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, Gérard, and Tzvetkov showed, for $\gamma$ a geodesic of $S^2$ (and indeed for more general surfaces), that $\mathrm{\mid }\mathrm{\mid }\psi {\mathrm{\mid }}_{\gamma }\mathrm{\mid }{\mathrm{\mid }}_{L^2}\ll {\lambda }^{1\mathrm{/}4}$ 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 $\mathrm{\mid }\mathrm{\mid }\psi {\mathrm{\mid }}_{{\mathcal{C}}_D}\mathrm{\mid }{\mathrm{\mid }}_{L^2}{\ll }_{D,\epsilon }{\lambda }^{\epsilon }$ for any $\epsilon >0$, which is essentially sharp.