The geometric sieve and the density of squarefree values of invariant polynomials

TL;DR

This paper introduces a new method to determine the density of squarefree values of certain invariant polynomials, with applications to number field discriminants and nonabelian extensions.

Contribution

The paper develops a novel geometric sieve method applicable to invariant polynomials with finite stabilizers, advancing the understanding of their squarefree value distribution.

Findings

Applicable to discriminant polynomials of prehomogeneous representations

Determines density of squarefree values for specific invariant polynomials

Supports applications in number field and extension distribution questions

Abstract

We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the discriminant polynomials of various prehomogeneous and coregular representations where generic stabilizers are finite. This has applications to a number of arithmetic distribution questions, e.g., to the density of small degree number fields having squarefree discriminant, and the density of certain unramified nonabelian extensions of quadratic fields. In separate works, the method forms an important ingredient in establishing lower bounds on the average orders of Selmer groups of elliptic curves.