$H^0$ of Igusa varieties via automorphic forms

Kret, Arno; Shin, Sug Woo

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1299-1390

Our main theorem describes the degree $0$ cohomology of non-basic Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod $p$ Hodge-type Shimura varieties with hyperspecial level at $p$. As an application we obtain a completely new approach to two geometric questions. Firstly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings–Chai, Hida, and others. Secondly, we verify the discrete part of the Hecke orbit conjecture, which amounts to the assertion that the irreducible components of a non-basic central leaf belong to a single prime-to-$p$ Hecke orbit, generalizing preceding works by Chai, Oort, Yu, et al. We also show purely local criteria for irreducibility of central leaves. Our proof is based on a Langlands–Kottwitz type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of $p$-adic groups due to Howe–Moore and Casselman.

Notre théorème principal décrit la cohomologie en degré $0$ des variétés d’Igusa non basiques en termes de représentations automorphes de dimension $1$ dans le cadre des réductions modulo $p$ des variétés de Shimura de type Hodge avec niveau hyper-spécial en $p$. Nous obtenons comme application une approche complètement nouvelle de deux questions géométriques. Premièrement, nous déduisons l’irréductibilité de la tour d’Igusa et sa généralisation aux variétés d’Igusa non basiques dans la même généralité, ce qui étend des résultats d’Igusa, Ribet, Falting-Chai, Hida, et d’autres. Deuxièmement, nous vérifions la partie discrète de la conjecture des orbites de Hecke, qui revient à l’assertion que les composantes irréductibles d’une feuille centrale non basique appartiennent à une unique orbite sous l’action de l’algèbre de Hecke première à $p$, ce qui généralise des travaux de Chai, Oort, Yu, entre autres. Nous démontrons aussi des critères purement locaux d’irréductibilité de la feuille centrale. Notre preuve est basée sur une formule de type Langlands-Kottwitz pour les variétés d’Igusa due à Mack-Crane, sur une étude asymptotique de la formule des traces, et sur une estimée pour les représentations unitaires et leurs modules de Jacquet en théorie des représentations des groupes $p$-adiques due à Howe-Moore et à Casselman.

Classification : 11G18
Keywords: Shimura varieties, Igusa varieties, central leaves, automorphic representations, trace formula
Mots-clés : Variétés de Shimura, variétés d’Igusa, feuilles centrales, représentations automorphes, formule des traces

[ACC + 23] Allen, Patrick B.; Calegari, Frank; Caraiani, Ana; Gee, Toby; Helm, David; Le Hung, Bao V.; Newton, James; Scholze, Peter; Taylor, Richard; Thorne, Jack A. Potential automorphy over CM fields, Ann. of Math. (2), Volume 197 (2023) no. 3, pp. 897-1113 | Zbl | MR | DOI

[AIP16] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent The adic, cuspidal, Hilbert eigenvarieties, Res. Math. Sci., Volume 3 (2016), 34, 36 pages | MR | Zbl | DOI

[AIP18] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent Le halo spectral, Ann. Sci. École Norm. Sup. (4), Volume 51 (2018) no. 3, pp. 603-655 | Zbl | MR | DOI

[Art89] Arthur, James The $L^2$-Lefschetz numbers of Hecke operators, Invent. Math., Volume 97 (1989) no. 2, pp. 257-290 | Zbl | MR | DOI

[Art96] Arthur, James On local character relations, Selecta Math. (N.S.), Volume 2 (1996) no. 4, pp. 501-579 | Zbl | MR | DOI

[BC17] Bergeron, Nicolas; Clozel, Laurent Sur la cohomologie des variétés hyperboliques de dimension $7$ trialitaires, Israel J. Math., Volume 222 (2017) no. 1, pp. 333-400 | Zbl | DOI

[BDK86] Bernstein, J.; Deligne, P.; Kazhdan, D. Trace Paley-Wiener theorem for reductive $p$-adic groups, J. Analyse Math., Volume 47 (1986), pp. 180-192 | Zbl | MR | DOI

[BMS22] Bertoloni Meli, Alexander; Shin, Sug Woo A stable trace formula for Igusa varieties, II, 2022 | arXiv

[Bor79] Borel, A. Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math.), Volume XXXIII, American Mathematical Society, Providence, R.I., 1979, pp. 27-61 | Zbl

[Bor91] Borel, Armand Linear algebraic groups, Graduate Texts in Math., 126, Springer-Verlag, New York, 1991, xii+288 pages | DOI

