Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intersecting geodesics on the modular surface   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.

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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex