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
Full metadata record
DC Field | Value | Language |
---|---|---|
dc.contributor.author | Bhargava, Manjul | - |
dc.contributor.author | Gross, Benedict H | - |
dc.contributor.author | Wang, Xiaoheng | - |
dc.date.accessioned | 2017-11-21T19:09:43Z | - |
dc.date.available | 2017-11-21T19:09:43Z | - |
dc.date.issued | 2017-04 | en_US |
dc.identifier.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 | en_US |
dc.identifier.uri | http://arks.princeton.edu/ark:/88435/pr1g35r | - |
dc.description.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. | en_US |
dc.format.extent | 451-493 | en_US |
dc.language.iso | en | en_US |
dc.relation.ispartof | JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY | en_US |
dc.rights | Author's manuscript | en_US |
dc.title | A positive proportion of locally soluble hyperelliptic curves over Q have no point over any odd degree extension | en_US |
dc.type | Journal Article | en_US |
dc.identifier.doi | 10.1090/jams/863 | - |
dc.date.eissued | 2016-07-27 | en_US |
pu.type.symplectic | http://www.symplectic.co.uk/publications/atom-terms/1.0/journal-article | en_US |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
1310.7692v2.pdf | 491.74 kB | Adobe PDF | View/Download |
Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.