Closed geodesics and periods of automorphic forms (Q5938990)

Let \(\gamma\) be a prime closed geodesic on a compact hyperbolic surface of genus \(g\) uniformized by \(\Gamma< \text{PSL} (2,\mathbb{R})\), and \(\ell(\gamma)\) be its length, while \([\gamma]\) is its homology class \(H_1(\mathbb{H}/ \Gamma,\mathbb{Z})\). Further, let \(\pi(T,\alpha):= \#\{\gamma: l(\gamma)\leq T\), \([\gamma]= \alpha\}\). NEWLINENEWLINENEWLINEFrom the paper: ``G. Shimura introduced a period of a weight \(2m\) automorphic form \(f\) over a closed geodesic \(\gamma\), which we denote by \(r_m(f,\gamma)\). Zelditch showed that \(f\) has an ``asymptotic period'' \(\varepsilon(f)\) over a fixed homology class \(\alpha\). More precisely, writing \({\mathcal C}(T,\alpha)= \{\gamma: l(\gamma)\leq T\), \([\gamma]= \alpha\}\), NEWLINE\[NEWLINE\lim_{T\to\infty} \frac{1}{\pi(T,\alpha)} \sum_{\gamma\in{\mathcal C}(T,\alpha)} r_m(f,\gamma)= \varepsilon(f), \tag{*}NEWLINE\]NEWLINE and \(\varepsilon(f)\) is independent of the choice of \(\alpha\). In this note we show that one cannot do better than averaging in this result, in the sense that the closed geodesics for which \(r_m(f,\gamma)\) is close to \(\varepsilon(f)\) have density zero in the set \(\{\gamma: [\gamma]= \alpha\}\). More precisely, we prove Theorem 1 below. NEWLINENEWLINENEWLINEFor a weight \(2m\) automorphic form \(f\), let \(T(f)\) denote the closed subgroup of \(\mathbb{C}\) generated by the periods \(r_m(f,\gamma)\). As we see, \(T(f)\) spans \(\mathbb{C}\). NEWLINENEWLINENEWLINETheorem 1. Let \(f\) be a nonzero automorphic form of weight \(2m\), \(m>1\). Then, for any \(\eta\in T(f)\) and \(\delta>0\), we have NEWLINE\[NEWLINE\#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;|r_m(f,\gamma)-\eta|\leq \delta\}\sim C\frac{e^T} {T^{{\mathfrak g}+2}},NEWLINE\]NEWLINE where \(C>0\) is a constant independent of \(\alpha\) and \(\eta\). NEWLINENEWLINENEWLINEThe most important special cases of this result are when \(T(f)\) is a lattice or when \(T(f)= \mathbb{C}\). In these cases we can make the slightly more precise statements below. In each case \(\beta_f: \mathbb{R}^{2{\mathfrak g}+2}\to \mathbb{R}\) is a certain ``thermodynamic'' function (depending only on \(f\)). NEWLINENEWLINENEWLINESpecial cases. (1) If \(T(f)\) is a lattice in \(\mathbb{C}\) then, for any \(\eta\in T(f)\), we have NEWLINE\[NEWLINE\#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;r_m(f,\gamma)= \eta\}\sim \frac{|\mathbb{C}/T(f)|} {(2\pi)^{{\mathfrak g}+1} \sqrt{\det \nabla^2\beta_f(0)}} \frac{e^T} {T^{{\mathfrak g}+2}},NEWLINE\]NEWLINE where \(|\mathbb{C}/ T(f)|\) denotes the area of a fundamental domain for \(T(f)\). NEWLINENEWLINENEWLINE(2) If \(T(f)= \mathbb{C}\) then, for any \(\eta\in \mathbb{C}\) and \(\delta>0\), we have NEWLINE\[NEWLINE\#\{\gamma: l(\gamma)\leq T,\;[\gamma]=\alpha,\;|r_m(f,\gamma)-\eta|\leq \delta\}\sim \frac{\pi\delta^2} {(2\pi)^{{\mathfrak g}+1} \sqrt{\det\nabla^2 \beta_f(0)}} \frac{e^T} {T^{{\mathfrak g}+2}}.NEWLINE\]NEWLINE Corollary. For any \(\delta>0\), we have NEWLINE\[NEWLINE\lim_{T\to\infty} \frac{1}{\pi(T,\alpha)} \#\{\gamma: l(\gamma)\leq T,\;[\gamma]= \alpha,\;|r_m(f,\gamma)- \varepsilon(f)|\leq \delta\}=0,NEWLINE\]NEWLINE i.e., the closed geodesics with period close to \(\varepsilon(f)\) have zero density in \(\{\gamma: [\gamma]= \alpha\}\). NEWLINENEWLINENEWLINERemark. The restriction to a fixed homology class \(\alpha\) in (*) is crucial, even though the result is independent of \(\alpha\). Without this restriction, the averages vanish''.