Lattice points counting and bounds on periods of Maass forms

TL;DR

This paper establishes new bounds on Fourier coefficients of Maass forms using amplification and lattice point counting techniques, advancing understanding of automorphic forms on hyperbolic spaces.

Contribution

It introduces a novel 'soft' proof method for bounding Fourier coefficients of Maass forms applicable to general co-finite lattices in PGL(2,R).

Findings

Derived non-trivial bounds on Fourier coefficients.

Applied amplification method with Airy phenomenon.

Utilized effective lattice point counting asymptotics.

Abstract

We provide a "soft" proof for non-trivial bounds on spherical, hyperbolic and unipotent Fourier coefficients of a fixed Maass form for a general co-finite lattice $\Gamma$ in $PGL(2,R)$. 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 $PGL(2,R)$.