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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(). Let Cd be the union of closed geodesics with discriminant d and let β 𝕏 be a compact geodesic segment. As an application of Duke’s theorem to the modular intersection kernel, we prove that {(p,𝜃p) : p β Cd} becomes equidistributed with respect to sin 𝜃dsd𝜃 on β × [0,π] with a power saving rate as d +. Here 𝜃p is the angle of intersection between β and Cd at p. This settles the main conjectures introduced by Rickards(2021).

We prove a similar result for the distribution of angles of intersections between Cd1 and Cd2 with a power-saving rate in d1 and d2 as d1 + d2 . 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()PSL 2() 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

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