The Lowest Eigenvalue for Congruence Groups

  • Henryk Iwaniec
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 20)


Let L 2 (Γ\ℍ) be the space of square-integrable automorphic functions with respect to a group Γ ⊂ SL 2 (ℝ) acting discontinuously on the hyperbolic plane ℍ such that the quotient space Γ\ℍ has finite volume. The Laplace-Beltrami operator on L 2 (Γ\ℍ) has a discrete spectrum λ0 = 0 < λ1 ≤ λ2 ≤ … and a continuous spectrum [1/4, ∞). The eigenpacket of con¬tinuous spectrum consists of Eisenstein series E a (z, s) on the line Re s = 1/2 and the eigenfunctions of the discrete spectrum are Maass cusp forms together with a finite number of residues of E a (z, s) at poles in the segment of 1/2 < s ≤ 1. If Γ is a congruence group the only pole of Eisenstein series is at s = 1 which yields a constant eigenfunction for eigenvalue λ0 = 0, the remaining subspace of discrete spectrum is cuspidal and infinite dimensional.


Discrete Spectrum Eisenstein Series Cusp Form Riemann Hypothesis Primitive Character 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [DBHI]
    D. Bump, W. Duke, J. Hoffstein and H. Iwaniec,An estimate for the Hecke eigenvalues of Maass forms, Inter.Math.Res.Notices 4 (1992), 75 – 81.MathSciNetCrossRefGoogle Scholar
  2. [DI]
    J.M. Deshouillers and H. Iwaniec,Kloosterman sums and Fourier coefficients of cusp forms, Invent.Math. 70 (1982), 219 – 288.MathSciNetMATHCrossRefGoogle Scholar
  3. [DI 1–3]
    W. Duke and H. Iwaniec,Estimates for coefficientsof L-functions, I (Montreal 1989 ), II (Amalfi 1992), III (Paris 1990 ).Google Scholar
  4. [GJ]
    S. Gelbart and H. Jacquet,A relation between automorphic representationsof GL(2) and GL(3), Ann.Sci.Ecole Norm.Sup. 4e serie 11 (1978), 471 – 552.MathSciNetMATHGoogle Scholar
  5. [I1]
    H. Iwaniec,Selberg’s lower bound of the first eigenvalue for congruence groupsy in Number Theory, Trace Formulas and Discrete Groups, Academic Press (1989), San Diego, 371–375.Google Scholar
  6. [I2]
    H. Iwaniec,Small eigenvalues of Laplacian forTQ(AT), Acta Arith. 56 (1990), 65 – 82.MathSciNetMATHGoogle Scholar
  7. [LRS]
    W. Luo, Z. Rudnick and P. Sarnak,On Selbergfs eigenvalue conjecture„Geom. Funct. Anal. 5 (1995), 387 – 401.MathSciNetMATHCrossRefGoogle Scholar
  8. [Se]
    A. Selberg,On the estimation of Fourier coefficients of modular forms, AMS. Proc.Symp. Pure Math. VII (1965), 1 – 15.MathSciNetGoogle Scholar
  9. [Sh]
    G. Shimura,On the holomorphy of certain Dirichlet series, Proc. London Math.Soc.(3) 31 (1975), 79 – 98.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Henryk Iwaniec
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations