We introduce the modular intersection kernel, and we use it to study how geodesics intersect on the full modular surface $\mathbb{𝕏}={\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_2(\mathbb{Z})\setminus ℍ$. Let $C_d$ be the union of closed geodesics with discriminant $d$ and let $\beta \subset \mathbb{𝕏}$ be a compact geodesic segment. As an application of Duke’s theorem to the modular intersection kernel, we prove that $\{(p,{𝜃}_p):p\in \beta \cap C_d\}$ becomes equidistributed with respect to $\sin <!--FUNCTION\; APPLICATION-->𝜃dsd𝜃$ on $\beta \times [0,\pi ]$ with a power saving rate as $d\to +\infty$. Here ${𝜃}_p$ is the angle of intersection between $\beta$ and $C_d$ at $p$. This settles the main conjectures introduced by Rickards(2021).

We prove a similar result for the distribution of angles of intersections between $C_{d_1}$ and $C_{d_2}$ with a power-saving rate in $d_1$ and $d_2$ as $d_1+d_2\to \infty$. Previous works on the corresponding problem for compact surfaces do not apply to $\mathbb{𝕏}$, because of the singular behavior of the modular intersection kernel near the cusp. We analyze the singular behavior of the modular intersection kernel by approximating it by general (not necessarily spherical) point-pair invariants on ${\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_2(\mathbb{Z})\setminus {\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_2(ℝ)$ and then by studying their full spectral expansion.

我们引入模交核，并用它来研究测地线如何在全模面上相交。$\mathbb{𝕏}={\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_{\mathrm{2个}}(\mathbb{Z})\setminus ℍ$. 让$C_d$是闭合测地线与判别式的联合$d$然后让$\beta \subset \mathbb{𝕏}$是一个紧凑的测地线段。作为杜克定理在模交核中的应用，我们证明了$\{(p,{𝜃}_p):p\in \beta \cap C_d\}$变得均匀分布$罪<!--FUNCTION\; APPLICATION-->𝜃d秒d𝜃$在$\beta \times [0,\pi ]$节电率为$d\to +\infty$. 这里${𝜃}_p$是之间的交角$\beta$和$C_d$在$p$. 这解决了 Rickards (2021) 引入的主要猜想。

我们证明了交叉点之间的角度分布的相似结果$C_{d_{\mathrm{1个}}}$和$C_{d_{\mathrm{2个}}}$节电率在$d_{\mathrm{1个}}$和$d_{\mathrm{2个}}$作为$d_{\mathrm{1个}}+d_{\mathrm{2个}}\to \infty$. 以前关于致密曲面相应问题的工作不适用于$\mathbb{𝕏}$，因为靠近尖点的模块化交集内核的奇异行为。我们通过用一般的（不一定是球形的）点对不变量在${\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_{\mathrm{2个}}(\mathbb{Z})\setminus {\mathrm{PSL}<!--FUNCTION\; APPLICATION-->}_{\mathrm{2个}}(ℝ)$然后通过研究它们的全光谱扩展。