Israel Journal of Mathematics

, Volume 225, Issue 1, pp 223–266 | Cite as

Periods and nonvanishing of central L-values for GL(2n)

  • Brooke Feigon
  • Kimball Martin
  • David Whitehouse


Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, πη) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.

We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AC89]
    J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, Vol. 120, Princeton University Press, Princeton, NJ, 1989.zbMATHGoogle Scholar
  2. [AG09]
    A. Aizenbud and D. Gourevitch, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet–Rallis’s theorem, Duke Mathematical Journal 149 (2009), 509–567.zbMATHGoogle Scholar
  3. [Art13]
    J. Arthur, The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications, Vol. 61, American Mathematical Society, Providence, RI, 2013.Google Scholar
  4. [BR10]
    A. I. Badulescu and D. Renard, Unitary dual of GL(n) at Archimedean places and global Jacquet–Langlands correspondence, Compositio Mathematica 146 (2010), 1115–1164.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [FJ93]
    S. Friedberg and H. Jacquet, Linear periods, Journal für die Reine und Angewandte Mathematik 443 (1993), 91–139.MathSciNetzbMATHGoogle Scholar
  6. [FLO12]
    B. Feigon, E. Lapid and O. Offen, On representations distinguished by unitary groups, Publications Mathématiques. Institut des Hautes ´Etudes Scientifiques 115 (2012), 185–323.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [FM14]
    M. Furusawa and K. Martin, On central critical values of the degree four L-functions for GSp(4): a simple trace formula, Mathematische Zeitschrift 277 (2014), 149–180.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [FM15]
    M. Furusawa and K. Martin, Local root numbers, Bessel models, and a conjecture of Guo and Jacquet, Journal of Number Theory 146 (2015), 150–170.CrossRefzbMATHGoogle Scholar
  9. [FM17]
    M. Furusawa and K. Morimoto, On special Bessel periods and the Gross–Prasad conjecture for SO(2n + 1) × SO(2), Mathematische Annalen 368 (2017), 561–586.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [GGP12]
    W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109.MathSciNetzbMATHGoogle Scholar
  11. [GP92]
    B. H. Gross and D. Prasad, On the decomposition of a representation of SOn when restricted to SOn−1, Canadian Journal of Mathematics 44 (1992), 974–1002.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [GP94]
    B. H. Gross and D. Prasad, On irreducible representations of SO2n+1 × SO2m, Canadian Journal of Mathematics 46 (1994), 930–950.MathSciNetCrossRefGoogle Scholar
  13. [Guo96]
    J. Guo, On a generalization of a result of Waldspurger, Canadian Journal of Mathematics 48 (1996), 105–142.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Guo97]
    J. Guo, Uniqueness of generalized Waldspurger model for GL(2n), Pacific Journal of Mathematics 180 (1997), 273–289.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Guo98]
    J. Guo, Spherical characters on certain p-adic symmetric spaces, MPI Preprints, No. 1998-24, Max-Planck-Institut für Mathematik, Bonn, 1998.Google Scholar
  16. [HM02a]
    J. Hakim and F. Murnaghan, Globalization of distinguished supercuspidal representations of GL(n), Canadian Mathematical Bulletin 45 (2002), 220–230.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [HM02b]
    J. Hakim and F. Murnaghan, Two types of distinguished supercuspidal representations, International Mathematics Research Notices 35 (2002), 1857–1889.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Jac86]
    H. Jacquet, Sur un résultat de Waldspurger, Annales Scientifiques de l’École Normale Supérieure 19 (1986), 185–229.CrossRefzbMATHGoogle Scholar
  19. [JC01]
    H. Jacquet and N. Chen, Positivity of quadratic base change L-functions, Bulletin de la Société Mathématique de France 129 (2001), 33–90.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [JM07]
    H. Jacquet and K. Martin, Shalika periods on GL2(D) and GL4, Pacific Journal of Mathematics 233 (2007), 341–370.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [JNQ08]
