×

zbMATH — the first resource for mathematics

The local Langlands conjecture for \(\text{GSp}(4)\). (English) Zbl 1230.11063
In the paper under review the authors establish the local Langlands correspondence for \(\text{GSp}(4)\) over a non-Archimedean field of characteristic zero. Their approach uses mainly the already established local Langlands correspondence for \(\text{GL}(n)\) and theta correspondence.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] J. Adams and D. Barbasch, ”Reductive dual pair correspondence for complex groups,” J. Funct. Anal., vol. 132, iss. 1, pp. 1-42, 1995. · Zbl 0841.22014
[2] J. D. Adler and D. Prasad, ”On certain multiplicity one theorems,” Israel J. Math., vol. 153, pp. 221-245, 2006. · Zbl 1137.22009
[3] M. Asgari and R. Schmidt, ”On the adjoint \(L\)-function of the \(p\)-adic \( GSp(4)\),” J. Number Theory, vol. 128, iss. 8, pp. 2340-2358, 2008. · Zbl 1213.11110
[4] M. Asgari and F. Shahidi, ”Generic transfer from \( GSp(4)\) to \( GL(4)\),” Compos. Math., vol. 142, iss. 3, pp. 541-550, 2006. · Zbl 1112.11027
[5] J. J. Chen, ”The \(n\times(n-2)\) local converse theorem for \({ GL}(n)\) over a \(p\)-adic field,” J. Number Theory, vol. 120, iss. 2, pp. 193-205, 2006. · Zbl 1193.11047
[6] P. S. Chan and W. T. Gan, The local Langlands conjecture for \({ GSp}(4)\) III: stability and (twisted) endoscopy. · Zbl 1366.11074
[7] S. DeBacker and M. Reeder, ”Depth-zero supercuspidal \(L\)-packets and their stability,” Ann. of Math., vol. 169, iss. 3, pp. 795-901, 2009. · Zbl 1193.11111
[8] D. Ginzburg, S. Rallis, and D. Soudry, ”Periods, poles of \(L\)-functions and symplectic-orthogonal theta lifts,” J. Reine Angew. Math., vol. 487, pp. 85-114, 1997. · Zbl 0928.11025
[9] W. T. Gan and S. Takeda, ”On Shalika periods and a theorem of Jacquet-Martin,” Amer. J. Math., vol. 132, iss. 2, pp. 475-528, 2010. · Zbl 1222.11067
[10] W. T. Gan and S. Takeda, ”The local Langlands conjecture for Sp(4),” Int. Math. Res. Not., vol. 2010, iss. 15, pp. 2987-3038, 2010. · Zbl 1239.11061
[11] W. T. Gan and S. Takeda, On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups. · Zbl 1291.11083
[12] W. T. Gan and S. Takeda, Theta correspondences for \({ GSp}_4\). · Zbl 1239.11062
[13] W. T. Gan and W. Tantono, The local Langlands conjecture for \({ GSp}(4)\) II: the case of inner forms. · Zbl 1298.11043
[14] 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
[15] G. Henniart, ”Une preuve simple des conjectures de Langlands pour \({ GL}(n)\) sur un corps \(p\)-adique,” Invent. Math., vol. 139, pp. 439-455, 2000. · Zbl 1048.11092
[16] G. Henniart, ”Correspondance de Langlands et fonctions \(L\) des carrés extérieur et symétrique,” Int. Math. Res. Not., vol. 2010, iss. 4, pp. 633-673, 2010. · Zbl 1184.22009
[17] 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
[18] H. Jacquet and J. Shalika, ”Exterior square \(L\)-functions,” in Automorphic Forms, Shimura Varieties, and \(L\)-Functions, Vol. II, Boston, MA, 1990, pp. 143-226. · Zbl 0695.10025
[19] D. Jiang and D. Soudry, ”Generic representations and local Langlands reciprocity law for \(p\)-adic \({ SO}_{2n+1}\),” in Contributions to Automorphic Forms, Geometry, and Number Theory, Baltimore, MD: Johns Hopkins Univ. Press, 2004, pp. 457-519. · Zbl 1062.11077
