Disparity in Selmer ranks of quadratic twists of elliptic curves

Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

The proof of Example 7.11 in the published version of this paper was incorrect, because we applied [5, Theorem 1.3] incorrectly. The statement and proof here have been corrected. We thank Lilybelle Cowland Kellock for pointing out the error

References in corpus (6)

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• Regulator constants and the parity conjecture
• Random maximal isotropic subspaces and Selmer groups
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Cited by in corpus (8)

• The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$
• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
• A Markov model for Selmer ranks in families of twists
• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• Quadratic twists of abelian varieties and disparity in Selmer ranks
• Second descent and rational points on Kummer varieties
• Ranks of abelian varieties in cyclotomic twist families
• On Conjectural Rank Parities of Quartic and Sextic Twists of Elliptic Curves