{"paper":{"title":"Local solubility of a family of ternary conics over a biprojective base I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Cameron Wilson","submitted_at":"2024-09-16T19:35:28Z","abstract_excerpt":"Let $f,g\\in\\mathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)\\in\\mathbb{P}^1(\\mathbb{Q})\\times\\mathbb{P}^1(\\mathbb{Q})$ such that the ternary conic\n  \\[\n  X_{(u,v)}: f(u_1,u_2)x^2 + g(v_1,v_2)y^2 = z^2\n  \\] has a rational point. We also give some conditions under which lower bounds exist."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.10688","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/2409.10688/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"}