### 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 language | English (US) |
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Pages (from-to) | 1077-1092 |

Number of pages | 16 |

Journal | International Journal of Number Theory |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Jun 1 2016 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Plane cubic curves
- local solubility
- random Diophantine equations

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## Cite this

Bhargava, M., Cremona, J., & Fisher, T. (2016). The proportion of plane cubic curves over ℚ that everywhere locally have a point.

*International Journal of Number Theory*,*12*(4), 1077-1092. https://doi.org/10.1142/S1793042116500664