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

TL;DR

This paper expresses the restriction norm of Maass cusp forms on closed geodesics as a sum of L-function values, leading to improved bounds unconditionally, advancing understanding of quantum restrictions on modular surfaces.

Contribution

It provides a new expression for the restriction norm in terms of L-functions, enabling unconditional bounds improvements for Maass forms on modular surfaces.

Findings

Unconditional bounds on restriction norms are improved.

Restriction norms are expressed via central L-values using Waldspurger's formula.

The approach advances quantum unique ergodicity studies on modular surfaces.

Abstract

Let $f$ be a Petersson normalized Hecke-Maass cusp form with spectral parameter $t\geq 2$ 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.