Intersecting geodesics on the modular surface
Intersecting geodesics on the modular surface
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DOI：
10.2140/ant.2023.17.1325
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发表时间：
2021-01
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影响因子：
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通讯作者：
Junehyuk Jung;Naser T. Sardari
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作者：
Junehyuk Jung;Naser T. Sardari
我们引入了\textit{模相交核}，并用它来研究测地线如何在全模曲面上相交$\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$。设$C_d$为封闭测地线与判别式$d$的并集，设$\beta\subset \mathbb{X}$为紧致测地线段。作为杜克定理在模交核中的一个应用，我们证明了$ \{\left(p,\theta_p\right)~:~p\in \beta \cap C_d\}$在$\beta \times [0,\pi]$上相对于$\sin \theta ds d\theta$是等分布的，并且节电率为$d \to +\infty$。这里$\theta_p$是$\beta$和$C_d$在$p$处的交角。这就解决了里卡兹提出的主要猜想\cite{rick}。对于$C_{d_1}$和$C_{d_2}$之间的交点角分布，我们证明了类似的结果，并且$d_1$和$d_2$的节电率与$d_1+d_2 \to \infty$相似。由于模交核在顶点附近的奇异性，以往关于紧曲面的相应问题的研究并不适用于$\mathbb{X}$。通过在$PSL_2\left(\mathbb{Z}\right) \backslash PSL_2\left(\mathbb{R}\right)$上用一般的（不一定是球形的）点对不变量逼近模交核，然后研究它们的全谱展开，分析了模交核的奇异性。
We introduce the \textit{modular intersection kernel}, and we use it to study how geodesics intersect on the full modular surface $\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$. 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 $ \{\left(p,\theta_p\right)~:~p\in \beta \cap C_d\}$ becomes equidistributed with respect to $\sin \theta ds d\theta$ on $\beta \times [0,\pi]$ with a power saving rate as $d \to +\infty$. Here $\theta_p$ is the angle of intersection between $\beta$ and $C_d$ at $p$. This settles the main conjectures introduced by Rickards \cite{rick}. 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 $PSL_2\left(\mathbb{Z}\right) \backslash PSL_2\left(\mathbb{R}\right)$ and then by studying their full spectral expansion.