zbMATH — the first resource for mathematics

Level raising and symmetric power functoriality. II. (English) Zbl 1339.11060
This paper is a sequel to the authors’ [Compos. Math. 150, No. 5, 729–748 (2014; Zbl 1304.11040)]. In that first paper, they had outlined a strategy to reduce a certain symmetric power lifting conjecture to two other conjectures about automorphic forms (concerning the existence of automorphic tensor products and the construction of level raising congruences between certain automorphic representations).
In the present paper, they carry out this strategy in certain cases. This allows them to prove that if \(\ell \geq 5\) is a prime number, then the implication \(\mathrm{SP}_{\ell-1}(K(\zeta_\ell)) \implies \mathrm{SP}_{\ell+1}(K(\zeta_\ell))\) holds. Here \(K\) is a finite Galois extension of \(\mathbb{Q}\), and \(\mathrm{SP}_{n+1}(K)\) is a conjecture (conjecture 1.1 of the paper) asserting the existence of the \(n\)th symmetric power lifting of certain automorphic representations of \(\mathrm{GL}_2(\mathbb{A}_F)\) with \(F\) a totally real number field linearly disjoint from \(K\).
Thanks to some previous work of H. H. Kim and F. Shahidi [Ann. Math. (2) 155, No. 3, 837–893 (2002; Zbl 1040.11036)], establishing \(\mathrm{SP}_4(\mathbb{Q})\), this allows the authors to prove \(\mathrm{SP}_6(\mathbb{Q}(\zeta_5))\) and \(\mathrm{SP}_8(\mathbb{Q}(\zeta_{35}))\). The proof of the paper’s main result relies on C. P. Mok’s study [Mem. Am. Math. Soc. 1108, iii-v, 248 p. (2015; Zbl 1316.22018)] of the space of automorphic forms on certain groups, and uses automorphy lifting methods.

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F80 Galois representations
Full Text: DOI
[1] J. Arthur, The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, Providence, RI: Amer. Math. Soc., 2013, vol. 61. · Zbl 1310.22014
[2] J. Bella”iche and P. Graftieaux, ”Augmentation du niveau pour \({ U}(3)\),” Amer. J. Math., vol. 128, iss. 2, pp. 271-309, 2006. · Zbl 1135.11025
[3] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, ”Potential automorphy and change of weight,” Ann. of Math., vol. 179, iss. 2, pp. 501-609, 2014. · Zbl 1310.11060
[4] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, ”A family of Calabi-Yau varieties and potential automorphy II,” Publ. Res. Inst. Math. Sci., vol. 47, iss. 1, pp. 29-98, 2011. · Zbl 1264.11044
[5] N. Bergeron, J. Millson, and C. Moeglin, Hodge type theorems for arithmetic manifolds associated to orthogonal groups. · Zbl 1349.14031
[6] A. Borel, ”Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup,” Invent. Math., vol. 35, pp. 233-259, 1976. · Zbl 0334.22012
[7] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local,” Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972. · Zbl 0254.14017
[8] P. Cartier, ”Representations of \(p\)-adic groups: a survey,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 111-155. · Zbl 0421.22010
[9] A. Caraiani, ”Local-global compatibility and the action of monodromy on nearby cycles,” Duke Math. J., vol. 161, iss. 12, pp. 2311-2413, 2012. · Zbl 1405.22028
[10] W. Casselman, Introduction to the theory of admissible representations of \(p\)-adic reductive groups.
