Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

Lafforgue, Vincent¹

¹CNRS et Institut Fourier, UMR 5582, Université Grenoble Alpes, 100 rue des Maths, 38610 Gières, France

Journal of the American Mathematical Society, Tome 31 (2018) no. 3, pp. 719-891

Résumé

For any reductive group $G$ over a global function field, we use the cohomology of $G$-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for $G$ in the “automorphic to Galois” direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Détail

DOI : 10.1090/jams/897

Bibliographie

[1] Revêtements étales et groupe fondamental (SGA 1) 2003

[2] Théorie des topos et cohomologie étale des schémas. Tome 3 1973

[3] Cohomologie 𝑙-adique et fonctions 𝐿 1977

[4] Arasteh Rad, Esmail, Hartl, Urs Uniformizing The Moduli Stacks of Global G-Shtukas

[5] Arthur, James Unipotent automorphic representations: conjectures Astérisque 1989 13 71

[6] Badulescu, Alexandru Ioan, Roche, Philippe Global Jacquet-Langlands correspondence for division algebras in characteristic 𝑝 Int. Math. Res. Not. IMRN 2017 2172 2206

[7] Bate, Michael, Martin, Benjamin, Röhrle, Gerhard A geometric approach to complete reducibility Invent. Math. 2005 177 218

[8] Beauville, Arnaud, Laszlo, Yves Un lemme de descente C. R. Acad. Sci. Paris Sér. I Math. 1995 335 340

[9] Behrend, Kai Achim The Lefschetz trace formula for the moduli stack of principal bundles 1991 148

[10] Behrend, Kai Moduli spaces for vector bundles with level structures on algebraic curves. Diplomarbeit, non published, 1996. Summary available

[11] Beĭlinson, A. A., Bernstein, J., Deligne, P. Faisceaux pervers 1982 5 171

[12] Beilinson, Alexander, Drinfeld, Vladimir Quantization of Hitchin’s integrable system and Hecke eigensheaves 1999

[13] Beilinson, Alexander, Drinfeld, Vladimir Chiral algebras 2004

[14] Beilinson, A. Langlands parameters for Heisenberg modules 2006 51 60

[15] Bellaïche, Joël, Chenevier, Gaëtan Families of Galois representations and Selmer groups Astérisque 2009

[16] Bezrukavnikov, Roman, Finkelberg, Michael Equivariant Satake category and Kostant-Whittaker reduction Mosc. Math. J. 2008

[17] Blasius, Don On multiplicities for 𝑆𝐿(𝑛) Israel J. Math. 1994 237 251

[18] Blasius, Don, Rogawski, Jonathan D. Zeta functions of Shimura varieties 1994 525 571

[19] Böckle, Gebhard, Harris, Michael, Khare, Chandrashekhar, Thorne, Jack ̂𝐺-local systems on smooth projective curves are potentially automorphic. 2016

[20] Borel, A. Automorphic 𝐿-functions 1979 27 61

[21] Borel, Armand Linear algebraic groups 1991

[22] Borel, A., Jacquet, H. Automorphic forms and automorphic representations 1979 189 207

[23] Bourbaki, N. Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples 2012

[24] Bruhat, F., Tits, J. Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée Inst. Hautes Études Sci. Publ. Math. 1984 197 376

[25] Braverman, A., Gaitsgory, D. Geometric Eisenstein series Invent. Math. 2002 287 384

[26] Braverman, Alexander, Varshavsky, Yakov From automorphic sheaves to the cohomology of the moduli spaces of 𝐹-bundles 2006

[27] Buzzard, Kevin, Gee, Toby The conjectural connections between automorphic representations and Galois representations 2014 135 187

[28] Cartier, P. Representations of 𝑝-adic groups: a survey 1979 111 155

[29] Chenevier, Gaëtan Représentations galoisiennes automorphes et conséquences arithmétiques des conjectures de Langlands et Arthur. Habilitation à diriger des recherches, 2013

[30] Chenevier, Gaëtan The 𝑝-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings 2014 221 285

[31] Chandler, Bruce, Magnus, Wilhelm The history of combinatorial group theory 1982

[32] Cvitanović, Predrag Group theory 2008

[33] Cogdell, J. W., Piatetski-Shapiro, I. I. Converse theorems for 𝐺𝐿_{𝑛} Inst. Hautes Études Sci. Publ. Math. 1994 157 214

[34] Deligne, Pierre Formes modulaires et représentations 𝑙-adiques 1971

[35] Deligne, Pierre La conjecture de Weil. II Inst. Hautes Études Sci. Publ. Math. 1980 137 252

