Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
Number Theory
2026-03-13 v2
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 , the density of discriminants of field extensions of degree at most 5 over .
Keywords
Cite
@article{arxiv.1512.03035,
title = {Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces},
author = {Manjul Bhargava and Arul Shankar and Xiaoheng Wang},
journal= {arXiv preprint arXiv:1512.03035},
year = {2026}
}
Comments
Main results have been extended to all global fields, including those of small positive characteristic. Several new results are also proven, including applications to relative class numbers of quadratic and cubic extensions, unramified nonabelian extensions of quadratic extensions, and the expected number of points in the fibers of degree 4 and 5 covers of curves over a finite field. 38 pages