Lattice points counting and bounds on periods of Maass forms

Abstract

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 \Gamma in PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {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 PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {R})} . This requires only L 2 L^2 -theory. We also show how to use the uniform bound for the L 4 L^4 -norm of K K -types in a fixed automorphic representation of PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {R})} in order to slightly improve these bounds.