Representations of p-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 p-adic groups: structure theory via types, Proc. London Math. Soc. 77 (1998), 582-634. | MR 1643417 | Zbl 0911.22014

[4] W. Casselman, The unramified principal series of 𝔭-adic groups. I. The spherical function, Compositio Math. 40 (1980), 387-406. | Numdam | MR 571057 | Zbl 0472.22004

[5] L. Clozel, The fundamental lemma for stable base change, Duke Math. J. 61 (1990), 255-302. | MR 1068388 | Zbl 0731.22011

[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. | Zbl 0258.00005

[7] P. Deligne, Sommes trigonométriques 1977, 168-232. | MR 463174 | Zbl 0349.10031

[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. | MR 337993 | Zbl 0281.14010

[9] V. Drinfeld, Elliptic modules, Math. USSR Sb. 23 (1974), 561-592. | MR 384707 | Zbl 0321.14014

[10] A. Genestier & J. Tilouine, Systèmes de Taylor-Wiles pour GSp 4 , Astérisque 302 (2005), 177-290. | MR 2234862 | Zbl 1142.11036

[11] D. J. Goldstein, Hecke algebra isomorphisms for tamely ramified characters, Thèse, The University of Chicago, 1990. | MR 2611915

[12] U. Görtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), 689-727. | MR 1871975 | Zbl 1073.14526

[13] U. Görtz, Alcove walks and nearby cycles on affine flag manifolds, J. Algebraic Combin. 26 (2007), 415-430. | MR 2341858 | Zbl 1152.20006

[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. | MR 2350783 | Zbl 1157.14013

[15] U. Görtz & C.-F. Yu, The supersingular locus in Siegel modular varieties with Iwahori level structure, Math. Annalen 353 (2012), 465-498. | MR 2915544 | Zbl 1257.14019

[16] T. J. Haines, The combinatorics of Bernstein functions, Trans. Amer. Math. Soc. 353 (2001), 1251-1278. | MR 1804418 | Zbl 0962.14018

[17] T. J. Haines, Test functions for Shimura varieties: the Drinfeld case, Duke Math. J. 106 (2001), 19-40. | MR 1810365 | Zbl 1014.20002

[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. | MR 2192017 | Zbl 1148.11028

[19] T. J. Haines, The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J. 149 (2009), 569-643. | MR 2553880 | Zbl 1194.22019

[20] T. J. Haines, Base change for Bernstein centers of depth zero principal series blocks, Ann. Sci. École Norm. Sup. 45 (2012), 681-718. | Numdam | MR 3053007 | Zbl pre06155584

[21] T. J. Haines, R. E. Kottwitz & A. Prasad, Iwahori-Hecke algebras, J. Ramanujan Math. Soc. 25 (2010), 113-145. | MR 2642451 | Zbl 1202.22013

[22] T. J. Haines & B. C. Ngô, Alcoves associated to special fibers of local models, Amer. J. Math. 124 (2002), 1125-1152. | MR 1939783 | Zbl 1047.20037

[23] T. J. Haines & A. Pettet, Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra, J. Algebra 252 (2002), 127-149. | MR 1922389 | Zbl 1056.20003

[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. | MR 1876802 | Zbl 1036.11027

[26] M. Harris & R. Taylor, Regular models of certain Shimura varieties, Asian J. Math. 6 (2002), 61-94. | MR 1902647 | Zbl 1008.11022

[27] R. B. Howlett & G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math. 58 (1980), 37-64. | MR 570873 | Zbl 0435.20023

[28] L. Illusie, Autour du théorème de monodromie locale, Astérisque 223 (1994), 9-57. | MR 1293970 | Zbl 0837.14013

[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 L-functions of the linear group, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., 1979, 63-86. | MR 546609 | Zbl 0413.12007

[31] N. M. Katz & B. Mazur, Arithmetic moduli of elliptic curves, Annals of Math. Studies 108, Princeton Univ. Press, 1985. | MR 772569 | Zbl 0576.14026

[32] R. E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785-806. | MR 683003 | Zbl 0506.20017

[33] R. E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287-300. | MR 761308 | Zbl 0533.14009

[34] R. E. Kottwitz, Base change for unit elements of Hecke algebras, Compositio Math. 60 (1986), 237-250. | Numdam | MR 868140

[35] R. E. Kottwitz, Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, 1990, 161-209. | MR 1044820 | Zbl 0743.14019

[36] R. E. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992), 653-665. | MR 1163241 | Zbl 0765.22011

[37] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444. | MR 1124982 | Zbl 0796.14014

[38] R. E. Kottwitz & M. Rapoport, Minuscule alcoves for GL n and GSp 2n , Manuscripta Math. 102 (2000), 403-428. | MR 1785323 | Zbl 0981.17003

[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. | MR 1074306 | Zbl 0731.22012

[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.

[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. | MR 354617 | Zbl 0279.14007

[44] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635. | MR 991016 | Zbl 0715.22020

[45] L. Morris, Tamely ramified intertwining algebras, Invent. Math. 114 (1993), 1-54. | MR 1235019 | Zbl 0854.22022

[46] B. C. Ngô & A. Genestier, Alcôves et p-rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble) 52 (2002), 1665-1680. | Numdam | MR 1952527 | Zbl 1046.14023

[47] G. Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom. 9 (2000), 577-605. | MR 1752014 | Zbl 0978.14023

[48] G. Pappas & M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), 118-198. | MR 2435422 | Zbl 1159.22010

[49] A. Ram, Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Pure Appl. Math. Q. 2 (2006), 963-1013. | MR 2282411 | Zbl 1127.20005

[50] M. Rapoport, On the bad reduction of Shimura varieties 1988), Perspect. Math. 11, Academic Press, 1990, 253-321. | MR 1044832 | Zbl 0716.14010

[51] M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271-318. | MR 2141705 | Zbl 1084.11029

[52] M. Rapoport & T. Zink, Period spaces for p-divisible groups, Annals of Math. Studies 141, Princeton Univ. Press, 1996. | MR 1393439 | Zbl 0873.14039

[53] A. Roche, Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. École Norm. Sup. 31 (1998), 361-413. | Numdam | MR 1621409 | Zbl 0903.22009

[54] F. Rodier, Représentations de GL (n,k)k est un corps p-adique, in Bourbaki Seminar, Vol. 1981/1982, Astérisque 92, Soc. Math. France, 1982, 201-218. | Numdam | MR 689531 | Zbl 0506.22019

[55] P. Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. 2011 (2011), 3368-3425. | MR 2822177 | Zbl pre05942001

[56] P. Scholze, The Langlands-Kottwitz approach for some simple Shimura varieties, to appear in Invent. Math. | MR 3049931 | Zbl pre06176661

[57] M. Strauch, Galois actions on torsion points of one-dimensional formal modules, J. Number Theory 130 (2010), 528-533. | MR 2584836 | Zbl 1221.11227

[58] J. Tate & F. Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. 3 (1970), 1-21. | Numdam | MR 265368 | Zbl 0195.50801

[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. | MR 1216197 | Zbl 0802.22005