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

TL;DR

The article generalizes Bhargava–Shankar–Wang’s squarefree-discriminant density results for polynomial families to ADE-type curve families over arbitrary number fields using Vinberg representations. It combines a precise invariant-theoretic parametrization via a Kostant section, a geometry-of-numbers framework over number fields, and a squarefree sieve to establish that the global density equals the product of local densities. Key technical components include a robust reduction theory with box-shaped fundamental domains, a tail-analysis separating strong vs. weak divisibility, and a case-by-case ADE analysis to compute cusp volumes. The findings provide a canonical local-global density principle for squarefree discriminants in ADE curve families, with potential implications for understanding Selmer groups and arithmetic statistics of these families across number fields.

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.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5: Graded Jacobson-Morozov
• proof