The proportion of plane cubic curves over ℚ that everywhere locally have a point

Manjul Bhargava, John Cremona, Tom Fisher

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over ℚ that have points everywhere locally; numerically, this density is shown to be ≈ 97.3%.

Original languageEnglish (US)
Pages (from-to)1077-1092
Number of pages16
JournalInternational Journal of Number Theory
Volume12
Issue number4
DOIs
StatePublished - Jun 1 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Plane cubic curves
  • local solubility
  • random Diophantine equations

Fingerprint Dive into the research topics of 'The proportion of plane cubic curves over ℚ that everywhere locally have a point'. Together they form a unique fingerprint.

  • Cite this