Arthur’s multiplicity formula for GSp 4 and restriction to Sp 4
Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 469-535.

We prove the classification of discrete automorphic representations of GSp 4 explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp 4 and Sp 4 .

Nous donnons une preuve de la classification des représentations automorphes discrètes de GSp 4 expliquée dans [Art04], ainsi que de la compatibilité avec les correspondances de Langlands locales pour GSp 4 et Sp 4 .

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DOI: 10.5802/jep.99
Classification: 11F72, 11F46, 11F55
Keywords: Automorphic forms, trace formula, endoscopy, Arthur multiplicity formula, Siegel-Hilbert modular forms
Mot clés : Formes automorphes, formule des traces, endoscopie, formule de multiplicité d’Arthur, formes modulaires de Siegel-Hilbert

Toby Gee 1; Olivier Taïbi 2

1 Department of Mathematics, Imperial College London London SW7 2AZ, UK
2 CNRS et Unité de Mathématiques Pures et Appliquées, ENS de Lyon
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Arthur{\textquoteright}s multiplicity formula for $\protect \mathbf{GSp}_4$ and restriction to $\protect \mathbf{Sp}_4$},
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Toby Gee; Olivier Taïbi. Arthur’s multiplicity formula for $\protect \mathbf{GSp}_4$ and restriction to $\protect \mathbf{Sp}_4$. Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 469-535. doi : 10.5802/jep.99. https://jep.centre-mersenne.org/articles/10.5802/jep.99/

[AMR18] N. Arancibia, C. Mœglin & D. Renard - “Paquets d’Arthur des groupes classiques et unitaires”, Ann. Fac. Sci. Toulouse Math. (6) 27 (2018) no. 5, p. 1023-1105 | DOI | MR | Zbl

[AP06] J. D. Adler & D. Prasad - “On certain multiplicity one theorems”, Israel J. Math. 153 (2006), p. 221-245 | DOI | MR | Zbl

[Art01] J. Arthur - “A stable trace formula. II. Global descent”, Invent. Math. 143 (2001) no. 1, p. 157-220 | DOI | MR | Zbl

[Art02] J. Arthur - “A stable trace formula. I. General expansions”, J. Inst. Math. Jussieu 1 (2002) no. 2, p. 175-277 | DOI | MR | Zbl

[Art03] J. Arthur - “A stable trace formula. III. Proof of the main theorems”, Ann. of Math. (2) 158 (2003) no. 3, p. 769-873 | DOI | MR | Zbl

[Art04] J. Arthur - “Automorphic representations of GSp(4), in Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, p. 65-81 | MR | Zbl

[Art13] J. Arthur - The endoscopic classification of representations. Orthogonal and symplectic groups, Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013 | DOI | Zbl

[AS14] M. Asgari & F. Shahidi - “Image of functoriality for general spin groups”, Manuscripta Math. 144 (2014) no. 3-4, p. 609-638 | DOI | MR | Zbl

[Aub95] A.-M. Aubert - “Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique”, Trans. Amer. Math. Soc. 347 (1995) no. 6, p. 2179-2189 | DOI | MR | Zbl

[BCGP] G. Boxer, F. Calegari, T. Gee & V. Pilloni - “Abelian surfaces over totally real fields are potentially modular”, in preparation

[Ber84] J. N. Bernstein - “P-invariant distributions on GL(N) and the classification of unitary representations of GL (N) (non-Archimedean case)”, in Lie group representations, II (College Park, Md., 1982/1983), Lect. Notes in Math., vol. 1041, Springer, Berlin, 1984, p. 50-102 | DOI | MR | Zbl

[Bor79] A. Borel - “Automorphic L-functions”, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, p. 27-61 | Zbl

[Bou05] N. Bourbaki - Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics, Springer-Verlag, Berlin, 2005 | Zbl

[BT65] A. Borel & J. Tits - “Groupes réductifs”, Publ. Math. Inst. Hautes Études Sci. (1965) no. 27, p. 55-150 | DOI | Zbl

[BW00] A. Borel & N. Wallach - Continuous cohomology, discrete subgroups, and representations of reductive groups, Math. Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000 | MR | Zbl

[BZ76] J. N. Bernstein & A. V. Zelevinskiĭ - “Induced representations of the group GL(n) over a p-adic field”, Funktsional. Anal. i Prilozhen. 10 (1976) no. 3, p. 74-75 | MR

[BZ77] J. N. Bernstein & A. V. Zelevinskiĭ - “Induced representations of reductive 𝔭-adic groups. I”, Ann. Sci. École Norm. Sup. (4) 10 (1977) no. 4, p. 441-472 | DOI | MR

