Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces

TL;DR

This paper develops geometry-of-numbers techniques to count orbits in prehomogeneous vector spaces over global fields, enabling the determination of discriminant densities of field extensions up to degree 5.

Contribution

It introduces a novel approach using geometry-of-numbers methods to analyze orbit counts in prehomogeneous vector spaces over global fields.

Findings

Derived density formulas for discriminants of degree ≤ 5 extensions

Established a framework applicable to any global field

Enhanced understanding of orbit distribution in prehomogeneous vector spaces

Abstract

We develop geometry-of-numbers methods to count orbits in prehomogeneous vector spaces having bounded invariants over any global field. As our primary example, we apply these techniques to determine, for any base global field $F$, the density of discriminants of field extensions of degree at most 5 over $F$.