The 𝐿²-norm of Maass wave functions

Abstract

Let D D denote the fundamental domain for the full modular group. Suppose that f ∈ L 2 ( D ) f \in {L^2}(D) satisfies the wave equation Δ f = λ f \Delta f = \lambda f , where Δ \Delta is the noneuclidean Laplacian, and further, assume that f f is a common eigenfunction for all the Hecke operators. Then upper and lower bounds for the L 2 {L^2} -norm of f f are determined which depend only on λ \lambda and the first Fourier coefficient of f f .