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

By Manjul Bhargava. 2014

82 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ab075dd90d5222adc0d167b16475bddfa8cfea2b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups (2017, background): “re are at most ˝ n;$n D $n ﬁeldsK2Z n(Q;S n) withD K= D. Inthiscase,˝= $ n,sothatforallsuﬃcientlysmall " 0 >0,thezerodensityexceptionalsetisat mostofsizeD 5X$n+" 0. Remark 1.14. Bhargava [Bha14] has conjectured (in the essentially equivalent setting of degree nS n-extensions with square”
Failures of weak approximation in families (2015, methodology): “This sieve was first introduced by Ekedahl [16], but was developed extensively by Poonen–Stoll [32] and Bhargava [2].”