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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; with an appendix by Vladimir Dokchitser
Journal: J. Amer. Math. Soc. 30 (2017), 451-493
MSC (2000): Primary 11G30; Secondary 14G05
DOI: https://doi.org/10.1090/jams/863
Published electronically: July 27, 2016
MathSciNet review: 3600041
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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
MR Author ID: 623882
Email: bhargava@math.princeton.edu

Benedict H. Gross
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
MR Author ID: 77400
Email: gross@math.harvard.edu

Xiaoheng Wang
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID: 1074838
Email: xw5@math.princeton.edu

Tim Dokchitser
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
MR Author ID: 733080
Email: tim.dokchitser@bristol.ac.uk

Vladimir Dokchitser
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
MR Author ID: 768165
Email: v.dokchitser@warwick.ac.uk

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