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Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces
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 ⟨ ϕ, φ2 ⟩ uniformly in tϕ→ ∞ and tφ→ ∞, where tϕ and tφ are the eigenparameters corresponding to Maass forms ϕ and φ on X.
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BibTeX
@misc{jung2026sharp,
title = {{Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces}},
author = {Junehyuk Jung and Insung Park and Peter Zenz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27071},
eprint = {2609.27071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27071},
}
RIS
TY - UNPB
TI - Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces
AU - Junehyuk Jung
AU - Insung Park
AU - Peter Zenz
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27071
UR - https://arxiv.org/abs/2609.27071
AB - 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 ⟨ ϕ, φ^2 ⟩ uniformly in t_ϕ→ ∞ and t_φ→ ∞, where t_ϕ and t_φ are the eigenparameters corresponding to Maass forms ϕ and φ on X.
ER -