### Abstract

A hyperelliptic curve over ℚ is called “locally soluble” if it has a point over every completion of ℚ. In this paper, we prove that a positive proportion of hyperelliptic curves over ℚ of genus g≥ 1 are locally soluble but have no points over any odd degree extension of ℚ. We also obtain a number of related results. For example, we prove that for any fixed odd integer k > 0, the proportion of locally soluble hyperelliptic curves over ℚ of genus g having no points over any odd degree extension of ℚ of degree at most k tends to 1 as g tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.

Original language | English (US) |
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Pages (from-to) | 451-493 |

Number of pages | 43 |

Journal | Journal of the American Mathematical Society |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - 2017 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Brauer-Manin obstruction
- Generalized Jacobian
- Hyperelliptic curves
- Points over extensions
- Rational points

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## Cite this

*Journal of the American Mathematical Society*,

*30*(2), 451-493. https://doi.org/10.1090/jams/863