[CG15] P.-S. Chan & W. T. Gan - “The local Langlands conjecture for GSp (4) III: Stability and twisted endoscopy”, J. Number Theory 146 (2015), p. 69-133 | DOI | MR | Zbl

[Che18] G. Chenevier - “On restrictions and extensions of cusp forms” (2018), preliminary draft available at http://gaetan.chenevier.perso.math.cnrs.fr/pub.html | Zbl

[CL10] P.-H. Chaudouard & G. Laumon - “Le lemme fondamental pondéré. I. Constructions géométriques”, Compositio Math. 146 (2010) no. 6, p. 1416-1506 | DOI | Zbl

[Clo84] L. Clozel - “Théorème d’Atiyah-Bott pour les variétés 𝔭-adiques et caractères des groupes réductifs”, in Harmonic analysis on Lie groups and symmetric spaces (Kleebach, 1983), Mém. Soc. Math. France (N.S.), vol. 15, Société Mathématique de France, Paris, 1984, p. 39-64 | Zbl

[Clo86] L. Clozel - “On limit multiplicities of discrete series representations in spaces of automorphic forms”, Invent. Math. 83 (1986) no. 2, p. 265-284 | DOI | MR | Zbl

[CS80] W. Casselman & J. Shalika - “The unramified principal series of p-adic groups. II. The Whittaker function”, Compositio Math. 41 (1980) no. 2, p. 207-231 | MR | Zbl

[GJ78] S. Gelbart & H. Jacquet - “A relation between automorphic representations of GL (2) and GL(3), Ann. Sci. École Norm. Sup. (4) 11 (1978) no. 4, p. 471-542 | DOI | MR | Zbl

[GK82] S. S. Gelbart & A. W. Knapp - “L-indistinguishability and R groups for the special linear group”, Adv. in Math. 43 (1982) no. 2, p. 101-121 | DOI | MR | Zbl

[GT10] W. T. Gan & S. Takeda - “The local Langlands conjecture for Sp(4)”, Internat. Math. Res. Notices (2010) no. 15, p. 2987-3038 | DOI | MR | Zbl

[GT11a] W. T. Gan & S. Takeda - “The local Langlands conjecture for GSp(4), Ann. of Math. (2) 173 (2011) no. 3, p. 1841-1882 | DOI | MR | Zbl

[GT11b] W. T. Gan & S. Takeda - “Theta correspondences for GSp(4), Represent. Theory 15 (2011), p. 670-718 | DOI | MR | Zbl

[Hal95] T. C. Hales - “On the fundamental lemma for standard endoscopy: reduction to unit elements”, Canad. J. Math. 47 (1995) no. 5, p. 974-994 | DOI | MR | Zbl

[Hen09] G. Henniart - “Sur la fonctorialité, pour GL (4), donnée par le carré extérieur”, Moscow Math. J. 9 (2009) no. 1, p. 33-45 | DOI | Zbl

[HS12] K. Hiraga & H. Saito - On L-packets for inner forms of SL n , Mem. Amer. Math. Soc., vol. 215, no. 1013, American Mathematical Society, Providence, RI, 2012 | DOI | Zbl

[HT01] M. Harris & R. Taylor - The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001 | MR | Zbl

[Joh84] J. F. Johnson - “Lie algebra cohomology and the resolution of certain Harish-Chandra modules”, Math. Ann. 267 (1984) no. 3, p. 377-393 | DOI | MR | Zbl

[JS77] H. Jacquet & J. A. Shalika - “A non-vanishing theorem for zeta functions of GL n , Invent. Math. 38 (1976/77) no. 1, p. 1-16 | DOI | MR | Zbl

[JS81] H. Jacquet & J. A. Shalika - “On Euler products and the classification of automorphic forms. II”, Amer. J. Math. 103 (1981) no. 4, p. 777-815 | DOI | MR | Zbl

[Kal15] T. Kaletha - “Global rigid inner forms and multiplicities of discrete automorphic representations”, 2015, arXiv:1501.01667 | Zbl

[Kim03] H. H. Kim - “Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2 , J. Amer. Math. Soc. 16 (2003) no. 1, p. 139-183 | DOI | MR

[Knu91] M.-A. Knus - Quadratic and Hermitian forms over rings, Grundlehren Math. Wiss., vol. 294, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[Kot86] R. Kottwitz - “Stable trace formula: elliptic singular terms”, Math. Ann. 275 (1986) no. 3, p. 365-399 | DOI | MR | Zbl

[Kri03] M. Krishnamurthy - “The Asai transfer to GL 4 via the Langlands-Shahidi method”, Internat. Math. Res. Notices (2003) no. 41, p. 2221-2254 | DOI | MR

