{"paper":{"title":"Twisted Teichm\\\"uller curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Christian Wei{\\ss}","submitted_at":"2012-08-09T13:04:20Z","abstract_excerpt":"Let $X_D$ denote the Hilbert modular surface $\\HH \\times \\HH^- / \\SL_2(\\OD)$. In \\cite{HZ76}, F. Hirzebruch and D. Zagier introduced Hirzebruch-Zagier cycles, that could also be called twisted diagonals. These are maps $\\HH \\to \\HH \\times \\HH^-$ given by $z \\mapsto (Mz,-M^\\sigma z)$ where $M \\in \\GL_2^+(K)$ and $\\sigma$ denotes the Galois conjugate. The projection of a twisted diagonal to $X_D$ yields a Kobayashi curve, i.e. an algebraic curve which is a geodesic for the Kobayashi metric on $X_D$. Properties of Hirzebruch-Zagier cycles have been abundantly studied in the literature.\\\\Teichm\\\"u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1895","kind":"arxiv","version":3},"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"}