Lattice points counting and bounds on periods of maass forms

We provide a “soft” proof for nontrivial bounds on spherical, hyperbolic, and unipotent Fourier coefficients of a fixed Maass form for a general cofinite lattice Γ in PGL₂(R). We use the amplification method based on the Airy type phenomenon for corresponding matrix coefficients and an effective Selberg type pointwise asymptotic for the lattice points counting in various homogeneous spaces for the group PGL₂(R). This requires only L²-theory. We also show how to use the uniform bound for the L⁴-norm of K-types in a fixed automorphic representation of PGL₂(R) in order to slightly improve these bounds.