Size of discriminants of periodic geodesics of the modular surface

Abstract

Pick a random matrix $\gamma$ in $\Gamma={\rm SL}(2,\mathbb{Z})$. Denote by $\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let $\Delta_K$, $\Delta_\gamma$ and $\Delta = {\rm Tr}(\gamma)^2-4$ 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%.