Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

Taïbi, Olivier

doi:10.24033/asens.2321

Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula
[Dimensions des espaces de formes automorphes en niveau un pour les groupes classiques déployés à l'aide de la formule des traces]

Taïbi, Olivier

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 2, pp. 269-344.

Résumé
Abstract

Nous démontrons des formules explicites pour le nombre de représentations automorphes cuspidales algébriques régulières et essentiellement auto-duales pour les groupes linéaires sur $ℚ$ , comme fonction des poids de Hodge. Nous en déduisons des formules explicites pour les dimensions des espaces de formes modulaires de Siegel cuspidales à valeurs vectorielles.

We derive explicit formulae for the number of level one, regular algebraic and essentially self-dual automorphic cuspidal representations of general linear groups over $ℚ$ , as a function of the Hodge weights. As a consequence, we obtain formulae for dimensions of spaces of vector-valued Siegel modular cusp forms.

Publié le : 2017-11-12

MR Zbl

DOI : 10.24033/asens.2321

Classification : 11F72, 11Y40, 11R39, 11F46, 22E47, 11-04.
Keywords: Arthur-Selberg trace formula, orbital integrals, algorithm, endoscopy, Adams-Johnson packets, Siegel modular forms.
Mot clés : Formule des traces d'Arthur-Selberg, intégrales orbitales, algorithme, endoscopie, paquets d'Adams-Johnson, formes modulaires de Siegel.

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     title = {Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {269--344},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Taïbi, Olivier. Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 2, pp. 269-344. doi : 10.24033/asens.2321. http://www.numdam.org/articles/10.24033/asens.2321/

Bibliographie
Cité par

Abrashkin, V. Galois modules of period p group schemes over the ring of Witt vectors, Math. USSR, Izv., Volume 31 (1988), pp. 1-46 | DOI | MR | Zbl

Adams, J.; Johnson, J. F. Endoscopic groups and packets of nontempered representations, Compos. math., Volume 64 (1987), pp. 271-309 | Numdam | MR | Zbl

Arancibia, N.; Moeglin, C.; Renard, D. Paquets d'Arthur des groupes classiques et unitaires (preprint arXiv:1507.01432 ) | Numdam | MR | Zbl

Arthur, J. A stable trace formula. I. General expansions, J. Inst. Math. Jussieu, Volume 1 (2002), pp. 175-277 | DOI | MR | Zbl

Arthur, J., American Mathematical Society Colloquium Publications, 61, Amer. Math. Soc., 2013 | MR | Zbl

Arthur, J. The invariant trace formula. II. Global theory, J. Amer. Math. Soc., Volume 1 (1988), pp. 501-554 | DOI | MR | Zbl

Arthur, J. The $L^{2}$ -Lefschetz numbers of Hecke operators, Invent. math., Volume 97 (1989), pp. 257-290 | DOI | MR | Zbl

Arthur, J. Unipotent automorphic representations: conjectures, Astérisque, Volume 171-172 (1989), pp. 13-71 | Numdam | MR | Zbl

Asgari, M.; Schmidt, R. Siegel modular forms and representations, Manuscripta Math., Volume 104 (2001), pp. 173-200 | DOI | MR | Zbl

Bergström, J.; Faber, C.; van der Geer, G. Siegel modular forms of degree three and the cohomology of local systems, Selecta Math. (N.S.), Volume 20 (2014), pp. 83-124 | DOI | MR | Zbl

Buzzard, K.; Gee, T. The conjectural connections between automorphic representations and Galois representations (preprint arXiv:1009.0785 ) | MR

Borel, A.; Jacquet, H., Automorphic forms, representations and $L$ -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 189-207 | MR | Zbl

Borel, A., 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., Providence, R.I., 1979, pp. 27-61 | MR | Zbl

Bourbaki, N., Actualités Scientifiques et Industrielles, 1337, Hermann, 1968 (; réimpression Springer, 2007) | MR | Zbl

Bruhat, F., Soc. Math. France, Paris, 1959, pp. 85-93 | Numdam | Zbl

Bruinier, J. H.; van der Geer, G.; Harder, G.; Zagier, D. The 1-2-3 of modular forms, Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid (Ranestad, K., ed.) (Universitext), Springer, Berlin (2008), 266 pages | MR | Zbl

Borel, A.; Wallach, N. R., Mathematical Surveys and Monographs, 67, Amer. Math. Soc., Providence, RI, 2000, 260 pages | MR | Zbl

Caraiani, A. Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., Volume 161 (2012), pp. 2311-2413 | DOI | MR | Zbl

