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BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY

Published online by Cambridge University Press:  24 April 2017

Yiannis Sakellaridis
Affiliation:
Department of Mathematics and Computer Science, Rutgers University at Newark, 101 Warren Street, Smith Hall 216, Newark, NJ 07102, USA (sakellar@rutgers.edu)
Corresponding
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Abstract

For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$ , where $T$ is a nontrivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to nonstandard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ ‘Beyond Endoscopy’ program.

Type
Research Article
Copyright
© Cambridge University Press 2017 

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References

Altuğ, S. A., Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations, Compos. Math. 151(10) (2015), 17911820. doi:10.1112/S0010437X15007320.CrossRefGoogle Scholar
Aubert, A.-M. and Plymen, R., Plancherel measure for GL(n, F) and GL(m, D): explicit formulas and Bernstein decomposition, J. Number Theory 112(1) (2005), 2666. doi:10.1016/j.jnt.2005.01.005.CrossRefGoogle Scholar
Bernstein, J. N., On the support of Plancherel measure, J. Geom. Phys. 5(4) (1988), 663710. doi:10.1016/0393-0440(88)90024-1.CrossRefGoogle Scholar
Bernstein, J. and Krötz, B., Smooth Fréchet globalizations of Harish-Chandra modules, Israel J. Math. 199(1) (2014), 45111. doi:10.1007/s11856-013-0056-1.CrossRefGoogle Scholar
Frenkel, E., Langlands, R. and Châu Ngô, B., Formule des traces et fonctorialité: le début d’un programme, Ann. Sci. Math. Québec 34(2) (2010), 199243.Google Scholar
Gaitsgory, D. and Nadler, D., Spherical varieties and Langlands duality, Mosc. Math. J. 10(1) (2010), 65137, 271.CrossRefGoogle Scholar
Gan, W. T., Gross, B. H. and Prasad, D., Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1109. Sur les conjectures de Gross et Prasad. I.Google Scholar
Herman, P. E., Beyond endoscopy for the Rankin-Selberg L-function, J. Number Theory 131(9) (2011), 16911722. doi:10.1016/j.jnt.2011.01.019.CrossRefGoogle Scholar
Herman, P. E., The functional equation and beyond endoscopy, Pacific J. Math. 260(2) (2012), 497513. doi:10.2140/pjm.2012.260.497.CrossRefGoogle Scholar
Hiraga, K., Ichino, A. and Ikeda, T., Formal degrees and adjoint 𝛾-factors, J. Amer. Math. Soc. 21(1) (2008), 283304. doi:10.1090/S0894-0347-07-00567-X.CrossRefGoogle Scholar
Ichino, A. and Ikeda, T., On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal. 19(5) (2010), 13781425. doi:10.1007/s00039-009-0040-4.CrossRefGoogle Scholar
Jacquet, H., Sur un résultat de Waldspurger, Ann. Sci. Éc. Norm. Supér (4) 19(2) (1986), 185229. URL: http://www.numdam.org/item?id=ASENS_1986_4_19_2_185_0.CrossRefGoogle Scholar
Jacquet, H., Sur un résultat de Waldspurger. II, Compos. Math. 63(3) (1987), 315389. URL: http://www.numdam.org/item?id=CM_1987__63_3_315_0.Google Scholar
Jacquet, H. and Langlands, R. P., Automorphic Forms on GL(2), Lecture Notes in Mathematics, Volume 114 (Springer, Berlin, 1970).CrossRefGoogle Scholar
Kottwitz, R. E., Stable trace formula: cuspidal tempered terms, Duke Math. J. 51(3) (1984), 611650. doi:10.1215/S0012-7094-84-05129-9.CrossRefGoogle Scholar
Langlands, R. P., Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611697 (Johns Hopkins University Press, Baltimore, MD, 2004).Google Scholar
Lapid, E. and Mao, Z., A conjecture on Whittaker-Fourier coefficients of cusp forms, J. Number Theory 146 (2015), 448505. doi:10.1016/j.jnt.2013.10.003.CrossRefGoogle Scholar
Mœglin, C. and Waldspurger, J.-L., Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, Volume 113 (Cambridge University Press, Cambridge, 1995). Une paraphrase de l’Écriture [A paraphrase of Scripture]. doi:10.1017/CBO9780511470905.CrossRefGoogle Scholar
Ono, T., On the Tamagawa number of algebraic tori, Ann. of Math. (2) 78 (1963), 4773.CrossRefGoogle Scholar
Prasad, D., Relating invariant linear form and local epsilon factors via global methods, Duke Math. J. 138(2) (2007), 233261. With an appendix by Hiroshi Saito. doi:10.1215/S0012-7094-07-13823-7.CrossRefGoogle Scholar
Rudnick, Z., Poincaré series, PhD thesis, Yale University (1990).Google Scholar
Saito, H., On Tunnell’s formula for characters of GL(2), Compos. Math. 85(1) (1993), 99108. URL: http://www.numdam.org/item?id=CM_1993__85_1_99_0.Google Scholar
Sakellaridis, Y., Beyond endoscopy for the relative trace formula I: local theory, in Automorphic Representations and L-functions, Tata Institute of Fundamental Research Studies in Mathematics, Volume 22, pp. 521590 (Tata Institute of Fundamental Research, Mumbai, 2013).Google Scholar
Sakellaridis, Y., Spherical functions on spherical varieties, Amer. J. Math. 135(5) (2013), 12911381. doi:10.1353/ajm.2013.0046.CrossRefGoogle Scholar
Sakellaridis, Y., The Schwartz space of a smooth semi-algebraic stack, Selecta Math. (N.S.) 22(4) (2016), 24012490. doi:10.1007/s00029-016-0285-3.CrossRefGoogle Scholar
Sakellaridis, Y. and Venkatesh, Akshay, Periods and harmonic analysis on spherical varieties. Astérisque, 291pp, to appear, arXiv:1203.0039.Google Scholar
Tunnell, J. B., Local 𝜖-factors and characters of GL(2), Amer. J. Math. 105(6) (1983), 12771307. doi:10.2307/2374441.CrossRefGoogle Scholar
Venkatesh, A., ‘Beyond endoscopy’ and special forms on GL(2), J. Reine Angew. Math. 577 (2004), 2380. doi:10.1515/crll.2004.2004.577.23.Google Scholar
Vogan, D. A. Jr., The local Langlands conjecture, in Representation Theory of Groups and Algebras, Contemporary Mathematics, Volume 145, pp. 305379 (American Mathematical Society, Providence, RI, 1993).CrossRefGoogle Scholar
Waldspurger, J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54(2) (1985), 173242. URL: http://www.numdam.org/item?id=CM_1985__54_2_173_0.Google Scholar
White, P.-J., The base change L-function for modular forms and Beyond Endoscopy, J. Number Theory 140 (2014), 1337. doi:10.1016/j.jnt.2014.01.006.CrossRefGoogle Scholar
Zhang, W., Automorphic period and the central value of Rankin–Selberg L-function, J. Amer. Math. Soc. 27(2) (2014), 541612. doi:10.1090/S0894-0347-2014-00784-0.CrossRefGoogle Scholar

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