On the $L^2$-restriction norm problem for closed geodesics on the modular surface

Abstract

Let $f$ be a Petersson normalized Hecke-Maass cusp form with spectral parameter $t\ge 2$ and let ${\mathcal{C}}_D$ be the union of closed geodesics in ${\text{Sl}}_2(\mathbb{Z})\setminus \mathbb{H}$ associated to a fundamental discriminant $D>0$. Following a suggestion by Sarnak in his letter to Reznikov, we express the restriction norm $||f{|}_{{\mathcal{C}}_D}|{|}_2^2$ as a weighted sum of central values of L-functions using Waldspurger's formula. This allows us to get an unconditional improvement over the current bounds.

18 pages