{"paper":{"title":"Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions","arxiv_id":"2504.20759","primary_cat":"math.NT","submitted_at":"2025-04-29T13:37:36+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v2","sentences":[{"i":0,"p":0.99,"id":"s0","own":true,"role":"contribution","text":"With the motivation to study the Selmer group of an elliptic curve, we improve the theory of Kolyvagin systems to describe the Fitting ideals of a Selmer group in the core rank zero situation.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.4,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":1,"p":0.94,"id":"s1","own":true,"role":"method","text":"By relaxing a Selmer structure of rank zero at certain prime, we can construct an auxiliary Kolyvagin system whose localisation determines most of the Fitting ideals of the Selmer group, and all of them when the Galois representation is not self-dual.","hedge":{"p":0.83,"label":"asserted"},"own_p":0.9,"main_claim_p":0.26,"quantitative":false,"quantitative_p":0.26,"role_probabilities":{"gap":0.0,"method":0.94,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":2,"p":0.73,"id":"s2","own":true,"role":"result","text":"With this new theory, one can describe, in terms of the modular symbols, the Galois structure of the Selmer group of an elliptic curve $E/\\mathbb{Q}$ over a finite abelian extension whose degree is coprime to $p$.","hedge":{"p":0.53,"label":"asserted"},"own_p":0.77,"main_claim_p":0.41,"quantitative":false,"quantitative_p":0.19,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.73,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.11,"interpretation":0.16}}]},"compiled":{"main_claim":{"p":0.53,"text":"With this new theory, one can describe, in terms of the modular symbols, the Galois structure of the Selmer group of an elliptic curve $E/\\mathbb{Q}$ over a finite abelian extension whose degree is coprime to $p$.","location":"abstract s2","hedge_strength":{"p":0.53,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":3143,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.7,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.08,"compares_with_prior_results":0.29,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":0.79,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.21,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.79,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.72,"text":"By relaxing a Selmer structure of rank zero at certain prime, we can construct an auxiliary Kolyvagin system whose localisation determines most of the Fitting ideals of the Selmer group, and all of them when the Galois representation is not self-dual.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v2"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":3,"sentences_in_abstract":3},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}