{"id":"W1976181015","doi":"10.1007/s00208-004-0517-2","title":"Reductions of an elliptic curve and their Tate-Shafarevich groups","year":2004,"lang":"en","type":"article","venue":"Mathematische Annalen","topic":"Algebraic Geometry and Number Theory","field":"Mathematics","cited_by":20,"is_retracted":false,"has_abstract":false,"ca_institutions":"Fields Institute for Research in Mathematical Sciences","funders":"","keywords":"Mathematics; Modulo; Elliptic curve; Irrational number; Order (exchange); Pure mathematics; Function (biology); Combinatorics; Geometry","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.000786567,0.0002398257,0.0004336568,0.0001296259,0.0001326072,0.00002887564,0.0002518358,0.0001120712,0.0002585808],"category_scores_gemma":[0.0002748649,0.000188122,0.0001032872,0.0002973177,0.0001894767,0.0003258402,0.00009038815,0.0001935076,0.00006169453],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00001662952,"about_ca_system_score_gemma":0.00003276265,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00001162222,"about_ca_topic_score_gemma":0.000006997094,"domain_scores_codex":[0.9987148,0.0001009393,0.000448874,0.0002874141,0.0001772836,0.0002707035],"domain_scores_gemma":[0.998722,0.0002497623,0.0002306366,0.0005699448,0.00009750133,0.0001301978],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00005435646,0.001122229,0.0001606337,0.001351623,0.0003130111,0.00001405998,0.01891672,0.00002621366,0.005756794,0.9672574,0.0006442817,0.00438261],"study_design_scores_gemma":[0.0004578483,0.0001502761,0.0003376882,0.0002083796,0.00006632322,0.0001918018,0.002456994,0.00004373013,0.02033524,0.9751838,0.0003275769,0.000240266],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9871302,0.0005787101,0.007677441,0.0002956708,0.00006939664,0.0002852957,0.00002819478,0.0001108643,0.003824286],"genre_scores_gemma":[0.9815559,0.00007370808,0.0174627,0.00004728957,0.00009830774,0.00002903858,0.00001156226,0.00004422592,0.0006772446],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.01645973,"threshold_uncertainty_score":0.7671393,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04101843430212915,"score_gpt":0.2956320791869715,"score_spread":0.2546136448848424,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}