Size of discriminants of periodic geodesics of the modular surface

Abstract

Pick a random matrix $\gamma$ in $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by ${\mathcal{O}}_K$ the Dedekind ring generated by its eigenvalues, and let ${\Delta }_K$, ${\Delta }_{\gamma }$ and $\Delta =\mathrm{Tr}(\gamma {)}^2-4$ be the respective discriminant of the rings ${\mathcal{O}}_K$, the multiplier ring $M(2,\mathbb{Z})\cap \mathbb{Q}[\gamma ]$ and $\mathbb{Z}[\gamma ]$. We show that their ratios admit probability limit distributions. In particular, 42% of the elements of $\Gamma$ have a fundamental discriminant, and $\mathbb{Z}[\gamma ]$ is a ring of integers with probability 32%.