On Whittaker models and the vanishing of Fourier coefficients of cusp forms

  • Stephen Gelbart
  • David Soudry


The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group.


Whittaker models cusp forms automorphic cuspidal representations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Coates J and Wiles A, On the conjecture of Birch and Swinnerton-Dyer,Invent. Math. 39 (1977) 223–251MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Cogdell J and Piatetski-Shapiro I, Base change for SL2,J. Number Theory (to appear)Google Scholar
  3. [3]
    Gelbart S and Piatetski-Shapiro I, Automorphic forms and L-functions for the unitary group, in:Lie group representations II, Lecture notes in mathematics No. 1041 (Berlin, Heidelberg: Springer Verlag) (1984) pp. 141–184Google Scholar
  4. [4]
    Gelbart S and Piatetski-Shapiro I, Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger,Isr. J. Math. 44 (1983) 97–126MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Howe R and Piatetski-Shapiro I, Some examples of automorphic forms on Sp(4),Duke Math. J. 50 (1983) 55–106MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Howe R, θ-series and invariant theory,Proc. Symp. Pure Math. No. 33, Part I (Providence, RI: Am. Math. Soc.) (1979) 275–286Google Scholar
  7. [7]
    Jacquet H, Piatetski-Shapiro I and Shalika J, The θ-correspondence from GSp(4) to GL(4) (in preparation)Google Scholar
  8. [8]
    Langlands R P, Base change for GL(2),Ann. Math. Stud. No. 96 (Princeton: University Press) (1980).Google Scholar
  9. [9]
    Piatetski-Shapiro I, Euler subgroups, in:Lie groups and their representations (ed.) I M Gelfand (New York: Halsted Press) (1975)Google Scholar
  10. [10]
    Piatetski-Shapiro I, Work of Waldspurger, in:Lie group representations II, Lecture notes in mathematics No. 1041 (Berlin, Heidelberg: Springer Verlag) (1984) pp. 280–302Google Scholar
  11. [11]
    Piatetski-Shapiro I, Multiplicity one theorems,Proc. Symp. Pure Math. No. 33, Part I (Providence, RI: Am. Math. Soc.) (1979) 209–212Google Scholar
  12. [12]
    Piatetski-Shapiro I, On the Saito-Kurokawa lifting,Invent. Math. 71 (1983) 309–338MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Shalika J, The multiplicity one theorem for GL(n), Ann. Math. 100 (1974) 171–193CrossRefMathSciNetGoogle Scholar
  14. [14]
    Shimura G, On modular forms of half-integral weight,Ann. Math. 97 (1973) 440–481CrossRefMathSciNetGoogle Scholar
  15. [15]
    Soudry D, A uniqueness theorem for representations of GSO(6) and the strong multiplicity one theorem for generic representations of GSp(4),Isr. J. Math. 58 (1987) 257–287MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Waldspurger J, Correspondence de Shimura,J. Math. Pure Appl. 59 (1980) 1–133MATHMathSciNetGoogle Scholar
  17. [17]
    Waldspurger J, Correspondences de Shimura et quaternions, (preprint) (1982)Google Scholar
  18. [18]
    Waldspurger J, Un exercise sur GSp(4, F) et les representations de Weil,Bull. Soc. Math. France 115 (1987) 35–69MATHMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1987

Authors and Affiliations

  • Stephen Gelbart
    • 1
  • David Soudry
    • 2
  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsTel-Aviv UniversityTel AvivIsrael

Personalised recommendations