[Boy07] Boyer, Pascal On the irreductibility of some Igusa varieties, 2007 | arXiv

[BS15] Bhatt, Bhargav; Scholze, Peter The pro-étale topology for schemes, Astérisque, 369, Société Mathématique de France, Paris, 2015, pp. 99-201

[BS17] Bhatt, Bhargav; Scholze, Peter Projectivity of the Witt vector affine Grassmannian, Invent. Math., Volume 209 (2017) no. 2, pp. 329-423 | Zbl | MR | DOI

[BZ77] Bernstein, I. N.; Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. I, Ann. Sci. École Norm. Sup. (4), Volume 10 (1977) no. 4, pp. 441-472 | Zbl | MR | DOI

[Cas95] Casselman, W. Introduction to the theory of admissible representations of $p$-adic reductive groups, 1995 (https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf)

[CD85] Clozel, L.; Delorme, P. Pseudo-coefficients et cohomologie des groupes de Lie réductifs réels, C. R. Acad. Sci. Paris Sér. I Math., Volume 300 (1985) no. 12, pp. 385-387 | Zbl

[CEF + 16] Caraiani, Ana; Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila $p$-adic $q$-expansion principles on unitary Shimura varieties, Directions in number theory (Assoc. Women Math. Ser.), Volume 3, Springer, Cham, 2016, pp. 197-243 | MR | Zbl | DOI

[Cha05] Chai, Ching-Li Hecke orbits on Siegel modular varieties, Geometric methods in algebra and number theory (Progress in Math.), Volume 235, Birkhäuser Boston, Boston, MA, 2005, pp. 71-107 | MR | Zbl | DOI

[Cha06] Chai, Ching-Li Hecke orbits as Shimura varieties in positive characteristic, International Congress of Mathematicians. Vol. II, European Mathematical Society, Zürich, 2006, pp. 295-312 | MR | Zbl

[Cha08] Chai, Ching-Li Methods for $p$-adic monodromy, J. Inst. Math. Jussieu, Volume 7 (2008) no. 2, pp. 247-268 | Zbl | MR | DOI

[CL10] Cluckers, Raf; Loeser, François Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2), Volume 171 (2010) no. 2, pp. 1011-1065 | MR | Zbl | DOI

[CO11] Chai, Ching-Li; Oort, Frans Monodromy and irreducibility of leaves, Ann. of Math. (2), Volume 173 (2011) no. 3, pp. 1359-1396 | MR | Zbl | DOI

[CO19] Chai, Ching-Li; Oort, Frans The Hecke orbit conjecture: a survey and outlook, Open problems in arithmetic algebraic geometry (Adv. Lect. Math. (ALM)), Volume 46, International Press, Somerville, MA, 2019, pp. 235-262 | MR | Zbl

[COU01] Clozel, Laurent; Oh, Hee; Ullmo, Emmanuel Hecke operators and equidistribution of Hecke points, Invent. Math., Volume 144 (2001) no. 2, pp. 327-351 | Zbl | MR | DOI

[CS17] Caraiani, Ana; Scholze, Peter On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2), Volume 186 (2017) no. 3, pp. 649-766 | Zbl | MR | DOI

[CS19] Caraiani, Ana; Scholze, Peter On the generic part of the cohomology of non-compact unitary Shimura varieties, 2019 | arXiv

[Dal22] Dalal, Rahul Sato-Tate equidistribution for families of automorphic representations through the stable trace formula, Algebra Number Theory, Volume 16 (2022) no. 1, pp. 59-137 | Zbl | MR | DOI

[DeB06] DeBacker, Stephen Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory, Michigan Math. J., Volume 54 (2006) no. 1, pp. 157-178 | Zbl | MR | DOI

[Del71] Deligne, Pierre Travaux de Shimura, Séminaire Bourbaki, (1970/71) (Lect. Notes in Math.), Volume 244, Springer, Berlin, 1971, pp. 123-165 (Exp. No. 389) | MR | Zbl | Numdam | DOI

[Del79] Deligne, P. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and $L$-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math.), Volume XXXIII, American Mathematical Society, Providence, R.I., 1979, pp. 247-289 | Zbl

[EM21] Eischen, Ellen; Mantovan, Elena $p$-adic families of automorphic forms in the $\mu$-ordinary setting, Amer. J. Math., Volume 143 (2021) no. 1, pp. 1-52 | Zbl | MR | DOI

[EMO01] Edixhoven, S. J.; Moonen, B. J. J.; Oort, F. Open problems in algebraic geometry, Bull. Sci. Math., Volume 125 (2001) no. 1, pp. 1-22 | Zbl | MR | DOI

