Geometric invariants for real quadratic fields (Q338429)

Let \(\mathbb K\) be a real quadratic field. Then \(\mathbb K={\mathbb Q}(\sqrt D)\) where \(D>1\) is the discriminant of \(\mathbb K\). Let \(\text{Cl}^+(\mathbb K)\) be its ideal class group in the narrow sense. Then \(\text{Gen}(\mathbb K)= \text{Cl}^+(\mathbb K)/(\text{Cl}^+(\mathbb K))^2\) is the genus group of \(\mathbb K\). The modular group \(\Gamma =\text{PSL}(2, {\mathbb Z})\) acts on the upper half plane \(\mathcal H\) with standard fundamental domain NEWLINE\[NEWLINE\begin{aligned}{\mathcal F}=\{z\in {\mathcal H}:-1/2\leq \text{Re} (x)\leq 0 \;\text{and}\;|z|\geq 1\}\;\\ \cup \;\{z \in {\mathcal H}:0< \text{Re} (x) <1/2 \;\text{and}\;|z|> 1\}.\end{aligned}NEWLINE\]NEWLINE To an element \(A\in \text{Cl}^+(\mathbb K)\) one can associate a modular closed geodesic \({\mathcal C}_A\) on \(\Gamma \backslash \mathcal H\). In the paper under review the authors associate to \(A\) a finite area hyperbolic surface \(\mathcal F_A\) whose boundary component is a simple closed geodesic whose image in \(\Gamma \backslash \mathcal H\) is \({\mathcal C}_A\). The main result of the paper is the following theorem.NEWLINENEWLINETheorem 2. Suppose that for each positive fundamental discriminant \(D>1\), we choose a genus \(G_D\in \text{Gen}({\mathbb K})\). Let \(\Omega\) be an open disc contained in the fundamental domain \({\mathcal F}\) for \(\Gamma =\text{PSL}(2, {\mathbb Z})\), and let \(\Gamma\Omega\) be its orbit under the action of \(\Gamma\). We have NEWLINE\[NEWLINE{\pi\over 3}\sum_{A\in G_D}\text{area}({\mathcal F}_A\cap \Gamma\Omega)\sim \text{area}(\Omega)\sum_{A\in G_D}\text{area}({\mathcal F}_A),NEWLINE\]NEWLINE as \(D\rightarrow \infty\) through fundamental discriminants.NEWLINENEWLINEThis result is closely related to the uniform distribution result for closed geodesics obtained in [\textit{W. Duke,} Invent. Math. 92, No. 1, 73--90 (1988; Zbl 0628.10029)]. As with the proof of the latter result the authors employ the analytic method to obtain their proof of Theorem 2. This requires them to establish extensions of formulas of Hecke and Katok-Sarnak to which much of the paper is dedicated.