{"paper":{"title":"On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture","arxiv_id":"2609.15328","primary_cat":"math.NT","submitted_at":"2026-09-14T10:17:19+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. 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