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%.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Plane cubic curves
- local solubility
- random Diophantine equations