Year: 2026
Type: preprint
Abstract: We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle ϕ, φ^2 \rangle$ uniformly in $t_ϕ\to \infty$ and $t_φ\to \infty$, where $t_ϕ$ and $t_φ$ are the eigenparameters corresponding to Maass forms $ϕ$ and $φ$ on $X$.
Source: arXiv (Cornell University)
Authors: Jung, Junehyuk, Park, Insung, Zenz, Peter
Institution: Stony Brook University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214209005