Intersecting geodesics on the modular surface

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 C_d 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, θ_p) : P ∊ β ∩ C_d} becomes equidistributed with respect to sin θ ds dθ on β ×[0, π] with a power saving rate as d → + ∞. 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_d1 and C_d2 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.