[36] Deligne, P. Catégories tannakiennes 1990 111 195

[37] Deligne, Pierre Finitude de l’extension de ℚ engendrée par des traces de Frobenius, en caractéristique finie Mosc. Math. J. 2012

[38] Drinfel′D, V. G. Langlands’ conjecture for 𝐺𝐿(2) over functional fields 1980 565 574

[39] Drinfel′D, V. G. Moduli varieties of 𝐹-sheaves Funktsional. Anal. i Prilozhen. 1987 23 41

[40] Drinfel′D, V. G. Cohomology of compactified moduli varieties of 𝐹-sheaves of rank 2 Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 1987

[41] Drinfel′D, V. G. Proof of the Petersson conjecture for 𝐺𝐿(2) over a global field of characteristic 𝑝 Funktsional. Anal. i Prilozhen. 1988

[42] Drinfel′D, V. G., Simpson, Carlos 𝐵-structures on 𝐺-bundles and local triviality Math. Res. Lett. 1995 823 829

[43] Drinfeld, Vladimir On a conjecture of Deligne Mosc. Math. J. 2012

[44] Drinfeld, Vladimir On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field 2015

[45] Eisenbud, David Commutative algebra 1995

[46] Esnault, Hélène, Kerz, Moritz A finiteness theorem for Galois representations of function fields over finite fields (after Deligne) Acta Math. Vietnam. 2012 531 562

[47] Faltings, Gerd, Chai, Ching-Li Degeneration of abelian varieties 1990

[48] Faltings, Gerd Group schemes with strict 𝒪-action Mosc. Math. J. 2002 249 279

[49] Finkelberg, Michael, Lysenko, Sergey Twisted geometric Satake equivalence J. Inst. Math. Jussieu 2010 719 739

[50] Frobenius, Ferdinand G. Über die Primfactoren der Gruppendeterminante Ges. Abh III (S’ber. Akad. Wiss. Berlin) 1343–1382 1896

[51] Gaitsgory, D. Construction of central elements in the affine Hecke algebra via nearby cycles Invent. Math. 2001 253 280

[52] Gaitsgory, D. On a vanishing conjecture appearing in the geometric Langlands correspondence Ann. of Math. (2) 2004 617 682

[53] Gaitsgory, D. On de Jong’s conjecture Israel J. Math. 2007 155 191

[54] Gaitsgory, Dennis Outline of the proof of the geometric Langlands conjecture for 𝐺𝐿₂ Astérisque 2015 1 112

[55] Gaitsgory, Dennis, Lysenko, Sergey Parameters and duality for the metaplectic geometric Langlands theory. 2016

[56] Gan, Wee Teck, Gao, Fan The Langlands-Weissman Program for Brylinski-Deligne extensions 2014

[57] Genestier, Alain, Tilouine, Jacques Systèmes de Taylor-Wiles pour 𝐺𝑆𝑝₄ Astérisque 2005 177 290

[58] Genestier, Alain, Lafforgue, Vincent Théorie de Fontaine en égales caractéristiques Ann. Sci. Éc. Norm. Supér. (4) 2011 263 360

[59] Genestier, Alain, Lafforgue, Vincent Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale 2017

[60] Ginzburg, Victor Perverse sheaves on a loop group and Langlands duality 1995

[61] Gross, Benedict H. On the Satake isomorphism 1998 223 237

[62] Harder, G. Chevalley groups over function fields and automorphic forms Ann. of Math. (2) 1974 249 306

[63] Heinloth, Jochen Uniformization of 𝒢-bundles Math. Ann. 2010 499 528

[64] Heinloth, Jochen, Schmitt, Alexander H. W. The cohomology rings of moduli stacks of principal bundles over curves Doc. Math. 2010 423 488

[65] Hilbert, David Ueber die vollen Invariantensysteme Math. Ann. 1893 313 373

[66] Hiraga, Kaoru, Saito, Hiroshi On 𝐿-packets for inner forms of 𝑆𝐿_{𝑛} Mem. Amer. Math. Soc. 2012

[67] Huneke, Craig, Swanson, Irena Integral closure of ideals, rings, and modules 2006

[68] Huybrechts, Daniel, Lehn, Manfred The geometry of moduli spaces of sheaves 2010

[69] Jacquet, H., Shalika, J. A. On Euler products and the classification of automorphic representations. I Amer. J. Math. 1981 499 558

[70] Jacquet, H., Shalika, J. A. On Euler products and the classification of automorphic forms. II Amer. J. Math. 1981 777 815

[71] Jacquet, H., Piatetskii-Shapiro, I. I., Shalika, J. A. Rankin-Selberg convolutions Amer. J. Math. 1983 367 464

[72] Jannsen, Uwe On finite-dimensional motives and Murre’s conjecture 2007 112 142