[Kri12] M. Krishnamurthy - “Determination of cusp forms on GL(2) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)”, J. Number Theory 132 (2012) no. 6, p. 1359-1384 | DOI | MR | Zbl

[KS99] R. E. Kottwitz & D. Shelstad - Foundations of twisted endoscopy, Astérisque, vol. 255, Société Mathématique de France, Paris, 1999 | Zbl

[KS12] R. E. Kottwitz & D. Shelstad - “On splitting invariants and sign conventions in endoscopic transfer”, 2012, arXiv:1201.5658

[Lab85] J.-P. Labesse - “Cohomologie, L-groupes et fonctorialité”, Compositio Math. 55 (1985) no. 2, p. 163-184 | MR | Zbl

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

[Lan79] R. P. Langlands - “Automorphic representations, Shimura varieties, and motives. Ein Märchen”, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, American Mathematical Society, Providence, RI, 1979, p. 205-246 | Zbl

[Lan80] R. P. Langlands - Base change for GL(2), Annals of Math. Studies, vol. 96, Princeton University Press, Princeton, N.J., 1980 | MR | Zbl

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

[Lan89] R. P. Langlands - “On the classification of irreducible representations of real algebraic groups”, in Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, American Mathematical Society, 1989, p. 101-170 | DOI | MR | Zbl

[Lem10] B. Lemaire - “Caractères tordus des représentations admissibles”, 2010, arXiv:1007.3576

[LL79] J.-P. Labesse & R. P. Langlands - “L-indistinguishability for SL(2), Canad. J. Math. 31 (1979) no. 4, p. 726-785 | DOI | MR | Zbl

[LMW15] B. Lemaire, C. Mœglin & J.-L. Waldspurger - “Le lemme fondamental pour l’endoscopie tordue: réduction aux éléments unités”, 2015, arXiv:1506.03383 | Zbl

[LW13] J.-P. Labesse & J.-L. Waldspurger - La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, vol. 31, American Mathematical Society, Providence, RI, 2013 | Zbl

[LW15] B. Lemaire & J.-L. Waldspurger - “Le lemme fondamental pour l’endoscopie tordue: le cas où le groupe endoscopique non ramifié est un tore”, 2015, arXiv:1511.08606 | Zbl

[Mez16] P. Mezo - “Tempered spectral transfer in the twisted endoscopy of real groups”, J. Inst. Math. Jussieu 15 (2016) no. 3, p. 569-612 | DOI | MR | Zbl

[Mok14] C. P. Mok - “Galois representations attached to automorphic forms on GL 2 over CM fields”, Compositio Math. 150 (2014) no. 4, p. 523-567 | DOI | MR | Zbl

[MR15] C. Mœglin & D. Renard - “Paquets d’Arthur des groupes classiques complexes”, 2015, arXiv:1507.01432 | Zbl

[MW89] C. Mœglin & J.-L. Waldspurger - “Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4) 22 (1989) no. 4, p. 605-674 | DOI | Zbl

[MW94] C. Mœglin & J.-L. Waldspurger - Décomposition spectrale et séries d’Eisenstein. Une paraphrase de l’Écriture, Progress in Math., vol. 113, Birkhäuser Verlag, Basel, 1994 | Zbl

[MW06] C. Mœglin & J.-L. Waldspurger - “Sur le transfert des traces d’un groupe classique p-adique à un groupe linéaire tordu”, Selecta Math. (N.S.) 12 (2006) no. 3-4, p. 433-515 | DOI | Zbl

[MW16a] C. Mœglin & J.-L. Waldspurger - Stabilisation de la formule des traces tordue. Vol. 1, Progress in Math., vol. 316, Birkhäuser/Springer, Cham, 2016 | MR | Zbl

[MW16b] C. Mœglin & J.-L. Waldspurger - Stabilisation de la formule des traces tordue. Vol. 2, Progress in Math., vol. 317, Birkhäuser/Springer, Cham, 2016 | MR | Zbl

[Mœg06] C. Mœglin - “Sur certains paquets d’Arthur et involution d’Aubert-Schneider-Stuhler généralisée”, Represent. Theory 10 (2006), p. 86-129 | DOI | Zbl

[Mœg11] C. Mœglin - “Multiplicité 1 dans les paquets d’Arthur aux places p-adiques”, in On certain L-functions, Clay Math. Proc., vol. 13, American Mathematical Society, Providence, RI, 2011, p. 333-374 | Zbl

[Ngô10] B. C. Ngô - “Le lemme fondamental pour les algèbres de Lie”, Publ. Math. Inst. Hautes Études Sci. (2010) no. 111, p. 1-169 | DOI | Zbl

