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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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.