[73] Kazhdan, D. A. An introduction to Drinfel′d’s “Shtuka” 1979 347 356

[74] Kazhdan, David, Varshavsky, Yakov On the cohomology of the moduli spaces of 𝐹-bundles: stable cuspidal Deligne-Lusztig part, in preparation 2013

[75] Kostant, Bertram The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group Amer. J. Math. 1959 973 1032

[76] Kottwitz, Robert E. Stable trace formula: cuspidal tempered terms Duke Math. J. 1984 611 650

[77] Kottwitz, Robert E. Stable trace formula: elliptic singular terms Math. Ann. 1986 365 399

[78] Kottwitz, Robert E. Shimura varieties and 𝜆-adic representations 1990 161 209

[79] Labesse, J.-P., Langlands, R. P. 𝐿-indistinguishability for 𝑆𝐿(2) Canadian J. Math. 1979 726 785

[80] Lafforgue, Laurent Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson Astérisque 1997

[81] Lafforgue, Laurent Une compactification des champs classifiant les chtoucas de Drinfeld J. Amer. Math. Soc. 1998 1001 1036

[82] Lafforgue, Laurent Chtoucas de Drinfeld et correspondance de Langlands Invent. Math. 2002 1 241

[83] Lafforgue, Laurent Chtoucas de Drinfeld, formule des traces d’Arthur-Selberg et correspondance de Langlands 2002 383 400

[84] Langlands, R. P. Problems in the theory of automorphic forms 1970 18 61

[85] Lapid, Erez M. Some results on multiplicities for 𝑆𝐿(𝑛) Israel J. Math. 1999 157 186

[86] Larsen, Michael On the conjugacy of element-conjugate homomorphisms Israel J. Math. 1994 253 277

[87] Larsen, Michael On the conjugacy of element-conjugate homomorphisms. II Quart. J. Math. Oxford Ser. (2) 1996 73 85

[88] Laszlo, Yves, Olsson, Martin The six operations for sheaves on Artin stacks. I. Finite coefficients Publ. Math. Inst. Hautes Études Sci. 2008 109 168

[89] Laszlo, Yves, Olsson, Martin The six operations for sheaves on Artin stacks. II. Adic coefficients Publ. Math. Inst. Hautes Études Sci. 2008 169 210

[90] Lau, Eike Sören On generalised 𝐷-shtukas 2004

[91] Lau, Eike On degenerations of 𝒟-shtukas Duke Math. J. 2007 351 389

[92] Laumon, G. Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil Inst. Hautes Études Sci. Publ. Math. 1987 131 210

[93] Laumon, Gérard Cohomology of Drinfeld modular varieties. Part I 1996

[94] Laumon, Gérard Cohomology of Drinfeld modular varieties. Part II 1997

[95] Laumon, G. Drinfeld shtukas 1997 50 109

[96] Laumon, Gérard, Moret-Bailly, Laurent Champs algébriques 2000

[97] Laumon, G., Rapoport, M., Stuhler, U. 𝒟-elliptic sheaves and the Langlands correspondence Invent. Math. 1993 217 338

[98] Lubotzky, Alexander, Magid, Andy R. Varieties of representations of finitely generated groups Mem. Amer. Math. Soc. 1985

[99] Lusztig, George Singularities, character formulas, and a 𝑞-analog of weight multiplicities 1983 208 229

[100] Lysenko, Sergey Moduli of metaplectic bundles on curves and theta-sheaves Ann. Sci. École Norm. Sup. (4) 2006 415 466

[101] Lysenko, Sergey Geometric theta-lifting for the dual pair 𝕊𝕆₂ₘ,𝕊𝕡_{2𝕟} Ann. Sci. Éc. Norm. Supér. (4) 2011 427 493

[102] Lysenko, Sergey Twisted geometric Satake equivalence: reductive case 2014

[103] Lysenko, Sergey Twisted geometric Langlands correspondence for a torus Int. Math. Res. Not. IMRN 2015 8680 8723

[104] Mcnamara, Peter J. Principal series representations of metaplectic groups over local fields 2012 299 327

[105] Mirković, I., Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings Ann. of Math. (2) 2007 95 143

[106] Mumford, D., Fogarty, J., Kirwan, F. Geometric invariant theory 1994

[107] Ngô, Báo Châu Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye Ann. Sci. École Norm. Sup. (4) 1999 619 679

[108] Ngô, Bao Châu 𝒟-chtoucas de Drinfeld à modifications symétriques et identité de changement de base Ann. Sci. École Norm. Sup. (4) 2006 197 243

[109] Ngô, Bao Châu Fibration de Hitchin et endoscopie Invent. Math. 2006 399 453

