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

Bhargava, Manjul; Gross, Benedict; Wang, Xiaoheng

doi:10.1090/jams/863

A positive proportion of locally soluble hyperelliptic curves over $\mathbb {Q}$ have no point over any odd degree extension
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by Manjul Bhargava, Benedict H. Gross and Xiaoheng Wang; with an appendix by Tim Dokchitser; with an appendix by Vladimir Dokchitser

J. Amer. Math. Soc. 30 (2017), 451-493

DOI: https://doi.org/10.1090/jams/863

Published electronically: July 27, 2016

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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.

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