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

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.

31 pages

References in corpus (2)

• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7

Cited by in corpus (19)

• On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
• Nonabelian Cohen-Lenstra Moments
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
• Odd degree number fields with odd class number
• A positive proportion of Thue equations fail the integral Hasse principle
• A predicted distribution for Galois groups of maximal unramified extensions
• Density of power-free values of polynomials
• $E_8$ and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve
• Malle's Conjecture for $G\times A$, with $G=S_3,S_4,S_5$
• The Hasse norm principle for abelian extensions
• On the representation of $k$-free integers by binary forms
• The distribution of $H_8$-extensions of quadratic fields
• Moments of unramified 2-group extensions of quadratic fields
• 2-Selmer groups of hyperelliptic curves with two marked points
• On the distribution of equivalence classes of random symmetric p-adic matrices
• On number fields with $k$-free discriminants
• Equidistribution of rational points and the geometric sieve for toric varieties
• Levels of distribution for sieve problems in prehomogeneous vector spaces
• Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations