A sum formula for a pair of closed geodesics on a hyperbolic surface
type: Item
instance of: scholarly article
title: A sum formula for a pair of closed geodesics on a hyperbolic surface
zbMATH Open document ID: 1195.11073
DOI: 10.1215/00127094-2008-024
author: Nigel J. E. Pitt
published in: Duke Mathematical Journal
publication date: 2008-08-05T00:00:00Z
review text: Let \(\Gamma<\text{PSL}_2(\mathbb R)\) be a cocompact Fuchsian group acting on the upper half-plane \(\mathbb H\). As is well known, the space \(L^2(\Gamma\setminus\mathbb H)\) is spanned by the eigenfunctions of the hyperbolic Laplacian \(\Delta\). Denote by \((\varphi_j)_{j\geq 0}\) a complete orthonormal system of such eigenfunctions with corresponding eigenvalues \(\lambda_j=\frac 14+t^2_j\geq 0\). The closed geodesics \(\gamma\) on \(\Gamma\setminus\mathbb H\) correspond bijectively to the conjugacy classes of hyperbolic elements of \(\Gamma\). Denote by \[ {\mathcal P}_\gamma(\varphi_j)=\int_\gamma\varphi_j\frac{|dz|}{y} \] the period integral of the eigenfunction \(\varphi_j\) along the closed geodesic \(\gamma\). The aim of the paper under review is to prove a summation formula (resembling the Selberg trace formula) that relates the periods for a fixed pair of primitive closed geodesics \(\gamma_1, \gamma_2\) to the geometry of these geodesics on \(\Gamma\setminus \mathbb H\). The geometric data embodied in this formula are the angles at which the two geodesics intersect and the lengths of those geodesic segments that meet \(\gamma_1,\gamma_2\) orthogonally. A notable feature of the summation formula is that the Mehler-Fock transforms of two test functions come up which are evaluated at the numbers \(t_j\) corresponding to the eigenvalues of \(\Delta\). The proof of the summation formula requires a computation of the Petersson inner product of two suitable chosen Poincaré series in two different ways, ``spectrally'' and ``geometrically''. As an application the author proves some growth estimates for the period integrals.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 5306889
zbMATH Keywords: Mehler-Fock transform
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Laplacian
zbMATH Keywords: geodesic
zbMATH Keywords: period integral
zbMATH Keywords: summation formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2008-024
OpenAlex ID: W2002239621
cites work: Q3310101
cites work: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
cites work: Fourier coefficients of the resolvent for a Fuchsian group.
cites work: Local analysis of Selberg's trace formula
cites work: Q4481373
cites work: Prime geodesic theorem.
cites work: Q4846158
cites work: Closed geodesics, periods and arithmetic of modular forms
cites work: Q5568135
cites work: Quantum ergodicity of eigenfunctions on \(\text{PSL}_ 2(\mathbb{Z}) \backslash H^ 2\)
cites work: On pairs of closed geodesics on hyperbolic surfaces
cites work: Q4845625
cites work: Q3482231
cites work: Q3993105
cites work: Q3735790
cites work: Q3290778
cites work: Q5846853
cites work: Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms
description: scientific article; zbMATH DE number 5306889