[PR94] V. Platonov & A. Rapinchuk - Algebraic groups and number theory, Pure and Applied Math., vol. 139, Academic Press, Inc., Boston, MA, 1994 | MR | Zbl

[Ram00] D. Ramakrishnan - “Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000) no. 1, p. 45-111 | DOI | MR | Zbl

[Ram02] D. Ramakrishnan - “Modularity of solvable Artin representations of GO (4)-type”, Internat. Math. Res. Notices (2002) no. 1, p. 1-54 | DOI | MR | Zbl

[Rod73] F. Rodier - “Whittaker models for admissible representations of reductive p-adic split groups”, in Harmonic analysis on homogeneous spaces (Williams Coll., Williamstown, Mass., 1972), Proc. Sympos. Pure Math., vol. XXVI, American Mathematical Society, Providence, RI, 1973, p. 425-430 | Zbl

[RV18] A. Roche & C. R. Vinroot - “A factorization result for classical and similitude groups”, Canad. Math. Bull. 61 (2018) no. 1, 174–190 pages | DOI | MR | Zbl

[Sat63] I. Satake - “Theory of spherical functions on reductive algebraic groups over 𝔭-adic fields”, Publ. Math. Inst. Hautes Études Sci. (1963) no. 18, p. 5-69 | DOI | MR | Zbl

[Ser97] J.-P. Serre - “Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p , J. Amer. Math. Soc. 10 (1997) no. 1, p. 75-102 | DOI

[Sha74] J. A. Shalika - “The multiplicity one theorem for GL n , Ann. of Math. (2) 100 (1974), p. 171-193 | DOI | MR | Zbl

[Sha81] F. Shahidi - “On certain L-functions”, Amer. J. Math. 103 (1981) no. 2, p. 297-355 | DOI | Zbl

[Sha97] F. Shahidi - “On non-vanishing of twisted symmetric and exterior square L-functions for GL(n), Pacific J. Math. (1997), p. 311-322, Special issue in memoriam Olga Taussky-Todd | DOI | MR | Zbl

[Sha10] F. Shahidi - Eisenstein series and automorphic L-functions, Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010 | DOI | MR | Zbl

[She08] D. Shelstad - “Tempered endoscopy for real groups. III. Inversion of transfer and L-packet structure”, Represent. Theory 12 (2008), p. 369-402 | DOI | MR | Zbl

[She10] D. Shelstad - “Tempered endoscopy for real groups. II. Spectral transfer factors”, in Automorphic forms and the Langlands program, Adv. Lect. Math. (ALM), vol. 9, Int. Press, Somerville, MA, 2010, p. 236-276 | MR | Zbl

[She12] D. Shelstad - “On geometric transfer in real twisted endoscopy”, Ann. of Math. (2) 176 (2012) no. 3, p. 1919-1985 | DOI | MR | Zbl

[Sil78] A. J. Silberger - “The Langlands quotient theorem for p-adic groups”, Math. Ann. 236 (1978) no. 2, p. 95-104 | DOI | MR | Zbl

[Spr98] T. A. Springer - Linear algebraic groups, Progress in Math., vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998 | MR | Zbl

[SS97] P. Schneider & U. Stuhler - “Representation theory and sheaves on the Bruhat-Tits building”, Publ. Math. Inst. Hautes Études Sci. (1997) no. 85, p. 97-191 | DOI | MR | Zbl

[SZ14] A. J. Silberger & E.-W. Zink - “Langlands classification for L-parameters”, 2014, arXiv:1007.3576 | Zbl

[Taï19] O. Taïbi - “Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups”, J. Eur. Math. Soc. (JEMS) 21 (2019) no. 3, 839–871 pages | MR | Zbl

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

[Vog86] D. A. Vogan - “The unitary dual of GL(n) over an Archimedean field”, Invent. Math. 83 (1986) no. 3, p. 449-505 | DOI | MR | Zbl

[VW90] D. A. Vogan & N. R. Wallach - “Intertwining operators for real reductive groups”, Adv. in Math. 82 (1990) no. 2, p. 203-243 | DOI | MR | Zbl

[Wal88] N. R. Wallach - Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988 | MR | Zbl

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

[Wal03] J.-L. Waldspurger - “La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra)”, J. Inst. Math. Jussieu 2 (2003) no. 2, p. 235-333 | DOI | MR | Zbl

[War72] G. Warner - Harmonic analysis on semi-simple Lie groups. I, Grundlehren Math. Wiss., vol. 188, Springer-Verlag, New York-Heidelberg, 1972 | MR | Zbl

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

[Xu18] B. Xu - “L-packets of quasisplit GSp(2n) and GO(2n)”, Math. Ann. 370 (2018) no. 1-2, p. 71-189 | DOI | MR | Zbl

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