This paper is concerned with parameterizing and counting of conjugacy classes of infinite maximal dihedral subgroups of \(\Gamma= \text{PSL}_2(\mathbb Z)\) and their connection to binary quadratic forms. Geometrically speaking, these classes correspond to closed geodesics on the modular surface which are equivalent to themselves when their orientation is reversed. The infinite maximal dihedral subgroups of \(\Gamma\) are of the form \(\langle\gamma,S\rangle\) where \(\gamma\) is a primitive hyperbolic element of \(\Gamma\) and \(S\in\Gamma\) satisfies \(S^{-1}\gamma JS=\gamma^{-1}\). Such elements \(\gamma\in\Gamma\) are called reciprocal. Of crucial importance for the author’s approach is an explicit parametrization of the set of reciprocal classes in terms of the set of pairs \((a,b)\in\mathbb Z^2\) such that \(d= 4a^2+ b^2\) is not a square and in terms of the least positive solution of the Pellian equation \(t^2- du^2= 4\). Moreover, the explicit correspondence between the set of primitive hyperbolic classes and classes of primitive binary quadratic forms is used in an essential way. The main result says that the number of primitive hyperbolic reciprocal classes with trace \(\leq x\) is asymptotically equal to \({3\over 8} x\) for \(x\to\infty\), and there are three more results of similar type.
The geodesics under consideration have recently arisen in a number of different contexts, and there are ample comments on these and on the question of equidistribution of closed geodesics (also in higher dimensions).
For the entire collection see [Zbl 1121.11003].