On Selberg’s Eigenvalue Conjecture

  • Wenzhi Luo
  • Zeév Rudnick
  • Peter Sarnak
Chapter

Abstract

Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 < λ1 < … be the eigenvalues of the non-euclidean Laplacian on L 2(Γ\H 2). A fundamental conjecture of Selberg ([Se]) asserts that the smallest nonzero eigenvalue λ1(Γ) ≥1/4 = 0.25. In the same paper Selberg proved that λ1(Γ) ≥3/16 = 0.1875. Gelbart and Jacquet ([GJ]), using very different methods, improved this to λ1(Γ) > 3/16. Iwaniec ([I]) showed that for almost all Hecke congruence groups Γ0(p) with a certain multiplier χ p , one has λ1(Γ0(p), χ p ) ≥ 44/225 = 0.19555…. In [I], he also established a density theorem for possible exceptional eigenvalues as above, which while not giving any improvement on 3/16 for an individual Γ is sufficiently strong to substitute for Selberg’s conjecture in many applications to number theory. Selberg’s conjecture is the archimedean analogue of the “Ramanujan Conjectures” on the Fourier coefficients of Maass forms. For these, much progress has been made in improving the relevant estimates, beginning with Serre ([Ser]) and later on Shahidi ([Sh2]) and Bump-Duke-Hoffstein-Iwaniec ([BDHI]). In this paper we restore the balance and establish in part for the archimedean place what is known at the finite places. The method on the face of it is quite different, but the quality of the results coincide (the reason will be made clear later).

Keywords

Congruence Subgroup Dirichlet Character Maass Form Cuspidal Automorphic Representation Archimedean Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BR]
    L. Barthel, D. Ramakrishnan, A nonvanishing result for twists of L-functions of GL(n), Duke Math. Jour. 74 (1994), 681–700.MathSciNetMATHCrossRefGoogle Scholar
  2. [BDHI]
    D. Bump, W. Duke, J. Hoffstein, H. Iwaniec, An estimate for the Hecke eigenvalues of Maass forms, Inter. Math. Res. Notices 4 (1992), 75–81.MathSciNetCrossRefGoogle Scholar
  3. [BuGi]
    D. Bump, D. Ginzburg, Symmetric square L-functions on GL(r), Ann. of Math. 136 (1992), 137–205.MathSciNetMATHCrossRefGoogle Scholar
  4. [D]
    P. Deligne, SGA 4\(\frac{1}{2}\) Cohomologie Etale, Lecture Notes in Math. 569, Springer-Verlag 1977.Google Scholar
  5. [DuI]
    W. Duke, H. Iwaniec, Estimates for coefficients of L-functions, Proc. C.R.M. Conf. on Automorphic Forms and Analytic Number Theory, Montréal (1989), 43–48.Google Scholar
  6. [GJ]
    S. Gelbart, H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. 4e série 11 (1978), 471–552.MathSciNetMATHGoogle Scholar
  7. [GoJ]
    R. Godement, H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer-Verlag, 1972.MATHGoogle Scholar
  8. [GolS]
    D. Goldfeld, P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983), 243–250.MathSciNetMATHCrossRefGoogle Scholar
  9. [I]
    H. Iwaniec, Small eigenvalues of the Laplacian on Г0(N), Acta Arith. 56 (1990), 65–82.MathSciNetMATHGoogle Scholar
  10. [J]
    H. Jacquet, Principal L-functions of the linear group, Proc. Symp. Pure Math. 33:2, 63–86, American Math. Soc., Providence 1979.Google Scholar
  11. [JP-SS]
    H. Jacquet, I.I. Piatetski-Shapiro, J.A. Shalika, Rankin-Selberg convolutions, Amer. Jour. of Math 105 (1983), 367–464.MathSciNetMATHCrossRefGoogle Scholar
  12. [JSha1]
    H. Jacquet, J.A. Shalika, On Euler products and the classification of automorphic representations I, Amer. Jour. of Math. 103 (1981), 499–558.MathSciNetMATHCrossRefGoogle Scholar
  13. [JSha2]
    H. Jacquet, J.A. Shalika, Rankin-Selberg convolutions: Archimedean theory, Israel Math. Conf. Proceedings 2 (1990), 125–208.MathSciNetGoogle Scholar
  14. [L]
    Yu.V. Linnik, Additive problems and eigenvalues of modular operators, Proc. I.C.M. Stockholm (1962), 270–284.Google Scholar
  15. [LuS]
    W. Luo, P. Sarnak, Quantum Ergodicity of Eigenfunctions on PSL 2(Z)\H 2, preprint (1994).Google Scholar
  16. [MW]
    C. Mœglin, J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. Ecole Norm. Sup. 4e série 22 (1989), 605–674.MATHGoogle Scholar
  17. [PP-S]
    S.J. Patterson, I.I. Piatetski-Shapiro, The symmetric square L-function attached to a cuspidal automorphic representation of GL(3), Math. Ann. 283 (1989), 551–572.MathSciNetMATHCrossRefGoogle Scholar
  18. [R]
    D. Rohrlich, Nonvanishing of L-functions for GL(2), Invent. Math. 97 (1989), 383–401.MathSciNetCrossRefGoogle Scholar
  19. [RuS]
    Z. Rudnick, P. Sarnak, Zeros of Principal L-functions and Random Matrix Theory, preprint.Google Scholar
  20. [S]
    P. Sarnak, Prime Geodesic Theorems, Ph.D. Thesis, Stanford University (1980).Google Scholar
  21. [Se]
    A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. in Pure Math. 8, Amer. Math. Soc., Providence (1965), 1–15.Google Scholar
  22. [Ser]
    J.-P. Serre, letter to J.-M. Deshouillers (1981).Google Scholar
  23. [Sh1]
    F. Shahidi, On certain L-functions, Amer. Jour. of Math. 103 (1981), 297–355.MathSciNetMATHCrossRefGoogle Scholar
  24. [Sh2]
    F. Shahidi, On the Ramanujan conjectures and finiteness of poles for certain L-functions, Ann. of Math. 127 (1988), 547–584.MathSciNetMATHCrossRefGoogle Scholar
  25. [Shi]
    G. Shimura, On the periods of modular forms, Math. Ann. 229 (1977), 211–221.MathSciNetMATHCrossRefGoogle Scholar
  26. [We]
    A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. 34 (1948), 204–207.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1995

Authors and Affiliations

  • Wenzhi Luo
    • 1
    • 3
  • Zeév Rudnick
    • 2
    • 4
  • Peter Sarnak
    • 2
  1. 1.School of MathematicsInstitute for Advanced StudiesPrincetonUSA
  2. 2.Dept. of MathPrinceton UniversityPrincetonUSA
  3. 3.MSRIBerkeleyUSA
  4. 4.School of Math. SciTel Aviv UniversityTel AvivIsrael

Personalised recommendations