We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound for the $L^2$-norm of the restriction to a fixed compact geodesic segment of $\eta=\{ iy~:~y>0\} \subset \mathbb{ H}$. We also obtain an upper bound of $O_\epsilon\left(t_\phi^{ 3/8+\epsilon}\right)$ for the $L^\infty$ norm along a density $1$ subsequence of Hecke--Maass cusp forms; for such forms, this is an improvement over the upper bound of $O_\epsilon\left(t_\phi^{ 5/12+\epsilon}\right)$ given by Iwaniec and Sarnak. In a recent work of Ghosh, Reznikov, and Sarnak, the authors proved for all even Hecke--Maass forms that the number of nodal domains, which intersect a geodesic segment of $\eta$, grows faster than $t_\phi^{ 1/12-\epsilon}$ for any $\epsilon>0$, under the assumption that the Lindel{ \"o}f Hypothesis is true and that the geodesic segment is long enough. Upon removing a density zero subset of even Hecke--Maass forms, we prove without making any assumptions that the number of nodal domains grows faster than $t_\phi^{ 1/8-\epsilon}$ for any $\epsilon>0$.
Categories: math.NT math.DG math.SP