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Detecting Large Tate-Shafarevich Groups via BSD Geometric Invariants: Lessons from a Computational Audit of 1.9 Million Elliptic Curves

Detecting Large Tate-Shafarevich Groups via BSD Geometric Invariants: Lessons from a Computational Audit of 1.9 Million Elliptic Curves

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

We investigate computational methods for identifying elliptic curves with anomalously large Tate-Shafarevich groups ($|Ш| ≫ 1$) among rank-0 curves over $ℚ$. After documenting and correcting circular reasoning in AI-assisted analysis, we find that the BSD geometric factor $α_{BSD}(E) = Ω_E^+ · ∏_p c_p(E) / |E(ℚ)_{tors}|^2$ achieves 99.5% precision at 98.4% recall for detecting $|Ш| > 1$ curves. We additionally report a power-law tail distribution for $|Ш|$ across 1.9 million curves with exponent $α̂ = 2.02 ± 0.07$, placing the distribution at the convergence threshold for $𝔼[|Ш|]$.