The Geodesic Restriction Problem for Arithmetic Spherical Harmonics

Given a Riemannian manifold $M$ and an $L^2$-normalized Laplacian eigenfunction $\psi$ on $M$ with eigenvalue $\lambda^2$, a general problem in analysis is to understand how the mass of $\psi$ distributes around $M$. There are different ways to attack this problem. One of them is to analyze the $L^p$-norm of $\psi$ restricted to a submanifold of $M$. Here, we concentrate on the case $M=S^2$, $p=2$, and we restrict to geodesics of the sphere. Burq, G\'erard, and Tzvetkov showed, for $\gamma$ a geodesic of $S^2$ (and indeed for more general surfaces), that $||\psi|_{\gamma}||_{L^2} \ll \lambda^{1/4}$ and that this bound is optimal in general. In this paper, we specialize to the case in which $\psi$ is an eigenfunction of all the Hecke operators on the sphere and consider the set of geodesics $\mathcal{C}_{D}$ of $S^2$ associated to fundamental discriminants $D<0$. By combining approaches of Ali and Magee, we improve the previous upper bound to $||\psi|_{\mathcal{C}_{D}}||_{L^2} \ll_{D,\varepsilon} \lambda^{\varepsilon}$ for any $\varepsilon>0$, which is essentially sharp.