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

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arxiv 1402.0031 v1 pith:5XLY3UCY submitted 2014-01-31 math.NT

Manjul Bhargava This is my paper · ORCID
classification math.NT
keywords densitymethodpolynomialssquarefreecertaindiscriminantfieldsnumber
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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.

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Cited by 6 Pith papers

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    The authors conjecture that the leading constant in Malle's conjecture is governed by a Tamagawa measure and a partially unramified Brauer group on the classifying stack BG, with explicit Euler products.

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  4. Lower bounds on heights of odd degree points of hyperelliptic curves

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    Authors conjecture an explicit leading constant for the number of number fields of bounded discriminant by transferring Manin philosophy to classifying stacks, plus related conjectures on multi-heights and local conditions.

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