## Abstract

Let F be a totally real number field and A/F an abelian variety with real multiplication (RM) by the ring of integers O of a totally real field. Assuming A admits an O-linear 3-isogeny over F, we prove that a positive proportion of the quadratic twists Ad have rank 0. If moreover A is principally polarized and III(Ad) is finite, then a positive proportion of Ad have O-rank 1. Our proofsmake use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for A/Q of prime level, using Mazur s study of the Eisenstein ideal. For example, suppose p ? 10 or 19 (mod 27), and let A be the unique optimal quotient of J0(p) with a rational point P of order 3. We prove that at least 25% of twists Ad have rank 0 and the average O-rank of Ad(F) is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly 1/8 of twists of the quotient A/(P) have nontrivial 3-Torsion in their Tate Shafarevich groups.

Original language | American English |
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Pages (from-to) | 3267-3298 |

Number of pages | 32 |

Journal | International Mathematics Research Notices |

Volume | 2021 |

Issue number | 5 |

DOIs | |

State | Published - 1 Mar 2021 |

### Bibliographical note

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