[EO06] Eskin, Alex; Oh, Hee Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems, Volume 26 (2006) no. 1, pp. 163-167 | Zbl | MR | DOI

[Far04] Fargues, Laurent Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales, Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales (Astérisque), Volume 291, Société Mathématique de France, Paris, 2004, pp. 1-199 | MR | Zbl

[FK88] Flicker, Yuval Z.; Kazhdan, David A. A simple trace formula, J. Analyse Math., Volume 50 (1988), pp. 189-200 | Zbl | MR | DOI

[FLN10] Frenkel, Edward; Langlands, Robert; Ngô, Báo Châu Formule des traces et fonctorialité: le début d’un programme, Ann. Sci. Math. Québec, Volume 34 (2010) no. 2, pp. 199-243 | Zbl

[Fra98] Franke, J. Harmonic analysis in weighted $L_2$-spaces, Ann. Sci. École Norm. Sup. (4), Volume 31 (1998) no. 2, pp. 181-279 | MR | DOI

[GK21] Goren, Eyal Z.; Kassaei, Payman L. $p$-adic dynamics of Hecke operators on modular curves, J. Théor. Nombres Bordeaux, Volume 33 (2021) no. 2, pp. 387-431 http://jtnb.cedram.org/... | Zbl | MR | Numdam | DOI

[GKM97] Goresky, M.; Kottwitz, R.; MacPherson, R. Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J., Volume 89 (1997) no. 3, pp. 477-554 | Zbl | MR | DOI

[Ham17] Hamacher, Paul The almost product structure of Newton strata in the deformation space of a Barsotti-Tate group with crystalline Tate tensors, Math. Z., Volume 287 (2017) no. 3-4, pp. 1255-1277 | Zbl | MR | DOI

[Ham19] Hamacher, Paul The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type, J. Algebraic Geom., Volume 28 (2019) no. 4, pp. 721-749 | Zbl | MR | DOI

[He14] He, Xuhua Geometric and homological properties of affine Deligne-Lusztig varieties, Ann. of Math. (2), Volume 179 (2014) no. 1, pp. 367-404 | Zbl | MR | DOI

[Hid04] Hida, Haruzo $p$-adic automorphic forms on Shimura varieties, Springer Monogr. in Math., Springer-Verlag, New York, 2004 | DOI

[Hid09] Hida, Haruzo Irreducibility of the Igusa tower, Acta Math. Sinica (N.S.), Volume 25 (2009) no. 1, pp. 1-20 | Zbl | MR | DOI

[Hid11] Hida, Haruzo Irreducibility of the Igusa tower over unitary Shimura varieties, On certain $L$-functions (Clay Math. Proc.), Volume 13, American Mathematical Society, Providence, RI, 2011, pp. 187-203 | Zbl | MR

[HK19] Hamacher, Paul; Kim, Wansu $\ell$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients, Math. Ann., Volume 375 (2019) no. 3-4, pp. 973-1044 | Zbl | DOI

[HM79] Howe, Roger E.; Moore, Calvin C. Asymptotic properties of unitary representations, J. Funct. Anal., Volume 32 (1979) no. 1, pp. 72-96 | Zbl | MR | DOI

[HMRL20] Herrero, Sebastián; Menares, Ricardo; Rivera-Letelier, Juan $p$-adic distribution of CM points and Hecke orbits I: Convergence towards the Gauss point, Algebra Number Theory, Volume 14 (2020) no. 5, pp. 1239-1290 | Zbl | MR | DOI

[How20] Howe, Sean A unipotent circle action on $p$-adic modular forms, Trans. Amer. Math. Soc. Ser. B, Volume 7 (2020), pp. 186-226 | Zbl | MR | DOI

[HT01] Harris, M.; Taylor, R. The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, 151, Princeton University Press, Princeton, NJ, 2001

[HZZ21] He, Xuhua; Zhou, Rong; Zhu, Yihang Stabilizers of irreducible components of affine Deligne–Lusztig varieties, 2021 | arXiv

[Igu68] Igusa, Jun-ichi On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan, Volume 20 (1968), pp. 96-106 | MR | DOI

[Kaz86] Kazhdan, David Cuspidal geometry of $p$-adic groups, J. Analyse Math., Volume 47 (1986), pp. 1-36 | MR | Zbl | DOI

[Kim19] Kim, Wansu On central leaves of Hodge-type Shimura varieties with parahoric level structure, Math. Z., Volume 291 (2019) no. 1-2, pp. 329-363 | Zbl | MR | DOI