[110] Ngô, Bao Châu, Ngô Dac, Tuân Comptage de 𝐺-chtoucas: la partie régulière elliptique J. Inst. Math. Jussieu 2008 181 203

[111] Tuan Ngo Dac Compactification des champs de chtoucas et théorie géométrique des invariants Astérisque 2007 124

[112] Ngô Dac, Tuân Sur le développement spectral de la formule des traces d’Arthur-Selberg sur les corps de fonctions Bull. Soc. Math. France 2009 545 586

[113] Ngô Dac, Tuân Formule des traces d’Arthur-Selberg pour les corps de fonctions II

[114] Pappas, G., Rapoport, M. Twisted loop groups and their affine flag varieties Adv. Math. 2008 118 198

[115] Peterson, Elisha On a Diagrammatic Proof of the Cayley-Hamilton Theorem 2009

[116] Pjateckij-Šapiro, I. I. Euler subgroups 1975 597 620

[117] Poincaré, H. Sur les groupes des équations linéaires Acta Math. 1884 201 312

[118] Procesi, C. The invariant theory of 𝑛×𝑛 matrices Advances in Math. 1976 306 381

[119] Procesi, Claudio A formal inverse to the Cayley-Hamilton theorem J. Algebra 1987 63 74

[120] Reich, Ryan Cohen Twisted geometric Satake equivalence via gerbes on the factorizable Grassmannian Represent. Theory 2012 345 449

[121] Richardson, R. W. Conjugacy classes of 𝑛-tuples in Lie algebras and algebraic groups Duke Math. J. 1988 1 35

[122] Richarz, Timo A new approach to the geometric Satake equivalence Doc. Math. 2014 209 246

[123] Rouquier, Raphaël Caractérisation des caractères et pseudo-caractères J. Algebra 1996 571 586

[124] Satake, Ichirô Theory of spherical functions on reductive algebraic groups over 𝔭-adic fields Inst. Hautes Études Sci. Publ. Math. 1963 5 69

[125] Schieder, Simon The Harder-Narasimhan stratification of the moduli stack of 𝐺-bundles via Drinfeld’s compactifications Selecta Math. (N.S.) 2015 763 831

[126] Serre, Jean-Pierre Complète réductibilité Astérisque 2005

[127] Seshadri, C. S. Fibrés vectoriels sur les courbes algébriques 1982 209

[128] Shalika, J. A. The multiplicity one theorem for 𝐺𝐿_{𝑛} Ann. of Math. (2) 1974 171 193

[129] Sorger, Christoph Lectures on moduli of principal 𝐺-bundles over algebraic curves 2000 1 57

[130] Taylor, Richard Galois representations associated to Siegel modular forms of low weight Duke Math. J. 1991 281 332

[131] Thang, Nguyen Quoc On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic Proc. Japan Acad. Ser. A Math. Sci. 2011 203 208

[132] Varshavsky, Yakov Moduli spaces of principal 𝐹-bundles Selecta Math. (N.S.) 2004 131 166

[133] Varshavsky, Yakov A proof of a generalization of Deligne’s conjecture Electron. Res. Announc. Amer. Math. Soc. 2005 78 88

[134] Varshavsky, Yakov Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara Geom. Funct. Anal. 2007 271 319

[135] Varshavsky, Yakov Towards Langlands correspondence over function fields for split reductive groups 2009 139 147

[136] Varshavsky, Yakov Communication personnelle, janvier 2016

[137] Vinberg, E. B. On invariants of a set of matrices J. Lie Theory 1996 249 269

[138] Wang, Jonathan The moduli stack of G-bundles 2011

[139] Wang, Jonathan Peiyu On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics 2017 172

[140] Wang, Song Dimension Data, Local and Global Conjugacy in Reductive Groups 2007

[141] Wang, Song On dimension data and local vs. global conjugacy 2012 365 382

[142] Wang, Song On local and global conjugacy J. Algebra 2015 334 359

[143] Weissman, Martin H. Split metaplectic groups and their L-groups J. Reine Angew. Math. 2014 89 141

[144] Weissman, Martin H. L-groups and parameters for covering groups 2015

[145] Wiles, A. On ordinary 𝜆-adic representations associated to modular forms Invent. Math. 1988 529 573

[146] Xiao, Liang, Zhu, Xinwen Cycles on Shimura varieties via geometric Satake 2017

[147] Xue, Cong Cohomologie cuspidale des champs de chtoucas Thèse de doctorat, Université Paris-Saclay 2017

[148] Zhu, Xinwen The geometric Satake correspondence for ramified groups Ann. Sci. Éc. Norm. Supér. (4) 2015 409 451