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

Manjul Bhargava, Benedict H. Gross, Xiaoheng Wang, Tim Dokchitser, Vladimir Dokchitser

Research output: Contribution to journalArticle

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Abstract

A hyperelliptic curve over ℚ is called “locally soluble” if it has a point over every completion of ℚ. In this paper, we prove that a positive proportion of hyperelliptic curves over ℚ of genus g≥ 1 are locally soluble but have no points over any odd degree extension of 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 ℚ of genus g having no points over any odd degree extension of 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.

Original languageEnglish (US)
Pages (from-to)451-493
Number of pages43
JournalJournal of the American Mathematical Society
Volume30
Issue number2
DOIs
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Brauer-Manin obstruction
  • Generalized Jacobian
  • Hyperelliptic curves
  • Points over extensions
  • Rational points

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