The Lowest Eigenvalue for Congruence Groups
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.
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