[11] W. Casselman, ”The unramified principal series of \({\mathfrak p}\)-adic groups. I. The spherical function,” Compositio Math., vol. 40, iss. 3, pp. 387-406, 1980. · Zbl 0472.22004
[12] G. Chenevier and M. Harris, ”Construction of automorphic Galois representations, II,” Cambridge Math J., vol. 1, pp. 53-73, 2013. · Zbl 1310.11062
[13] L. Clozel, M. Harris, and J. Labesse, ”Endoscopic transfer,” in On the Stabilization of the Trace Formula, Somerville, MA: Int. Press, 2011, vol. 1, pp. 475-496. · Zbl 1255.11027
[14] L. Clozel, M. Harris, and R. Taylor, ”Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations,” Publ. Math. Inst. Hautes Études Sci., vol. 108, pp. 1-181, 2008. · Zbl 1169.11020
[15] P. Chaudouard and G. Laumon, ”Le lemme fondamental pondéré. I. Constructions géométriques,” Compos. Math., vol. 146, iss. 6, pp. 1416-1506, 2010. · Zbl 1206.14026
[16] P. Chaudouard and G. Laumon, ”Le lemme fondamental pondéré. II. Énoncés cohomologiques,” Ann. of Math., vol. 176, iss. 3, pp. 1647-1781, 2012. · Zbl 1264.11043
[17] L. Clozel, ”The fundamental lemma for stable base change,” Duke Math. J., vol. 61, iss. 1, pp. 255-302, 1990. · Zbl 0731.22011
[18] L. Clozel, ”Motifs et formes automorphes: applications du principe de fonctorialité,” in Automorphic Forms, Shimura varieties, and \(L\)-Functions, Vol. I, Academic Press, Boston, 1990, vol. 10, pp. 77-159. · Zbl 0705.11029
[19] L. Clozel, ”Identités de caractères en la place archimédienne,” in On the Stabilization of the Trace Formula, Somerville, MA: Int. Press, 2011, vol. 1, pp. 351-367. · Zbl 1255.11027
[20] L. Clozel and J. A. Thorne, ”Level-raising and symmetric power functoriality, I,” Compos. Math., vol. 150, iss. 5, pp. 729-748, 2014. · Zbl 1024.11027
[21] T. Gee, ”Automorphic lifts of prescribed types,” Math. Ann., vol. 350, iss. 1, pp. 107-144, 2011. · Zbl 1276.11085
[22] D. Geraghty, Modularity lifting theorems for ordinary Galois representations. · Zbl 1418.11098
[23] D. Goldberg, ”Reducibility of generalized principal series representations of \({ U}(2,2)\) via base change,” Compositio Math., vol. 86, iss. 3, pp. 245-264, 1993. · Zbl 0788.22021
[24] R. Guralnick, Adequacy of representations of finite groups of Lie type.
[25] G. Henniart, ”Une caractérisation de la correspondance de Langlands locale pour \({ GL}(n)\),” Bull. Soc. Math. France, vol. 130, iss. 4, pp. 587-602, 2002. · Zbl 1029.22023
[26] M. Harris and R. Taylor, The Geometry and Cohomology of some Simple Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2001, vol. 151. · Zbl 1036.11027
[27] N. Iwahori, ”Generalized Tits system (Bruhat decompostition) on \(p\)-adic semisimple groups,” in Algebraic Groups and Discontinuous Subgroups, Providence, RI: Amer. Math. Soc., 1966, pp. 71-83. · Zbl 0199.06901
[28] R. E. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, Paris: Math. Soc. France, 1999, vol. 255. · Zbl 0958.22013
[29] H. H. Kim and F. Shahidi, ”Functorial products for \({ GL}_2\times{ GL}_3\) and the symmetric cube for \({ GL}_2\),” Ann. of Math., vol. 155, iss. 3, pp. 837-893, 2002. · Zbl 1040.11036
[30] J. Labesse, ”Changement de base CM et séries discrètes,” in On the Stabilization of the Trace Formula, Somerville, MA: Int. Press, 2011, vol. 1, pp. 429-470. · Zbl 1255.11027
[31] J. -P. Labesse, ”Les facteurs de transfert pour les groupes unitaires,” in On the Stabilization of the Trace Formula, Somerville, MA: Int. Press, 2011, vol. 1, pp. 411-421. · Zbl 1255.11027
