# A positive proportion of locally soluble hyperelliptic curves over Q have no point over any odd degree extension

## Author(s): Bhargava, Manjul; Gross, Benedict H; Wang, Xiaoheng

DownloadTo refer to this page use:

`http://arks.princeton.edu/ark:/88435/pr1g35r`

Abstract: | A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. 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 $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ 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. |

Publication Date: | Apr-2017 |

Electronic Publication Date: | 27-Jul-2016 |

Citation: | Bhargava, Manjul, Gross, Benedict H, Wang, Xiaoheng. A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 30 (2017), no. 2, 451-493 , DOI 10.1090/jams/863 |

DOI: | 10.1090/jams/863 |

Pages: | 451-493 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY |

Version: | Author's manuscript |

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.