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 $\approx 97.3\%$.