Carter, R. W. Conjugacy classes in the Weyl group, Compos. math., Volume 25 (1972), pp. 1-59 | Numdam | MR | Zbl

Chenevier, G.; Clozel, L. Corps de nombres peu ramifiés et formes automorphes autoduales, J. Amer. Math. Soc., Volume 22 (2009), pp. 467-519 | DOI | MR | Zbl

Clozel, L.; Delorme, P. Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II, Ann. Sci. Éc. Norm. Sup., Volume 23 (1990), pp. 193-228 | DOI | Numdam | MR | Zbl

Cho, S. Group schemes and local densities of quadratic lattices in residue characteristic 2, Compos. Math., Volume 151 (2015), pp. 793-827 | DOI | MR | Zbl

Cho, S. Group schemes and local densities of ramified Hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory, Volume 10 (2016), pp. 451-532 | DOI | MR | Zbl

Chenevier, G.; Lannes, J. Formes automorphes et voisins de Kneser des réseaux de Niemeier (preprint arXiv:1409.7616 )

Chaudouard, P.-H.; Laumon, G. Le lemme fondamental pondéré. I. Constructions géométriques, Compos. Math., Volume 146 (2010), pp. 1416-1506 | DOI | MR | Zbl

Chaudouard, P.-H.; Laumon, G. Le lemme fondamental pondéré. II. Énoncés cohomologiques, Ann. of Math., Volume 176 (2012), pp. 1647-1781 | DOI | MR | Zbl

Clozel, L. Purity reigns supreme, Int. Math. Res. Not. IMRN (2013), pp. 328-346 | DOI | MR | Zbl

Clozel, L. Motifs et Formes Automorphes: Applications du Principe de Fonctorialité, Automorphic Forms, Shimura Varieties, and $L$ -functions (Clozel, L.; Milne, J. S., eds.), Volume 1 (1988) | MR | Zbl

Chenevier, G.; Renard, D. Level one algebraic cusp forms of classical groups of small rank, Mem. Amer. Math. Soc., Volume 237 (2015) | MR | Zbl

Casselman, W.; Shalika, J. The unramified principal series of p-adic groups. II: The Whittaker function, Compos. Math., Volume 41 (1980), pp. 207-231 | Numdam | MR | Zbl

(Demazure, M.; Grothendieck, A., eds.), Lecture Notes in Math., 153, Springer, Berlin-New York, 1970 | Zbl

(Deligne, P.; Katz, N., eds.), Lecture Notes in Math., 340, Springer, 1973, 438 pages | MR | Zbl

Enright, T. J.; Parthasarathy, R., Noncommutative harmonic analysis and Lie groups (Marseille, 1980) (Lecture Notes in Math.), Volume 880, Springer, Berlin-New York, 1981, pp. 74-90 | DOI | MR | Zbl

Fontaine, J.-M. Il n'y a pas de variété abélienne sur $ℤ$ , Invent. math., Volume 81 (1985), pp. 515-538 | DOI | MR | Zbl

Fermigier, S. Annulation de la cohomologie cuspidale de sous-groupes de congruence de ${GL}_{n} (𝐙)$ , Math. Ann., Volume 306 (1996), pp. 247-256 | DOI | MR | Zbl

Freitag, E., Grundl. math. Wiss., 254, Springer, Berlin, 1983, 341 pages | MR | Zbl

Goresky, M.; Kottwitz, R. E.; MacPherson, R. Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J., Volume 89 (1997), pp. 477-554 | DOI | MR | Zbl

Gross, B. H.; Pollack, D. On the Euler characteristic of the discrete spectrum, J. Number Theory, Volume 110 (2005), pp. 136-163 | DOI | MR | Zbl

Gross, B. H. On the motive of a reductive group, Invent. math., Volume 130 (1997), pp. 287-313 | DOI | MR | Zbl

Gross, B. H. Algebraic modular forms, Israel J. Math., Volume 113 (1999), pp. 61-93 | DOI | MR | Zbl

Gan, W. T.; Yu, J.-K. Group schemes and local densities, Duke Math. J., Volume 105 (2000), pp. 497-524 | MR | Zbl

Harish-Chandra Representations of semisimple Lie groups. IV, Amer. J. Math., Volume 77 (1955), pp. 743-777 | DOI | MR | Zbl

Harish-Chandra Representations of semisimple Lie groups. V, Amer. J. Math., Volume 78 (1956), pp. 1-41 | DOI | MR | Zbl

Harish-Chandra Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math., Volume 78 (1956), pp. 564-628 | DOI | MR | Zbl

Harish-Chandra, Notes by J. G. M. Mars. Lecture Notes in Math., No. 62, Springer, Berlin-New York, 1968, 138 pages | MR | Zbl

