{"paper":{"title":"When the BSD Real Period Remembers Elliptic Curves over $\\mathbb{Q}$","arxiv_id":"2610.06015","primary_cat":"math.NT","submitted_at":"2026-10-05T09:12:57+00:00","version_compiled":"v1"},"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 v1","sentences":[{"i":0,"p":0.98,"id":"s0","own":true,"role":"contribution","text":"We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.92,"main_claim_p":0.63,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.02,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":1,"p":0.97,"id":"s1","own":true,"role":"result","text":"For example, the real period $\\Omega_E =\\frac{\\Gamma(1/4)^2}{2\\sqrt{\\pi}\\,17^{1/4}}$ and the Tate--Shafarevich group $ \\Sha(E/\\mathbb{Q})\\cong \\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ remember the elliptic curve $E: y^2=x^3+17x$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.75,"main_claim_p":0.18,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.01}},{"i":2,"p":0.66,"id":"s2","own":false,"role":"background","text":"Let $K$ be a number field admitting a real embedding, and fix one such embedding $\\sigma\\colon K\\hookrightarrow\\mathbb{R}$.","hedge":{"p":0.64,"label":"unclear"},"own_p":0.22,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.27,"result":0.0,"background":0.66,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":3,"p":0.93,"id":"s3","own":true,"role":"result","text":"We first prove that, for an elliptic curve $E$ over $K$, the real period of $E$ attached to $\\sigma$ determines the $K$-isogeny class of $E$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.26,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":4,"p":0.98,"id":"s4","own":true,"role":"result","text":"We then prove that an elliptic curve $E$ over $\\mathbb{Q}$ is determined up to isomorphism over $\\mathbb{Q}$ by its real period, the number of connected components of $E(\\mathbb{R})$, and the real period of the quadratic twist $E^{D}$ by a negative square-free integer $D\\equiv 1 \\bmod 4$ coprime to the minimal discriminant of $E$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.31,"quantitative":false,"quantitative_p":0.37,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":5,"p":0.97,"id":"s5","own":true,"role":"result","text":"Moreover, over $\\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).","hedge":{"p":0.99,"label":"asserted"},"own_p":0.88,"main_claim_p":0.16,"quantitative":false,"quantitative_p":0.29,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.02}}]},"compiled":{"main_claim":{"p":0.84,"text":"We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative 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