Geodesic
On the Langlands correspondence for symplectic motives
Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 678-692.

Voir la notice de l'article provenant de la source Math-Net.Ru

We present a refinement of the global Langlands correspondence for symplectic motives. Using the local theory of generic representations of odd orthogonal groups, we define a new vector in the associated automorphic representation, which is the tensor product of test vectors for the Whittaker functionals.
Keywords: symplectic motive, generic representation, Langlands correspondence. 
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B. H. Gross. On the Langlands correspondence for symplectic motives. Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 678-692. http://geodesic.mathdoc.fr/item/IM2_2016_80_4_a4/

[1] B. H. Gross, D. Prasad, “Test vectors for linear forms”, Math. Ann., 291:1 (1991), 343–355 | DOI | MR | Zbl

[2] A. Weil, “Über die Bestimmung Dirichletscher Reihen durch Functionalgleichungen”, Math. Ann., 168:1 (1967), 149–156 | DOI | MR | Zbl

[3] N. R. Wallach, “On the constant term of a square integrable automorphic form”, Operator algebras and group representations (Neptun, 1980), v. II, Monogr. Stud. Math., 18, Pitman, Boston, MA, 1984, 227–237 | MR | Zbl

[4] P. Deligne, “Valeurs de fonctions $L$ et périodes d'intégrales”, Automorphic forms, representations and $L$-functions, Part 2 (Oregon State Univ., Corvallis, OR, 1977), Proc. Sympos. Pure Math., 33.2, Amer. Math. Soc., Providence, RI, 1979, 313–346 | DOI | MR | Zbl

[5] B. H. Gross, “$L$-functions at the central critical point”, Motives, Part 1 (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55.1, Amer. Math. Soc., Providence, RI, 1994, 527–535 | DOI | MR | Zbl

[6] D. A. Vogan, Jr., “Gelfand–Kirillov dimension for Harish-Chandra modules”, Invent. Math., 48:1 (1978), 75–98 | DOI | MR | Zbl

[7] W. Casselman, “Canonical extensions of Harish-Chandra modules to representations of $G$”, Canad. J. Math., 41:3 (1989), 385–438 | DOI | MR | Zbl

[8] W. Schmid, “Some properties of square-integrable representations of semisimple Lie groups”, Ann. of Math. (2), 102:3 (1975), 535–564 | DOI | MR | Zbl

[9] A. A. Popa, “Whittaker newforms for Archimedean representations of $\operatorname{GL}(2)$”, J. Number Theory, 128:6 (2008), 1637–1645 | DOI | MR | Zbl

[10] Dihua Jiang, D. Soudry, “Generic representations and local Langlands reciprocity law for $p$-adic $\operatorname{SO}_{2n+1}$”, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, 457–519 | MR | Zbl

[11] Wee Teck Gan, B. H. Gross, D. Prasad, “Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups”, Sur les conjectures de Gross et Prasad, v. I, Astérisque, 346, Soc. Math. France, Paris, 2012, 1–109 | MR | Zbl

[12] B. Roberts, R. Schmidt, Local newforms for $\operatorname{GSp}(4)$, Lecture Notes in Math., 1918, Springer, Berlin, 2007, viii+307 pp. | DOI | MR | Zbl

[13] A. Brumer, K. Kramer, “Paramodular abelian varieties of odd conductor”, Trans. Amer. Math. Soc., 366:5 (2014), 2463–2516 | DOI | MR | Zbl

[14] W. Casselman, “On some results of Atkin and Lehner”, Math. Ann., 201:4 (1973), 301–314 | DOI | MR | Zbl

[15] Pei-Yu Tsai, On newforms for split special odd orthogonal groups, Thesis (Ph.D.), Harvard Univ., 2013, 143 pp. | MR

[16] R. P. Langlands, “Automorphic representations, Shimura varieties, and motives”, Automorphic forms, representations and $L$-functions, Part 2 (Oregon State Univ., Corvallis, OR, 1977), Proc. Sympos. Pure Math., 33.2, Amer. Math. Soc., Providence, RI, 1979, 205–246 | DOI | MR | Zbl

[17] J.-P. Labesse, R. P. Langlands, “$L$-indistinguishability for $\operatorname{SL}(2)$”, Canad. J. Math., 31:4 (1979), 726–785 | DOI | MR | Zbl

[18] T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, “Potential automorphy and change of weight”, Ann. of Math. (2), 179:2 (2014), 501–609 | DOI | MR | Zbl

[19] G. Shimura, “On the factors of the jacobian variety of a modular function field”, J. Math. Soc. Japan, 25:3 (1973), 523–544 | DOI | MR | Zbl

[20] H. Jacquet, R. P. Langlands, Automorphic forms on $\operatorname{GL}(2)$, Lecture Notes in Math., 114, Springer-Verlag, Berlin–New York, 1970, vii+548 pp. | DOI | MR | MR | Zbl | Zbl

[21] G. Faltings, “Endlichkeitssätze für abelsche {V}arietäten über {Z}ahlkörpern”, Invent. Math., 73:3 (1983), 349–366 ; “Erratum”, Invent. Math., 75:2 (1984), 381 | DOI | MR | Zbl | DOI | MR

[22] D. Mumford, Abelian varieties, Tata Inst. Fund. Res. Stud. Math., 5, Oxford Univ. Press, London, 1970, viii+242 pp. | MR | Zbl | Zbl

[23] C. Poor, D. S. Yuen, “Paramodular cusp forms”, Math. Comp., 84:293 (2015), 1401–1438 | DOI | MR | Zbl

[24] H. Yoshida, “On Siegel modular forms obtained from theta series”, J. Reine Angew. Math., 352 (1984), 184–219 | DOI | MR | Zbl