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The Lowest Eigenvalue for Congruence Groups

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

Abstract

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.

Keywords

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.

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Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

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

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