On an extreme value law for the unipotent flow on ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})\mathrm{/}{\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$

We study an extreme value distribution for the unipotent flow on the modular surface ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})\mathrm{/}{\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r\in [-\frac{1}{2}\log 2,\mathrm{\infty })$, and establish asymptotic behavior of $F(r)$ as $r\to -\mathrm{\infty }$.