[Kis10] Kisin, Mark Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc., Volume 23 (2010) no. 4, pp. 967-1012 | Zbl | MR | DOI

[Kis17] Kisin, Mark $\mathrm{Mod}p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc., Volume 30 (2017) no. 3, pp. 819-914 | Zbl | MR | DOI

[KM85] Katz, Nicholas M.; Mazur, Barry Arithmetic moduli of elliptic curves, Annals of Math. Studies, 108, Princeton University Press, Princeton, NJ, 1985 | DOI

[KMP16] Kim, Wansu; Madapusi Pera, Keerthi 2-adic integral canonical models, Forum Math. Sigma, Volume 4 (2016), e28, 34 pages | Zbl | MR | DOI

[KMPS22] Kisin, Mark; Madapusi Pera, Keerthi; Shin, Sug Woo Honda-Tate theory for Shimura varieties, Duke Math. J., Volume 171 (2022) no. 7, pp. 1559-1614 | Zbl | MR | DOI

[Kot82] Kottwitz, Robert E. Rational conjugacy classes in reductive groups, Duke Math. J., Volume 49 (1982) no. 4, pp. 785-806 http://projecteuclid.org/euclid.dmj/1077315531 | MR | Zbl

[Kot84a] Kottwitz, Robert E. Shimura varieties and twisted orbital integrals, Math. Ann., Volume 269 (1984) no. 3, pp. 287-300 | Zbl | MR | DOI

[Kot84b] Kottwitz, Robert E. Stable trace formula: cuspidal tempered terms, Duke Math. J., Volume 51 (1984) no. 3, pp. 611-650 | Zbl | MR | DOI

[Kot85] Kottwitz, Robert E. Isocrystals with additional structure, Compositio Math., Volume 56 (1985) no. 2, pp. 201-220 | Numdam | MR | Zbl

[Kot86] Kottwitz, Robert E. Stable trace formula: elliptic singular terms, Math. Ann., Volume 275 (1986) no. 3, pp. 365-399 | MR | Zbl | DOI

[Kot88] Kottwitz, Robert E. Tamagawa numbers, Ann. of Math. (2), Volume 127 (1988) no. 3, pp. 629-646 | Zbl | MR | DOI

[Kot90] Kottwitz, Robert E. Shimura varieties and $\lambda$-adic representations, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) (Perspect. Math.), Volume 10, Academic Press, Boston, MA, 1990, pp. 161-209 | Zbl | MR

[Kot92a] Kottwitz, Robert E. On the $\lambda$-adic representations associated to some simple Shimura varieties, Invent. Math., Volume 108 (1992) no. 3, pp. 653-665 | Zbl | MR | DOI

[Kot92b] Kottwitz, Robert E. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444 | Zbl | MR | DOI

[Kot97] Kottwitz, Robert E. Isocrystals with additional structure. II, Compositio Math., Volume 109 (1997) no. 3, pp. 255-339 | Zbl | MR | DOI

[KS99] Kottwitz, Robert E.; Shelstad, D. Foundations of twisted endoscopy, Astérisque, 255, Société Mathématique de France, Paris, 1999 | Numdam

[KST20] Kim, Ju-Lee; Shin, Sug Woo; Templier, Nicolas Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families, Adv. Math., Volume 362 (2020), 106955, 57 pages | Zbl | MR | DOI

[KSZ] Kisin, Mark; Shin, Sug Woo; Zhu, Yihang The stable trace formula for certain Shimura varieties of abelian type

[KV16] Kazhdan, David; Varshavsky, Yakov Geometric approach to parabolic induction, Selecta Math. (N.S.), Volume 22 (2016) no. 4, pp. 2243-2269 | Zbl | MR | DOI

[Lab99] Labesse, J.-P. Cohomologie, stabilisation et changement de base, Astérisque, 257, Société Mathématique de France, Paris, 1999 | Numdam

[Lan79] Langlands, R. P. Stable conjugacy: definitions and lemmas, Canad. Appl. Math. Quart., Volume 31 (1979) no. 4, pp. 700-725 | Zbl | MR | DOI

[Lan83] Langlands, R. P. Les débuts d’une formule des traces stable, Publications Mathématiques de l’Université Paris VII, 13, Université de Paris VII, U.E.R. de Mathématiques, Paris, 1983

[Lan89] Langlands, R. P. On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups (Math. Surveys Monogr.), Volume 31, American Mathematical Society, Providence, RI, 1989, pp. 101-170 | Zbl | MR | DOI

