## Abstract

Let A be an abelian variety over a number field F and let p be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group III(A_{s}) should contain an element of order p for a positive proportion of quadratic twists A_{s} of A. We give a general method to prove instances of this conjecture by exploiting independent isogenies of A. For each prime p, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial p-torsion in their Tate–Shafarevich groups. In particular, when the modular curve X_{0}(3p) has infinitely many F-rational points, the method applies to “most” elliptic curves E having a cyclic 3p-isogeny. It also applies in certain cases when X_{0}(3p) has only finitely many rational points. For example, we find an elliptic curve over for which a positive proportion of quadratic twists have an element of order 5 in their Tate–Shafarevich groups. The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime p ≡ 1 (mod 9), examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order p in their Tate–Shafarevich groups.

Original language | American English |
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Pages (from-to) | 627-655 |

Number of pages | 29 |

Journal | Algebra and Number Theory |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:The authors would like to thank the anonymous referees for their many detailed and thoughtful comments. Bhargava was supported by a Simons Investigator Grant and NSF grant DMS-1001828. Lemke Oliver was partially supported by NSF grant DMS-1601398.

Publisher Copyright:

© 2021, Mathematical Science Publishers. All rights reserved.

## Keywords

- Abelian varieties
- Elliptic curves
- Selmer groups
- Tate–Shafarevich groups