## 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 ⊂ 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 d_{1} and d_{2} as d_{1} + d_{2} → ∞. 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.

Original language | English (US) |
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Pages (from-to) | 1325-1357 |

Number of pages | 33 |

Journal | Algebra and Number Theory |

Volume | 17 |

Issue number | 7 |

DOIs | |

State | Published - 2023 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory