Finite-Size Effects in the Delaunay Heuristics for Tate–Shafarevich Groups: A Systematic Verification Across Ranks

The Cohen–Lenstra–Delaunay heuristics predict the asymptotic distribution of the Tate–Shafarevich group III(E/Q) for elliptic curves of rank u, ordered by conductor. Using the complete Cremona database (conductor ≤ 400,000; over 2.4 million curves), we perform the first systematic quantitative comparison of these predictions against empirical data across ranks 0, 1, 2, and 3. We find that: 1. The qualitative prediction—that the proportion of curves with nontrivial III decreases monotonically with rank—is confirmed. 2. The quantitative discrepancy between the asymptotic predictions and finite-conductor data is enormous: for rank 0, the observed P(|III| = 1) ≈ 0.79 versus the predicted 0.08, a factor of ∼10. 3. The cross-rank suppression ratio P(|III| > 1 | rank = 0)/P(|III| > 1 | rank = 2) is ∼ 1600 in the data, versus ∼ 20 predicted asymptotically—an 80-fold discrepancy attributable to rank-dependent finite-size effects. 4. For rank 0, the growth rate P(III[2] > 0) ∝ N0.22 implies convergence to the asymptotic regime requires conductor N ≳ 108. 5. Nontrivial III for rank 2 curves first appears at conductor ≈ 194,000; all 21 such cases have |III| = 4. These results provide the first empirical constraints on the “distance to asymptotics” for the Delaunay heuristics and reveal that finite-size effects are strongly rank-dependent. All |III| values reported are analytic orders |IIIan| computed via the BSD formula; for rank ≥ 2, equality with the algebraic |III| remains conjectural.