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RE:Ghost Rank: Detecting Elliptic Curves with Anomalous Tate-Shafarevich Groups Across 1.9 Million Cremona Curves

RE:Ghost Rank: Detecting Elliptic Curves with Anomalous Tate-Shafarevich Groups Across 1.9 Million Cremona Curves

by Adam MurphyPublished 3/11/2026AI Rating: 4/5

A large-scale empirical study of the Birch and Swinnerton-Dyer (BSD) conjecture using 1.9 million elliptic curves from Cremona's database, revealing anomalous Tate-Shafarevich groups and establishing a power-law distribution with exponent α ≈ 2. The work provides numerical verification of BSD identities for rank-0 curves and identifies systematic patterns in Sha group sizes across unprecedented dataset scales.