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arXiv · 2211.04983

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

Dana Abou Ali
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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.

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Dana Abou Ali. 2022-11-09. On the $L^{2}$-restriction norm problem for closed geodesics on the modular surface. https://arxiv.org/abs/2211.04983

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