Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant

For families of curves arising from a Dynkin diagram of type ADE, we show that the density of such curves having squarefree discriminant is equal to the product of local densities. We do so using the framework of Thorne and Laga’s PhD theses and geometry-of-numbers techniques developed by Bhargava, here expanded over number fields.