The density of ADE families of curves having squarefree discriminant

Abstract

We determine the density of curves having squarefree discriminant in some families of curves that arise from Vinberg representations, showing that the global density is the product of the local densities. We do so using the framework of Thorne and Laga's PhD theses and Bhargava's orbit-counting techniques. This paper generalises a previous result by Bhargava, Shankar and Wang. As an application, we give an asymptotic for the number of reducible orbits of our representations.