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

TL;DR

This paper computes the exact proportion of plane cubic curves over the rationals that have points everywhere locally, revealing a universal rational function in p and estimating the global density at approximately 97.3%.

Contribution

It explicitly determines the rational function describing local solubility of plane cubic curves over p-adic fields, and computes the global density over Q.

Findings

Proportion over ${f Q}_p$ is a p-independent rational function.

Global density of locally soluble plane cubics is approximately 97.3%.

Explicit rational function for local solubility is derived.

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 $\approx 97.3\%$.