2013/01/26 by Junehyuk Jung, Jung, Junehyuk
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1301.6211
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 L2-norm of the restriction to a fixed compact geodesic segment of η=\iy~:~y>0\ ⊂ ℍ. We also obtain an upper bound of Oε(tϕ3/8+ε) for the L^∞ norm along a density 1 subsequence of Hecke--Maass cusp forms; for such forms, this is an improvement over the upper bound of Oε(tϕ5/12+ε) 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 η, grows faster than tϕ1/12-ε for any ε>0, under the assumption that the Lindelöf 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 tϕ1/8-ε for any ε>0.