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A positive proportion of locally soluble hyperelliptic curves over $ \mathbb{Q}$ have no point over any odd degree extension

Authors: Manjul Bhargava, Benedict H. Gross and Xiaoheng Wang; with an appendix by Tim Dokchitser; Vladimir Dokchitser
Journal: J. Amer. Math. Soc. 30 (2017), 451-493
MSC (2000): Primary 11G30; Secondary 14G05
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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Additional Information

Manjul Bhargava
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Benedict H. Gross
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Xiaoheng Wang
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Tim Dokchitser
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Vladimir Dokchitser
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Keywords: Rational points, hyperelliptic curves, Brauer-Manin obstruction, generalized Jacobian, points over extensions
Received by editor(s): November 14, 2013
Received by editor(s) in revised form: December 31, 2015, and April 20, 2016
Published electronically: July 27, 2016
Additional Notes: The first and third authors were supported by a Simons Investigator Grant and NSF grant DMS-1001828.
The second author was supported by NSF grant DMS-0901102.
The authors of the appendix were supported by Royal Society University Research Fellowships.
Article copyright: © Copyright 2016 American Mathematical Society

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