Intersecting geodesics on the modular surface

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We introduce the modular intersection kernel, and we use it to study how geodesics intersect on the full modular surface X = PSL2(Z)\H. 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 θ ds dθ 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 PSL2(Z)\ PSL2(ℝ) and then by studying their full spectral expansion.

Original languageEnglish (US)
Pages (from-to)1325-1357
Number of pages33
JournalAlgebra and Number Theory
Issue number7
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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