Published April 10, 2026 | Version 1.0
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Finite-Size Effects in the Delaunay Heuristics for Tate-Shafarevich Groups: A Systematic Verification Across Ranks

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  • 1. Independent Researcher

Description

The Cohen-Lenstra-Delaunay heuristics predict the asymptotic distribution of the Tate-Shafarevich group Sha(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 Sha decreases monotonically with rank is confirmed. (2) The quantitative discrepancy is enormous: for rank 0, observed P(|Sha|=1) ~ 0.79 versus predicted 0.08, a factor of ~10. (3) The cross-rank suppression ratio P(|Sha|>1|rank=0)/P(|Sha|>1|rank=2) is ~1600 in the data versus ~20 predicted, an 80-fold discrepancy from rank-dependent finite-size effects. (4) For rank 0, P(Sha[2]>0) proportional to N^0.22 implies convergence requires conductor N >> 10^8. (5) Nontrivial Sha for rank-2 curves first appears at conductor 194,040; all 21 such cases have |Sha|=4. These results provide the first empirical constraints on the distance to asymptotics for the Delaunay heuristics.

Data: Cremona database via SageMath 10.8. Computations in Python 3.11/SQLite. The 21 rank-2 curves with nontrivial Sha are listed in full in Table 9. This investigation originated as an auxiliary study during AI-assisted scientific discovery research (companion paper tao2026c).

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Preprint: 10.5281/zenodo.19446364 (DOI)

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