Skip to main content
SpringerLink
Log in

On the nonvanishing of someL-functions

  • Published:
Proceedings of the Indian Academy of Sciences - Mathematical Sciences Aims and scope Submit manuscript

Abstract

The non-vanishing, at the centre of symmetry, of theL-function attached to an automorphic representation of GL(2) or its twists by quadratic characters has been extensively investigated, in particular by Waldspurger. The purpose of this paper is to outline a new proof of Waldspurger’s results. The automorphic representations of GL(2) and its metaplectic cover are compared in two different ways; one way is by means of a “relative trace formula”; the relative trace formula presented here is actually a generalization of the work of Iwaniec.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bruggeman R, Fourier coefficients of cusps forms,Invent. Math. 45 (1978) 1–18

    Article  MATH  MathSciNet  Google Scholar 

  2. Deshouillers J M and Iwaniec H, Kloosterman sums and Fourier coefficients of cusp forms,Invent. Math. 70 (1982) 219–288

    Article  MATH  MathSciNet  Google Scholar 

  3. Flicker Y, Automorphic forms on covering groups of GL(2),Invent. Math. 57 (1980) 119–182

    Article  MATH  MathSciNet  Google Scholar 

  4. Flicker Y and Kazdhan D, Metaplectic correspondence,Publ. Math. IHES No. 64 (1986) pp. 53–110

  5. Gelbart S, Weil’s representations and the spectrum of the metaplectic group,Lecture notes in Mathematics (Heidelberg, Berlin, New York: Springer-Verlag) vol. 530 (1976)

    Google Scholar 

  6. Gelbart S and Piatetski Shapiro I, Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger,Isr. J. Math. 44 (1983) 97–126

    Article  MATH  Google Scholar 

  7. Gelbart S and Soudry D, On Whittaker models and the vanishing of Fourier coefficients of cusp formsProc. Indian Acad. Sci. (Math. Sci.) 97 (1987) 67–74

    MathSciNet  Google Scholar 

  8. Goldfeld D, Hoffstein D and Patterson J, On automorphic functions of half-integral weight with applications to elliptic curves, in:Number theory related to Fermais last theorem (Boston: Birkhauser) (1982) pp. 153–193

    Google Scholar 

  9. Iwaniec H, On Waldspurger theorem (preprint)

  10. Iwaniec H, Fourier coefficients of modular forms of half-integral weight,Invent. Math. 87 (1987) 385–401

    Article  MATH  MathSciNet  Google Scholar 

  11. Jacquet H, Sur un résultat de Waldspurger,Ann. Scient. Ec. Norm. Sup. ′4Series, t19 (1986) 185–229

    MATH  MathSciNet  Google Scholar 

  12. Kazdhan D and Patterson S, Metaplectic forms,Publ. Math. No. 59 (1982) 35–142

  13. Kazdhan D and Patterson S, Towards a generalized Shimura correspondence,Adv. Math. 60 (1986) 161–234

    Article  Google Scholar 

  14. Kuznietsov N V, Peterson hypothesis for parabolic forms of weight zero and Llinnik hypothesis,Math. Shorn. 111 (1980) 334–383

    Google Scholar 

  15. Kohnen W, Fourier coefficients of modular forms of half integral weight,Math. Ann. 271 (1985) 237–268

    Article  MATH  MathSciNet  Google Scholar 

  16. Labesse J-P and Langlands R, L-indistinguishability for SL(2),Can. J. Math. 31 (1979) 726–785

    MATH  MathSciNet  Google Scholar 

  17. Piatetski Shapiro I, Work of Waldspurger, in:Lie group representations II, Lecture notes in mathematics (Springer-Verlag) vol. 1041 (1984) pp. 280–301

    Google Scholar 

  18. Salié H, Uber die Kloostermannschen summen S(u,v,q),Math. Z. 34 (1931) 91–109

    Article  MATH  Google Scholar 

  19. Waldspurger J-L, Correspondence de Shimura,J. Math. Pure Appl. 59 (1980) 1–113

    MATH  MathSciNet  Google Scholar 

  20. Waldspurger J-L, Sur les coefficients de Fourier des formes modulaires de poids demi-entier,J. Math. Pure Appt. 60 (1981) 375–484

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, 10027, New York, NY, USA

    Hervé Jacquet

Authors
  1. Hervé Jacquet
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This is the text of a lecture delivered in June 1987 in Paris at the Symposium in honour of R Godement. This work was supported in part by NSF grant DMS-85-02789.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jacquet, H. On the nonvanishing of someL-functions. Proc. Indian Acad. Sci. (Math. Sci.) 97, 117–155 (1987). https://doi.org/10.1007/BF02837819

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02837819

Keywords

Search

Navigation