Size of discriminants of periodic geodesics of the modular surface

Abstract

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

Theorems & Definitions (55)

• Proposition 1.1
• Theorem 1
• Theorem 2
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 3.1