A sum formula for a pair of closed geodesics on a hyperbolic surface
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.
scientific article; zbMATH DE number 5306889
scientific article