{"paper":{"title":"Local data of elliptic curves under quadratic twist","arxiv_id":"2501.03209","primary_cat":"math.NT","submitted_at":"2025-01-06T18:41:56+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":1.0,"id":"s0","own":false,"role":"background","text":"Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field.","hedge":{"p":0.88,"label":"asserted"},"own_p":0.05,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.86,"id":"s1","own":true,"role":"contribution","text":"In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist.","hedge":{"p":0.8,"label":"asserted"},"own_p":0.73,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.14,"limitation":0.0,"future_work":0.0,"contribution":0.86,"interpretation":0.0}},{"i":2,"p":0.85,"id":"s2","own":true,"role":"method","text":"To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.85,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.15,"interpretation":0.0}},{"i":3,"p":0.87,"id":"s3","own":true,"role":"result","text":"Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.93,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.87,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":true,"role":"limitation","text":"We note that when the residue field has characteristic $2$, we only consider the special case $K=\\mathbb{Q}_{2}$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.58,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.47,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":1.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.92,"id":"s5","own":true,"role":"result","text":"In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$.","location":"abstract s3","hedge_strength":{"p":1.0,"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 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