Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A positive proportion of locally soluble hyperelliptic curves over $\mathbb {Q}$ have no point over any odd degree extension
HTML articles powered by AMS MathViewer

by Manjul Bhargava, Benedict H. Gross and Xiaoheng Wang; with an appendix by Tim Dokchitser; with an appendix by Vladimir Dokchitser PDF
J. Amer. Math. Soc. 30 (2017), 451-493 Request permission

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.
References
Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11G30, 14G05
  • Retrieve articles in all journals with MSC (2000): 11G30, 14G05
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
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 451-493
  • MSC (2000): Primary 11G30; Secondary 14G05
  • DOI: https://doi.org/10.1090/jams/863
  • MathSciNet review: 3600041