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April 14, 2026Open Access

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

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Authors

TRTao Rui

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Overview

Computational analysis reveals a concentration phenomenon in the Tate-Shafarevich group linked to isogenies.

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.

Cite This Study

Tao Rui (2026) studied this question.

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