Intersecting geodesics on the modular surface

Jung, Junehyuk; Sardari, Naser Talebizadeh

doi:10.2140/ant.2023.17.1325

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Intersecting geodesics on the modular surface

Junehyuk Jung and Naser Talebizadeh Sardari

Vol. 17 (2023), No. 7, 1325–1357

DOI: 10.2140/ant.2023.17.1325

Abstract

We introduce the modular intersection kernel, and we use it to study how geodesics intersect on the full modular surface $𝕏 = {PSL}_{2} (ℤ) ∖ ℍ$ . Let $C_{d}$ be the union of closed geodesics with discriminant $d$ and let $β \subset 𝕏$ be a compact geodesic segment. As an application of Duke’s theorem to the modular intersection kernel, we prove that ${(p, 𝜃_{p}) : p \in β \cap C_{d}}$ becomes equidistributed with respect to $\sin 𝜃 d s d 𝜃$ on $β \times [0, π]$ with a power saving rate as $d \to + \infty$ . Here $𝜃_{p}$ is the angle of intersection between $β$ 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 $𝕏$ , 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 ${PSL}_{2} (ℤ) ∖ {PSL}_{2} (ℝ)$ and then by studying their full spectral expansion.

Keywords

Closed geodesics, Modular forms, Intersection angles

Mathematical Subject Classification

Primary: 11F03, 11F70, 11F72

Milestones

Received: 9 December 2021

Revised: 18 May 2022

Accepted: 6 July 2022

Published: 30 May 2023

Authors

Junehyuk Jung
Department of Mathematics Brown University Providence, RI United States

Naser Talebizadeh Sardari
Department of Mathematics Pennsylvania State University State College, PA United States

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 17, No. 7, 2023