## Abstract

We consider the family of hyperelliptic curves over Q of fixed genus along with a marked rational Weierstrass point and a marked rational nonWeierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2, and that most such curves have only three rational points. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also consider another related family of curves and study the interplay between these two families. There is a family φ of isogenies between these two families, and we prove that the average size of the φ-Selmer groups is exactly 2.

Original language | English (US) |
---|---|

Pages (from-to) | 267-304 |

Number of pages | 38 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics