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

TL;DR

This paper extends geometry-of-numbers methods over number fields to analyze the density of ADE family curves with squarefree discriminants, showing it equals the product of local densities.

Contribution

It introduces a framework combining Thorne and Laga's approaches with Bhargava's techniques over number fields for studying discriminant densities.

Findings

Density of ADE family curves with squarefree discriminant equals product of local densities.

Framework successfully extends geometry-of-numbers techniques to number fields.

Provides a new method for analyzing discriminant properties in algebraic geometry.

Abstract

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.