The proportion of plane cubic curves over \(\mathbb{Q}\) that everywhere locally have a point (Q2805971)

A plane cubic curve over \(\mathbb Q\) is defined by an equation of the form \(C(X,Y,Z)=0\) where NEWLINE\[NEWLINEC(X,Y,Z)=aX^3 + bX^2Y + cX^2Z + dXY^2 + eXYZ + fXZ^2 + gY^3 + hY^2Z + iY Z^2 + jZ^3,NEWLINE\]NEWLINE and the coefficients are lying in \(\mathbb Z\). A plane cubic is \textit{everywhere locally soluble} if it has a point over every completion of \(\mathbb Q\). In this article, the authors compute the probability \(\rho\) that a random plane cubic has a point everywhere locally soluble. This probability is obtained as the product of local probabilities \(\rho(p)\), \(p\) is a prime integer, where \(\rho(p)\) is the probability that the plane cubic has a point over \(\mathbb{Q}_p\). In fact, \(\rho(p)\) is given explicitly as a rational function in \(p\) which is independent of \(p\). This is performed by analyzing the possibilities for the reduction of a plane cubic modulo \(p\). This enables the authors to show that \(\rho\approx97.256\%\).