Geodesic restrictions of arithmetic eigenfunctions

TL;DR

This paper improves bounds on the restriction of arithmetic eigenfunctions to geodesic segments on hyperbolic surfaces by extending arithmetic amplification techniques, leading to power savings over previous local bounds.

Contribution

It introduces an extension of arithmetic amplification methods to obtain stronger bounds on eigenfunction restrictions and Fourier coefficients along geodesics.

Findings

Achieved power savings over previous bounds for eigenfunction restrictions

Improved bounds for Fourier coefficients along geodesics

Extended arithmetic amplification techniques to new settings

Abstract

Let X be an arithmetic hyperbolic surface, \psi a Hecke-Maass form, and l a geodesic segment on X. We obtain a power saving over the local bound of Burq-G\'erard-Tzvetkov for the L^2 norm of \psi restricted to l, by extending the technique of arithmetic amplification developed by Iwaniec and Sarnak. We also improve the local bounds for various Fourier coefficients of \psi along l.