Intersecting geodesics on the modular surface

We introduce the \textit{modular intersection kernel}, and we use it to study how geodesics intersect on full modular surface $\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$. Let $C_d$ be union of closed with discriminant $d$ let $\beta\subset \mathbb{X}$ a compact geodesic segment. As an application Duke's theorem kernel, prove that $ \{\left(p,\theta_p\right)~:~p\in \beta \cap C_d\}$ becomes equidistributed respect $\sin \theta ds d\theta$ $\beta \times [0,\pi]$ power saving rate as $d \to +\infty$. Here $\theta_p$ is angle between $\beta$ at $p$. This settles main conjectures introduced by Rickards \cite{rick}. similar result for distribution angles intersections $C_{d_1}$ $C_{d_2}$ power-saving in $d_1$ $d_2$ $d_1+d_2 \infty$. Previous works corresponding problem surfaces do not apply $\mathbb{X}$, because singular behavior kernel near cusp. analyze approximating general (not necessarily spherical) point-pair invariants $PSL_2\left(\mathbb{Z}\right) PSL_2\left(\mathbb{R}\right)$ then studying their spectral expansion.