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

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.