Harish-Chandra, Lecture Notes in Math., 162, Springer, 1970, 125 pages | MR | Zbl

Hecht, H. The characters of some representations of Harish-Chandra, Math. Ann., Volume 219 (1976), pp. 213-226 | DOI | MR | Zbl

Igusa, J.-i. On Siegel modular forms of genus two, Amer. J. Math., Volume 84 (1962), pp. 175-200 | DOI | MR | Zbl

Ikeda, T. On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2 n$ , Ann. of Math., Volume 154 (2001), pp. 641-681 | DOI | MR | Zbl

Jacobson, N. A note on Hermitian forms, Bull. Amer. Math. Soc., Volume 46 (1940), pp. 264-268 | DOI | JFM | MR

Jacobowitz, R. Hermitian forms over local fields, Amer. J. Math., Volume 84 (1962), pp. 441-465 | DOI | MR | Zbl

Johnson, J. F. Lie algebra cohomology and the resolution of certain Harish-Chandra modules, Math. Ann., Volume 267 (1984), pp. 377-393 | DOI | MR | Zbl

Kaletha, T. Rigid inner forms of real and $p$ -adic groups, Ann. of Math., Volume 184 (2016), pp. 559-632 | DOI | MR | Zbl

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

Knapp, A. W. Commutativity of intertwining operators for semisimple groups, Compos. math., Volume 46 (1982), pp. 33-84 | Numdam | MR | Zbl

Knapp, A. W., Princeton Mathematical Series, 36, Princeton Univ. Press, Princeton, NJ, 1986, 774 pages | MR | Zbl

Kostant, B. On Whittaker vectors and representation theory, Invent. math., Volume 48 (1978), pp. 101-184 | DOI | MR | Zbl

Kottwitz, R. Stable trace formula: elliptic singular terms, Math. Annalen, Volume 275 (1986), pp. 365-399 | DOI | MR | Zbl

Kottwitz, R. E. Tamagawa numbers, Ann. of Math., Volume 127 (1988), pp. 629-646 | DOI | MR | Zbl

Kottwitz, R. E., 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 | MR | Zbl

Langlands, R. P., Representation theory and harmonic analysis on semisimple Lie groups (Math. Surveys Monogr.), Volume 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101-170 | DOI | MR | Zbl

Lansky, J.; Pollack, D. Hecke algebras and automorphic forms, Compos. math., Volume 130 (2002), pp. 21-48 | DOI | MR | Zbl

Langlands, R. P.; Shelstad, D. On the definition of transfer factors, Math. Ann., Volume 278 (1987), pp. 219-271 | DOI | MR | Zbl

Langlands, R. P.; Shelstad, D., The Grothendieck Festschrift, Vol. II (Progr. Math.), Volume 87, Birkhäuser, 1990, pp. 485-563 | MR | Zbl

Labesse, J.-P.; Waldspurger, J.-L., CRM Monograph Series, 31, Amer. Math. Soc., Providence, RI, 2013, 234 pages | MR | Zbl

Martens, S. The characters of the holomorphic discrete series, Proc. Nat. Acad. Sci. U.S.A., Volume 72 (1975), pp. 3275-3276 | DOI | MR | Zbl

Mestre, J.-F. Formules explicites et minorations de conducteurs de variétés algébriques, Compos. math., Volume 58 (1986), pp. 209-232 | Numdam | MR | Zbl

Mezo, P. Tempered spectral transfer in the twisted endoscopy of real groups (available at http://people.math.carleton.ca/~mezo/research.html )

Mezo, P. Character identities in the twisted endoscopy of real reductive groups, Mem. Amer. Math. Soc., Volume 222 (2013), 94 pages | MR | Zbl

Miller, S. D. The highest lowest zero and other applications of positivity, Duke Math. J., Volume 112 (2002), pp. 83-116 | DOI | MR | Zbl

Moeglin, C.; Waldspurger, J.-L. Stabilisation de la formule des traces tordue VI, X (preprints available at https://webusers.imj-prg.fr/~jean-loup.waldspurger/ ) | MR

Moeglin, C.; Waldspurger, J.-L. Le spectre résiduel de $GL (n)$ , Ann. sci. Éc. Norm. Sup., Volume 22 (1989), pp. 605-674 | DOI | Numdam | MR | Zbl

Nebe, G.; Venkov, B. On Siegel modular forms of weight 12, J. reine angew. Math., Volume 531 (2001), pp. 49-60 | MR | Zbl

