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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

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dc.contributor.authorBhargava, Manjul-
dc.contributor.authorGross, Benedict H-
dc.contributor.authorWang, Xiaoheng-
dc.date.accessioned2017-11-21T19:09:43Z-
dc.date.available2017-11-21T19:09:43Z-
dc.date.issued2017-04en_US
dc.identifier.citationBhargava, 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/863en_US
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/pr1g35r-
dc.description.abstractA 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.extent451-493en_US
dc.language.isoenen_US
dc.relation.ispartofJOURNAL OF THE AMERICAN MATHEMATICAL SOCIETYen_US
dc.rightsAuthor's manuscripten_US
dc.titleA positive proportion of locally soluble hyperelliptic curves over Q have no point over any odd degree extensionen_US
dc.typeJournal Articleen_US
dc.identifier.doi10.1090/jams/863-
dc.date.eissued2016-07-27en_US
pu.type.symplectichttp://www.symplectic.co.uk/publications/atom-terms/1.0/journal-articleen_US

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