Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, http://arxiv.org/abs/1402.0031
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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.
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math.NT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
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.
citing papers explorer
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Geometry-of-numbers methods over global fields II: Coregular representations
Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
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The leading constant in Malle's conjecture
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.