Geodesic restrictions of arithmetic eigenfunctions

Abstract

Let $X$ be an arithmetic hyperbolic surface arising from a quaternion division algebra over $\mathbb{Q}$. Let $\psi$ be a Hecke–Maass form on $X$, and let $\ell$ be a geodesic segment. We obtain a power saving over the local bound of Burq, Gérard, and Tzvetkov for the $L^2$-norm of $\psi$ restricted to $\ell$, 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 $\ell$.