
joint with Manjul Bhargava and John Cremona
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, 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 appoximately 97.3%.