A positive proportion of locally soluble hyperelliptic curves over 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
DOI:
https://doi.org/10.1090/jams/863
Published electronically:
July 27, 2016
MathSciNet review:
3600041
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A hyperelliptic curve over is called ``locally soluble'' if it has a point over every completion of
. In this paper, we prove that a positive proportion of hyperelliptic curves over
of genus
are locally soluble but have no points over any odd degree extension of
. We also obtain a number of related results. For example, we prove that for any fixed odd integer
, the proportion of locally soluble hyperelliptic curves over
of genus
having no points over any odd degree extension of
of degree at most
tends to
as
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
, 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
Email:
bhargava@math.princeton.edu
Benedict H. Gross
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email:
gross@math.harvard.edu
Xiaoheng Wang
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email:
xw5@math.princeton.edu
Tim Dokchitser
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email:
tim.dokchitser@bristol.ac.uk
Vladimir Dokchitser
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email:
v.dokchitser@warwick.ac.uk
DOI:
https://doi.org/10.1090/jams/863
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