[20] S. S. Kudla, ”On the local theta-correspondence,” Invent. Math., vol. 83, iss. 2, pp. 229-255, 1986. · Zbl 0583.22010
[21] S. S. Kudla and S. Rallis, ”On first occurrence in the local theta correspondence,” in Automorphic Representations, \(L\)-Functions and Applications: Progress and Prospects, Berlin: de Gruyter, 2005, vol. 11, pp. 273-308. · Zbl 1109.22012
[22] E. M. Lapid and S. Rallis, ”On the local factors of representations of classical groups,” in Automorphic Representations, \(L\)-Functions and Applications: Progress and Prospects, Berlin: de Gruyter, 2005, vol. 11, pp. 309-359. · Zbl 1188.11023
[23] G. Muić and G. Savin, ”Symplectic-orthogonal theta lifts of generic discrete series,” Duke Math. J., vol. 101, iss. 2, pp. 317-333, 2000. · Zbl 0955.22014
[24] G. Muić and G. Savin, ”Complementary series for Hermitian quaternionic groups,” Canad. Math. Bull., vol. 43, iss. 1, pp. 90-99, 2000. · Zbl 0945.22005
[25] D. Prasad, ”On the local Howe duality correspondence,” Internat. Math. Res. Notices, iss. 11, pp. 279-287, 1993. · Zbl 0804.22009
[26] D. Prasad, ”Invariant forms for representations of \({ GL}_2\) over a local field,” Amer. J. Math., vol. 114, iss. 6, pp. 1317-1363, 1992. · Zbl 0780.22004
[27] D. Prasad, ”Some remarks on representations of a division algebra and of the Galois group of a local field,” J. Number Theory, vol. 74, iss. 1, pp. 73-97, 1999. · Zbl 0931.11054
[28] A. Paul, ”On the Howe correspondence for symplectic-orthogonal dual pairs,” J. Funct. Anal., vol. 228, iss. 2, pp. 270-310, 2005. · Zbl 1084.22008
[29] S. Rallis, ”On the Howe duality conjecture,” Compositio Math., vol. 51, iss. 3, pp. 333-399, 1984. · Zbl 0624.22011
[30] B. Roberts, ”The theta correspondence for similitudes,” Israel J. Math., vol. 94, pp. 285-317, 1996. · Zbl 0870.22011
[31] B. Roberts, ”Global \(L\)-packets for \({ GSp}(2)\) and theta lifts,” Doc. Math., vol. 6, pp. 247-314, 2001. · Zbl 1056.11029
[32] B. Roberts and R. Schmidt, Local Newforms for \({ GSp}(4)\), New York: Springer-Verlag, 2007, vol. 1918. · Zbl 1126.11027
[33] F. Shahidi, ”A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups,” Ann. of Math., vol. 132, iss. 2, pp. 273-330, 1990. · Zbl 0780.22005
[34] A. J. Silberger, Introduction to Harmonic Analysis on Reductive \(p\)-adic Groups, Princeton, N.J.: Princeton Univ. Press, 1979, vol. 23. · Zbl 0458.22006
[35] D. Soudry, ”A uniqueness theorem for representations of \({ GSO}(6)\) and the strong multiplicity one theorem for generic representations of \({ GSp}(4)\),” Israel J. Math., vol. 58, iss. 3, pp. 257-287, 1987. · Zbl 0642.22003
[36] P. J. Sally Jr. and M. Tadić, ”Induced representations and classifications for \({ GSp}(2,F)\) and \({ Sp}(2,F)\),” Mém. Soc. Math. France, iss. 52, pp. 75-133, 1993. · Zbl 0784.22008
[37] M. Vignéras, ”Correspondances entre representations automorphes de \({ GL}(2)\) sur une extension quadratique de \({ GSp}(4)\) sur \({\mathbf Q}\), conjecture locale de Langlands pour \({ GSp}(4)\),” in The Selberg Trace Formula and Related Topics, Providence, RI, 1986, pp. 463-527. · Zbl 0595.12009
[38] W. Zink, Letter to M.-F. Vigneras (1984).
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.