The proportion of plane cubic curves over $\mathbb{Q}$ that everywhere locally have a point

Abstract

We show that the proportion of plane cubic curves over ${\mathbb{Q}}_p$ that have a ${\mathbb{Q}}_p$-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 $\mathbb{Q}$ that have points everywhere locally; numerically, this density is shown to be $\approx 97.3\%$.

13 pages

References in corpus (2)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• What is the probability that a random integral quadratic form in $n$ variables is isotropic?