    D. Jiang, C. Nien and Y. Qin, Local Shalika models and functoriality, Manuscripta Mathematica 127 (2008), 187–217.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [JR96]
    H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Mathematica 102 (1996), 65–123.MathSciNetzbMATHGoogle Scholar
  23. [JS90]
    H. Jacquet and J. Shalika, Exterior square L-functions, in Automorphic Forms, Shimura Varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 11, Academic Press, Boston, MA, 1990, pp. 143–226.Google Scholar
  24. [KL15]
    A. Knightly and C. Li, Simple supercuspidal representations of GL(n), Taiwanese Journal of Mathematics 19 (2015), 995–1029.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [LR00]
    E. Lapid and J. Rogawski, Stabilization of periods of Eisenstein series and Bessel distributions on GL(3) relative to U(3), Documenta Mathematica 5 (2000), 317–350.MathSciNetzbMATHGoogle Scholar
  26. [Mur08]
    F. Murnaghan, Spherical characters: the supercuspidal case, in Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey, Contemporary Mathematics, Vol. 449, American Mathematical Society, Providence, RI, 2008, pp. 301–313.Google Scholar
  27. [Mur11]
    F. Murnaghan, Regularity and distinction of supercuspidal representations, in Harmonic Analysis on Reductive, p-adic Groups, Contemporary Mathematics, Vol. 543, American Mathematical Society, Providence, RI, 2011, pp. 155–183.MathSciNetzbMATHGoogle Scholar
  28. [MW09]
    K. Martin and D. Whitehouse, Central L-values and toric periods for GL(2), International Mathematics Research Notices 1 (2009), 141–191.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Pra07]
    D. Prasad, Relating invariant linear form and local epsilon factors via global methods, Duke Mathematical Journal 138 (2007), 233–261.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [PSP08]
    D. Prasad and R. Schulze-Pillot, Generalised form of a conjecture of Jacquet and a local consequence, Journal für die Reine und Angewandte Mathematik 616 (2008), 219–236.MathSciNetzbMATHGoogle Scholar
  31. [PTB11]
    D. Prasad and R. Takloo-Bighash, Bessel models for GSp(4), Journal für die Reine und Angewandte Mathematik 655 (2011), 189–243.MathSciNetzbMATHGoogle Scholar
  32. [Ram15]
    D. Ramakrishnan, A mild Tchebotarev theorem for GL(n), Journal of Number Theory 146 (2015), 519–533.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Rog83]
    J. D. Rogawski, Representations of GL(n) and division algebras over a p-adic field, Duke Mathematical Journal 50 (1983), 161–196.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [RR96]
    C. Rader and S. Rallis, Spherical characters on p-adic symmetric spaces, American Journal of Mathematics 118 (1996), 91–178.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Wal85]
    J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Mathematica 54 (1985), 173–242.MathSciNetzbMATHGoogle Scholar
  36. [Zha14a]
    W. Zhang, Automorphic period and the central value of Rankin–Selberg L-function, Journal of the American Mathematical Society 27 (2014), 541–612.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [Zha14b]
    W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Annals of Mathematics 180 (2014), 971–1049.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [Zha15]
    C. Zhang, On the smooth transfer for Guo–Jacquet relative trace formulae, Compositio Mathematica 151 (2015), 1821–1877.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Zha16]
    C. Zhang, Local periods for discrete series representations, Journal of Functional Analysis 271 (2016), 1525–1543.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Zha17]
    C. Zhang, An orthogonality relation for spherical characters of supercuspidal representations, Pacific Journal of Mathematics 270 (2017), 247–255.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Brooke Feigon
    • 1
  • Kimball Martin
    • 2
  • David Whitehouse
  1. 1.Department of MathematicsThe City College of New YorkNew YorkUSA
  2. 2.Department of MathematicsUniversity of OklahomaNormanUSA

Personalised recommendations