O'Meara, O. T., Classics in Mathematics, Springer, 2000, 342 pages (reprint of the 1973 edition) | MR | Zbl

Petersen, D. Cohomology of local systems on the moduli of principally polarized abelian surfaces, Pacific J. Math., Volume 275 (2015), pp. 39-61 | DOI | MR | Zbl

Prasad, G. Volumes of $S$ -arithmetic quotients of semi-simple groups, Publ. Math. IHÉS, Volume 69 (1989), pp. 91-117 | DOI | Numdam | MR | Zbl

Poor, C.; Yuen, D. S. Computations of spaces of Siegel modular cusp forms, J. Math. Soc. Japan, Volume 59 (2007), pp. 185-222 | MR | Zbl

Stein, W. A. et al. Sage Mathematics Software (Version 6.1.1) (2014) ( http://www.sagemath.org )

Schmid, W. On the characters of the discrete series. The Hermitian symmetric case, Invent. math., Volume 30 (1975), pp. 47-144 | DOI | MR | Zbl

Serre, J.-P., Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) (Ann. of Math. Studies), Volume 70, Princeton Univ. Press, 1971, pp. 77-169 | MR | Zbl

Shahidi, F. A proof of Langlands' conjecture on Plancherel measures; complementary series for $p$ -adic groups, Ann. of Math., Volume 132 (1990), pp. 273-330 | DOI | MR | Zbl

Shelstad, D., Representation theory of real reductive Lie groups (Contemp. Math.), Volume 472, Amer. Math. Soc., Providence, RI, 2008, pp. 215-246 | DOI | MR | Zbl

Shelstad, D. Tempered endoscopy for real groups. III. Inversion of transfer and $L$ -packet structure, Represent. Theory, Volume 12 (2008), pp. 369-402 | DOI | MR | Zbl

Shelstad, D., Automorphic forms and the Langlands program (Adv. Lect. Math. (ALM)), Volume 9, Int. Press, Somerville, MA, 2010, pp. 236-276 | MR | Zbl

Siegel, C. L. Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, Volume 1969 (1969), pp. 87-102 | MR | Zbl

Springer, T. A., Progr. Math., 9, Birkhäuser, 1998, 334 pages | MR

Taylor, R. Galois representations, Ann. Fac. Sci. Toulouse Math., Volume 13 (2004), pp. 73-119 | DOI | Numdam | MR | Zbl

Tsushima, R. An explicit dimension formula for the spaces of generalized automorphic forms with respect to $Sp (2, 𝐙)$ , Proc. Japan Acad. Ser. A Math. Sci., Volume 59 (1983), pp. 139-142 | DOI | MR | Zbl

Tsushima, R., Automorphic forms of several variables (Katata, 1983) (Progr. Math.), Volume 46, Birkhäuser, 1984, pp. 378-383 | MR | Zbl

Tsuyumine, S. On Siegel modular forms of degree three, Amer. J. Math., Volume 108 (1986), pp. 755-862 | DOI | MR | Zbl

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

Vogan, D. A. J. Gel'fand-Kirillov dimension for Harish-Chandra modules, Invent. math., Volume 48 (1978), pp. 75-98 | DOI | MR | Zbl

Vogan, D. A. J.; Zuckerman, G. J. Unitary representations with nonzero cohomology, Compos. math., Volume 53 (1984), pp. 51-90 | Numdam | MR | Zbl

Waldspurger, J.-L. Stabilisation de la formule des traces tordue I, II, III, IV, V, VII, VIII, IX (preprints available at https://webusers.imj-prg.fr/~jean-loup.waldspurger/ )

Waldspurger, J.-L. Stabilisation de la formule des traces tordue I: endoscopie tordue sur un corps local (preprint arXiv:1401.4569 )

Waldspurger, J.-L. Les facteurs de transfert pour les groupes classiques: un formulaire, Manuscripta Math., Volume 133 (2010), pp. 41-82 | DOI | MR | Zbl

Wall, G. E. On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc., Volume 3 (1963), pp. 1-62 | MR | Zbl

Wallach, N. R., Operator algebras and group representations, Vol. II (Neptun, 1980) (Monogr. Stud. Math.), Volume 18, Pitman, Boston, MA, 1984, pp. 227-237 | MR | Zbl

Washington, L. C., Graduate Texts in Math., 83, Springer, New York, 1997, 487 pages | MR | Zbl

Weil, A., Progr. Math., 23, Birkhäuser, 1982, 126 pages | MR | Zbl

Weissauer, R. Vektorwertige Siegelsche Modulformen kleinen Gewichtes, J. reine angew. Math., Volume 343 (1983), pp. 184-202 | MR | Zbl

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