[32] R. P. Langlands and D. Shelstad, ”On the definition of transfer factors,” Math. Ann., vol. 278, iss. 1-4, pp. 219-271, 1987. · Zbl 0644.22005
[33] G. Lusztig, ”Affine Hecke algebras and their graded version,” J. Amer. Math. Soc., vol. 2, iss. 3, pp. 599-635, 1989. · Zbl 0715.22020
[34] A. M’inguez, ”Unramified representations of unitary groups,” in On the stabilization of the trace formula, Int. Press, Somerville, MA, 2011, vol. 1, pp. 389-422. · Zbl 1255.11027
[35] C. Moeglin, ”Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands,” in Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski-Shapiro, Providence, RI: Amer. Math. Soc., 2014, vol. 614, pp. 295-336. · Zbl 1298.22019
[36] C. Moeglin, ”Classification et changement de base pour les séries discrètes des groupes unitaires \(p\)-adiques,” Pacific J. Math., vol. 233, iss. 1, pp. 159-204, 2007. · Zbl 1157.22010
[37] C. P. Mok, Endoscopic classification of representations of quasi-split unitary groups. · Zbl 1316.22018
[38] L. Morris, ”Level zero \(\mathbf G\)-types,” Compositio Math., vol. 118, iss. 2, pp. 135-157, 1999. · Zbl 0937.22011
[39] B. C. Ngô, ”Le lemme fondamental pour les algèbres de Lie,” Publ. Math. Inst. Hautes Études Sci., vol. 111, pp. 1-169, 2010. · Zbl 1200.22011
[40] M. Reeder, ”Nonstandard intertwining operators and the structure of unramified principal series representations,” Forum Math., vol. 9, iss. 4, pp. 457-516, 1997. · Zbl 0882.22020
[41] M. Reeder, ”Matrices for affine Hecke modules,” J. Algebra, vol. 231, iss. 2, pp. 758-785, 2000. · Zbl 0979.20008
[42] K. A. Ribet, ”Congruence relations between modular forms,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Warsaw, 1984, pp. 503-514. · Zbl 0575.10024
[43] J. D. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Princeton, NJ: Princeton Univ. Press, 1990, vol. 123. · Zbl 0724.11031
[44] M. Suzuki, Group Theory. I, New York: Springer-Verlag, 1982, vol. 247. · Zbl 0472.20001
[45] J. Tate, ”Number theoretic background,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 3-26. · Zbl 0422.12007
[46] R. Taylor, ”On Galois representations associated to Hilbert modular forms,” Invent. Math., vol. 98, iss. 2, pp. 265-280, 1989. · Zbl 0705.11031
[47] J. A. Thorne, Automorphy lifting for residually reducible \(l\)-adic Galois representations. · Zbl 1396.11087
[48] J. Thorne, ”On the automorphy of \(l\)-adic Galois representations with small residual image,” J. Inst. Math. Jussieu, vol. 11, iss. 4, pp. 855-920, 2012. · Zbl 1269.11054
[49] J. Tits, ”Reductive groups over local fields,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 29-69. · Zbl 0415.20035
[50] J. -L. Waldspurger, Stabilisation de la formule des traces tordue I: endoscopie tordue sur un corps local.
[51] J. -L. Waldspurger, Stabilisation de la formule des traces tordue II: intégrales orbitales et endoscopie sur un corps local non-archimédien; définitions et énoncés des résultats.
[52] J. -L. Waldspurger, ”Le lemme fondamental implique le transfert,” Compositio Math., vol. 105, iss. 2, pp. 153-236, 1997. · Zbl 0871.22005
[53] J. -L. Waldspurger, ”À propos du lemme fondamental pondéré tordu,” Math. Ann., vol. 343, iss. 1, pp. 103-174, 2009. · Zbl 1158.22011
[54] J. Waldspurger, ”Endoscopie et changement de caractéristique: intégrales orbitales pondérées,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 59, iss. 5, pp. 1753-1818, 2009. · Zbl 1184.22003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.