{"paper":{"title":"Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeanine Van Order","submitted_at":"2025-10-11T18:03:15Z","abstract_excerpt":"Let $E$ be an elliptic curve parametrized by a newform $\\phi \\in S_2(\\Gamma_0(N))$. Let $k$ be a quadratic field of discriminant $d_k$ prime to $N$. Given a character $\\chi$ of the ideal class group of $k$ with theta series $\\theta(\\chi)$, we relate the central derivative values $\\Lambda'(E/k, \\chi, 1) = \\Lambda'(1/2, \\phi \\times \\theta(\\chi))$ of the $\\chi$-twisted $L$-function of $E/k$ to arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N)\\times X_0(N)$ when $k$ is imaginary quadratic, and to sums of Green's functions of Hirzebruch-Zagier divisors along real geodesic cycles of $X_0(N"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.10303","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.10303/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}