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April 14, 20260 citationsOpen Access

Isogeny-Dependence of Non-Trivial Sha: A Rank-Graded Concentration Phenomenon for Elliptic Curves over Q

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TRTao Rui

Key Points

  • This study investigates how rational isogenies relate to the Tate-Shafarevich group of elliptic curves over Q, focusing on their rank dependence.
  • Systematic computational analysis using the Cremona database of elliptic curves.
  • Evaluation of conditional probabilities regarding isogenies and non-trivial Tate-Shafarevich groups.
  • Investigation of rank-2 curves with specific focus on 2-isogenies and Kodaira types.
  • Application of BSD (Birch and Swinnerton-Dyer) conjectures and budget analysis.
  • The probability of obtaining a rational p-isogeny increases with Mordell-Weil rank.
  • All rank-2 curves with |III|>1 have a partial-split 2-isogeny.
  • Significant Tamagawa product differences noted among Kodaira types.
  • BSD redistribution leads to zero probability of non-trivial Tate-Shafarevich groups for certain ranks.

Abstract

We report a systematic computational study of the relationship between rational isogenies and the Tate-Shafarevich group III(E/Q) for elliptic curves over Q, using the Cremona database (2,483,649 curves with conductor 1 possess a partial-split 2-isogeny (exactly one rational 2-torsion point, p 1 as rank -> infinity, and provide an isogeny-corrected Delaunay heuristic with empirical parameters. Data: Cremona elliptic curve database (conductor split type -> Kodaira types -> period); (4) BSD redistribution within isogeny classes; (5) BSD budget constraint explaining absence of Sha3 at rank 2.

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Cite This Study

Tao Rui (2026) studied this question.

synapsesocial.com/papers/69ddd9e1e195c95cdefd750dhttps://doi.org/10.5281/zenodo.19521938
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