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

Abstract

We show that the proportion of plane cubic curves over Q p that have a Q p -rational point is a rational function in p , where the rational function is independent of p ,andwe 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%.