[Lan13] Lan, Kai-Wen Arithmetic compactifications of PEL-type Shimura varieties, Lond. Math. Soc. Monogr. Ser., 36, Princeton University Press, Princeton, NJ, 2013 | DOI

[Lau97] Laumon, Gérard Sur la cohomologie à supports compacts des variétés de Shimura pour $\mathrm{GSp}{\left(4\right)}_{\mathbf{Q}}$, Compositio Math., Volume 105 (1997) no. 3, pp. 267-359 | Zbl | DOI

[Lau05] Laumon, Gérard Fonctions zêtas des variétés de Siegel de dimension trois, Formes automorphes. II. Le cas du groupe $\mathrm{G}\mathrm{Sp}\left(4\right)$ (Astérisque), Volume 302, Société Mathématique de France, Paris, 2005, pp. 1-66 | Zbl | Numdam

[Lov17] Lovering, T. Filtered F-crystals on Shimura varieties of abelian type, 2017 | arXiv

[LR87] Langlands, R. P.; Rapoport, M. Shimuravarietäten und Gerben, J. reine angew. Math., Volume 378 (1987), pp. 113-220 | Zbl

[LS87] Langlands, R. P.; Shelstad, D. On the definition of transfer factors, Math. Ann., Volume 278 (1987) no. 1-4, pp. 219-271 | Zbl | MR | DOI

[Man05] Mantovan, Elena On the cohomology of certain PEL-type Shimura varieties, Duke Math. J., Volume 129 (2005) no. 3, pp. 573-610 | Zbl | MR | DOI

[MC22] Mack-Crane, Sander Counting points on Igusa varieties of Hodge type, 2022 | arXiv

[Mil05] Milne, J. S. Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties (Clay Math. Proc.), Volume 4, American Mathematical Society, Providence, RI, 2005, pp. 265-378 | MR | Zbl

[MP19] Madapusi Pera, Keerthi Toroidal compactifications of integral models of Shimura varieties of Hodge type, Ann. Sci. École Norm. Sup. (4), Volume 52 (2019) no. 2, pp. 393-514 | Zbl | MR | DOI

[MST22] Maulik, Davesh; Shankar, Ananth N.; Tang, Yunqing Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture, Invent. Math., Volume 228 (2022) no. 3, pp. 1075-1143 | MR | Zbl | DOI

[Ngô10] Ngô, Bao Châu Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. (2010) no. 111, pp. 1-169 | Zbl | Numdam | DOI

[Oki23] Oki, Yasuhiro On the connected components of Shimura varieties for CM unitary groups in odd variables, J. Number Theory, Volume 245 (2023), pp. 1-64 | Zbl | MR | DOI

[Oor19] Oort, Frans Appendix 1: Some questions in algebraic geometry, Open problems in arithmetic algebraic geometry (Adv. Lect. Math. (ALM)), Volume 46, International Press, Somerville, MA, 2019, pp. 263-283 | Zbl | MR

[PR94] Platonov, V.; Rapinchuk, A. Algebraic groups and number theory, Pure and Applied Math., 139, Academic Press, Inc., Boston, MA, 1994

[Pra82] Prasad, Gopal Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. math. France, Volume 110 (1982) no. 2, pp. 197-202 | Zbl | Numdam | MR | DOI

[PST21] Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob Canonical Heights on Shimura Varieties and the André-Oort Conjecture, 2021 | arXiv

[Rie70] Riehm, Carl The norm $1$ group of a $𝔓$-adic division algebra, Amer. J. Math., Volume 92 (1970), pp. 499-523 | MR | DOI

[Rog83] Rogawski, Jonathan D. Representations of $\mathrm{GL}\left(n\right)$ and division algebras over a $p$-adic field, Duke Math. J., Volume 50 (1983) no. 1, pp. 161-196 http://projecteuclid.org/... | MR | Zbl

[RR96] Rapoport, M.; Richartz, M. On the classification and specialization of $F$-isocrystals with additional structure, Compositio Math., Volume 103 (1996) no. 2, pp. 153-181 | Zbl | Numdam | MR

[RZ96] Rapoport, M.; Zink, Th. Period spaces for $p$-divisible groups, Annals of Math. Studies, 141, Princeton University Press, Princeton, NJ, 1996 | DOI

[Ser96] Serre, J.-P. Two letters on quaternions and modular forms (mod $p$), Israel J. Math., Volume 95 (1996), pp. 281-299 (With introduction, appendix and references by R. Livné) | MR | DOI

