{"paper":{"title":"Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Alberto Verjovsky","submitted_at":"2017-11-09T20:40:38Z","abstract_excerpt":"Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=\\text{PSL}(2,\\mathbb{R})/\\text{PSL}(2,\\mathbb{Z})$ of the modular orbifold $\\text{PSL}(2,\\mathbb Z)$. In fact, the canonical projection $\\mathfrak{p}:M\\to\\mathbb{H}/\\text{PSL}(2,\\mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $y\\to0$: for every smooth fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03593","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}