Geodesics in homology classes and periods of automorphic forms
Let \(\Gamma\) be a cocompact purely hyperbolic discrete subgroup of \(PSL_ 2({\mathbb{R}})\) acting on the upper half-plane H and \(X=\Gamma \setminus H\) the associated compact Riemann surface of genus \(g\geq 2\). Each closed geodesic \(\gamma\) on X belongs to a homology class \(h(\gamma)\in H_ 1(X,{\mathbb{Z}})\). For a homology class \(\beta \in H_ 1(X,{\mathbb{Z}})\) let \(\pi_{\beta}(T):=\#\{\gamma: h(\gamma)=\beta\), L(\(\gamma\))\(\leq T\}\) (T\(\geq 0)\) be the counting function of closed geodesics of length L(\(\gamma\)) at most T with \(h(\gamma)=\beta\). The analogue of the prime geodesic theorem for \(\pi_{\beta}(T)\) is known from work of Adachi/Sunada and Phillips/Sarnak: \[ \pi_{\beta}(T)=(g- 1)^ g\frac{e^ T}{T^{g+1}}(1+O(\frac{1}{T}))\text{ for } T\to \infty. \] The key ingredient in the proof of this theorem is the trace formula. The aim of the work under review is to extend the latter result to the summatory functions \[ \psi_{\beta}(\sigma,T)=\sum_{h(\gamma)=\beta,\quad L(\gamma)\leq T}\int_{\gamma}\sigma, \] where \(\sigma\) is an automorphic form. The main result is a theorem describing the asymptotic behaviour of \(\psi_{\beta}(\sigma,T)/\psi_{\beta}(1,T)\) (where \(\sigma\perp 1)\) as \(T\to \infty\). This implies uniform distribution of closed geodesics in a fixed homology class relative to Haar measure on \(\Gamma \setminus PSL_ 2({\mathbb{R}})\) as the length tends to infinity. - For holomorphic forms \(\sigma\) the author reformulates his theorem in terms of Eichler cohomology and Poincaré series.
scientific article; zbMATH DE number 4210240
scientific article