[SGA4-3] Artin, M.; Grothendieck, A.; Verdier, J.-L. Théorie des topos et cohomologie étale des schémas. Tome 3, Lect. Notes in Math., 305, Springer-Verlag, Berlin-New York, 1972–1973 Séminaire de Géométrie Algébrique du Bois–Marie 1963–64 (SGA 4)

[Shi09] Shin, Sug Woo Counting points on Igusa varieties, Duke Math. J., Volume 146 (2009) no. 3, pp. 509-568 | Zbl | MR | DOI

[Shi10] Shin, Sug Woo A stable trace formula for Igusa varieties, J. Inst. Math. Jussieu, Volume 9 (2010) no. 4, pp. 847-895 | Zbl | MR | DOI

[Shi11] Shin, S. W. Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), Volume 173 (2011) no. 3, pp. 1645-1741 | Zbl | MR | DOI

[Shi12] Shin, Sug Woo On the cohomology of Rapoport-Zink spaces of EL-type, Amer. J. Math., Volume 134 (2012) no. 2, pp. 407-452 | Zbl | MR | DOI

[Sta21] Stacks project authors The Stacks project, https://stacks.math.columbia.edu, 2021

[Ste65] Steinberg, Robert Regular elements of semisimple algebraic groups, Publ. Math. Inst. Hautes Études Sci. (1965) no. 25, pp. 49-80 | MR | DOI

[SZ22] Shen, Xu; Zhang, Chao Stratifications in good reductions of Shimura varieties of abelian type, Asian J. Math., Volume 26 (2022) no. 2, pp. 167-226 | Zbl | MR | DOI

[Tsi18] Tsimerman, Jacob The André-Oort conjecture for ${𝒜}_g$, Ann. of Math. (2), Volume 187 (2018) no. 2, pp. 379-390 | Zbl | DOI

[vD72] van Dijk, G. Computation of certain induced characters of $p$-adic groups, Math. Ann., Volume 199 (1972), pp. 229-240 | Zbl | MR | DOI

[vH10] van Hoften, Pol Mod $p$ points on Shimura varieties of parahoric level, 2010 | arXiv

[vHX21] van Hoften, Pol; Xiao, Luciena Xiao Monodromy and irreducibility of Igusa varieties, 2021 | arXiv

[Wal97] Waldspurger, J.-L. Le lemme fondamental implique le transfert, Compositio Math., Volume 105 (1997) no. 2, pp. 153-236 | Zbl | MR | DOI

[Wal06] Waldspurger, J.-L. Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu, Volume 5 (2006) no. 3, pp. 423-525 | Zbl | DOI

[Wal08] Waldspurger, J.-L. L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc., 194, no. 908, American Mathematical Society, Providence, RI, 2008, x+261 pages | DOI

[Wor13] Wortmann, Daniel The $\mu$-ordinary locus for Shimura varieties of Hodge type, 2013 | arXiv

[Xia20] Xiao, Luciena On the Hecke orbit conjecture for PEL type Shimura varieties, 2020 | arXiv

[Xu16] Xu, Bin On a lifting problem of L-packets, Compositio Math., Volume 152 (2016) no. 9, pp. 1800-1850 | Zbl | MR | DOI

[Xu17] Xu, Bin On the cuspidal support of discrete series for $p$-adic quasisplit $Sp\left(N\right)$ and $SO\left(N\right)$, Manuscripta Math., Volume 154 (2017) no. 3-4, pp. 441-502 | Zbl | MR | DOI

[YCO20] Yu, Chia-Fu; Chai, Ching-Li; Oort, Frans Stratifying Lie strata of Hilbert modular varieties, Taiwanese J. Math., Volume 24 (2020) no. 6, pp. 1307-1352 | Zbl | MR | DOI

[Zha15] Zhang, Chao Stratifications and foliations for good reductions of Shimura varieties of Hodge type, 2015 | arXiv

[Zho19] Zhou, R. Motivic cohomology of quaternionic Shimura varieties and level raising, 2019 | arXiv

[Zho20] Zhou, R. Coherent sheaves on the stack of Langlands parameters, 2020 | arXiv

[Zhu17] Zhu, Xinwen Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2), Volume 185 (2017) no. 2, pp. 403-492 | Zbl | MR | DOI

[ZZ20] Zhou, Rong; Zhu, Yihang Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties, Camb. J. Math., Volume 8 (2020) no. 1, pp. 149-241 | Zbl | MR | DOI